Algebraic Methods in Philosophical Logic (Oxford Logic Guides, 41)

MATRICES

MATRICES AND ATLASES

234

actually was written after Meyer's result (despite the date) and provides a more directly algebraic proof of this result, as well as obtaining additional results, the most important of which is the "pretabularity" of RM, which roughly means that every extension is such that its tautologies are characterized by a finite characteristic matrix. A version of it may be found in Anderson and Belnap (1975, Section 29.4). Dunn and Meyer (1971) also showed that LC is pretabular (cf. Section 11.10 below). In this section we have mentioned "tautology hood" in a matrix from time to time, implicitly focusing on unary assertional logics. It is worth remarking that things can change if one looks at consequence. For example, Dunn (1970) shows that the Sugihara matrix SIQI defined on the rationals is strongly characteristic for RM, meaning that ¢ is deducible from r using the axioms and rules of RM just when every interpretation in SIQI that assigns every member ofr a designated value also assigns ¢ a designated value (in symbols: r I-RM ¢ iftT FSiI)l ¢). All of Sz, Sz±, SIQI± are tautology-equivalent to SIQI. But it is shown that none of Sz-, Sz±, SIQI± is strongly characteristic for RM. It is an interesting question as to when a logic is characterized by a finite matrix. Ulrich (1968,1986) gives an interesting answer for unary assertionallogics. Sometimes while there is no single finite matrix characterizing a logic, there is nonetheless a system of finite matrices that jointly characterize it. This is true of the Lukasiewicz infinite logic, and ofLC and RM. This illustrates the finite model property (Definition 6.16.4). 7.1.8

Interpretation

How should we best interpret all of these different many-valued matrices? Something like degrees of truth might be appropriate given Lukasiewicz's motivation, but there is another possibility, namely that they be interpreted as "propositions." This is most easily illustrated in terms of the Gtidel matrices. Thus let U2 = {ao, al }. Think of ao and al as states of information, and accordingly introduce an "information order" postulating that ao bal. In the present circumstance we can actually identify these information states with "moments of time." Of course we also postulate ao b ao and al bal. A proposition can then be understood to be a subset of U2 that is closed upward under b. Let us indicate the collection of these as ~ t (U2) (the mnemonics are ~(U2) for the power set of U2 and t for upward closed). Thus the propositions are 0, {al }, {ao, al }. The reader will immediately recognize the fortuitous circumstance that these are three, and that they are linearly ordered by ~. For p, q E ~ t (U2), define p /\ q = p n q, p V q = p U q. These clearly agree with the Lukasiewicz definition of conjunction and disjunction as minimum and maximum. We next define implication and negation as follows: (1) (2)

p -+ q

= {X : VfJ ;;;] X, ""p = {X : VfJ

if fJ

;;;]

E

P then

fJ

E q};

x, fJ ¢ p}.

The knowledgeable reader will recognize that these definitions correspond to the socalled Kripke-Grzegorczyk semantics for intuitionistic logic. More can be found on this in Chapter 11. The reader can easily check that tlle following tables are obtained

235

using the above definitions, and that these tables are isomorphic to the Gi::idel tables given previously (we omit conjunction and disjunction since their match with min and max has been already discussed): -+

{ao,ad*

{ad

tao, a]} {ao,ad tao, ad

{ad tao, ad tao, ad

tao, ad* {ad

0 0

tao, ad* {ad

0

tao, ad

0

0 0 0 tao, a]}

Exercise 7.1.9 Show tllat this can be extended to an (n+ I)-valued Gi::idel matrix whose elements are {O, ... , n} by using n + 1 linearly ordered states ao b al, ... , ai-] b ai, ... , an-] b an E Un. It is easy to see that the propositions correspond to the principal cones [ai) plus the empty cone 0. Given this one concrete illustration, we shall quickly discuss how something similar can be provided for the Lukasiewicz tables. Again we shall use a linearly ordered set U, whose elements are again thought of as moments of time. For concreteness let us again consider in effect U2, but this time we shall drop the a notation, and just set U2 = {O, I}, with 0 thought of as "the present." The reader should not be confused by the fact that we are using natural numbers both for the elements of the three-valued matrix and for the states in its interpretation. The propositions on U2 are then the three upward closed cones: {{O, I}, {I}, 0}, with to, I} being a proposition that is true at the present (and so will continue to have been true), {I} being a proposition that will become true at the next (and final) moment, and 0 being a proposition that is never true. We again take conjunction to be intersection, and disjunction to be union, but we have the following definitions of implication and negation: (3) (4)

p -+ q = {i :

Vj

E

U such that i + j

E

U, if j

E

P then i

+j

E q};

""p = {i : n - i ¢ p} .

The reader will now see why we chose to use natural numbers for the states, nan1ely because it makes sense to add and subtract them. The idea above comes from Urquhart (1973). There is a different interpretation of essentially the San1e mathematics that comes from Scott (1974) in terms of "degrees of precision.,,5 We shall not go into either the philosophy or the mathematics needed to make more sense of this idea, but let us point out that we at least have a basis for seeing what Scott has meant by his slogan of "replacing many values by many valuations" (the many valuations being given by relativizing the truth of a proposition to a degree of precision, or, in terms of the Urquhart semantics, a point in time). How do we interpret the Sugihara matrices? Again we think of U as moments of time. But we complicate matters by saying that a proposition can be both true and false at a given moment. SCf.

Urquhart (1985) for a discussion and comparison of the two interpretations.

RELATIONS AMONG MATRICES

MATRICES AND ATLASES

236

This leads to the construal of a proposition p as a pair of sets of moments (p+, p-), where p+ consists of the moments that p is true, and p- is the moments that p is false. We require that both p+, p- be cones in the information order: -,(p+,p-) = (p-,p+), (p+,p-) /\ (q+, q-) = (p+ n q+, p- U q-), (p+,p-) V (q+, q-) = (p+ U q+, p- n q-).

The definition of -+ is slightly complicated. Put intuitively, it is a variation on (l), the definition of intuitionistic implication, that treats falsity independently from truth. We begin by setting (p+,p-) -+ (q+, q-) = ((p -+ q)+, (p -+ q)-). We then define (p -+ q)+ to be

And we define (p -+ q)- to be

7.2

Relations Among Matrices: Submatrices, Homomorphic Images, and Direct Products

There are relations among matrices which conespond to certain relations among algebras discussed in Chapter 2. Since a matrix M = (A, D) is just an algebra with a designated subset, a matrix always has an "algebraic part" A (in symbols, alg(M)). It is thus to be expected that the conesponding relations among matrices will be defined by firs t requiring the relation among the algebraic parts, and then adding on some requirement about how the designated elements are to be treated. Given two matrices M and M', we say that M is a submatrix ofM' iff (1) alg(M) is a subalgebra of alg(M'), and

(2) D = D'nM.

An example might be to let M' be all the propositions of biology, and let M be all the propositions in some subfield of biology, like cytology. Then D' might be all the true propositions of biology, and D would be all the true propositions of cytology. We would always expect (2) to hold for any example such as the one above. However, there is also the weaker relationship (2') D

Readers who wish to learn more about the semantics of RM and how it relates to other semantical approaches to relevance logic can consult Anderson et al. (1992). Exercise 7.1.10 Show that one gets the three-element Sugihara matrix when one has just two moments of time ao and al. Go on to show that one gets the (n + I)-element Sugihara matrix by having n moments of time. In the rest of the book we shall not directly discuss Lukasiewicz's approach to manyvalued logic, and we shall say little about relevance logic. But we brought these subjects up here to make the point that a matrix understood abstractly as an algebra with some designated elements can often be given a more intuitive interpretation ("representation") where the elements are understood as sets of "states." In the examples above the states are taken to be moments in time, but they can also be taken to be states of information about an evolving system. When matrices are so interpreted, it is best to think of a matrix element as a "proposition," and not as a "truth value." This is because the most natural way to think of its representation is as the set of states in which it is true. This is one way of understanding Scott's (1974) idea of replacing "many values" by (sets of) "many valuations." A valuation is just a mapping of sentences into {t, f} and such a valuation can be regarded as a state. 6 6There is a subtlety here. Once each sentence tjJ has been assigned a proposition II tjJ II in the sense of a set of states, a state a uniquely determines a valuation: v(tjJ) = t iff a E IItjJll. But the converse does not hold. There can be two states that determine the same valuation. In this sense states are more abstract than valuations, and recognize that there can be more to a state that can be expressed by the particular sentences of the formal language.

237

~

D',

which should not be confused with (2). We speak of weak submatrices if (2) is replaced with (2'). We get a conesponding strong and weak sense of homomorphism for matrices. In both cases h must, of course, satisfy (1) h is an (operational) homomorphism of alg(M) into the alg(M'), and in the strong case it must satisfy (2) a E D only if h(a) ED',

as well as (2') a E D if h(a) ED',

whereas in the weak case it must satisfy only (2). This last is just the requirement that h be absolutely faithful, with respect to preserving the "unary relation" D (cf. Section 2.5). Some alternative terminology that we find useful is to call an operational homomorphism satisfying (2) positive (for it preserves designation), and one satisfying (2') negative (for it preserves non-designation). A (weak) homomorphism is then just a positive homomorphism, and a strong homomorphism is then a homomorphism that is both positive and negative. The literature, somewhat inconsistently with its practice with regard to "submatrix," seems to ascribe the term "homomorphism" simpliciter to the weak case, bringing out the terminology "strong homomorphism" when it wants something satisfying (2'). (A notable exception is to be found in Czelakowski (l980a, p. 16).) We shall follow this practice, but shall avail ourselves of the "strong" and "weak" epithets when we want to be perfectly clear.

238

MATRICES AND ATLASES

How can one give some intuitive logical force to these two different notions of homomorphism? Let us think about interpretations of a sentential language into an algebra of propositions. These are operational homomorphisms (from a universally free algebra), and as such are not required to preserve anything about designation. A natural way to think of designation of sentences is to think that there is some theory that asserts a certain set of the sentences as theorems. A natural way to think of designation of propositions is to think that there is the set of true propositions. A positive homomorphism is then just a sound interpretation (the sentences asserted are true), and corresponds to the well-known principle of charity in interpretation. A negative homomorphism is then a complete interpretation (all the true sentences are asserted). Let h be a (strong/weak) homomorphism from Minto M'. When h is onto we shall call M' a (strong/weak) homomOlphic image of M, and, conversely, M an inverse (strong/weak) homomorphic image of M'. Clearly the first relation is many-one, whereas the second, converse relation is one-many. When h is one-one we call it a (strong/weak) isomorphism, and call M' an isomorphic image and M an inverse isomorphic image (both or either the weak or strong kind). Note that if h is a strong isomorphism, then its inverse is also a strong isomorphism. Thus if M' is a strong isomorphic image of M, then M is also a strong isomorphic image of M', and symmetrically for inverse strong isomorphic images. But the same does not necessarily hold for weak isomorphisms. Normally, when we speak just of an "isomorphism" (or "isomorphic image" or "inverse isomorphic image") we shall mean these in their strong senses. Finally, we shall define the notions of direct and subdirect products of matrices. A direct product I1EI Mj is a structure (I1EI alg(Mj), D), where D = XiEIDi, i.e., an element (di)iEI of the direct product is designated iff each component di E Di. The subdirect product is just a submatrix of a direct product in which all the projection homomorphisms are onto. Clearly subdirect products admit of strong and weak versions depending on whether the homomorphisms are strong or weak. Our default convention will be that the term refers to the weak version, just as with homomorphisms. Indeed, there are very few instances of the strong version, because, if one thinks about it, one sees that it rules out "mixed elements" of the subdirect product, i.e., elements such as (Xl, X2), where Xl ED] and yet x2 ¢ D2. Since matrices are fundamentally just algebras, it is sometimes useful to speak of subalgebras, homomorphic images, isomorphic images, direct products, etc. with reference to the algebras (ignoring the sets of designated values). Context usually makes clear when we mean, say, "homomorphism," in the algebraic sense or in the matrix sense (and hopefully in the latter case whether we mean the weak or strong sense). But when context is not so clear we shall speak, say, of an algebraic homomorphism or matrix (strong or weak) homomorphism, and similarly with algebraic direct product, matrix (strong or weak) direct product, etc.

PROTO-PRESERVATION THEOREMS

7.3

239

Proto-preservation Theorems

In a later section, we discuss how various logical properties, such as validity, are preserv~d by t?e various relationships among matrices discussed in the previous section. In thIS sectIOn we shall do some preliminary work toward this end, examining to what extent designation and non-designation of sentences are carried up from submatrices homomorphic images, and (sub)direct products to the original matrices. One of ou; chief to?ls will be the invocation of weak and strong versions of these relationships as appropnate. Throughout all of the following lemmas, we shall assume some fixed language L, a~d M and M'. will be matrices of the same similarity type as L. We shall follow our preVIOusly establIshed convention of letting D and D' be the respective sets of designated elements, but we shall extend this convention by letting U and U' be the respective sets of undesignated elements. Each lemma is followed by a diagram that depicts it.

Lemma 7.3.1 Let M' be a weak submatrix ofM, and let z' be an interpretation in M'. Then I' is also an interpretation in M. Furthel; ijl'(cjJ) ED', then z'(cjJ) ED. IfM' is a strong submatrix ofM, then also ijz'(cjJ)

E

U', then 1'(cjJ)

E

U.

Proof It is clear that I' is also an interpretation in M, since it is clear that a homomorphism into a subalgebra is also a homomorphism into the algebra itself. Now the first pmt of the lemma follows from the fact that weak submatrices require that D' ~ D. L

L

I'

I'

M'

:S

I'

D'

M

w

D

FIG. 7.1. First part of Lemma 7.3.1 L

L

z'

I'

M'

:S s

z'

M

D' U'

FIG. 7.2. Second part of Lemma 7.3.1

D

U

PROTO-PRESERVATION THEOREMS

MATRICES AND ATLASES

240

For the second part, since strong submatrices require that D' = M' n D, it follows for an element a E M' that if a is not in D', then a is not in D, i.e., U' ~ U. The lemma is now immediate. 0 Lemma 7.3.2 Let h be a (weak) homol1101phism from M onto M' and I' be an interpretation in M'. Then for some intelpretation 1 in M, for every sentence ¢, if I( ¢) E D, then I'(¢) ED'. If h is a strong homomorphism from M onto M', then ifl(¢) E U, then I'(¢) E U' (and also ifl(¢) ED, then I'(¢) ED').

z' h w

The above lemmas all relate a given interpretation in a homomorphic image to some other interpretation in its source matrix. The next lemma treats the other direction.

31

I'

M' ..... , f - - - - - M

Then since our hypothesis is that I(¢) E D, we know, by (2) and the fact that h preserves designation, that I' (¢) E D'. For the second part, the desired interpretation is produced just as above, and clearly has the properties mentioned, since a strong homomorphism preserves both designation and undesignation. 0

Lemma 7.3.3 Let I be an interpretation in M and let h be a (weak) hOl11ol1101phism fivmMontoM'. Then the composition loh, w/zereloh(x) = h(/(x)), isanintelpretation ofL in M'. Further, for all sentences ¢, if I( ¢) E D, then (10 h)( ¢) ED'. If h is a strong homol1101phismfimn M onto M', then also ifl(¢) E U then (10 h)(¢) E U'.

L

L

241

L

L

D' ""'.,f----- D h

10

h

10

h

FIG. 7.3. First part of Lemma 7.3.2 D' ....E f - - - - - D h

M' ""'.,f----- M w

L

L

FIG. 7.5. First part of Lemma 7.3.3

31

I'

h M' ..... , f - - - - - M s

L

L

D' .... , f - - - - - D h

U'"

10

h

10

U

FIG. 7.4. Second part of Lemma 7.3.2

h

M' ""'Ef----- M s Proo.f Define I on atomic sentences so that I(p) E h- 1(I' (p)). This actually requires the Axiom of Choice, but, speaking intuitively, we know from the fact that h is onto that each element in M', including I'(p), comes via h from possibly many elements in M. It is "merely" the matter of selecting in each case one of them to "play the role" of l' (p). Extend this definition inductively so that (1) I(Oi(¢I,···, ¢m)) = Oi(I(¢l),···, I(¢m))·

Clearly h(/(p)) = I'(p) for each atomic sentence p.1t is then a transparent induction to show that (2) h(I(¢)) = I'(¢).

h

D'

""'.,f-----

U'"

h

D U

FIG. 7.6. Second part of Lemma 7.3.3 Proo.f The composition of any two homomorphisms is a homomorphism, and an interpretation is just a homomorphism from an algebra of sentences. 0

The next lemmas relate to direct products. Lemma 7.3.4 Let ItEI Mj be a direct product and let Ii be an intnpretation ofL in one of the component matrices Mj. Then there is an intelpretation I in I1EI Mi such thatforevery sentence ¢, ifl(¢) ED, then li(¢) E D i .

L

L

L

L

3/

3/

D/·

~Eo------

1[i

D/·

D

Proof This is a simple application of Lemma 7.3.4, given the obvious fact that the D projection 1[i(ai)iEI) = ai is a weak homomorphism.

interpretation Di.

Ii

== / 0

1[i

such thatfor every sentence p, ifl(p)

Proof Apply Lemma 7.3.3.

Remark 7.3.5 It might be thought that we would next prove the natural strengthening of Lemma 7.3.4, that is, if we knew that the projection 1[i were a strong homomorphism, we would know that the interpretation Ii carried over from the direct product not just designation, but also undesignation. But a little reflection shows that except in the most degenerate of cases there are no such "strong direct products." The problem is that an undesignated "sequence" (ai )iEI of the direct product can still have some designated components. We might flirt with the idea of defining a "strong direct product" on the union of the Cartesian product of the designated sets and the Cartesian product of their complements (XiEI Di U XiEIUi). Such a construction (though not so labeled) actually occurs in Belnap (1967). But a little reflection shows that these are also rare fauna (although not quite so rare as on the first alternative above). The problem is that an operation applied on two "pure sequences," (do,dl)' (UO,llj) = (do' UO,dl' UI),

can very likely produce a "mixed sequence," since there is very little hope that do . lIO and dl . III will be either both designated or both undesignated. To require that these mixed cases do not occur would be in effect to require "quasi-truth-functionality," i.e., to require of all the matrices under consideration that when elements have the same status of designation, then operations on them produce elements all having the same status of designation (in short, that being both designated or both undesignated is a congruence in the sense of the next section). This is normality in the sense of Church (1956), and can correctly be guessed to have too close an affinity to classical logic to be of any general utility. We can, however, prove a result for direct products related to Lemma 7.3.3 regarding homomorphisms. Mi be a direct product of the (indexed set of) matrices (Mi )iEI, and let / be an interpretation of L in IIiE! Mi. Then for each i E I, there exists an

~E"------

1[i

D

FIG. 7.8. Lemma 7.3.6

FIG. 7.7. Lemma 7.3.4

Lemma 7.3.6 Let IIiEI

243

PRESERVATION THEOREMS

MATRICES AND ATLASES

242

E

(Di)iEI, then li(P)

E

D

Lemma 7.3.7 Let IIiEI Mi be a direct product of the (indexed set of) matrices (Mi )iEI, let / be an interpretation ofL in IIiEI M i , and let If! be some particular sentence such that l(lf!) is undesignated in the direct product. Then for some i E I, there exists an intelpretation Ii such that for eVelY sentence p, if I(p) is designated in IIiEI M i , then li(P) E Di, and /i(lf!) 5i D;. Proof We know from Lemma 7.3.6 that Ii = 10 1[i has all but the last mentioned property with respect to any of the component matrices Mi. But since I(lf!) is undesignated in the direct product, there must be some component at which it is un designated, namely (/(If!))i. The matrix Mi is then the desired matrix for the lemma. D

7.4

Preservation Theorems

The "proto-preservation" theorems of the preceding section allow us to quickly establish a number of results concerning the preservation of "matrix validity" in various senses. We shall focus on the preservation of validity in the unary, asymmetric, and symmetric consequence senses. Recall from Chapter 5 that each interpretation I in a matrix M gives rise to a valuation VI> where v/(p) = tiff l(p) E D. The set of admissible valuations according to M, VM, is then defined as the set of all such VI' Where K is a class of (similar) matrices, we can then define the set of admissible valuations according to K as U{VM : M E K}. Using these sets of admissible valuations, we can define validity in M in various senses. These definitions make explicit notions that were implicit in the previous chapter and the previous section. First we define (unmy assertional) validity ("tautologyhood") in a matrix M as v(p) = t for every v E VM, i.e., /(p) E D for every interpretation in M. This is customarily denoted by some notation such as I=M p. This can be extended to a class K of (similar) mahices in the obvious way, requiring that I=M p for every ME K, which is the same as requiring that v( p) = t for each v in the set VM of admissible valuations according to K. We write I=K p.

PRESERVATION THEOREMS

MATRICES AND ATLASES

244

Asymmetric matrix consequence is defined in a corresponding manner. r I=M cjJ iff for every v E VM, if v(y) = t for every y E r, then v(cjJ) = t, i.e., iff for every interpretation I in M, if I(Y) E D for all y E r, then l(cjJ) E D. Again this can be extended to a class of matrices K, r I=K cjJ iff for every v E VK, if v(y) = t for all y E r, then v(cjJ) = t, i.e., iff for every ME K and interpretation I in M, if I(Y) ED for all y E r, then l(cjJ) E D. Symmetric matrix consequence is defined analogously. r I=M A iff for every v E VM, if v(y) = t for every y E r, then v(8) = t for some 8 E A, i.e., iff for every interpretation I in M, if I(Y) E D for all y E r, then 1(8) ED for some 8 E A. Again this can be extended to a class of matrices K, r 1= K A iff for every v E VK, if v(y) = t for all y E r, then v(8) = t for some 8 E A, i.e., iff for every M E K and interpretation I in M, if I(Y) ED for all y E r, then 1(8) ED for some 8 EA. lt is easy now to see, for example, that unary assertional validity in a matrix is preserved under both weak and strong submatrix. Thus, suppose that M' is a weak submatrix ofM and I=M cjJ. Consider an arbitrary interpretation t' in M'. By Lemma 7.3.1, I' is also an interpretation in M and so t'(cjJ) ED' since t'(cjJ) ED. Thus I=M' cjJ. We shall leave to the reader the routine derivation of other preservation theorems from their corresponding "proto-lemmas." The results are summarized in the following table. A check of course means that validity is preserved, and a cross means it is not. A small check (./) is an immediate consequence of a large check (v') in its vicinity, just as a small cross (x) is an immediate consequence of a large one ( X)· Note that there are only negative results in nine places, and because of the immediate consequences noted, these boil down to just four, which are labeled in order. We shall address these below.

245

We can produce an actual counter-example by letting M' be the three-valued Lukasiewicz matlix and letting M be a "designation expansion," i.e., the same except that we extend the set of designated values to be {l, ~} (in effect "designated" means "nonfalse"). It may seem odd that M' and M are the same on the algebraic component, but an algebra surely counts as a subalgebra of itself. Consider the sentence p V "p, and assign p the value ~. The interpretation of p V .,p is itself then ~, and thus p V "p is rejected in M' by this interpretation. But this interpretation no longer rejects p V "p in M, and indeed no interpretation does since the assignment of either 1 or 0 always gives p V .,p the value 1. Counter-example 2. Asymmetric consequence is not preserved under weak homomorphic images. We in fact show this for singleton left- and right-hand sides. Let us see where the argument for preservation breaks down. Suppose h is a matrix homomorphism of M onto M', and that cjJ f!:M' lJf. Then there is an interpretation t' in M' such that t' (cjJ) E D' and t' (lJf) ¢ D'. One can then argue that if one constructs an assignment IO(p) E h- 1(t' (p)) for each atomic sentence p, that the resulting interpretation in M will have the property that I(X) E h- 1 (t' (X)) for all sentences cjJ. So far, so good. But then we have just been arguing the algebraic situation, we have not yet turned our attention to the question of the designated values. Consider now l(cjJ) and l(lJf). We know that l(lJf) is not designated, for if l(lJf) E D, then since h preserves designation h(l(lJf)) = 1'(lJf) ED', but this is false. But there is no way to show that z( cjJ) is designated. We would try to argue that if l(cjJ) ¢ D, then h(z(cjJ)) = 1'(cjJ) ¢ D'. But this assumes that h preserves non-designation. Obviously what is required is that h is a strong homomorphism. We now produce an actual counter-example, again by fussing with the three-valued logic. For this purpose we add a new unary connective \/ with the table:

Unary Asymmetric Symmetric Submatrix

Homomorphic image

Inverse hom. image

Direct product

Weak Strong Weak Strong Weak Strong Weak Strong

Xl

x

x

./

v'

v'

X2

x

./

v'

v'

x:

x

x

./

./

v'

v'

v'

X4

v'

./

./

v'

Counter-example 1. Unary assertional consequence is not preserved under weak submatrices. Suppose that M' is a weak submatrix of M and (contrapositively) that f!:M' cjJ. Suppose that I' is an interpretation in M' such that I' (cjJ) ¢ D'. In terms of the relationship of the designated sets, all that we know is D' ~ D. For a strong submatrix we would have D n M' = D', and could argue then that t'(cjJ) ¢ D, as required. But this breaks down in the first step when we only have the inclusion the one way.

1 1

'2 2

1

'2

1

'2

1

'2

On a Bochvar reading of the value ~ as "garbage" this can be interpreted as "anything in, garbage out." We again use the trick of letting M and M' agree on their algebraic component (identity is an isomorphism and hence a homomorphism), and let this algebra have simp.ly the operation \/ defined by the above table. Thus M = M' = {I, ~, O}. Again we SImply expand the designated set (but in the converse direction). Thus D = {I} and D' = {I, ~ }. It is clear that \/p f!:M' q since we can assign q the value 0, and no matter what value is assigned to p, \/ p will take the designated value ~. But in M this value is not designated, and so there is no invalidating assignment. Incidentally, the reader may be bothered by the "artificiality" of the operation \/, which is just the constant ~ function. Such a reader may take the three-valued Lukasiewicz matrix for -+ but make ~ the only designated value. Call this M. M' is the same except that we make the value 1 designated as well. Clearly p -+ p I=M q since p -+ p

246

always takes the undesignated value 1, and yet p -'; P ft:M' q since in M' 1 is designated and q can be assigned the undesignated value 0. . Counter-example 3. Unary assertional consequence is not preserved under weak znverse homomorphic image. This is a simple reinterpretation of the situation of counterexample 2. M has {I} as the set of designated elements, and M' has {I, ~ }. M' is of course a weak homomorphic image of M under the identity homomorphism, and p V "'p is valid in M' and yet not in M. Counter-example 4. Symmetric consequence is not preserved under direct product. Indeed, we can show this for an empty left-hand side. Consider the direct product 22 of the two element Boolean algebra 2, with 1 as its only designated value. Its elements are {(1, 1), (1,0), (0, 1), (O,O)}, and the only designated v~lue is (1, 1). The cons~quence f- p,"'P is clearly valid in 2, since ..,p takes the Opposl~e value ~ro~ Pi as~urm~ th~t always one of p, ..,p will take the value 1. But f- p,"'P IS not vahd m 2 , smce If p IS assigned (1, 0) then ..,p is interpreted as (0, 1), and so neither ends up designated. Exercise 7.4.1 Complete all of the reasoning needed to justify the various checks and crosses in the table above. Exercise 7.4.2 Prove the claims about the matrices in the three examples in Section 5.12. (Hint: First relate the notions of strict, strong and weak equivalence of matrices to the notions at the beginning of this section.)

7.5

Varieties Theorem Analogs for Matrices

In Chapter 2 we stated Birkhoff's Theorem 2.10.3 and presented its proof. Recall .that this theorem links a proof-theoretical characterization of a class of algebras (equatIOnally definable) with a model-theoretic characterization of the class (closure under subalgebra, homomorphic image, and direct product). This gives a purely algebraic answer to the question: when is a class of algebras axiomatizable? Obviously the same t~pe ~f question can be asked of classes of matrices. We can ask when a ~lass of matr~ces IS characteristic for a "logic." This question is actually several questIOns, dependmg on what one takes a "logic" to be. As we saw in Section 6.2 there are various alternatives. We shall present answers to this question for unary assertional, asymmetric, and symmetric consequence logics. The first was shown by Blok and Pigozzi (1986), a~d th~ latter two by Czelakowski (1980a, 1983). Czelakowski (1980b), and Blok and Plgozz1 (1992) are also relevant. We omit proofs, which can be found in the works cited. A couple of preservation theorems not proven in Section 7.4 are given as exercises. 7.5.1

VARIETIES THEOREM ANALOGS FOR MATRICES

MATRICES AND ATLASES

247

Note that designation extension is related to our notion of A being a weak submatrix, except the two underlying algebras moe required to be the same. 7 Theorem 7.5.2 (Blok and Pigozzi 1986) A class of matrices K is the class of all matrices satisfying a unmy assertionallogic iff K is closed under (weak) direct products, strong submatrices, inverse strong homomorphic images, designation expansions, and strong homomOlphic images. Applying these operations finitely often, and in a certain order, suffices, as is summarized in obvious symbols: Matr(f-) = HstD HstSstP(Matr(f-».

We know from Section 7.4 that unary assertional consequence is closed under inverse strong homomorphic image (expansion), strong homomorphic image (contraction). We also know from Section 7.4 that since it is closed under weak submatrix, then it is closed under designation expansion. Designation extension plays a role for matrices similar to that of homomorphic image in equational systems. Exercise 7.5.3 Show that every weak homomorphic image of a matrix M can be obtained as a strong homomorphic image of a designation extension of M. Blok and Pigozzi actually state their theorem using notions of "contraction" and "expansion," which are equivalent to strong homomorphic and inverse strong homomorphic images: Definition 7.5.4 (Blok and Pigozzi, 1986) A matrix M = (A, D) is an expansion of a matrix M' = (A', D') (If there exists an onto algebraic homo117Olphism h : A I--l- A' such that h- I (D') = D. Conversely, M' is said to be a contraction of M. Blok and Pigozzi define M' to be a relative of M iff M' can be obtained from M by a finite number of contractions and expansions. This defines an equivalence relation, and, as Blok and Pigozzi observe, plays a role in the model theory of unary assertional logics similar to that played by isomorphism in equational systems. Blok and Pigozzi show that every relative of M can be obtained as an expansion of a contraction of M. Exercise 7.5.5 Show that if M is a relative of M', then f-M 7.5.2

= f-~.

Asymmetric consequence logics

We turn now to the case of an asymmetric consequence relation. It turns out that in characterizing the closure conditions on a class of matrices we have to work with a notion more complicated than direct product, namely an m-reduced product of matrices. We first introduce the simpler notion of a reduced product of matrices.

Unmyassertionallogics

We turn first to the question of characterizing the "vmieties" of matrices for unary assertionallogic. We first provide a needed definition: Definition 7.5.1 A matrix M' = (A', D') is a designation extension of a matrix M = (A, D) (fl(l) A' = A and (2) D ~ D'.

7Blok and Pigozzi actually speak of "filter extensions," but they do not necessarily mean a filter in the lattice sense. They mean any designated set of elements that satisfies the axioms and inference rules of some given underlying logic. (In a lattice this would most naturally form a filter, hence the terminology.) In fact they do not need this for their proof of the varieties theorem analog, and so, for the sake of both simplification and comparison to other results, we do not here impose any such requirement on the designated elements. This is why we prefer the term "designation expansion."

248

MATRICES AND ATLASES

Definition 7.5.6 Let I be a set of indices, and let ~ be a filter of sets of its power set rtJ(I). The product of matrices reduced by ~ (in symbols IT!l MiEl) is defined as the quotient matrix of the ordinary direct product ITiEl MiEl' produced by the congruence relation =;5 induced by ~ as follows: (ai liEf =;5 (b i liEl iff {i : ai = bi} E ~. An element [(ai liEl] =;5 is designated (ff {i : ai E Di} E~. Remark 7.5.7 This is to say that we first form the reduced product of the underlying algebras (as in Section 2.16) and then characterize designation. Note that just as the congruence relation can be understood as saying that (ailiEf and (biliEI are "almost everywhere" identical, designation of [(ai )iE[] =;5 can be understood as saying (ai liEf is "almost everywhere" designated in the underlying direct product. There are two examples of special interest. First, when ~ is just the power set of I, we obtain the ordinary direct product. Second, when ~ is a maximal filter, we obtain what is called an ultraplVduct (~ is usually called an ultrafilter in this context). Definition 7.5.8 An ultraproduct of a similarity class of matrices is a plVduct of these matrices reduced by a maximal filter. We also have use for the following abstraction: Definition 7.5.9 ~ is said to be an m-filter if besides being closed under upward inclusion and finite intersections, it is also closed under infinite intersections as long as the cardinality of the collection of sets being intersected has cardinality less than m. An m-reduced product of matrices is just a product of matrices reduced by m-filter~. We can now state the theorem of Czelakowski, subject to certain technical conditions on the cardinals m, which we shall subsequently explain. Any reader who wants or needs to skip over these technical considerations should go immediate to the corollary, which applies to most "real-life" logics. Theorem 7.5.10 (Czelakowski, 1980a) A class of matrices K is the class of all matrices satisfying an asymmetric consequence relation f- iff K is closed under (strong) submatrices, strong homomorphic images, inverse stlVng homomorphic images, and m-reduced plVducts, where m is an infinite regular cardinal weakly bounded by the cardinality of f- and the successor of the cardinal of the set of sentences in the underlying language. In obvious symbols:

We now tum to the technical conditions on m. First we explain the notion of the cardinality of an asymmetric consequence relation. It is the least infinite cardinal n such that the consequences of any set of sentences r can be obtained by taking the union of the consequences of sets of sentences each of whose cardinality is strictly smaller than n. The cardinality of f- is always less than or equal to the successor cardinal of the set of sentences in the underlying language. Observe that when the language is denumerable and f- is compact, its cardinality is ~o.

CONGRUENCES AND QUOTIENT MATRICES

249

A regular cardinal m is one such that it "cannot be surpassed from below," i.e., given a family of cardinals {mi} iEl such that each member mi < m and such that the family itself has a cardinal less than m, then LiEl mi < m. This is a fairly technical notion but let us note that the first infinite cardinal ~o is regular. This last observation, together with observation of the preceding paragraph, leads to the following, much less technical version of the theorem for the ordinary case where the underlying language is denumerable. Corollary 7.5.11 (Czelakowski, 1980a) Let f- be a compact asymmetric consequence relation on a denumerable sentential language. Then the theorem above can be simplified by replacing m-reduced plVducts with reduced plVducts. The following suffices: Matr(f-) = HstHstSstPR(Matr(f-».

PIVO! The observations already noted show that ~o satisfies the technical conditions, so we simply add that an ~o-reduced product is simply a reduced product, since any filter is closed under finite intersections. D Exercise 7.5.12 Show that an asymmetric consequence relation is preserved under reduced products. 7.5.3

Symmetric consequence logics

Czelakowski extended his theorem to apply to symmetric consequence relations. The conditions are similar, except closure under strong homomorphic images is dropped, and "Ultra-product" is substituted for "m-reduced product," which means the technical condition disappears. This gives a prettier statement: Theorem 7.5.13 (Czelakowski, 1983) A class of matrices K is the class of all matrices satisfying a symmetric consequence relation f- iff K is closed under ultraproduct, (strong) submatrices, strong matrix homomorphisms, and inverse strong homomorphic images. The following suffices: Matr(f-) = HstHstSst Pu(Matr(f-».

Exercise 7.5.14 Show that any symmetric consequence relation is closed under ultraproduct. We close this section by simply raising the question about how to extend the results above to a larger class of similar results. There various ways of presenting logics which we have not considered in this section-for example, unary refutational systems (cjJ f-). Not only are there a lot of ways of presenting logics, but there are also a lot of closure conditions on classes of matIices floating around, and it would be interesting to explore which combinations of them correspond to which "vmieties" oflogics (thus completing the pun set up by the title of Section 6.2: "The Varieties of Logical Experience"). 7.6

Congruences and Quotient Matrices

We recall from Section 2.6 that a congruence on an algebra is just an equivalence relation = that respects the operations of the algebra. In defining a congruence on a matIix

MATRICES AND ATLASES

250

M we obviously want it to be a congruence on the underlying algebra alg(M), but the question is how much the relation should respect designation. The natural requirement is (1) if a

E D

and a == b, then bED.

Because of the symmetry of ==, this is equivalent to requiIing that if a == b, then either both a and b are designated or else both a and bare undesignated. We shall call a congruence satisfying (1) a strong congruence, and one that is merely a congruence on the underlying algebra a weak congruence. We shall single out neither for the epithet congruence simpliciter, the problem being that we feel a certain tension: the strong congruence is certainly the one with the greater claim to naturalness, and yet (as we shall see) it has an affinity with the so-called strong homomorphism. Given a congruence == of either type, we can define a quotient matrix Mj == as follows: The algebraic part of Mj == will be the quotient algebra alg(M) j == as defined in Section 2.6, i.e., its elements will be the equivalence classes [a] detelmined by a E M, and operations are defined on them by way of representatives. So, for example, taking a binary operation: (2) [a]

* [b] = [a * b].

The important question is which equivalence classes we are to count as designated. For a weak quotient matrix, its set of designated elements, D j ==, will consist of those cliques [a] such that there is some a' E [a] (i.e., a' == a) such that a' E D. (This is in effect a special case of (Q) of Section 2.6, when R is a "unary relation".) For a strong quotient matrix we shall define the set of designated elements exactly the same way, but the difference between the weak and strong notions comes out in our requiring for a strong quotient matrix that the equivalence relation == be strong. It just turns out (because of (1)) that a clique [a] is designated iff all a' E [a] (i.e., all a' == a) are such that a' E D. How can we describe the notion of congruence with some intuitive logical vocabulary? Well, one way to think of congruence is as provable equivalence in some theory, and since theories should be closed under provable equivalence, this quickly motivates the requirement (I) above. The story can go on to give some intuitive force to the notion of a quotient matrix. Many times, propositions that are non-equivalent in some "inner theory" are said to be equivalent in some "outer theory" (proper extension). Thus, for example, propositions that are not logically equivalent might be said to be mathematically equivalent. And sentences that are not mathematically equivalent, might be said to be physically (in physics) equivalent. And sentences that are not physically equivalent might be said to be biologically equivalent, etc. And at the very beginning of the chain, we can have sentences that are not identical (say p and p 1\ p) even though they are logically equivalent. Now if one wants, one can just leave the situation at this level of description. But if one is interested in reifying what it is that equivalent propositions have in common (what makes them the "same"), one can go on to say that they express the same proposition "really," i.e., in some more

CONGRUENCES AND QUOTIENT MATRICES

251

elaborate theory. Thus it is a familiar quasi-nominalistic move to identify propositions with classes of logically equivalent sentences, and it is certainly not too funny a way of talking to say that two logically distinct propositions express mathematically the same proposition, etc. The partitioning into equivalence classes to form quotient matrices can be understood to be just a convenient set-theoretical device to reify what it is that equivalent propositions (or sentences) have in common. With some story like the above, how could we ever motivate some requirement weaker than (l)? Possibly the designated set could be the theorems of the "inner theory." Maybe D is some set of mathematical theorems (or truths) and == is physical equivalence. Assuming some standard reductionist imagery, we could imagine that a proposition a is equivalent to some mathematical fact a', and on that account the physical proposition [a] ought to be designated. It is clear that every strong homomorphism determines a strong congruence. Thus, count two elements as congruent when they are carried into the same element. We know from Section 2.6 that this is a congruence on the algebraic part of the matrix. But since a strong homomorphism can never carry a designated element and an undesignated element to the same value, it is clear that this congruence also respects designation in the sense of (2) above, and that we have then a strong congruence. Conversely, every strong congruence determines a strong homomorphism. As we know, once again from Section 2.6, the canonical homomorphism, which carries an element a into the class [a] of the quotient matrix, is an algebraic homomorphism. It is also clear that the canonical homomorphism respects designation, since if a is designated, then [a] is designated by virtue of our decision on how to designate cliques in the quotient. And if a is undesignated, then, by (2), all elements congruent to a are undesignated, and thus each element of [a] is undesignated and so [a] is undesignated. Playing with these facts, one can establish the following theorem. Theorem 7.6.1 (Strong homomorphism theorem for matrices) Every strong 11omomOlphic image ofM is isol1ZOIphic to a strong quotient matrix ofM. Exercise 7.6.2 Give a detailed proof of the above. What, then, of weak homomorphisms? Clearly weak congruences determine weak homomorphisms, since the canonical homomorphism again carries a designated element a to the designated clique [a]. (Designation of the clique requires only that olle member be designated.) But it is not necessarily true that weak homomorphisms determine even weak congruences. The problem is that a weak homomorphism can carry an undesignated element a to a designated element a', even though no other designated element is carried to a'. An obvious fix to this problem is to restrict our attention to homomorphisms that are (minimally) faithful in the sense of Section 2.5 (thinking of "designation" as a unar·y relation). What this means is that in the problem case described above, a' must also have some other designated element b carried to a. Incidentally, notice that the canonical homomorphism onto a weak quotient matlix is always faithful, since [a] is made designated only when there is some member b that is designated. But then b is the desired designated pre-image. The following is easily proven from the discussion above.

MATRICES AND ATLASES

252

Theorem 7.6.3 (Weak homomorphism theorem for matrices) EvelY weak faithful homomorphic image ofM is isomorphic to a weak quotient matrix ofM. Exercise 7.6.4 Fill in the details. Consider the set Cst(M) of all strong congruences on a matrix M = (A, D). As with algebras, the smallest congruence is just the identity relation restlicted to A. But this time the largest "natural" congruence cannot in general be the universal relation A x A (unlike the case with algebras), for if A has at least two elements and D f:. A, then the universal relation would identify a designated element with a non-designated element. Still, we have: Theorem 7.6.5 The set Cst(M)of all strong congruences on a matrix M = (A, D) forms a complete lattice. Proof We leave this to the reader. In virtue of Corollary 3.7.5, all that needs to be checked is that the intersection of relations () "compatible" with D (if a(}b, then a E D only if b E D)8 is also a relation compatible D, and similarly with the transitive closure of the union of relations compatible with D. 0 Blok and Pigozzi answer the question as to how to characterize the largest strong congruence on a matrix. As they point out, their solution stems from Leibniz's principle of the identity of indiscernibles. It is better rephrased in this context as congruence of indiscernibles, because the idea is to define two elements to be congruent just when they are extl'insically indiscernible in terms of their roles in the matrix. The Leibniz congruence has to do with indiscernibility by way of predicates (by which is meant a relation). There are just two natural atomic predicate symbols in the first-order language used to describe a matrix: one is a unary predicate for membership in the designated set, and the other is the binary predicate for identity. This discussion assumes that we have only the first, which we denote by D[x]. Definition 7.6.6 An n-ary predicate (relation) P is first-order definable over a matrix M = (A, D) iff there is afonnula (XI, ... , Xll, YI, .. ·, Yk) offirst-order logic containing only the predicate D and function symbols corresponding to the various operations of A, and there are elements C], ... , Ck E A, such that for all ai, ... , all E A,

Definition 7.6.7 Let A be an algebra and let D over D is defined by QAD

= {(a, b) : Pea)

~

~

A. The Leibniz congruence on A

PCb) for every definable predicate P}.

CONGRUENCES AND QUOTIENT MATRICES

253

Exercise 7.6.8 Prove that QAD is a strong congruence. Remark 7.6.9 Note that the first-order formula (XI, ... , Xn , YI, ... , Yk) in Definition 7.6.6 can have all of the usual connectives and quantifiers in it, as well as various occurrences of the predicate D. We signal this by using the capital letter <1>, reserving the lower case p, as has been our practice, for sentences (really terms) in the sentential language appropriate to M. There is a fussy point to be made about this. Given a sentence of the sentential language, we have been writing it as P(PI, ... , Pn, ql, ... , qk), and we have been writing the corresponding first-order term as P(XI, ... , XI!, YI, ... , Yk). As the reader can easily see, the two terms m'e "isomorphic" except for the choice of the symbols (variables and operation symbols). For convenience, let us assume that the terms are written in the same language, with "x I" and "PI" just being two names in our metalanguage for the same symbol, and similarly with the other matching symbols. Let us consider just atomic formulas, i.e., those first-order formulas of the form D[P(XI, ... , X Il , YI,···, Yk)]. Remember that such a formula says that

Definition 7.6.10 We shall say that an n-ary predicate (relation) P is atomically definable over a matrix M = (A, D) iff there is an atomic formula D [p(x I, ... , x n , YI, ... , Yk)] offirst-order logic containing only the displayed occurrence of the predicate D and function symbols corresponding to the various operations of A, and there are elements C], ... , Ck E A, such thatfor all al, ... , all E A,

Definition 7.6.11 Let A be an algebra and let D ~ A. The atomic Leibniz congruence on A over D is defined by

Q% D =

{(a, b) : Pea) ~ PCb) for every atomically definable predicate P}.

It turns out that the restriction to atomic formulas in Definition 7.6.11 makes no difference. The predicates in Definition 7.6.7 can without loss be restricted to those definable by atomic formulas, i.e., formulas of the form D[P(XI, YI, ... , Yk)].

Lemma 7.6.12 QAD

= Q% D.

Proof This is an immediate consequence of the fact that the atomic replacement is equivalent to complex replacement (cf. Theorem 2.6.5). 0

The function QA(D) = QAD is defined on all subsets of A and is called the Leibniz operator on A.

Blok and Pigozzi use Lemma 7.6.12 to prove the following characterization of the Leibniz congruence in terms of other strong congruences:

SNote that when () is a congruence, it is symmetric, and so this is the same as requiring "two-way respect": if a(}b, then a E D iff bED.

Theorem 7.6.13 For any matrix M = (A, D), QAD is the largest strong congruence in the lattice of strong congruences on M.

Proof Let 0 be any strong congruence on M, and assume (a, b) EO. Let pep], q], ... , qk) be any sentence, and let Z be any interpretation in A. Since 0 is a congruence, we have for C] , ... ,Ck E A,

Since 0 is a strong congruence, this means: pA(a, C], ... , Ck) ED<=> pA(b, C], ... , Ck) ED.

Hence (using Lemma 7.6.12) we have (a, b)

E

QAD.

o

Let us consider the special case of a Leibniz congruence on an algebra of sentences. We know from Section 6.12 that theories correspond to the designated subsets, so we can write QsT to obtain a certain congruence on the algebra of sentences (the subscript is often omitted), namely the largest congruence that is compatible with T. Dividing the set S of sentences by this congruence gives the quotient algebra S I nT, and then dividing out the theory T as well gives us the matrix (S I nT, TinT). We .n~ed a na~e for this matrix. Note that this is neither the Lindenbaum algebra (because It IS a matnx, and because QT is generated from "above" rather than below), nor is it the Lindenbaum matrix (because the elements of the Lindenbaum matrix are sentences, not equivalence classes of sentences). We shall call it the Blok-Pigozzi matrix (detennined by T).

Theorem 7.6.14 p QT lJI (!ffor all intelpretations z in the Blok-Pigozzi matrix (SI nT, TinT), z(p) = Z(lJI)·

Proof The direction from right to left is a kind of completeness. Instantiate I to be the canonical interpretation I(X) = [X]nT. Then assuming z(p) = l(lJI), we have [p]nT = [lJI] nT, and hence p QT lJI· The direction from left to right is a kind of soundness result. Let 1(p) = [p'] nT and l(lJI) = [lJI'] nT. We first observe that we can choose p' and lJI' to be substitution instances of p and lJI. The reason is that we can consider just the interpretation of the atomic sentences I(pj) = [xIlnT, ... ,z(Pi) = [Xi], ... , and pick a sentence P' from each equivalence class, being careful to pick the same sentence for identic~l equivalence classes. This induces a substitution (J : CJ(Pi) ~ (p;), and in gen~ral (J(X) = x(p]1 p']' ... , Plllp;l)' where P], ... , Pil are all the atomIC sentences occurrmg in X. It is easy to prove by induction that z(X) = [x(p]lp'l' ... ,Plllp;l)] = [(JX]. N ow we can complete the proof of soundness. Since p QT lJI, we have p' QTlJI', and 0 this means [ifJ']nT = [lJI'] nT, i.e., I(p) = z(lJI).

7.7

255

THE STRUCTURE OF CONGRUENCES

MATRICES AND ATLASES

254

The Structure of Congruences

Whether we are talking of weak or strong congruences, they can be regarded as sets of ordered pairs, and ordered by set inclusion. Then =] ~ =2 means intuitively that =] is a "stronger" (stlicter) relation than =2. Somewhat in the face of English usage, the "stronger" relation is the "smaller" of the two, the idea being thatfewer pairs satisfy the stronger relation.

The inclusion relation clearly has the following properties, for arbitrary sets x, y, and z: (l) x (2) x (3) x

~

x

~

y and y ~ x imply x

~

y and y ~ z imply x ~ z

(reflexivity);

=y

(antisymmetry); (transitivity).

Any relation with these properties is called a partial order. Partial orders, and some related notions, were introduced more fully in Chapter 3, but we shall review pertinent facts about them as needed. Let £(M) be the set of all equivalence relations on (the carrier set of) M, and similarly let Cw(M) and Cst(M) be, respectively, the sets of all weak and strong congruences on M. It is easy to see that given any non-empty subset E of £(M), the intersection E is also an equivalence relation on M. It is also easy to see that the relation a E) b holds just when aOb for all 0 E E. Then, it is straightforward that E is reflexive since each 0 E E is reflexive, and similarly for symmetry and transitivity. If the members of E happen to be either all weak or all strong congruences, it is similarly easy to see that E will inherit the respective property. Thus £(M), Cw(M), and Cst(M) are all such that they are closed under non-vacuous intersections. Whenever U is a set with a partial order ~, given any subset S, we can ask whether S has a greatest lower bound (glb), i.e., whether there is an element 1\ S which is such that:

n

(n

n

n

(4) "Ix E S, 1\ S ~ x (lower bound). (5) Given any element u such that "Ix E S, II then u ~ 1\ S) (greatest lower bound).

~

x (i.e., given any lower bound

II

of S,

n

It is easy to see that for non-empty sets S, S has just these properties. We also have the dual notion of the least upper bound (lllb) of S, which is an

VS satisfying: "Ix E S, x ~ VS (upper bound).

element

(6) (7) Given any element II such that "Ix E S, x then V S ~ u) (least upper bound).

~

u (i.e., given any upper bound u of S,

It is easy to see that £(M) (and also Cw(M) and Cst(M» always contains the glb of any subset and that this is just intersection. We now address the question of whether £(M) (and also Cv(M) and Cst(M» always contains all the lubs of its subsets. In answering this question, it is important to note that the union of a bunch of equivalence relations is rarely itself an equivalence relation. This is because we may have a =] band b =2 c, and yet have no equivalence relation in the bunch so that a = c. The obvious answer to this problem is to take the transitive closure, i.e., the smallest transitive relation that includes the union. What this amounts to in practical tenns is that we shall say that a =E b iff there is some sequence of elements (possibly null) X], ... ,Xi,Xi+], ... ,XJc, and of equivalence relations 0], ... , Oi, Oi+], ... , Ok+] E E, such that (8) aO]x], ... ,XiOi+]Xi+], ... ,XkOk+]b.

256

THE CANCELLATION PROPERTY

MATRICES AND ATLASES

It is then easy to see that =E is an equivalence relation. It is equally easy to verify that if each e E E respects D, then =E respects D (respect for the operations or for designation just transmits itself across the chain (8)). Thus it is clear that each of [(M), Cw(M), and Cst(M) is closed with respect to the operation =E on non-empty subsets E. It is clear that =E is the lub of E, VE, in any of [(M), Cw(M), or Cst(M). So far we have been talking about taking glbs and lubs of non-empty sets. What happens when E = 0? Then, among the equivalence relations [(M), the lub V0 must be an equivalence relation that is included in every equivalence relation which is an upper bound of every equivalence relation in 0. But since there are no equivalence relations in 0, this means that every equivalence relation is such an upper bound, and so V 0 must be included in every equivalence relation on M. But this is just the identity relation (restricted to M), since each equivalence relation must be reflexive. Similar considerations give the same conclusion for either Cw(M) or Cst(M), the point being that identity clearly respects both operations and designation (indeed, presumably indiscemibility in all respects). Identity (restricted to the elements of the matrix) is of course the strictest equivalence or congruence, in any sensible sense of equivalence or congruence, and as such is at the very bottom of any of [(M), Cw(M), or Cst(M). But what of the largest element? We can quickly see that in the cases of [(M) and Cw(M) the largest element is just the universal relation on M, M x M (the relation that holds between any two elements of M). But this relation obviously does not respect designation (assuming that the matrix has at least one designated and one undesignated element), and so does not count as a strong congruence. Is there, then, a weakest strong congruence? The answer is clearly yes, since we know that every non-empty subset of Cst(M) has a lub, and so in particular, if we take the lub VCst(M) of the whole set of strong congruences we obtain our desired weakest strong congruence. Let us denote it by /-l. It should now be clear what happens when we take the glb of 0. In the environment of each of [(M), Cw(M), and Cst(M) we obtain their top element. In [(M) and Cw(M), this is the universal relation, whereas in Cst(M) it is something stronger. A partially ordered set that contains glbs and lubs for all of its subsets is called a complete lattice. We summarize all of the above discussion in the following theorem.

Theorem 7.7.1 Given a matrix M, its set of equivalence relations [(M), set of weak congruences Cw(M), and set of strong congruences Cst(M) are all complete lattices (with ~ the partial order). In each case, for non-empty subsets, the glb is intersection and the lub is transitive closure. In each case the identity relation (restricted to M) is the bottom element (and the lub of 0). In the case of [(M) and Cw(M) the top element is the universal relation (restricted to M), M x M. Remark 7.7.2 Clearly [(M) ~ Cw(M) ~ Cst(M). The identity map thus gives a kind of embedding of the complete lattice of equivalence relations into the complete lattice of weak congruences, and that, in tum, into the complete lattice of strong congruences. Since intersections and transitive closures of non-empty sets depend only on the elements of the sets, and not on other elements in the environment (unlike the general case

257

of glbs and lubs), we have that this embedding preserves glbs and lubs of non-empty sets. Also clearly it preserves V0. Also 1\ 0 is preserved by the first embedding, but not the second. The reader may have some sense of mystery as to just what the top element "looks like" in the case of Cst(M). Let us introduce the notation I(a/ p) to indicate the "semantic substitution of the element a for the atomic sentence p in the interpretation I," i.e., the interpretation just like I except (perhaps) for assigning a to p. Theorem 7.7.3 (Czelakowski 1980a) Given a matrix M, let L be a language (with infinitely many generators) appropriate to M. Let /-l be a relation on M defined so that aJlb ({ffor all sentences cjJ ofL and all atomic sentences p ofL and all interpret ations I, l(a/p)(cjJ) = l(b/p)(cjJ). Then /-l is the weakest congruence on M. Exercise 7.7.4 Prove the above theorem. 7.8

The Cancellation Property

Recall from Section 6.11 that a unary assertionallogic always has a characteristic matrix, namely its Lindenbaum matrix. We saw in Sections 6.12 and 6.13 that formal asymmetric and symmetric consequence logics also always have a characteristic semantics that can be defined in terms of a given set of "propositions," but our proof had us looking at the Lindenbaum atlas (with many different designated subsets) rather than just at the Lindenbaum matrix (with its single designated subset). In this section we discuss whether, and under what circumstances, we are forced to an atlas instead of just a matrix. We prove a theorem due to Shoesmith and Smiley (1978) giving a necessary and sufficient condition for a broad class of symmetric logics having a charactelistic matrix. There is an analogous (and simpler) result of Shoesmith and Smiley (1971), proven for asymmetric logics, that we shall examine after we look at the symmetric version. Before stating the theorem, we need to explain a key notion called cancellation. Following Shoesmith and Smiley, we say of two sentences cjJ and Iff that they are disconnected if they share no atomic sentences, and we shall say of two sets of sentences r and I:::.. that they are disconnected if for each cjJ E r and each Iff E 1:::.., cjJ and Iff are disconnected. Finally, we shall say of a family of sets of sentences (ri ) iEI that it is disconnected if for each j, k E I with j =f. Ie, r j is disconnected from r Ie. We then say of a symmetric logic that it has the cancellation property if and only if whenever (ri U I:::.. i ) is a disconnected family of sets of sentences such that Uri f- U I:::..i, then for some i, ri f- I:::..i. The cancellation property is a quite natural condition for a logic. The quick intuitive idea is that there can be no real logical interaction between formulas in sets rj and I:::..Ie with different indices, since the formulas share no content (except for degenerate cases such as when some sentences of rj are contradictory, or some sentences of I:::..j are valid), so that all of the "action" can be "localized" at some pair (ri, I:::..i) which

258

MATRICES AND ATLASES

THE CANCELLATION PROPERTY

presumably share content (or else we are back in one of the degenerate cases mentioned above, in which case it can be degenerately localized). It is easy to prove the following lemma.

Lemma 7.8.1 If a symmetric logic has a characteristic matrix, then it has the cancellation property. Proof Let L be a symmetric logic characterized by the matrix M, and let be a disconnected family. Suppose for each i, not (ri I- Ji i ). Then for each i there is some interpretation Ii that assigns a designated value to each sentence in r i and yet assigns an undesignated value to each sentence in Jii. Since the value assigned to a sentence depends only on the interpretations of the atomic sentences that occur in it, the valuations Ii can be combined into a single interpretation I so that I(¢) = li(¢) for ¢ E r i UJi i . (If ¢ is not in any ri UJii, define I on the atomic sentences in ¢ arbitrarily.) It is clear that I invalidates uri I- UJii as desired. D It turns out that under a suitable hypothesis about "stability" (the definition of which will be provided in the course of the proof) the converse of Lemma 7.8.1 also holds.

Theorem 7.8.2 (Shoesmith and Smiley) For a stable symmetric (formal) logic, a necessary and sufficient condition for it to have a characteristic matrix is for it to have the cancellation property. Proof Necessity is of course provided by Lemma 7.8.1. Recall that a symmetric logic is just a formal symmetric consequence relation 1-. We shall now sketch a strategy for proving sufficiency (somewhat different than that of Shoesmith and Smiley), and in the process uncover the needed definition of "stability." Let us collect together all of the pairs such that not eri I- Jii). The key idea is for each index i to make a copy of the set of sentences Si (for later convenience when invoking substitutions, we let one of these be S itself). We then let the elements of the matrix be the union of all these sets, and (as a first approximation) let the set of designated elements D be the union of the copies of the ris. We draw an appropriate picture (for the simple case of two pairs) in Figure 7.9.

:--.. r'1.' .. ' . '.

r'2

259

Before proceeding, we must correct the first approximation. In the final construction D will not simply be the union of the ris, but shall be a somewhat larger set. We shall have to show, reverting to the picture above, that Ji'1 UJi; is not a consequence of r'1 ur;, and then invoke the global cut property to partition S' (the union of all the copies-in the picture S1 US2) into the desired D and its complement (as the horizontal dotted line suggests). But consequence was defined only on the original set of sentences S, and so we must extend the definition to the set of copies S'. We do this as follows. The first thought is to define a new relation on S' so that r' 1-' Ji' iff there is a substitution cr so that r' = crer), Ji' = cr(Ji) , and r I- Ji. The problem with this is that it is pretty plain that 1-' does not satisfy dilution (the problem is that if we try to dilute before we substitute, the new items then become subject to substitution whether we want them to or not). So we just build into the definition of 1-' that it is the closure of I- under substitution and dilution, i.e., r' 1-' Ji' iff there exist r and Ji (subsets of S) and a substitution cr (defined on S') so that r' ;2 cr(r), Ji' ;2 cr(Ji), and r I- Ji. It is still not necessary that 1-' so amended be a consequence relation, but it might be. Clearly it satisfies overlap, but there is still the question of global cut. This brings us to the promised definition: Shoe smith and Smiley call a symmetric consequence logic Istable when it has the property that 1-' is always a symmetric consequence relation (for an arbitrary extension of the original language by new atomic sentences). We state some simple relationships between 1-' and 1-.

Fact 7.8.3 If rand Ji are sets of sentences of S, and r' and Ji' are the respective results of applying some one-one substitution cr (in S'), then r I- Ji iff r' 1-' Ji'. Fact 7.8.4 1-' has the cancellation property (given that I- has). Proof We leave the straightforward proof of Fact 7.8.3 to the reader. For Fact 7.8.4, we suppose that is a disconnected family of sets of sentences of S' and also that U r; 1-' U Ji;. By the definition of 1-', there exist sets rand Ji of sentences of S so that cr(r) ~

U r;,

cr(Ji) ~

U Ji;,

and r I- Ji.

Let r i = {¢: ¢ E rand cr(¢) E r;}, and let Jii be defined analogously. Then is a disconnected family since is. Note that U ri = rand U Jii = Ji, and so we can apply cancellation (on r I- Ji) to obtain ri I- Jii (for some i). And so (again by definition of 1-') r; 1-' Ji;, as required. D Returning now to the proof of Theorem 7.8.2 and reverting to the picture above, we must show that it is not the case that r'1 U r; 1-' Ji'1 U Ji;. Suppose to the contrary that

.. ' Ji'

Ji'1 S "Before"

.'

"After" FIG. 7.9. Illustration for i = 2

2

r'1' r; 1-' Ji~, Ji; . Since is clearly a disconnected family, this means that by the cancellation property (and Fact 7.8.4, which entitles us to apply it), we must have either r'1 1-' Ji'1 or r'2 1-' Ji'2' But Fact 7.8.3 entitles us to remove the primes, obtaining

260

THE CANCELLATION PROPERTY

MATRICES AND ATLASES

f] f- al or f2 f- a2.

But this is contrary to our original choice of the pairs (fi' ai) such that not (fi f- ai). The argument above, although carried out for the simple case of two pairs, is completely general, and so we know that it is not the case that U f-' U The penultimate step of the construction is then to invoke the global cut property so as to partition S' into two sets, D' and - D', so that U ~ D', U ~ D', and not (D' f- -D'). The final stage of the construction is to consider the algebra of sentences S' defined on the sentences S', and outfit it with the designated set D' so as to obtain "the Shoesmith-Smiley matlix" (S', D'). (Note that despite the definite article, it is not unique, depending as it does on a choice of D'.) Interpretations in the matlix are just substitutions, so soundness follows as with the constructions from the previous chapter of the Lindenbaum matlix and the Scott atlas. (We leave to the reader to check that f-' is formal, given that f- is.) And completeness is guaranteed by the construction, for if there are sets of sentences fi and ai such that not (fi f- ai), then we know that there is a substitution (J so that (J(fi) ~ D' and O'(ai) ~ - D' ((J just assigns to each sentence its ith copy). D

f;

f;

a;. a; -

Remark 7.8.5 Note that the construction above leads to a larger cardinality than does the construction of the Lindenbaum matrix. Since one copy S; has to be made for each pair of sets (fi, ai), it is reasonably clear that even when S is denumerable, most generally one will be constructing "continuum many" copies of a denumerable set, and so the union will be non-countable (indeed, of the power of the continuum). The following is an easy application of Theorem 7.8.2. Corollary 7.8.6 (Shoesmith and Smiley) For a compact symmetric logic, a necessary and sufficient condition for it to have a characteristic matrix isfor it to have the cancellation property. Proof It clearly suffices to show that a compact symmetlic logic is stable, and this boils down to showing that f-' (as defined in the proof above) satisfies the cut property. We leave the details to the reader. D

We now briefly discuss the case of an asymmetric consequence relation. First, the cancellation property specialized to the asymmetlic case comes down to the following: (ACP) If f, U fi f- p, the family (fi) is disconnected, and f and {p} are both disconnected from each n, then f f- P unless some fi is "absolutely inconsistent" in the sense that n f- If/ for all sentences If/. Thus consider the family that has in it the pair (f, {p} ) as well as the pairs (fi' 0). To say that this family is disconnected is precisely to give the hypotheses about disconnection above in (ACP). And to say that the union of the first components has as a consequence the union of the second components is precisely to give the hypothesis that f, Ufi f- p. So applying the symmetric version of the cancellation property, we obtain that either f f- P or else some n f- 0. But this last is the same as fi f- a for all

261

sets of sentences a (dilution). The closest we can come to saying this for the case of an asymmetric logic is that fi f- If/ for all sentences If/, but this is just the definition of absolute inconsistency. We can now state and prove the analog of Theorem 7.8.2 for asymmetric logics. Theorem 7.8.7 (Shoesmith and Smiley) Given a stable asymmetric (formal) logic, a necessary and sufficient condition for it to have a characteristic matrix is that it have the property (ACP). Proof Of course an asymmetric logic is just a formal asymmetric consequence relation f-, and to say that f- is stable is just to analogize the definition for a symmetric consequence relation. This requires that if we extend the original set of sentences S over which f- is defined by arbitrary new atomic sentences (obtaining a new set of sentences S' :2 S), and define f' f-' p' iff there exist f and p such that f ~ Sand pES, and a substitution (J (defined on S') so that r' :2 (J(r), p' = (J(p), and f f- p, then the relation f-' is an asymmetlic consequence relation. The rest of the proof is entirely analogous to that of Theorem 7.8.2, except that the set D can be just the closure under f-' of all the sets D

f;.

Formal asymmetlic consequence relations differ markedly from their symmetlic cousins, as shown by the following somewhat surprising, but nonetheless trivial fact (due to Shoesmith and Smiley). Fact 7.8.8 All formal asymmetric consequence relations (compact or otherwise) defined on countable languages are stable. Proof Let f- be a formal asymmetric consequence relation. Let f-' be defined as in the definition of "stability." We must show that f-' is a formal asymmetric consequence relation. As with symmetric consequence, the properties of overlap, closure under substitution, and dilution are easy. We concentrate our attention then on the infinitary cut. Let us suppose then that f f-' 8, for each 8 E a, and that a, p f-' If/. By the definition of f-', there exists a countable set of sentences a' ~ a so that a', p f-' If/. (A substitution performed on a countable set leaves a countable set.) Now for each 8' E a', there exists (similarly) a countable set f' ~ f so that r' f-' 8'. Considering all the sentences in a' and in the sets f', we are considering a countable union of countable sets, which set theory tells us is again countable. It is clear that only a countable number of atomic sentences occur as generators of these countably many sentences, and so we can find a substitution (J that rewrites these in a one-one fashion as atomic sentences from the originally given countable language, and we can apply the in finitary cut there using f-. Reversing the substitution (and possibly diluting) gives us the desired infinitary cut for f-'. D

Corollary 7.8.9 For a countable asymmetric logic, a necessmy and sufficient condition for it to have a characteristic matrix isfor it to have the property (ACP).

262

7.9

MATRICES AND ATLASES

Normal Matrices

A matrix is normal (in the sense of Church 1956) if the set of designated elements forms a "truth set" with respect to the classical logical operations, i.e., (i) -a

E

D iff a ¢ D,

(ii) a 1\ bED iff a

E

D and bED.

Clearly this definition presupposes a particular choice of the primitive logical connectives, and would have to be appropliately modified to provide for disjunction, the material conditional, Sheffer stroke, or whatever. But the particular choice above is convenient from our point of view (because of the association with the lattice notation), and of course, is well known for the fact that all of the other classical truth-functional operations can be defined from these. In particular, we shall assume the definitions of <jJ V lff = ~(~<jJ & ~lff) and <jJ :J lff = ~(<jJ & ~lff). A normal matrix is one that can be viewed as semantically "OK" in the sense that its elements (regarded as propositions) can be divided up into "the true" and "the false" in such a way as to respect the classical logical operations. Algebraically, a normal matrix is such that if we just consider the operations 1\, V, and - (that is, if we consider its "reduct" (M, D, 1\, V, - )), then there exists a strong homomorphism of it into the twoelement Boolean algebra 2 (recall that a strong homomorphism will have to carryall elements of D into 1, and all other elements to 0, and so in this instance is unique). Another way of looking at the situation is that if we "divide out" the reduct by counting two elements "equivalent" if either they are both designated or both undesignated, then this relation is in fact a congruence (on the reduct), and of course the quotient algebra determined by this congruence is just 2. In this section we prove a result stated in Kripke (1965) giving necessary and sufficient conditions for a unary assertionallogic to have a normal characteristic matrix. The proof that we give is based on a reconstruction worked out with Nuel Belnap and Peter Woodruff many years ago. Before stating the theorem, we define some requisite notions. We presuppose some antecedent knowledge of the notion of a classical tautology (a sentence <jJ is a tautology iff evelY interpretation of it in the two-element Boolean algebra assigns it the value 1). A set of sentences f tautologically implies a set of sentences l:!.. (in symbols, f 1= l:!..) iff there exist n, ... , Ym E f, and 01, ... ,On E l:!.. such that the sentence (n & ... & Ym) :J (01 V ... V On) is a tautology.

Here we presuppose the customary definition of <jJ :J lff as ~<jJ V lff, parentheses in the conjunction and disjunction are to be associated to the left, and either or both of the conjunction and disjunction may be missing (when the ys are missing the consequent must be a tautology, and when the os are missing the antecedent must be such that its negation is a tautology). As a special case when l:!.. = {<jJ}, we have that f tautologically implies the sentence <jJ iff there exist sentences n, ... ,Yn E f such that (n & ... & Yn) :J <jJ is a tautology,

or, when f is empty, <jJ all by itself is a tautology.

NORMAL MATRICES

263

Recall also that it is part of our notion of a logic that its theorems are closed under substitution. A logic is said to be consistent if / is not a theorem of 12, where we define / to be <jJ & ~<jJ, for some fixed sentence <jJ. A set of sentences f is said to be tautologically consistent if / is not tautologically implied by f. We need the notion of a sentence <jJ' being an alphabetic variant of a sentence <jJ, which means that there is a one-one substitution (Y that assigns atomic sentences to atomic sentences and such that (Y(<jJ) = <jJ'.

A logic is said to be complete in the sense 0/ Hallden (see Hallden 1951) iff whenever <jJ V lff is a theorem and <jJ and lff are disconnected from each other (share no atomic formulas), then either <jJ is a theorem or lff is a theorem. Hallden completeness is intimately related to the cancellation property, and indeed is equivalent to it, given quite usual assumptions. We shall show that Hallden completeness implies the symmetric cancellation property, under natural assumptions which we shall develop in the course of the proof. Let us. suppose the hypothesis of the symmetric cancellation property, that Ufi I- Ul:!..i. If I- IS compact, then there are finitely many finite subsets f'. and l:!..'. (of fi and l:!..i respec'1y?, ~o thU' . ~ of all ) the sentences in fi, tIve at . fj 1-. U" l:!..j" Lettmg Yj be the conjunctIOn and SImIlarly WIth OJ, If I- supports the Ketonen rules of "conjunction on the left" and "disjunction on the right," we have

n

& ... & Yj & ... & Yn I- 01 V ... V OJ V ... V On,

and if I- has the deduction theorem property (the Gentzen rule of "conditional on the right"), we have I- (n & ... & Yj & ... & Yn) :J (01 V ... V OJ V ... V 011)'

Assuming for the moment that we have closure under tautological implication, upon rearranging terms we obtain I- (~n V Dr) V ... V (~Yj V OJ) V ... V (~YIl V on).

The other hypothesis of the cancellation property tells us that each disjunction is discon~ected from every other disjunction in the above. So by repeated applIcatIOns of Hallden completeness we will obtain for some index j, (~Yj. V ~j)

I-

(~Yj V

OJ),

or upon rewriting, I-Yj:Joj.

If we have the natural properties that amount to the converse of the deduction theorem property, and the converse of the Ketonen rules, we can conclude

and hence fj I- l:!..j,

as desired for the cancellation property.

264

MATRICES AND ATLASES

265

NORMAL MATRICES

We have thus seen how, on very usual assumptions, the cancellation property (at least the symmetric version) falls out of Hallden completeness. It turns out, as the reader can easily verify, that, except for compactness, all of the assumptions above can be justified on the basis of one general assumption, to wit, that the relation f- includes the relation of tautological implication. Further, as the reader can verify, on this same assumption, Hallden completeness follows from the cancellation property. Further, the reader can verify that this equivalence holds, on the same assumption, for the asymmetric case. We record these facts in the following theorem.

finally, we define T to be just the union of all of the stages (T will be a set since we begin with a set of sentences). D

Theorem 7.9.1 Let f- be a compact symmetric (or asymmetric) consequence relation including the relation of tautological implication (in symbols, 1= ~ f-). Then f- has the cancellation property iff f- satisfies Hallden completeness, i.e., whenever f- ¢ V lfI, then if ¢ and lfI are disconnected from each other, then either f- ¢ or f- lfI·

Proof To is tautologically consistent, since it is just L, which we shall now show is not only consistent (as is given) but also tautologically consistent. It is easy to see that L is closed under tautological implication (because of conditions (i) and (ii)). Thus L is tautologically consistent if it is consistent, and of course this is just condition (iii). Let us next consider the successor case Ta+!, and suppose, contrary to what we want to show, that it is not tautologically consistent. This means that

Having investigated Hallden completeness, we return to the conditions of Kripke for normality:

Theorem 7.9.2 (Kripke) Let L be a unary assertional logic. Then L has a normal characteristic matrix iff all of the following conditions hold: (i) All truth-functional tautologies (in &, V, and ~) are theorems of L. (ii) If ¢ and ¢ ~ lfI (=~ ¢ V lfI) are theorems of L, then lfI is a theorem of L.

The construction defined above is just an interesting variation on the usual Henkinstyle proof of the completeness of classical propositional logic.

Lemma 7.9.3 At each stage a in the construction defined above, the set Ta is tautologically consistent. Hence the set T, defined as the union of all stages, must be tautologically consistent.

Ta+1

1=

f·

Since (by inductive hypothesis) Ta is tautologically consistent, a formula ¢ (and a negated alphabetic variant) must have been added at stage a + I, and so the inconsistency must be laid at their feet. This means

(iii) L is consistent. (iv) L is complete in the sense of Hallden. Proof Let 1= represent tautological implication. Then it is easy to check that this relation has the deduction theorem property and its converse, i.e.,

r, ¢ 1= lfI

iff r 1= ¢ ~ lfI· We start with the set L of theorems of L as a basis, and inductively build up a set of sentences T that behaves as a "truth set," i.e., (i) ~¢ E T iff ¢ if. T, (ii) ¢ & lfI E T iff ¢ E T and lfI E T. The desired matrix will be of the "parasitic" variety, i.e., its elements will be sentences and T will be the designated set. We "enumerate" the set of sentences S: ¢I, ¢2, ... , ¢a+! .... In the usual case where S is denumerable, the indices will be just the positive integers; in the more general case the indices will be ordinals, and in particular, for convenience, will always be successor ordinals-we skip over the limit ordinals. We define To = L. At each successor stage a + 1, if ¢a+ I is not a theorem of L, we define Ta+! to be the result of adding ¢a+ I to Ta if the result is tautologically consistent, and at the same time, adding ~¢~+ I (where ¢~+I is the first alphabetical variant of ¢a+1 in the enumeration disconnected from all sentences already in Ta+! - L). If the result of adding ¢a+1 is not tautologically consistent, or ¢a+1 is a theorem of L (in which case it got in at stage 0), we let Ta+1 be just Ta. At limit stages (of course there will be none in the usual case when S is denumerable) we just take unions of all the sets introduced at earlier stages. And

By the finitary definition of tautological implication and the deduction theorem property, we know that there exist sentences Yl, ... , Yll E Ta so that (Yl & ... & YIl)

~ (¢ ~ (~¢' ~ f))

is a tautology.

Some of the y;s come from L and others were introduced at later stages. Let us denote the conjunction of the first of these by A and the conjunction of the latter by 'C. Then (rearranging terms and using the fact that ~lfI ~ f is truth-functionally equivalen t to lfI) A~

('C ~

(¢ ~ ¢')) is a tautology,

and hence, by condition (i), a theorem of L. Clearly A is a theorem of L (since, as observed above, L is closed under tautological implication). So by condition (ii), 'C ~

(¢ ~ ¢') is a theorem of L.

Hence (by closure under tautological implication), (~'C V ~¢) V ¢'

is a theorem of L.

By the construction, the sentence ¢' is disconnected from (~'C V ~¢). By condition (iv) (the Hallden completeness property), at least one of (~'C V ~¢) or ¢' is a theorem of L. If it is ¢', then since L is closed under substitution, ¢ is also a theorem of L. But in that case ¢ was not added at stage a + 1, contrary to our assumption above that it was this addition (along with ~¢') that led to inconsistency. So then 'C ~ ~¢ must be a theorem of L. This contradicts the supposition that ¢ could be added with tautological consistency to Ta.

At limit stages TfJ, any tautological inconsistency (since it comes from only a finite number of sentences) must already exist at some previous stage Ta, contrary to inductive hypothesis. D

Lemma 7.9.4 T is a truth set. Proof We first observe as a sub lemma that if T 1= lfI, then lfI E T. The point is that lfI is some sentence ¢a+! in the enumeration, and when its turn came up in the construction of T, it would at that point have been added, since it could in no way interfere with the tautological consistency of T, being already a tautological consequence. We next observe that we do not have lfI, ~lfI E T, for otherwise T is not tautologically consistent. We next show that we have at least one of lfI, ~lfI E T. If neither is in T, then we consider T U {lfI} and T U {~lfI}. In this case, each of lfI and ~ lfI must have led to tautological inconsistency when an attempt was made to add them to the construction when their turn came up. So we have T, lfI

1=

j, and T, ~lfI

1=

j.

By obvious moves licensed by tautological implication, we then have T

1=

j,

contradicting that T is tautologically consistent. We have thus established (i)

NORMAL ATLASES

MATRICES AND ATLASES

266

~lfI E

T iff lfI

~

T.

But it is easy to see as well that (ii) lfI/\ X

E T

iff lfI

E T

and X

E T,

using the sub lemma that T is closed under tautological implication together with the obvious facts that lfI /\ X 1= lfI, lfI /\ X 1= X, and lfI, X 1= lfI /\ X· Turning back now to completing the main lines of the proof, we finally have to verify that "the Kripke matrix" (S, T) is characteristic for L. It should be clear that if ¢ is a theorem of L, since L is closed under substitution, any substitution instance (J(¢) will be a member of T. And since interpretations are just substitutions, this establishes soundness, since ¢ will then always be assigned a designated value in every interpretation. Turning now to completeness, if ¢ is not a theorem of L, then either ¢ was never added in the construction of T, or if it was, at the same time a substitution instance ~¢' of ~¢ was added. Since T is consistent, this means that ¢' is not in T. In either case it is then possible to find a substitution instance of ¢ that is not designated, i.e., an interpretation that falsifies ¢. D

7.10

Normal Atlases

Generalizing the idea of a nonnal matrix from the previous section, an atlas (A, (Di») will be said to be normal if for every index i, the matrix (A, Di) is nonnal. Talking infonnally, this means that the atlas can be understood as a collection of possible worlds

267

realized as sets of true propositions. Since we know that it is atlases that are key to the characterization of logics understood as consequence relations (of either the asymmetric or symmetric stripe), it is natural to raise the question as to when such logics have a characteristic nonnal atlas. Let us first observe that if an asymmetric logic tolerates inconsistency in the sense that there exist sentences ¢, ~¢, and lfI so that it is not the case that ¢, ~¢ I- lfI, then it is clearly impossible that it has a normal characteristic atlas. For to falsify such an implication we would need to designate both an element a and also -a (realizing an "impossible world"). For a symmetric logic, we would talk about the existence of sentences ¢, ~¢, and a set of sentences r such that it is not the case that ¢, ~¢ I- r, and we would also want to bring to attention the dual situation when it is not the case that r I- ¢, ~¢ (which would require the realization of an "incomplete world"). One quick way of ruling out such situations is to require that the consequence relation I- include tautological implication, i.e., that 1= ~ 1-.

Theorem 7.10.1 Any symmetric, or compact asymmetric logic I- has a characteristic normal atlas iff it satisfies the following conditions: (i) The consequence relation includes the relation of tautological consequence (in symbols, 1= ~ I-). (ii) r I- ¢ :J lfI, fl iff r, ¢ I- lfI, fl (fl is of course empty for an asymmetric logic).

Remark 7.10.2 The reader may desire a comparison of the above conditions with those for the corresponding theorem for unary assertional logics of Kripke in Section 7.9. Condition (i) above is the straightforward generalization of condition (i) of the Kripke theorem. Also, it is easy to see (because of the cut property) that (ii) above implies (ii') if r I- ¢, fl and r I- ¢ :J lfI, fl, then r I- lfI, fl, which is the obvious generalization of the corresponding condition (ii) for unary assertionallogics given in the statement of the theorem by Kripke in Section 7.9. Looking at the converse direction, it is easy to see (using dilution and cut) that (ii') implies half of (ii) (the direction from left to right), but we believe that the other direction does not follow. An informal argument for this goes as follows. Notice that (ii') does not affect the left-hand side of 1-. Since neither dilution nor overlap can move formulas from the left to the right, these rules cannot lead from r, ¢ I- lfI, fl to the premises of (ii'), nor to r I- ¢ :J lfI, fl. The only hope would be to cut ¢, but dilution and overlap cannot produce an appropriate premise, since r I- fl is not given. The theorem of Kriplce had two additional conditions: (iii) consistency; and (iv) Hallden completeness. We have no need for a generalization of consistency, (say) that I- 0 (or 0 I- f) does not hold, because in the construction below we are always trying to find a counter-example for some consequence. So we are given a set r and a sentence ¢ such that it is not the case that r I- ¢, and from this we can argue that r is a consistent set and then use it as the basis of our construction of a truth set. Thus we can drop (iii) as a condition. This construction is even easier than in the Kripke theorem, because we do not have to find counter-examples for all non-theses in relation to the same set of designated values. This means that we can drop the condition of Hallden completeness.

o

268

NORMAL ATLASES

MATRICES AND ATLASES

Proof The proof differs for (1) the asymmetlic and (2) the symmetric case. (1) The proof for the compact asymmetric consequence relation is much like that for the unary assertional case (cf. Remark 7.10.2). Again, necessity is straightforward and is left to the reader as an exercise. For sufficiency, let us suppose that not (r f- p). We are going to inductively expand r u {",p} to a truth set. Thus we set To = r u {",p}. We next verify that To is consistent. If not, then r u {",p} f- f, and so (by (ii)), r f- "'P :::> f. But "'P :::> f F P, and so by (i) and the cut property we have r f- p, contrary to our hypothesis. We then enumerate the new set of sentences S : PI, P2, ... , Pa+l, .... In the usual case where S is denumerable, the indices will be just the positive integers; in the more general case the indices will be ordinals, and in particular, for convenience, will always be successor ordinals. (We skip over the limit ordinals.) We define To = r', and at each successor stage a + I, we define Ta+ I to be the result of adding Pa+l to Ta if the result is tautologically 'consistent, and closing the result under f-. Otherwise Ta+ I is defined to be just Ta. At limit stages (of course, there will be none in the usual case when S is denumerable) we just take unions of all the sets introduced at earlier stages. And finally, we define T to be just the union of all of the stages.

Lemma 7.10.3 T is closed under f-. Proof This follows trivially from the fact that each successor stage is closed under f-, given that f- is compact. That each successor stage is closed under f- is easily seen to be the case, for if T f- I(f, then when I(f'S tum came up in the enumeration, it would have been added, since adding it could not interfere with T's consistency, it being already a consequence of T. (The reader can add the details, using dilution and cut.) 0

Lemma 7.10.4 At each stage a in the construction defined above, the set Ta is consistent. Proof We first give the proof for the case of a compact asymmeUic logic. That To is consistent was shown above. Let us suppose that some successor stage Ta+l is inconsistent. The reader can easily see that Ta then must have been inconsistent, contrary to inductive hypothesis. At limit stages T{3, any inconsistency (since, by compactness, it comes from only a finite number of sentences) must already exist at some previous stage Ta , contrary to inductive hypothesis. 0

Corollary 7.10.5 T (defined as the union of all the stages in the construction defined above) is consistent. Proof Similar to that for limit stages in Lemma 7.10.4.

o

Lemma 7.10.6 T is a truth set. Proof We first observe that we do not have I(f, "'I(f E T, for otherwise T is not consistent (this uses fact that I(f, "'I(f F f, and (i)). We next show that we have at least one

269

I(f, "'I(f E T. If neither is in T, then we consider T U {I(f} and T U {"'I(f}. Each of and "'I(f must have led to inconsistency when an attempt was made to add it at the appropriate stage in the construction. So we have (using dilution)

of

I(f

T, I(f f- f, and T, "'I(f f- f.

By obvious moves (using (ii), (i) and facts that have T f-

"'I(f,

I(f

and T f-

:::>

f F

"'I(f

and "'I(f :::> f

F I(f), we

I(f.

But (using (ii)) then T f-

"'I(f

:::>

(I(f

:::> f),

and so (using (ii') twice), we have T f- f,

contradicting that T is consistent. We have thus established (i)

"'I(f E

T iff

I(f

Ii! T.

But it is easy to see as well that (ii)

I(f /\

X E T iff

I(f E

T and X E T,

using the lemma that T is closed under f-, together with the obvious facts (derived using (i)) that I(f /\ X f- I(f, I(f /\ X F X, and I(f, X f- I(f /\ X. 0 We finally construct the desired "Kripke atlas" by so constructing a truth set Ti for each pair (r, p) such that it is not the case that r f- p, and by letting the desired atlas be (S, For soundness, if 1::J.. f- I(f, then 0-*(1::J..) f- o-(I(f) (for any substitution 0-), and thus if 0-*(1::J..) ~ h then o-(I(f) E Ti (since each T; is closed under f-). This means that any substitution, i.e., interpretation, that designates all members of 1::J.. also designates I(f, and so we have soundness. Turning now to completeness, if not (r f- p), then it is easy to see from the construction that there exists a truth set Ti such that r ~ Ti, and yet P is not in Ti. The identity substitution (interpretation) thus designates all members of r in some designated set of the Kripke atlas, but fails to designate p. (This completes the proof of Theorem 7.10.1 for case (1).) (2) The proof for a symmeUic consequence relation is a bit easier, since the global cut property allows us to drop the hypothesis of compactness, and the construction of a truth set T does not involve "Lindenbaumizing." (Again, we leave the necessity part of the proof to the reader.) If it is not the case that r f- 1::J.., then we can simply invoke the global cut property to obtain a partition of the sentences into two sets of sentences (T, F) such that it is not the case that T f- F. It is easy to verify the following hold:

m».

(0) Not (T f- F).

p, pET or P E F (2) For no sentence p, pET and P E F

(1) For each sentence

(exhaustiveness). (exclusiveness).

270

(3a) (3b) (4a) (4b) (5a) (5b)

MATRICES AND ATLASES

For no sentence cjJ, cjJ E T and ",CjJ E T. For no sentence cjJ, cjJ E F and ",CjJ E F. For each sentence cjJ, cjJ E Tor ",CjJ E T. For each sentence cjJ, cjJ E F or ",CjJ E F. cjJ 1\ If/ E T iff cjJ E T and If/ E T. cjJ V If/ E F iff cjJ E Tor If/ E T.

(0)-(2) restate the global cut property. As for (3a), if cjJ, ",CjJ E T, then since cjJ, ",CjJ F Ll, by (i), we would have (using dilution) that r I- Ll, contrary to hypothesis. «3b) follows similarly.) (4a) follows from the fact that a sentence has to end up on one side of the partition (T, F), or the other. If cjJ ends up in T, fine; if it ends up in F, then by (3b) we know that ",CjJ cannot also be in F, and so ",CjJ ETas needed. «4b) is argued dually.) As for (5), let us suppose that cjJ 1\ If/ E T, but that, say, cjJ Ii T. Then by exhaustiveness cjJ E F, and since cjJ 1\ If/ F cjJ, we have by (i) (and dilution) that T I- F, contrary to (0). As for the converse, if both cjJ, If/ E T, and yet cjJ 1\ If/ Ii T, then (again by exhaustiveness), cjJ 1\ If/ E F. And since cjJ, If/ F cjJ 1\ If/, we would have (using (i) and dilution) T I- F, contrary to (0). (Again (5b) is argued dually.) Using (0)-(5) it is easy to see that T is a truth set. Further, then for each pair (n, Lli) such that not (r i I- Lli) there exists such a pair (Ti' Fi ), and so we can use the "parasitical" atlas defined on the algebra of sentences S, (S, (1j»), as a normal atlas in which every non-consequence can be falsified (completeness). It is also clear that no conect consequence is falsified this way, for if II I- B, and yet this consequence could be falsified, then there would have to exist a pair (Ti, Fi) and a substitution u so that u* (II) ~ Ti and u*(B) ~ F i . But because I- is formal, we know that u*(II) I- u*(B), and so by dilution we would have Ti I- Fi, contrary to specifications. D

Corollary 7.10.7 (Symmetric case) Any compact symmetric pre-consequence relation has a normal characteristic atlas. Proof Immediate from the second half of the proof of Theorem 7.10.1, and from Lemma 6.8.4 showing that a compact symmetric pre-consequence relation is a symmetric consequence. D

7.11

Normal Characteristic Matrices for Consequence Logics

In this section we in effect combine the results of the previous three sections. The background for all of this work is the fact that every unary assertionallogic has a characteristic matrix (the Lindenbaum matrix). In Section 7.8 we examined the conesponding property for consequence logics, providing necessary and sufficient conditions (due to Shoesmith and Smiley) for a consequence logic (of either the asymmetric or symmetlic flavor) having a characteristic matrix. In Section 7.9 we returned to unary assertional logics, and raised the question of a stronger, normal, characteristic matrix, again providing necessary and sufficient conditions (due to Kripke) for a unary assertional logic to have a normal characteristic matrix. In Section 7.10 we came back to consequence logics, raising the normality issue, but this time in the context of atlases instead of matrices

MATRICES AND ALGEBRAS

271

(just as every unary assertionallogic has a characteristic matrix, so every consequence logic has a characteristic atlas; the interest then extends to the normal ones in each case). Now in the present section we attack the question of when a consequence logic has a normal characteristic matrix, combining all of the issues of strength and generality at once.

Theorem 7.11.1 For a stable symmetric logic I- the following constitute necessmy and sufficient conditions for it to have a normal characteristic matrix: (i) I- has the cancellation property; (ii) F ~ I- (F is tautological implication); (iii) r I- cjJ :J If/, Ll iff r, cjJ I- If/, Ll.

Proof Before we begin, let us recall the observation made in Section 7.9 in discussing the conditions of the theorem of Kripke. There it was remarked that in the presence of condition (ii) above, Hallden completeness and the Cancellation Property amount to the same thing, so we can freely substitute them below in our reasoning. Necessity is straightforward and is left to the reader. As for sufficiency, we begin as with the theorem of Shoesmith and Smiley to collect together all the pairs (n, Lli) such that it is not the case that ri I- Lli' and then for each such pair make a disjoint copy Si of the set of sentences S (we let one of these be S itself). We then union these together to get a new set of sentences S', and define a new consequence relation 1-' on the subsets of S', making 1-' the closure of I- under substitution and dilution. That 1-' is a symmetric consequence relation is just the content of the hypothesis of "stability." We then verify as before that Ur; ¥ U Ll;, and then invoke the global cut property so as to partition S' into two sets T and F so that U r i ~ T and U Lli ~ F. We now have only to verify that T is a truth set, and this velification proceeds exactly like the verification for the symmetric case of the conesponding theorem for atlases of Section 7.10. D We can also prove the following results, by techniques familiar by now (and left to the reader).

Corollary 7.11.2 A compact symmetric pre-consequence relation has a normal characteristic matrix under precisely the same set of necessmy and sufficient conditions as in Theorem 7.1l.l. Theorem 7.11.3 Let I- be a compact asymmetric logic. Then the following are necessmy and sufficient conditions for I- to have a normal characteristic matrix: (i) I- has the asymmetric cancellation property; F ~ I- (F is asymmetric tautological implication); r I- cjJ :J If/ (ff r, cjJ I- If/.

(ii) (iii)

7.12

Matrices and Algebras

Although a matlix is an algebra, it is not just an algebra because of the need of singling out a designated subset. This is unfortunate in that it means in general that we

272

MATRICES AND ATLASES

cannot just cmT)' over results from universal algebra9 and apply them to matrices without thought (although often close analogs can be obtained). But in this section we shall discuss how it is that often "in real life" matrices can be viewed (without loss) as just algebras. Often a matrix M will have an "implication" operation --+, so that if we define an "implication" relation a :S b iff a --+ bED, it turns out that :S is a partial ordering. If in addition D is a (positive) cone under this partial order (a E D and a :S b only if bED), we shall call the matrix standard. There is a slightly weaker notion that is also of interest. If:s turns out only to be a pre-order (not necessarily anti-symmetric), then, given one more condition which we shall next desclibe, we shall call the matrix pre-standard. Thus define a == b iff both a :S band b :S a. The additional requirement is that == must be a congruence (when :S is anti-symmetric == is just identity and we have no need for this requirement). Note that == is a strong congruence given the requirement above that D be a cone. It is easy to see that the quotient matrix of a pre-standard matrix is itself standard. Now for standard matrices a trick for throwing away D is to find some distinguished element e so that D = {x E M: e :S x}. The element e can be understood as just a nullary operation added to the underlying algebra. Unfortunately, this does not by itself suffice to let us consider M as just an algebra, since we still have the partial order with which to contend. But when there is a semi-lattice operation 1\ so that a :S b holds just when a 1\ b = a, then the reduction of a matrix to a plain algebra can be complete. (Clearly it would also suffice to have the dual semi-lattice operation V with a :S b iff a V b = b.) This allows us to give an entirely equational characterization of D as the set of elements satisfying the equation e 1\ x = e. So let us call a standard matrix with an operation that interacts in the required way with :S (and hence ultimately with --+) a completely standard matrix. Another tlick is to find an equation of the form sea) = tea) that holds of exactly the designated elements D. Elok and Pigozzi (1989) observe that this can be done for the relevance logic R using the fact that a is designated iff a --+ a :S a. And, of course, this last can be restated equationally as (a --+ a) 1\ a = a. There are a few things that need to be checked about the coincidence of matrix notions and algebraic notions for completely standard matrices. Since equations are preserved under algebraic homomorphisms, it is easy to see that a (weak) matl'ix homomorphism is just an ordinm'y algebraic homomorphism. Also, it is easy to see that a submatrix is just a subalgebra. First notice that if A' is a subalgebra of A, then (the nullary operation) e' has to be identical to e. But then the cone of elements in A' determined bye' is just the same as the cone of elements in A determined bye, i.e., D' = A' n D, as required. It is left as an exercise for the reader to verify that direct (and subdirect) products of matrices are just direct (and subdirect) products in the algebraic sense. (The verification falls back on the componentwise definitions of designation and 9Matrices can be regarded as many-sorted algebras in a natural way, and we could then appeal to results of many-sorted universal algebra. We have thought it best to leave the latter subject outside the scope of this monograph, however.

WHEN IS A LOGIC "ALGEBRAIZABLE"?

273

:S, once it is noted that the distinguished element of the direct product is just the indexed set of the distinguished elements of the component algebras.) The above ideas are perhaps most familim' in the context of classical logic and Boolean algebras, where the distinguished element e can be picked as the greatest element 1, and so D = {I}. The particular choice of e depends, of course, on the posit that all logical truths co-imply one another (an assumption common to a number of logics other than classical logic, e.g., the usual modal and intuitionistic logics). It turns out that the general idea is of much wider utility, and can be applied even to the usual relevance logics (which certainly have no such posit), although the logics have to be extended conservatively with a constant sentence t with the property that it is a theorem that implies all theorems. 7.13

When is a Logic "Algebraizable"?

We have been examining logics using the tools of algebra. It is natural to ask: how far does this methodology extend? We here give some rough answers to this question. Our answers are rough because we feel that a more precise answer may actually get in the way of further research. We follow Chairman Mao in wanting a hundred flowers to bloom. An algebra is simply a set with some operations. A logic is a set of sentences with some kind of consequence relation. For the sake of clarity we shall focus on asymmetl'ic consequence. The applications to symmetlic consequence and unary assertional systems are often left to the reader. We start by recalling some results of the previous chapter. For a unary assertional logic we showed that its Lindenbaum matrix characterizes its assertions, whereas for an asymmetric consequence logic its Lindenbaum atlas characterizes its consequences, and for a symmetric consequence logic it is the corresponding Scott atlas that does the trick. An atlas is just an algebra with many designated sets, and it is easy to see that an atlas (A, (Di) iEJ) can be "unfolded" into an equivalent indexed set of matrices ((A, Di)) iEJ (equivalent in the sense of validating the same consequences, whether they be unm'y, asymmetric, or symmetric). When a collection of matrices all have the same underlying algebra, then this is sometimes called a bundle. Summarizing, we have shown in Chapter 5 that formal logics, whether they be unary, asymmetric, or symmetric, all have a "matrix semantics" in the following sense.

Theorem 7.13.1 For eVelY unary assertional logic L there is a class of matrices M such that for every sentence cp: I- L cp iff 1= M cp. For evelY asymmetric consequence logic L there is a class of matrices M sLlch that for evelY set of sentences r and sentence cp : r I-L cp (If 1=M cp. For eVelY symmetric consequence logic L there is a class of matrices M sLlch that for every set of sentences rand l::!. : r I- L l::!. (If r 1= M l::!.. A matrix is an algebra with a designated subset, so this result shows that algebras figure centrally in the semantics of logic. On the other hand a matrix is more than an algebra because of its designated subset D. In addressing which logics are "very algebraizable" we must address the question when D can be defined "algebraically."

274

WHEN IS A LOGIC "ALGEBRAIZABLE"?

MATRICES AND ATLASES

275

Czelakowski (1981) defined an algebraic semantics for a logic to be a class of matrices charactelizing the consequence relation of the logic, with each matrix having just one designated element d. Without the restriction to just one designated element, we shall call this a matrix semantics. Let us introduce a predicate T(cjJ) which intuitively means "cjJ is true." This means that we can add a constant ad to the language of the logic, with the interpretation rule that lead) = d. We can now define T(cjJ) as cjJ = ad. The idea is that an inference licensed in classical logic, e.g., {p, p -+ q} f- q, can be translated into a statement about Boolean algebras, namely, if x = 1, and x -+ y = 1, then y = l. Note that the relettering is unnecessary if we use the same variables and operation symbols in the language of the algebra that we use in the language of the logic, and we shall adopt this simplifying convention for this discussion. Having a single designated element works fine for many logics, including classical and intuitionistic logics, where any theorem is implied by any sentence whatsoever, and so all theorems are logically equivalent. For these logics we can then look at the equivalence class of theorems as the greatest element 1 in the natural partial order defined by [cjJ] ~ [vr] iff f- cjJ -+ vr. For logics where a theorem is not implied by any sentence whatsoever, e.g., relevance logic and some other substructural logics, we have to resort to another device, briefly described in the previous section. The device introduced in Dunn (1966) (cf. 1975, Section 28.2 Anderson and Belnap) was to add to the language of the logic R a constant t conceived of as the conjunction of all theorems. Anderson and Belnap had shown that t can be added conservatively to R. In the Lindenbaum algebra [t] ~ [cjJ] is then a way of saying that cjJ is a theorem, and this can be abstracted out by having an "identity" element e and defining T(a) as e ~ a. The idea is that the "true" elements can be viewed as forming a principal cone, and it is just an "accident" that with, say, Boolean algebras this cone is degenerate and contains only 1. Cf. Remark 6.15.13. Note incidentally that e ~ a can be rephrased as an equation:

Remark 7.13.5 Just what would be the the "Blok-and-Piggoziesque" criterion for algebraizability of a symmetric consequence logic? The cliterion for symmemc consequ~nce would be of the same form as their criterion for an asymmetric consequence loglc, but K would be weaker than quasi-equationally definable. It would instead have to do with definability by "symmetric quasi-equations," i.e., formulas of the form: a conjunction of equations implies a disjunction of equations.

e /\ a = a.

Blok and Pigozzi show that theoremhood in the relevance logic R can be characterized without use of t, using the equivalence

But this does involve adding a constant to the language. A less contrived answer is to say that it is when we can find a set of equations of the form sex) = t(x) (containing only the vmiable x) which holds precisely of the elements of D. Blok and Pigozzi (1989) call these defining equations. They introduce this generalization in their definition of what it is for a logic to have an algebraic semantics. 10 It roughly amounts to saying that the logic has a matrix semantics, and the designated set D of each matrix can be uniformly characterized by equations. This is not quite right, since they also require that the set of defining equations be finite. This turns out to be no problem since they only consider logics that satisfy compactness, so an infinite set of premises r can always be traded for a finite subset r'. It would be interesting to investigate logics without this reshiction to compactness. IOThough they also implicitly introduce the requirement that the set of premises r in r I- rjJ is always finite. This obviously does not hurt for a system that satisfies compactness, but otherwise seems arbitrary.

Definition 7.13.2 (Blok and Pigozzi 1989) An asymmetric consequence logic L has an algebraic semantics iff there exists a finite set of equations with a single variable p, Sl (p) = tl (p), ... , sn(P) = tn(P)' such that for all i (1 ~ i ~ n);

r

f- L cjJ iff {Sl(vr/P) = tl(vr/P),· .. ,sn(vr/p) = tn(vr/p):

vr E r}

F=K Si(vr/p) = ti(vr/p).

The definition of when a unary assertional logic has an algebraic semantics is just the sp.eci~l case of this when r is empty. We leave to the reader to state the obvious generahzatlOn for a symmetric consequence logic. Blok and Pigozzi go on to give as their criterion for when an asymmetric consequence logic is "algebraizable": Definition 7.13.3 An asymmetric consequence logic is algebraizable iff there is a quasiequationally definable class of algebras K such that K is an algebraic semantics for the logic. Their cliterion for a unary consequence logic would presumably be essentially the same, except that the class is required to be equationally definable. Definition 7.13.4 A unary assertionallogic is algebraizable iff there is an equationally definable class of algebras K such that K is an algebraic semantics for the logic.

(cjJ /\ (cjJ

cjJ))

-+

(cjJ

¢;>

-+

cjJ),

which is fundamentally based on the R-theorem cjJ

-+

[(cjJ

-+

cjJ)

-+

cjJ]

(Demodalizer),

whose converse also holds. This means that the theorems can be characterized as those sentences cjJ such that (cjJ

-+

cjJ)

-+

cjJ

is a theorem. This may seem to be a circular definition, since "theorem" occurs in both the definiens and the definiendum, but the previous formula can be rephrased algebraicallyas (a -+ a) ~ a,

and obviously this can be reexpressed as the identity that Blok and Pigozzi require: a /\ (a

-+

a)

= (a -+ a).

276

MATRICES AND ATLASES

Whichever is chosen, R has a (characteristic) algebraic semantics. Blok and Pigozzi go on to show that the relevance logic E has no algebraic semantics. As the name "demodalizer" suggests, E lacks it since E is a modal logic in addition to being a relevance logic. In E, the necessity operator D¢ can be defined as (¢ --+ ¢) --+ ¢, and so (the demodalizer) has the effect of making mere truths necessary. Blok and Pigozzi also show that the implicational fragment of R (R... ) has no algebraic semantics. This relies on the fact that R-+ lacks conjunction (and disjunction, since a 1\ (a --+ a) = a --+ a can be dualized to a V (a --+ a) = a). It is well worth noting that this depends upon the lack of appropriate equation, since the inequality a --+ a ::; a would suit the bill admirably if inequations were allowed. Its surrogate e 1\ a = a works just as well for the extension of R with a logical constant t (W). We do not quarrel with the work of Blok and Pigozzi. Indeed, we praise it. Their criterion of algebraizability has a certain "philosophical" naturalness, and generalizes a large class of motivating logics. Also they can prove a number of interesting theorems, one of which we state. Recall that OCT) is the Leibniz congruence determined by a theory T on the underlying algebra of formulas of some given logic. Blok and Pigozzi establish the following interesting relationship between algebraizability and the Leibniz operator. Theorem 7.13.6 (Blok and Pigozzi 1989) An asymmetric consequence logic L is algebraizable ifffor all theories T and S, ifT ~ S then OCT) ~ O(S). But we think the restriction to equalities is too restrictive, and we propose another. Definition 7.13.7 An asymmetric consequence logic L is partially algebraizable iff there is a quasi-inequationally definable class of tonoids K such that K is a (sound and complete) semantics for the logic. Their criterion for a unary consequence logic is essentially the same, except that the class is required to be inequationally definable. Definition 7.13.8 A unary assertional logic is partially algebraizable iff there is an inequationally definable class of tonoids K such that K is a (sound and complete) semantics for the logic.

8 REPRESENTATION THEOREMS 8.1 8.1.1

Partially Ordered Sets with Implication(s) Partially ordered sets

We annunciated a theme in Remark 6.5.3 which we want to reemphasize here. Propositions can be understood as sets of "possible worlds" or, as we now stress, more generally as sets of "information states." The latter is more general in that there can be states of information that are inconsistent, incomplete, or both, and so do not correspond to possible worlds. An "information frame" will always consist of at least a set U whose elements are regarded as "states of information." It may have additional features, for example a binary relation b on U thought of as an "information order." a b p is to be read as "P contains at least the information a." This order is to be understood "qualitatively" and not "quantitatively" and is thus to be contrasted with Shannon's information theory. The information order arises quite naturally in a number of places, but we will content ourselves with assigning its origins to the Kripke-Grzegorczyk semantics for intuitionistic logic (cf. Chapter 11). Other features that an information frame might possess include accessibility relations and/or operations combining pieces of information. The most familiar example of the first is Rap (P is possible relative to a), which comes from the Kripke semantics for modal logic (cf. Chapter 10). As a less familiar example of the first we give Rapy, understood as something like "a and p are compatible from the standpoint of y," and as an example of the second we have something like "the combination of a and p." These can sometimes be parsed in terms of each other, e.g., Rapy can be understood as a • p b y. These last examples arise from the semantics of relevance logic (and more generally substructural logics), as developed by Routley, Meyer, Fine, and Urquhart. See Dunn (1986) or Anderson et al. (1992) for details and history. Routley and Meyer (1972, 1973) are the key references, along with Meyer and Routley (1972), which is even more important in the context of algebraic logic. We refer generally to sets whose elements are regarded to be states as "UCLA propositions." As the reader saw by working through Exercise 6.5.2, binary consequence, understood "mathematically" as a partial order between propositions, can be fully represented as inclusion between sets, and the latter can be understood "philosophically" as consequence between UCLA propositions. We run through the proof of this in slow motion, since grasping its essence is of major importance. Let P = (P, ::;) be a poset. We think of P as a set of propositions and::; as (binary) consequence. These "propositions" are conceived of abstractly. They could be anything;

PARTIALLY ORDERED SETS WITH IMPLICATION(S)

REPRESENTATION THEOREMS

278

we know nothing about their internal structure. Recall that a cone C is a subset of P which is closed upward under S;, i.e., if x E C and x S; y, then y E C. Let C be the set of all cones of P. A cone is a kind of primitive "theory" (at least it is closed under binary consequence) and as such can be regarded as an information state. So it is natural to interpret an "abstract" proposition a E P as the set of information states, i.e., theories (cones) that contain it. Thus, we define (1) h(a) = {CEC:aEC}. This is the desired representation function. We need to show that it preserves preting it as s on C):

S;

to be the residual of some fusion operator. Let us arbitrarily take implication to be the right residual-it will tum out, as we shall see, that there is no way of distinguishing the left residual from the right residual in the absence of fusion. So we assume that we have a po set with a binary operation (S, S;, -+). The only properties of -+ that we assume are that it is antitonic in its first position (antecedent) and isotonic in its second position (consequent), i.e., rule suffixing and rule prefixing. We shall call such a structure an implicational poset, and as we saw in Section 3.10 it is an example of a more general structure, discussed in Dunn (1993a), called a tonoid.

(inter-

s

(2) a S; b iff h(a) h(b), i.e., (3) a S; b iff VC E C, a E C implies b E C.

The left-to-right half follows immediately from the definition of a cone. The rightto-left half follows by instantiating C to be the cone determined by a, i.e., the smallest cone containing a. This involves a small "existence" proof since we have to show that there is indeed such a cone. In this case it is easy, since it explicitly constructed as [a) = {x : a S; x}. As we shall see, there are other cases, as with the representation of distributive lattices and Boolean algebras, when we have to go through some maximalization argument, but here all we need is a cone and it is easy to show that [a) is a cone. Thus suppose that x E [a) and x S; y. The first conjunct just means a S; x and so by transitivity with the second conjunct we obtain as; y, i.e., y E [a) as needed. But what we have is: (4) a E [a) implies b E [a).

Since a E [a) follows from the reflexivity of S;, we have bE [a) by the argument above, i.e., a S; b as desired. We still have to show that h is one-one, but this follows easily from anti-symmetry and (2). Thus if h(a) = h(b), then h(a) s h(b) and h(b) s h(a). But then by (2) a S; b and b S; a, and so by anti-symmetry a = b. This concludes the proof. Before we pass on, let us observe that sets of cones of the form h(a) have a special property. This is confusing as to type level, but not only are they sets of cones, they themselves are cones. Thus if C E h(a), i.e., a E C, and C s C', then a E C', i.e., C' E h(a). This is to say that we can require UCLA propositions to be not just any sets of states, but restrict them to sets of states closed upward under the information order, i.e., if a E p and a b fJ, then fJ E p. This is sometimes useful, and plays a role, for example, in the Kripke-Grzegorczyk semantics for intuitionistic logic (cf. Section 11.4). 8.1.2

279

Implication structures

In the previous section we regarded implication as a relation. We now tum to examining structures with an implication operation. While we take the point of view that useful properties of implication can be sorted out by relating them to properties of fusing premises, it must be acknowledged that fusion is (regrettably) still a relatively arcane notion. Accordingly it is interesting to wonder just what are the minimal properties of implication that are needed to allow it

Representation of Implicational Posets We shall now prove the following, which illustrates a more general result for tonoids. Theorem 8.1.1 EvelY implicational poset (S,

S;, -+) can be embedded in a right residuated partially ordered (p.o.) groupoid (S', S;, 0, -+).

Proof We prove this by way of a representation result. Let U be a non-empty set, and let R be a ternary relation on U. We call the structure (U, R) a ternary frame. We define the following operations on subsets of U: A

0

B

= {X:

3a E A,3fJ E B,RafJx};

A -+ B = {X: Va, VfJ, if RaxfJ & a E A, then fJ E B}; B +- A = {X: Va, VfJ, if RxafJ & a E A, then fJ E B}.

It is easy to verify that ('(.J(U), 0, -+, +-) is a residuated p.o. groupoid. Indeed, any subcollection S' closed under the operations 0, -+, +- is a residuated p.o. groupoid. We call these concrete residuated p.o. groupoids. 0

We can now state and prove the following lemma. Lemma 8.1.2 Every implicational poset (S,

S;, -+) can be embedded in a concrete re-

siduated p.o. groupoid. Proof Let U be the set of all cones C on S. Define the relation R as RCI C2C3

iff Vx, y, if x E CI and x -+ y E C2 then y E C3.

Using R we can now define the operation -+ on subsets of U as in the concrete residuated poset above. Next define the map h(a) = {C : C is a cone and a E C}. It is easy to see that h is one-one, since if a f:. b, then a S; b or b S; a or all b (a and b are unrelated). Without loss of generality we may choose as; band allb to be the cases considered. (1) If as; b then the principal cone [b) = {x : b S; x} is not in h(a), but certainly [b) E h(b) as well as [a) E h(b). (2) If allb, then neither [a) E h(b) nor [b) E h(a). Next we show that h preserves -+, i.e., C E h(a -+ b) iff C E h(a) -+ h(b). To facilitate this we first translate the left- and right-hand sides via their definitions: C E h(a -+ b) iff a -+ b E C; C E h(a) -+ h(b) iff VCI, C2, RCI CC2 & a E CI implies b E C2.

REPRESENTATION THEOREMS

280

It only remains to show then that

But the left-to-right half is immediate, given the canonical definition of R, which simply says that if an implication is in C and its antecedent is in Cl then its consequent is in C2. For the right-to-Ieft half, we proceed contrapositively, assuming that a --+ b ¢ C. We then show that 3CI, C2, RCI CC2 & a E CI yet b ¢ C2. We simply let CI = [a). To obtain C2, we first consider the principal dual cone determined by b, (b] = {x : x :s; b} . We then set C2 = U - (b] (it is easy to check that this is a cone). It is clear that a E C I and b ¢ C2. What needs argument is that RCI CC2. Recalling the canonical definition of R, this amounts to assuming that x E CI, x --+ Y E C and showing y E C2· For reductio let us suppose then that y ¢ C2. Our hypotheses that x E CI and y ¢ C2 become a :s; x, and y :s; b. Using the rule forms of prefixing and suffixing we can easily derive x --+ y :s; a --+ b (y :s; b implies x --+ y :s; x --+ b, a :s; x implies x --+ b :s; a --+ b, and apply transitivity). Then our hypothesis that x --+ y E C gives that a --+ b E C, contrary to our initial supposition. 0 Representation of Donble Implicational Posets Let us now consider the case where we have both "residuals" but no fusion operation to connect them. To this end we will define a double implicational poset to be a structure (S,:S;, --+, ~), where both (S,:S;, --+) and (S,:S;, ~) are implicational posets, and the arrows interact by "pseudo-assertion": (1) a:S; (b ~ a) --+ b;

(2) a:S; b

~

(a --+ b).

The following shows that we have again correctly axiomatized (in this case both) residuals without residuation. Theorem 8.1.3 Every double implicational poset can be embedded in a concrete residuated p.o. groupoid.

Proof From the theorem above we know tlrat every implicational poset is so embeddable, using the canonical relation R defined above, but let us now subscript it:

By a symmetric argument every implicational poset is also so embeddable using the canonical relation:

PARTIALLY ORDERED SETS WITH IMPLICATION(S)

that R-+CI C2C3 says '
We now consider the ternary frame (U, R o ), and the operation above: A

0

B

= {X:

0

0

y E C3.

on subsets U defined

3a E A, 3fJ E B,RoafJx}

Using the same function h as used in tlre representation above (that is, h(a) CD we show h(a

a

0

0

b)

= h(a)

0

= {C : a E

h(b), i.e.,

b E C iff 3Cj,C2,a E Cl,b E C2, & RoCIC2C.

As with the canonical definitions of the relations R-+, R+- above, one direction of this is immediate. Here, however, it is tlre direction from right to left. We tlren only consider the other direction. To this end we assume that a 0 b E C, define CI = [a), C2 = [b), and show RoCIC2C. Thus suppose x E Cl, Y E C2. Then a :s; x, b :s; y, and so by isotonicity we get a 0 b :s; x 0 y. Since tlre first of these is in C, which as a cone is closed upward, then x 0 y E C as needed. We are not yet through, because if we were to just add this representation of 0 to tlre representations of the operations --+, and +- above, we would end up using three different relations R o, R-+, R+-, when we only want one. But just as above we can prove that the three relations in fact coincide, this time in virtue of residuation. We leave details to the reader. Remark 8.1.5 Let us consider how we might in fact have been more subtle in our representations of residuated p.o. groupoids and related structures above. Let (U, R) be a ternary frame and let [;;; be a partial order on U. We shall call tlre structure (U, [;;;, R) an articulated frame if it satisfies the following "monotonicity" conditions: RafJx & X [;;; X'

The only problem is that we seem to have two canonical relations instead of one. But it is easy to see that in fact the two relations coincide. Thus suppose R-+Cj C2 C 3. In order to show R+-CIC2C3 we suppose b ~ a E Cl and a E C2 and show b E C3. But in virtue of a E C2 and pseudo-assertion, we obtain (b +- a) --+ b E C2. But our hypothesis

281

implies RafJx';

Ra fJ X & a' [;;; a implies Ra' fJ X; RafJx & fJ' [;;; fJ implies RafJ' X·

We shall say that a subset of U is hereditary when it satisfies the following:

282

REPRESENTATION THEOREMS

If a

E

fJ then fJ

A & a !:

E

PARTIALLY ORDERED SETS WITH IMPLICATION(S)

A.

(3) e'

Let ~ t (U) be the class of all hereditary subsets. It is easy to see that ~ t (U) is closed under the concrete operations defined above (that the resultant sets are hereditary follows from the "monotonicity" conditions), and so we get another concrete example of a residuated p.o. groupoid, which we shall call an articulated concrete residuated p.o. groupoid. It is easy to see that but a slight modification of the canonical frame of all cones produces an articulated frame. Again U is the set of all cones of S, and Ro, R--+, and R+- are defined as before and are all equivalent (we shall thus denote them ambiguously as R). The difference is that we add as well the natural partial order!: on cones, i.e., set inclusion restricted to U. Thus we get an articulated frame (U,~, R). We verify that the "monotonicity" conditions above hold when R = R o . For the first, let us suppose that RoC] C2C3 & C3 ~ C~, and show R oC1 C2C~, But this is obvious, since our first supposition says that the hypotheses x E C1 and Y E C2 imply x 0 Y E C3, and so a fortiori these same hypotheses imply x 0 Y E C~ (since it includes C3). For the second "isotonicity" condition, suppose R oC1 C2C3 & ~ C1. We need to show that RoCi C2C3. So assume x E and Y E C2. Then afortiori x E C], and since RoC] C2C3 we conclude x 0 Y E C3 as needed. A similar argument takes care of the third "monotonicity" condition. We still have to show that h canies each element a E S into a hereditary subset of the cones of S. But h(a) = {C : a E C} is clearly hereditary, for if a E C and C ~ C' then a E C'.

Ci

Ci

We now consider more explicitly what happens when the residuated p.o. groupoid has a "right identity element" e satisfying a

0

e

= a.

It is the presence of this element that allows us to consider the algebraic analog of

"theorems." Thus by residuation we get e::; a -+ a,

which can be interpreted as saying that "a implies a" is a theorem. The element e can be interpreted as "empty" or "void," but it should not be interpreted as "nothing." It is better to think of it as the conjunction of all theorems. In general, the presence of e allows us to "push the metalinguistic deducibility relation into the object language," giving us for each "deducibility relation" a ::; b a corresponding "object language theorem" a -+ b ~ e. Thus a ::; b iff a

0

e ::; b iff e::; a -+ b.

We shall talk of "pushing" and "popping" for the left and right directions of this equivalence, respectively. Note incidentally, that all we need for pushing is that e is a lower right identity, i.e., a 0 e ::; a. Note that if we assume a left identity e' with the usual law

0

a= a

283

(left identity),

as well as a right identity it can be shown that e = e', since e = e' 0 e = e'. This is pleasant because we do not have two classes of "theorems" (the principal cone of e and also that of e'). But it is not absolutely unavoidable since for many purposes (including the most general representation theorem, as well as dealing with some subtleties in application to certain weak modal logics) we might have just "lower right and left identities," a 0 e ::; a, e' 0 a ::; a, and the uniqueness proof would no longer go through. (As we mentioned, all we need for "pushing" is that e is a lower right identity, but the assumption of full equality is far from unusual, and certainly simplifies things.) Remark 8.1.6 Amplifying the above, recall that our proof above of "push and pop" for the right residual depends on the axiom for the right identity being stated with an equality and not just the inequality, and, of course, similarly for the left residual and the left identity. Indeed, from push and pop for the right residual we can infer that e is a full right identity, and similarly for the left residual and e'. Showing just the first we start with e ::; a -+ (a 0 e) (fusing), and by "popping" we obtain a ::; a 0 e. But we get a 0 e ::; a by residuation from e ::; a -+ a, which comes from a ::; a by "pushing." Again we return to the topic of "residuals without residuation." By an assertional implicational poset we shall mean a structure (S,::;, -+, t), where (S,::;, -+) is an implicational po set and t (thought of as the "conjunction of all logical implications") satisfies a::; b iff

t::;

a -+ b

(push and pop).

The element t functions in the role of the right identity element in a right residuated p.o. groupoid, except for the embarrassment of there being no fusion operator present for it to be the right identity of. But it still allows us to characterize "theorems" ("assertions") by the relationship t ::; a. We next examine what happens when we have both residuals without residuation. By an assertional double implicational poset we shall mean a structure (S,::;, -+, +-, t), where each of (S, ::;, -+, t) and (S,::;, +-, t) is an implicational po set. Note that it is easy to prove that if we had not required the same t, but instead allowed that we have t' in place of t in the second implicational poset, then we could prove t = t' (and thus we have lost no generality by building uniqueness into our definition). Thus we choose to show that t ::; t'. To start, we derive from t' ::; t', by "push" for t, that t ::; t' -+ t'. But next we shall derive t' -+ t' ::; t' (and so t ::; t' results from transitivity). By pseudo-assertion we obtain t' ::; t' +- (t' -+ t'), and by "pop" for t' we obtain t' -+ t' ::; t'. The thing that glues t and t' together is that they "push and pop" the same partial ordering ::;. Remark 8.1.7 "Push" can be easily seen to be equivalent to the following inequality: (4) e::; a -+ a

(self-implication thesis).

This inequality follows immediately from "push" using a ::; a. Conversely, we obtain push by the following sequence of moves:

284

1. 2. 3. 4.

REPRESENTATION THEOREMS

PARTIALLY ORDERED SETS WITH IMPLICATION(S)

a ~ b

(hypothesis) a ...... a ~ a ...... b (1, rule prefixing) e ~ a ...... a (self-implication thesis) e ~ a ...... b (2, 3, transitivity).

It would be nice to find some inequality equivalent to "pop," but the best we have been able to come up with assumes either rule permutation or the presence of the other (left) residual (with the equivalent properties of rule pseudo-permutation or of pseudoassertion). Let us work through the case where we have the addition of the left residual (the case where we have just the right residual but have rule permutation is just a special case, since rule permutation gives that the right residual is also the left residual). In this case one can postulate (5) a +- e ~ a

Proof Recall that in the canonical frame RCI C2C3 iff 'ix, y, if x E CI and x ...... y E C2, then y E C3. Let us suppose that C b C', i.e., 3Ct E T, RCCtC', that is, 'ix, y, if x E C & x ...... Y E C t , then y E C'. Thus for any x E C, since t E C t and t ~ x ...... x, then x ...... x E Ct. And so we have x E C', and so C ~ C'. Conversely, if C ~ C', we show RC[ t )C', i.e., 'ix, y, if x E C & x ...... Y E [t), then y E C'. So we suppose x E C & x ...... Y E [t). This last means t ~ x ...... y, and so by "pop" x ~ y. But since x E C (which is a cone), then y E C ~ C', and so y E C' as needed. D

The above suggests that the way to define a concrete assertional implicational implicational poset is to take a structure (U, b, R, T), where (U, b, R) is an articulated frame (as defined above), T ~ U, and for a, fJ E U, a b fJ iff 3r E T, RarfJ. We then consider the hereditary subsets ~t (U), and on these define ...... , +-, 0 as above. We must close T itself upward under b. Calling the result Z, we have

(specialized pseudo-assertion).

Z~A

From this we derive "pop" as follows: 1. e

~

2. a 3. a

~ ~

~

2. a

+-

Z

(a +- e) ...... a

e

~

a

(pseudo-assertion) (1, pop).

The dual form of specialized pseudo-assertion e' ...... a to "pop" for the left-residual.

~ a

would similarly be equivalent

Representations of assertional implicational posets The element t causes more problems that one might have thought. We shall first work through the representation when we have only one implication ...... , and then work through the case of assertional double implicational posets. Then, we shall add fusion to obtain a full assertional residuated p.o. groupoid. The issue is how to verify "push and pop." As we saw above, t

~

...... A.

Our definition of a concrete assertional implicational poset is not yet complete, for we still need a natural condition that ensures "pop":

a ...... b (hypothesis) b +- e (1, rule pseudo-permutation) b (2, specialized pseudo-assertion, transitivity).

Conversely, from "pop" we can derive specialized pseudo-assertion: 1. e

285

a ...... a

is equivalent to "push," and so we shall start there. In the representation, t and a will be assigned certain sets T, A ~ U, and so we need T ~ A ...... A, i.e.,

A ...... B implies A ~ B.

Thus suppose that Z ~ A ...... B and a E A. Clearly what is needed to ensure that a E B is that 3r E Z, Ram. But this is just to require that b be reflexive. But this is built into our definition, since we required that (U, b, R) be an articulated frame, which then requires b to be a partial order. There is a lot more required of an articulated frame that is strictly needed, but all of this information can be read off of the canonical frame and so there is, certainly, no harm in requiring it. Thus, not only are transitivity and antisymmetry not strictly needed, but not all of the "monotonicity" conditions on Rare needed. Indeed, we strictly need only the "monotonicity" condition that Rxyz & y' ~ y' implies Rxy' z, which guarantees that A ...... B is hereditary given that both A and Bare. However, it is convenient to require all of the "monotonicity" conditions (they clearly hold in the canonical frame) just when we have the operations +- and 0 around, since the other conditions are needed to verify that hereditary subsets are closed under these operations. Thus, let us now consider the case of an assertional double implicational poset (S,~, ...... , +-, t). Unfortunately, since we also want Z ~ A +- A, i.e., forallr E T, a E A, RmfJ impliesfJ E A,

for rET, a E A, RarfJ implies fJ E A. This suggests that we define a relation a b fJ iff 3r E T, RarfJ, and that we work with only the subsets of U that are hereditary with respect to b. In the canonical frame, T = {C : t E C}, and in particular [t) E T. Moreover, in the canonical frame b is just inclusion between sets: Lemma 8.1.8 (Inclusion lemma) In the canonical frame, C b C' (ffC

~

~

C'.

this last suggests that we also define a relation a b' fJ iff 3r E T, RrafJ, and that we work with the subsets of U that are hereditary with respect to b' (as well as of course b). Things are getting a bit complicated, but the good news is that we can show that in the canonical frame band b' are in fact identical. We already know from the Lemma 8.1.8 that b is just ~ on cones. The argument can be run through symmetrically to obtain that b' is also just ~ on cones, and so b = b'.

SEMI-LATTICES

REPRESENTATION THEOREMS

286

The above suggests that the way to define a concrete assertional double implicationa I poset is to take a structure (U, 1:, R, T), where (U, 1:, R) is an articulated frame (as defined above), T ~ U, and require for a, f3 E U, a I: f3 iff 3r

E

T, Rarf3 iff 3r

E

T, Rmf3.

We then consider the set U, define Z as the closure of T under 1:, and define -+ and +on the hereditary subsets as above. For a concrete assertional residuated p.o. groupoid, we proceed as above, but this time we want to have 0 as well as -+ and +-, and we need to verify Z

A

0

= A,

A

Z

0

= A.

Again we use an articulated frame, consider the hereditary subsets g-J t (U), and on these define -+, +-, 0 as above. Again we close T itself upward under 1:, calling the result Z. Canonical frames are defined as before. The term "correspondence" was coined by van Benthem to label the fact that in modal logic certain conditions on the accessibility relation correspond to particular modal laws. The same phenomenon is alive here, in that postulating a certain property of the ternary relation R of a frame forces a certain inequality to hold, and vice versa. In Dunn (1986) one can find a list of such correspondences derived from works of Routley and Meyer, but properties of the relation R are linked to implicational theses. One needs to further link the implicational theses to properties of residuation (as we did above) to obtain correspondences to properties of fusion. Thus, to illustrate with the simplest example, commutativity of 0 (a 0 b = boa) corresponds to the commutativity of R in its first two positions (Raf3 X implies Rf3ax), and Routley and Meyer link this last to assertion. It is easy to see that requiring R to be commutative in its first two places forces 0 as defined on a frame to be commutative, since A

0

B

= {X:

3a

E

A,f3

E

B,Raf3x}

= {X:

3f3

E

B,a

E

A,Rf3ax}

= BoA.

Conversely, if 0 is commutative, then in the canonical frame Ra f3 X implies Rf3a X. (In particular, if we have cones in the canonical frame as we had so far then RCI C2 C 3 implies RC2CI C3.) For if a E a and b E f3, then (since Raf3 X) we have a 0 b E X, and so by commutativity of 0 we have that boa E X. Other correspondences are not quite so straightforward. It is possible to develop a framework where one does not have "pop," but only "push," though things get a bit hard to keep track of since then one has both t and t', I: and 1:', and one must look at sets that are hereditary simultaneously under both relations. When fusion is present this is equivalent to having just e

0

a ::; a,

but not the converse of these relations.

a

0

e' ::; a,

287

There are logics where such a generalization is useful, e.g., "ticket entailment" -+ a ::; a, and "non-alethic" modal logics, lacking Oa ::; a. The relationship between implication and residuation has been known for years (cf. Birkhoff 1940). And one can find many correspondences between laws for fusion and laws for residuation by browsing through, say, Birkhoff's Lattice Theory (especially the 1967 edition), Certaine (1943), Fuchs (1963), Dunn (1966) (relevant portions of which are reprinted in Anderson and Belnap 1975), etc. Perhaps the single best source is Meyer and Routley (1972), who provide a long list of such correspondences. But since they assume fusion is commutative (as it is in most usual logics), they have no distinction between the left and right residual. But the reader should be aware that there are applications where one would not want this assumption, e.g., the Lambek calculus and "quantales" (cf. Vickers 1989). More recent work connecting residuation and implication is that of Dosen (1988, 1989b), which has many affinities with the material presented here. The representation results above are modeled after those of Meyer and Routley (1972), which in tum are based on their semantics of relevance logic. We used cones whereas they use prime filters because they are assuming an underlying distributive lattice. (cf. Anderson and Belnap 1975), which lacks t

8.2

Semi-lattices

Theorem 8.2.1 (Representation for semi-lattices) If (A, /\) is a semi-lattice, then h : A -+ g-J(A) such that h(a) = {x E A : x ::; a} is an isomorphism of A into (g-J(A), n), i.e., every semi-lattice is isomorphic to a semi-lattice of sets. Proof. Note that unlike in the previous section h maps here each element a of the algebra into the principal dual cone generated by this element. We show that h(a/\b) = h(a)nh(b). Assume that r E h(a/\b). Then by the definition of h, r ::; a/\b. But then r ::; a and r ::; b (since a/\b ::; a, b). Then r E h(a) and r E h(b), and so r E h(a) n h(b) as desired. For the converse, note that /\ is glb, and so all the steps are reversible. Also, h is one-one, since each element generates a unique principal dual cone. (In other words, if a b then there is an element a E A, such that a ¢. h(b) (but, of course, a E h(a». 0

i

We cannot hope to simply extend the above result to lattices, that is, we cannot have every lattice isomorphic to a lattice of sets where /\ is n and V is U. (Such a lattice of sets is called a ring of sets.) This is because a ring of sets is such that the distributive laws hold:

= (a /\ b) V (a /\ c); a V (b /\ c) = (a V b) /\ (a V c).

(DLl) a /\ (b V c)

(DL2)

And not every lattice is distributive. Figure 8.1 contains a counter-example.

288

a

b

c

o FIG. 8.1. A non-distributive lattice

8.3

Lattices

The question arises as to how to extend the previous section's representation of semilattices in terms of sets to full lattices. This turns out to be more complicated and/or less intuitive than one might naively think. The problem, put quickly, is that while the meet of a lattice is nicely interpreted as intersection, there is no corresponding way to simultaneously understand lattice join which does not over-interpret it. The most natural way to think of join is as union, but since intersection distributes over union this restricts us to distributive lattices. As we shall see in the next sections, this is in fact the way that Stone represented distributive lattices, but what can be done for lattices generally? We talk above of the simultaneous understanding of meet and join, because there is no problem in interpreting each separately. Indeed, they can both be interpreted as intersection, since there is no intrinsic difference between a meet-semi-lattice and a join-semi-lattice. The representation of Theorem 8.2.1 applies just as well to (A, v) as it does to (A, 1\). The problem arises when we simultaneously attempt to represent a lattice (L, 1\, v). We cannot have 1\ and veach at the same time interpreted simply as n. In Section 13.5 we will briefly discuss two other representations of lattices, one by Urquhart, the other by Hartonas and Dunn. Both of these representations may be regarded as building on the idea that a lattice is somehow the "gluing together" of its meet-semi-lattice and its join-semi-lattice, and that the semi-lattice representations can be similarly glued together to get a representation of the whole lattice. But those representations need further ideas. For the moment we content ourselves with a much simpler representation of lattices due to Dosen, and suggested to him by an earlier representation result of Birkhoff and Frink (1948). We shall say more about the relationship between these results below. The central notion used by Birkhoff and Frink is that of a join-irreducible filter, which is a proper filter that is not an intersection of two other filters. We define in the next section the notion of a join-irreducible element of an algebra; it will be easy to see then

289

LATTICES

REPRESENTATION THEOREMS

that the principal filter generated by a join-irreducible element is an example of a joinirreducible filter. (Join-irreducible filters generalize prime filters - see Definition 3.18.3 - which are used in a representation of distributive lattices.) Mapping each element of the algebra into the set of joint-irreducible filters containing this element gives rise to a lattice. Birkhoff and Frink show that this lattice is isomorphic to the original one, with meet being intersection, and therefore called by them a "meet-representation." Note that they do not provide an explicit definition of the join operation. This becomes important in the comparison to Dosen. To make our presentation a bit more formal we prove here a separation principle for join-irreducible filters similar to the separation principle we proved in the previous section representing semi-lattices.

Lemma 8.3.1 (Join-irreducible filter separation principle) Let L be a lattice, and let a b. There is a join-irreducible filter J containing a but not b.

i

Proof. The principal filter generated by a certainly contains a, but b rf. [a). Now define a set of filters E as E = {F : a E F & b rf. F}. Due to the previous observation, [a) E E. It is routine to show that the union of any non-empty chain of E is a filter, and enjoys the defining property of E, hence it is a member of E. Using Zorn's lemma E has a maximal element, say, J. That is, a E J, b rf. J, and J is maximal with respect to these properties. We still have to show that J is join-irreducible. Assume to the contrary that J is join-reducible, that is, for some Fl and F2 (J i= Fl i= F2 i= J) J = Fl n F2. Then, J is a proper subset of both of these filters (J C Fl, J C F2). Were b E Fl and also b E F2, then one would have that b E Fl n F2 contradicting b rf. J. Thus, let us assume that b rf. Fl (the argument with the assumption b rf. F2 is similar). Since J C Fl there is acE Fl which is not in J. a, c E Fl implies a 1\ c E Fl, since Fl is a filter. Furthermore, a 1\ c b, because b rf. Fl. Clearly this contradicts J's maximality. Thus, J is join-irreducible. 0

i

Theorem 8.3.2 (Meet representation of lattices) Every lattice regarded just as a meet-semi-Iattice is isomOlphic to a meet-semi-lattice of sets (with intersection as the meet). Proof. The crucial insight to this proof is to use sets of join-irreducible filters. Define the map h as h(a) = {F: F is join-irreducible and a E F}.

To show that h is a homomorphism for 1\ assume F E h(a 1\ b). The following series of iffs is justified by the definition of h, by filter properties, and by the common understanding of set intersection: FE h(al\b) iff al\b E F iff a E F&b E F iff FE h(a)&F E h(b) iff FE h(a)nh(b).

The map is one-one, and hence an isomorphism, by the previous lemma.

o

This representation treats a lattice as an ordered algebra with intersection being meet, but without an operation corresponding to join. The set of filters, or of principal

REPRESENTATION THEOREMS

290

LATTICES

filters, or of join-irreducible filters (filters which are not the intersection of two other filters) is suitable for this representation. The canonical map h in each case assigns to each element of the lattice the set of filters (of the particular kind) that contain this element. Ono and Komori (1985), and independently Dosen (1989b), provided Kripke-style semantics for substructural logics with non-distributive conjuction and disjunction. I Dosen references the Birkhoff-Frink representation as his source, but he defines ajoin operation as they do not. As we shall see below, Dosen is overly generous in his citation because one cannot extend the Birkhoff and Frink result to represent join in addition to meet. We have recently found out from Dosen that he explicitly proved the representation of lattices that we give below, and it is forthcoming in as a paper in a volume edited by A. Krapez and published by the Mathematical Institute of Belgrade: A Tribute to S.B. Presic: Papers Celebrating His 65th Birthday. In this paper too he attributed what is really his result to Birkhoff and Frink. Ono, Komori, and Dosen give a Kripke-style semantics for substructural logics based on a semi-lattice-ordered monoid, in which conjuction has the usual truth condition x F cP A lfJ iff X F cP and X F lfJ, and in which disjunction has the seemingly unusual truth condition: X

F cP V lfJ

iff :la, 13, [a n 13 :::; X, and (a

F cP or a F lfJ)

and (13

F cP or 13 F lfJ)].

For our immediate purposes there is no need for the monoid (since it is used to define implication and fusion), and so we consider that we have just a meet-semi-Iattice. Things get a bit confusing as to level, since we are representing a lattice using a semi-lattice. To keep the levels clear we shall introduce separate notations and a philosophical reading. We shall denote a semi-lattice which is a "Kripke frame" as (U, b, n). We think of the elements of U as information states, and denote them by lower-case Greek letters a, 13, X. a b 13 says that 13 contains at least the information of a. So far all is familiar from Section 8.1. What is new is the operation an 13, which is something like the intersection of the two pieces of information. 2 With the operation n, we can in fact define a b 13 in the usual way as an 13 = a. It is familiar from intuitionistic logic that a "proposition" A is not just any arbitrary subset of U, but rather one that satisfies the "hereditary condition," which says: (her) If a E A and a b 13, then 13 E A. This fits with the idea that b is the information order. Ono and Komori, and Dosen require this in the form: If a I Cf.

F cP and a

b 13, then 13

F cp.

also Ono (1993).

2Note well that this is not the "conjunction." The conjunction would contain all of the information in both a and {J, but this contains only the information that is common to the two. We want a n {J k a, {J.

291

They also require a kind of "downward" hereditary condition: If a

F cP and 13 F cP, then a n 13 F cp.

The "algebraic form" of this is: (cl) If a

E A

and 13

E A,

then a n 13

E A.

In standard terminology of lattice theory, (her) and (cl) together are the definition of A being afilter (with the small, technical difference that we allow A to be empty in this section, whereas in our Definition 3.18.2 we excluded the "empty filter"). The upward hereditary condition is much more intuitive than the downward one. Let a and 13 be as independent pieces of information as one can imagine. Then one might think that a n 13 would be empty. Suppose now that a and 13 both make a proposition C true. How can this be since a and 13 are supposed to have so little in common? Well, perhaps C is the disjunction of two propositions A and B, with a making A true and 13 making B true. Why should a n 13 make C true? The answer is, we guess, that a and 13 have more commonality that we might have thought. They at least have in common the (disjunctive) information in C. We are not totally satisfied with this answer, but it is at least a story. There are probably others that do not have closure under n. Returning to the mathematics, we let A, B, C etc. range over filters of U. We think of these as "propositions." A A B = An B, A V B = {X : :la, 13 such that a n 13 b X and a E A u Band 13 E A u B}. The latter corresponds to the "unusual" definition of disjunction, but in fact it is not so unusual. Those familiar with lattice theory will recognize it as just the filter generated by A u B. We observe that this gives a lattice with the lattice ordering being ~. Since it is well known that n gives glbs, it suffices to show that A V B is the lLib of A,B. We first show that A ~ A V B. Suppose that X E A. Set a = 13 = X and the result is clear. The argument that B ~ A V B is symmetric. So all that remains is to show that A V B is the least among upper bounds. Let C 2 A, B. We must show that A V B ~ C. Thus suppose that X E A V B, i.e., :la, 13 such that an 13 b X and a E Au Band 13 E Au B. Clearly, Au B ~ C, and clearly a, 13 E Au B, and hence a, 13 E C. Since C is a filter, then a n 13 E C, and so X E C. Fact 8.3.3 A u B

Proof If x

E

~

A vB.

Au B, then set a = 13 =

x.

Fact 8.3.4 It is not always the case that A V B

D ~

Au B.

Proof Follows from the fact we can represent non-distributive lattices. Here is a "concrete" counter-example, which also proves the claim. D Example 8.3.5 Consider the free meet-semi-lattice generated by {a, b, c} as a frame, b being:::; and n being A. Take A to be [a A b), that is, the principal filter generated by a A b, and B to be [b A c). Clearly, a A c, a A b A c ¢ Au B, but A V B is the whole carrier set. (The minimal (but perhaps less revealing) example can be obtained over {a, b} in a similar way.)

REPRESENTATION THEOREMS

292

We shall denote an abstract lattice which we are trying to represent by the notation (L, /\, v), and denote its elements by a, b, c. We let :S denote its partial order (definable in the usual way as a :S b iff a /\ b = a).

Theorem 8.3.6 Every lattice (L, /\, V) is embeddable in the lattice offilters of a semilattice. Proof. Let (L, /\, V) be an arbitrary lattice, and let F, G, H range over the filters of L. The key to the proof is that the collection of filters of L forms a semi-lattice in a natural way. Define the canonical frame (U L, !:LnL) so that UL is the set of filters of L, and !:L is just inclusion among these filters, and nL is intersection of these filters. We define h(a)

FINITE DISTRIBUTIVE LATTICES

8.4

Finite Distributive Lattices

Let us try now to extend the result for semi-lattices from Section 8.2 to distributive lattices (lattices in which the distributive laws hold). But even if (A, /\, V) is a distributive lattice, h : A -+ \{.:i(A) such that h(a) = {x E A : x :S a} is not necessarily a homomorphism of A into (\{.:i(A), n , U). Figure 8.2 contains a counter-example in which h(a V b) =f. h(a) U h(b). The point is that 1 :S a V b, yet 1 a and 1 b, so 1 E h(a V b), yet 1 ¢ h(a) and 1 ¢ h(b).

i

i

{1,a,b,O}

= {F : a E F}.

We first show that h(a) is a proposition. This is clear, since if a E F and F ~ G, then a E G; and if a E F and a E G, then a E F n G. We show that h is an isomorphism. That h is one-one and preserves /\ is familiar. So it suffices to show that h preserves V, i.e.,

a

F E h(a V b) ¢}

3G, H[G

nH

~

¢}

Going from right to left, there are four cases: (1) a E G, a E H; (2) a E G, b E H; (3) bEG, a E H; (4) bEG, b E H. In each case we have a V bEG n H since a :S a V b and b :S a V b. But since G n H ~ F, then a V b E F as required. Now going from left to right, suppose a V b E F. Set G = [a) (the principal filter determined by a) and set H = [b). Then G n H ~ F, for if x E G n H then a :S x and b :S x. It follows that a V b :S x and so x E F. It is immediate that a E G, and so a E G or bEG. Similarly, it is immediate that bE H, and so a E H or bE H. D The reader is surely asking by now whether we can prove the representation above by using join-irreducible filters in place of arbitrary filters. An ingenious counter-example was produced by Katalin BimbO, and it has been pointed out by Tatsutoshi Takenaka that the essential feature of the counter-example can be found in the lattice of Figure 3.17, as can be seen by the following exercise.

Exercise 8.3.7 Consider the lattice of Figure 3.17. For an element x, let h(x) be the set of join-irreducible filters that contain x. Consider the join-irreducible filter [d). a- vb - E [d) and so it is required that there exist join-irreducible filters G and H with a- E G, b- E H such that G n H ~ [d). Show that this is not the case. It turns out that one can extend the representation of lattices as given by Theorem 8.3.6 to lattice-ordered residuated groupoids of various kinds, interpreting the additional operators 0, -+, and +- by way of a ternary accessibility relation R ~ U 3 as in Section 8.1. This representation theorem is implicit in the Kripke semantics of substructural logics as developed by Ono and Komori (1985) and Dosen (1989b). See also Ono (1993). We do not pursue this here.

{b,O}

{O} FIG. 8.2. Counter-example

F E h(a) V h(b),

F & (a E G or bEG) & (a E H or bE H)].

{a,O}

b

o

h(a V b) = h(a) V h(b),

a V bE F

293

Now what if we send a E A into just those elements x :S a such that for all elements y and z, x :S y V z 9 x :S y or x :S z? We call elements satisfying this property joinprime (or just prime when there is no confusion with meet-primeness, defined dually). The terminology is motivated by the fact that if we consider the lattice (:~~+, /\, V), where Z+ is the set of positive integers, m /\ n is the greatest common divisor of m and n, and m V n is the least common multiple of m and n (then m :S n iff min), then p is join-prime iff it is prime in the usual sense. Every element of a chain is prime. So we define h : A -+ \{.:i(A) so that h(a) = {p E A I p:S a and p is prime}. Clearly h is a homomorphism of A into (\{.:i(A), n, U). We already know from the representation theorems for semi-lattices that h preserves /\. And we know from the way we defined primeness that h preserves V. The question is whether h is one-one. Clearly what we need is the following. Separation principle: Va, bE A, if a b, then 3p E A such that p is prime, p :S a, and pi b. We next investigate when the separation principle holds. We first study prime elements. An element a is join-reducible (or just reducible when there is no confusion with the dual notion) iff for some elements band c, a = b V c and yet a =f. b and a =f. c.

i

Exercise 8.4.1 (1) Show that every prime element is irreducible. (2) Show for a distributive lattice that every irreducible element is prime. (3) Find a non-distributive lattice with an irreducible element that is not prime.

GENERAL REPRESENTATION

REPRESENTATION THEOREMS

294

(4) Recall from the previous section that a filter is said to be join-reducible iff it is proper and is not the intersection of two other filters. Show that a principal filter is join-irreducible iff it is determined by a non-zero (join-)irreducible element.

Theorem 8.4.3 Every finite distributive lattice satisfies the separation principle. Proof The claim follows from the next lemma together with Exercise 8.4.1 (2).

Lemma 8.4.4 In any finite lattice, every (non-zero) element a = are all of the irreducible elements less than or equal to a.

Solution (1) Suppose a is prime, and a = b V c. Then a S b or a S c. But since b, c S a, a = b

or a = c. (2) Suppose a is irreducible. Suppose as b V c. Then a = a A (b V c) = (a A b) V (a A c). Then a = a A b or a = a A c, i.e., a S b or a S c. (3) In Figure 8.3 a is irreducible, but not prime since a S b V c, but a b and a c.

i

i

(4) Left safely to the reader.

295

0

Vi<11 Pi, where the PiS -

Proof If a is irreducible we are through. Otherwise a = al Vaz (a ~ aI, a ~ az). If al and a2 are irreducible, we are through. Otherwise, say al = all Va12 and az = a2l VaZ2. If aJ J, a12, aZJ, a22 are irreducible we are through, otherwise, etc. Now this process 0 cannot go on indefinitely, since the lattice is finite.

Exercise 8.4.5 (1) Remove the "etc." from the above proof. (2) Find a non-distributive lattice in which the separation principle does not hold.

1

Solution

a

b

(1) Let H be the set of elements that cannot be represented as above. If the lemma is false, 3a E H. Since A is finite, there must be a minimal such element a. a = b V c (a ~ b, a ~ c). Since a is minimal, b = ViSJ qi, where the qiS are the irreducible elements S b. And c = Vi
c

i

o FIG. 8.3. A non-distributive lattice with an irreducible, non-prime element

Observe that 0 (if it exists) is always prime (and irreducible), so there is not much point in saying that it is. We henceforth for the sake of convenience exclude 0 as prime (or irreducible). Notice that this does not interfere with the fact that h : A --+ \?J(A) defined above is a homomorphism of (A, A, V) into (\?J(A), n, u). {p E A : p S a V band

=

pisprime(andp~O)} {pEA:Psaandpisprime(andp~O)} U {pEA:PSb and p is prime (and p ~ O)}. Adding p ~ 0 just gives you more information. Further, the separation principle will be true in a lattice just when it is true with p

relativized to non-zero prime elements. Exercise 8.4.2 Establish the above. Solution Suppose the separation principle holds. Then suppose a thatp S a andp b. Thenp ~ O.

i

i

i

Theorem 8.4.3 can be extended to those distributive lattices satisfying the so-called minimum condition: there are no infinite descending chains of the lattice al > az > a3 > ... ai > ai+l ... , but it unfortunately fails generally. Counter-example: Let X be an infinite set. We define an equivalence relation on \?J(X) as follows: for x, y S;;; X, define x == y iff x and y differ in only in a finite number of elements, i.e., if (x - y) U (y - x) is finite. Let [x] = {y S;;; X : x == y}. Define [x]A[Y] = [xny], and [x]v[y] = [xuy]. These are well-defined (single-valued) operations (i.e., == is a congruence), and we obtain a distributive lattice. Now let x be an infinite set, and y a finite set. Then [x] [y]. There are no irreducible elements. [0] is the class of all finite sets, and if z is infinite, then z can be partitioned so that z = ZI U zz where ZI n zz = 0 and ZI and zz are both infinite. Hence [z] = [zIl V [zz] ([z] ~ [zIl, [z] ~ [Z2]), and hence [z] is reducible.

i

Exercise 8.4.6 Verify all details in the above example. b. Then 3p such

In fact, nothing substantial is affected by our consideration of just non-zero prime elements. "Zeros" are always nuisances in math, sometimes we want to include them, sometimes exclude them, and almost always our choices are dictated by prevailing fashions.

8.5

The Prohlem of a General Representation for Distributive Lattices

N ow we appear to be in the soup as far as a general representation for distributive lattices is concerned. But Stone (1937) had the idea of considering not just the elements of a distributive lattice, but also certain "ideal elements."

STONE'S REPRESENTATION THEOREM FOR DISTRIBUTIVE LATTICES

REPRESENTATION THEOREMS

296

297

We now set down the abstract properties which a set of elements P must have for Stone's argument. Stone's conditions are:

{a, b}

(1) x /\ yEP iff x E P and yEP.

a

h

b

--+

{b}

{a}

o

o

FIG. 8.4. A lattice and the result of the application of h In order to understand his method, we consider a prime element p of a distributive lattice A, and let [p) = {x E A: p S x}. We thereby establish a one-one correspondence between the prime elements p and the sets [P). If p = q, then obviously [P) = [q). And if [P) = [q), then since p S p, then p S q, and similarly q S p. Instead of considering the function h defined above (h(a) = {p E A : p S a and p is prime}), we can consider a function h' that associates with each a E A the class h' (a) of all the sets [p) where p is a prime element less than or equal to a. h' is one-one since hand [ ) were one-one. And h' preserves /\ and V since h preserved /\ and V. Consider the example in Figure 8.4. It shows a four-element distributive lattice with h applied to it. Figure 8.5 shows the application of h' to the same lattice. It might be thought that in passing from the prime elements p to the sets [p), nothing is gained. And straightforwardly this is so, because of the one-one correspondence noted above. But Stone saw that the demonstration that h' is a homomorphism could be made so as not to depend upon the primeness of the elements p determining the sets [p), but instead on certain abstract properties of the sets themselves. We should be reminded at this point of Dedekind' s famed completion of the rationals so as to obtain the reals. He identified real numbers with certain sets of rationals called "cuts," and although every rational determined a cut, not every cut was determined by a rational. Indeed, the so-called "upper cut" determined by a rational p is the set of rationals greater than or equal to p, and is often denoted by [P), making the analogy even more closely with what Stone did.

a

b

o

h' --+

{{b, I}}

{{ a, I} }

o

FIG. 8.5. The result of the application of h'

V

yEP iff x

E

P or yEP.

Recall «Fl+) from Section 3.18) that a (non-empty) set satisfying (1) is called a filter (or dual ideal). Recall also (Definition 3.18.3) that a (non-empty) set satisfying both (l) and (2) is called a prime filter. We also noted there that the intersection of a family F of filters (all of which contain some common member) is also a filter, and that for this reason we may speak of the filter generated by a (non-empty) set X, denoted by [X), and mean the least filter which includes the set. That is, [X) = {F : F is a filter and F :2 X}. (This filter need not be proper, it may be the lattice itself.) It is memorable to regard a filter as the set of provable propositions and a prime filter as the set of true propositions (where /\ is and, V is or). We could have dualized all of the above discussion, utilizing ideals. We can think of an ideal as the set of refutable propositions (those whose negations are provable), and a prime ideal as the set of false propositions. The lattice itself furnishes a trivial example of an ideal, but we adopt a convention similar to the one we have for filters so as to exclude it except when we indicate otherwise.

n

Exercise 8.5.1 P is a prime filter iff P is a prime ideal. Solution P is a prime filter iff P satisfies Stone's conditions: (1) x /\ yEP iff x E P and yEP; (2) x V yEP iff x E POI' yEP. Dually, P is a prime ideal iff P satisfies the duals of Stone's conditions: (1') x V yEP iff x E P and yEP;

(2') x /\ yEP iff x

E

P or yEP.

(2') is just the contrapositive of (1). And (1') is just the contrapositive of (2).

8.6

{ {a, 1}, {b, 1} }

1

(2) x

Stone's Representation Theorem for Distributive Lattices

By a ring of sets is meant a non-empty collection R of sets that is closed under (binary) intersection and union, i.e., for X, Y E R, X n Y E R and X U Y E R. The power set of all subsets of some set is an example. Obviously, every ring of sets is a distributive lattice. We have, conversely, the following theorem. Theorem 8.6.1 (Stone's representation for distributive lattices) EvelY distributive lattice is isol170/phic to a ring of sets. In order to establish Stone's representation we shall use Zorn's lemma to establish the following lemma:

REPRESENTATION THEOREMS

298

Lemma 8.6.2 (Prime filter separation principle) Let L be a distributive lattice with a, bEL such that a b. Then there exists a prime filter P of L such that a E P and

i

b

¢ P.

Proof Let L, a, and b be as in the hypothesis. Then [a) is a filter separating a from b. But it may not be prime. So the proof consists of extending [a) until it is prime, but in such a way so as not to capture b. We let E be the family of filters of L which have a as a member but not b. E is nonempty, since it contains [a). Now let C be any non-empty chain of E. Then U C E E. For clearly a E U C. And clearly, b ¢ U C. It remains only to show U C is a filter. So we suppose c, dE U C, but then there are F, F' E C such that c E F and d E F'. But either F ~ F' or F' ~ F, so either both c, d E F' or both c, d E F. But since both F and F' are filters, then either c 1\ d E F' or c 1\ d E F. But then in either case c 1\ d E U C. Next we suppose that c E U C and c ~ d. But then there is F E C such that c E F. And since F is a filter, d E F. So d E U C. We have just established the hypothesis for (Zorn's lemma), so we conclude that E has some maximal member P. Recalling our definition of E, we have that P is a filter such that a E P and b ¢ P. It remains to show that P is prime. So we suppose that for some c, dEL, c V d E P but c ¢ P. By Theorem 3.18.12 [P, c) = {y E L : (::Ix) such that x E P and x 1\ c ~ y}, and [P, d) = {y E L : (::Ix) such that x E P and x 1\ d ~ y}. Since [P, c) is a filter, [P, c) ¢ E, since P is maximal in E and [P, c) is a proper superset of P. Thus P ~ [P, c), for if YEP, then for some x E P (namely y), x 1\ c ~ y. And P :j:. [P, c), since c E [P, c), but by hypothesis c ¢ P. But since [P, c) contains a (since a E P), and is a filter, it can only fail to be in E by containing b. Similarly, we show that [P, d) must contain b. Then for some x E P, x 1\ c ~ b, and for some yEP, Y 1\ d ~ b. But then let z = x 1\ y, Z 1\ c ~ b, and z 1\ d ~ b. Then (z 1\ c) V (z 1\ d) ~ b. But since L is distributive, then z 1\ (c V d) ~ b. But Z E P, since P is a lattice, and c V d E P by hypothesis. So then z 1\ (c V d) E P (P a lattice), and then bE P (again, P a lattice). But this contradicts our proof that b ¢ P, so contrary to hyp.othesis, either c E P or dE P, and so P is plime. D

Exercise 8.6.3 Show that in a distributive lattice a filter is join-irreducible iff it is plime. Conclude that the prime filter separation principle (Lemma 8.6.2) is actually a special case of the join-irreducible separation principle (Lemma 8.3.1). Having established the prime filter separation principle, we proceed to the proof of Stone's representation theorem. Proof Let L be a distributive lattice. For a E L, define h(a) = {P : P is a prime filter of L and a E P}. The function h is obviously one-one in virtue of the prime filter separation principle. We next show that h is a homomorphism onto the ring of the sets h(a) (a E L). We recall that a prime filter P can be characterized by the following two conditions:

STONE'S REPRESENTATION THEOREM FOR DISTRIBUTIVE LATTICES

299

(I) a 1\ b E P iff a E P and b E P;

(2) a V b E P iff a E P or b E P.

ThenP E h(al\b) iff al\b E Piff(by 1) a E P and bE Piff P E h(a) and P E h(b) iff P E h(a) n h(b). So h(a 1\ b) = h(a) n h(b). And P E h(a V b) iff a V b E P iff (by 2) a E P or b E P iff P E h(a) or P E h(b) iff P E h(a) U h(b). So h(a V b) = h(a) U h(b). These two chains of "iffs" establish not only that h is a homomorphism, but also something that might be overlooked in the scuffle, namely, that the sets h(a) are actually closed under (binary) intersection and union. So Stone's representation is complete. D

For distributive lattices, there is a close connection between prime filters and homomorphisms into the two-element distributive lattice 2-they co-determine each other. Proposition 8.6.4 Let A be a distributive lattice and let P be a (proper) prime filter of A. For x E A, define h(x) = 1 if x E P, and hex) = 0 if x ¢ P. Then h is a homomorphism onto 2. Conversely, let h be a homomorphism of A onto 2. Define , P = {x E A : x = I}. Then P is a prime filte!: Proof We leave this as an exercise for the reader. For those who want to cheat, look ahead at the proof of the similar connections for Boolean algebras (Theorem 8.10.1). D

Stone's representation can be somewhat trivially rephrased as: Theorem 8.6.5 EvelY distributive lattice is isomOlphic to a direct product oflliEI 2 i . The trick is simply to realize that each element of XiEI2i is just a characteristic function determining a subset of J, and of course each subset determines a characteristic function. We shall go into this in more detail later with Boolean algebras. But this gives another route to the proof of Stone's representation for distributive lattices using Birkhoff's prime factorization theorem (Theorem 2.8.3). We show: Lemma 8.6.6 A (non-degenerate) distributive lattice A is subdirectly irreducible only if it is isomOlphic to 2. Proof Suppose, contrapositively, that A is not isomorphic to 2. Then there is an element a which is neither the top or the bottom of the lattice. We use a to define two congruences: X -=1'1 Y iff x x -=v y iff x

1\ a V

a

= y 1\ a; = y V a.

We can safely leave to the reader that these are congruences, but we note that distribution enters in when showing that -=1'1 preserves v, and -=v preserves 1\.3 These two congruences have as their intersection the equality relation restricted to A, which we denote by E. We know from Lemma 2.8.7 that this means that A is isomorphic to a subdirect product of A/=.A and A/.=v, and so A is not subdirectly irreducible. D 3Hence =A determines a congruence on a meet-semi-Iattice, and similarly for =v on ajoin-semi-Iattice.

REPRESENTATION THEOREMS

300

BOOLEAN ALGEBRAS

301

x = 0 V X = (x /\ x') V x =

Exercise 8.6.7 Show the converse of the above lemma.

argument, interchanging the roles ofx and x', we have (x V x) /\ (x'V x) = 1 /\ (x'V x) = x' V x, i.e., x' ::; X.

8.7

Complemented lattices do not in general have unique complements.

Boolean Algebras

Recall from Section 3.9 that a lattice A is said to be complemented iff'
E

In the example in Figure 8.6, both band c are complements of a. And in the example in Figure 8.7 both a and b are complements of c.

A, 3X' E A

(1) x /\ x' ::; y, (2) y::;XVX'.

Any complemented lattice A has a least element (x /\ x') and a greatest element (x Vx'), and, of course, these elements are unique. We denote the least element by "0" and the greatest element by "1." We define a Boolean algebra as a complemented distributive lattice. Some good references on Boolean algebras include Halmos (1963) and Sikorski (1964). The motivating' example of a Boolean algebra is a collection of subsets of some set S, where the collection is closed under (binary) intersection and union, and also under (relative) complementation. (So if Y is in the collection, so is S - Y, which we customarily denote by Y.) Such a (non-empty) collection of sets is called afield of sets. A field of sets is thus a ring of sets closed under complementation. Recall that Corollary 3.14.10 ensures that each element of a Boolean algebra has but one complement. In virtue of this we let x be the complement of x (which we shall sometimes denote by variants such as -x). From an operational point of view, we may define a Boolean algebra as a quadruple (B, /\, v, -), where (B, /\, V) is a distributive lattice (operationally defined), and where - is a unary operation on B satisfying the complementation laws:

a

o FIG. 8.6. A non-uniquely complemented lattice

a

c b

o FIG. 8.7. Another non-uniquely complemented lattice

(C1) (x /\ x) V Y = y; (C2) (x V x) /\ Y = y. Notice that (C1) and (C2) are dual to one another, so the duality principle for distributive lattices may be extended to Boolean algebras.

Exercise 8.7.1

(ii) (iii) (iv) (v)

(2)

(i)

ais the complemeEt ofa. But also a is the complement of a, since a/\a = 0 and aV a = 1. Hence a= a, since complements are unique.

(ii) We show that

(1) Verify that every element of a complemented distributive lattice has a unique complement. Is this true of complemented lattices in general? (2) Establish that the following laws are true of Boolean algebras:

=a

(double complementation); a /\ b = a V b, a V b = a /\ b (De Morgan laws); ::::; b ~ b ::; (antitonic law); 0 = 1, 1 = 0; a::; b iff vb = 1 a::; b iff a /\ b = O.

(i) a

c

a

a

Solution (1) Suppose x /\ x = 0 and x V x = I. Then x V x' = 1. x' = 0 V x' = (x /\ x) V x' = (x V x') /\ (x V x') = 1 /\ (x V x') = X V x', i.e., x ::; x'. Running through the same

aV bis the complement of a /\ b:

= (a /\ b /\ a) V (a /\ b /\ b) = 0 V 0 = 0; = (a Va V b) /\ (b V a V b) = 1/\ 1 = 1. By duality, we obtain a V b = a/\ b. (iii) a ::; b =? a /\ b = a =? a /\ b = a =? (by ii) a V b = a =? b V a = a =? b ::; a. (iv) Since 0/\1 = 0 and 0 V 1 = 1,0 = 1 (and]" = 0). (v) a ::; b =? (a /\ b) = a =? a vb = (a V b) V b = 1. Conversely, a Vb = 1 =? a = a /\ 1 = a /\ (a V b) ~ (a /\ a) V (a /\ b) = a /\ b. And a /\ b = a =? a ::; b. By (a /\ b) /\ (a V ~) (a /\ b) V (a V b)

duality, a ::; b ¢} a /\ b = O.

302

8.8

REPRESENTATION THEOREMS

STONE'S REPRESENTATION THEOREM FOR BOOLEAN ALGEBRAS

Filters and Homomorphisms

It is common in algebra to find special subsets of algebras determining congruences, e.g., normal subgroups in groups, ideals in rings, etc. A filter F of a Boolean algebra B determines a congruence =F on B as follows. Define a +-7 b = (Ci V b) /\ (b Va), and then define a F b iff a +-7 b E F. Furthermore, every congruence is determined by a filter F, defining F = {x E B : x I} = [1].

=

=

=

Exercise 8.8.1 Show that =F is a congruence on B. Show also that for every congruence on B, there is a filter F ofB such that for a, bE B, a b iff a =F b.

=

=

Proof Suppose M is maximal. Then M is prime, so since x E B, x vi = 1 EM, M contains at least one of x or i. Further, M cannot contain for some x E B both x and i, for then x /\ i = 0 EM, and M is not proper. If F is a filter such that F :J M, then 3x E F such that x ¢ M. But then i EM and 0 hence i E F, but then x /\ i = 0 E F, and hence F is not proper.

Exercise 8.9.3 (1) Show that the following statements are equivalent without using any form of the

Axiom of Choice. Note that for the purposes of the exercise, a Boolean algebra is a non-degenerate one (i.e., one with more than a single element), and a filter is a proper one (i.e., one not identical with the whole algebra). (Hint: For (iv) :::} (i), consider B/=.r) (i) For any Boolean algebra B, if F is a filter of B, then there is a maximal filter M ofB such that F ~ M. (ii) For any Boolean algebra B, for any a, bE B, if a i b, then there is a maximal filter M of B such that a E M and b ¢ M. (iii) For any Boolean algebra B, for any a E B, if a 1'= 0, then there is a maximal filter M of B such that a EM. (iv) For any Boolean algebra B, there is a maximal filter M of B.

In review of the universal connection between homomorphic images and congruences established in Chapter 2, we have that filters detelmine homomorphic images, and that any homomorphic image (or an isomorphic copy thereof) is determined by some filter. Instead of writing "B /='F'" we often write "B / F." 8.9

Maximal Filters and Prime Filters

A filter M is maximal in a Boolean algebra B iff (1 ) M is a proper filter of B; (2) there is no other proper filter F of B such that M c F. Note well that by definition, a maximal filter is a proper filter. We stick to this convention even when we find it convenient to lapse from our previous convention that by a "filter" we mean a proper filter. For Boolean algebras, we have the following identification of maximal filters and prime filters (but the implication from left to right holds also for distributive lattices). Theorem 8.9.1 M is prime.

If M is a (proper) filter of a Boolean algebra B,

then M is maximal iff

Proof Suppose a V b E M, but a ¢ M and b ¢ M. Consider [M, a) and [M, b). Both properly include M, so M will not be maximal if we can show either [M, a) 1'= B or [M, b) 1'= B. Suppose to the contrary. Then for x E B, 3m] E M such that m] /\ a ::::; x and 3m2 E M such that m2/\b ::::; x. But then (ml /\m2)/\a ::::; x and (ml /\m2)/\b ::::; x. But then ((ml /\m2)/\a)V((ml /\m2)/\b) ::::; x. But then, by distribution, (ml /\m2)/\(aV b) ::::; x, and hence x E M. So x E B, x E M, and hence M is not proper. (Note the similarity of this proof to the proof of the prime filter separation principle.) Suppose M is plime. But then x E B, x vi = 1 E M. So x E B, x E M or i E M. Suppose F is a filter and M C F. Then 3x E F such that x ¢ M. Then i E M, and hence i E F. But then x /\ i = 0 E F, and so F is not proper. 0

We also have the following handy characterization of maximal filters in Boolean algebras. Theorem 8.9.2 If M is a (proper) filter of a Boolean algebra B, then M is maximal iff for any x E B, M contains exactly one of x and i.

303

(2) Devise a proof of Stone's representation for distributive lattice for the special case of denumerable distributive lattices that does not use Zorn's lemma (or any other equivalent of the Axiom of Choice). (Hint: Prove the denumerable case of the prime filter separation principle by induction.) 8.10

Stone's Representation Theorem for Boolean Algebras

Remember that the motivating example for a Boolean algebra was a field of sets, which is a ring of subsets of some set X closed not only under (binary) intersection and union, but also under complementation (relative to X). Just as Stone proved that every distributive lattice is isomorphic to a ring of sets, so also he proved the following representation theorem in Stone (1936). Theorem 8.10.1 Every Boolean algebra is isomOlphic to a field of sets. Proof Let B be a Boolean algebra. By Stone's representation for distributive lattices, we know that the mapping h such that for a E B, h(a) = {P : a E P and P is a (proper) prime filter of B} is a one-one mapping preserving both /\ and V in the sense that h(a /\ b) = h(a) /\ h(b) and h(a V b) = h(a) V h(b). It only remains to show that where P is the set of all (proper) prime filters of B, h(a) = P - h(a). This follows immediately from the already mentioned identification of prime and maximal filters of a Boolean algebra, together with the also already mentioned fact that a maximal filter of a Boolean algebra contains exactly one of every element and its complement. Thus, where P is a (proper) prime filter, P

E

h(a) iff - a

E

P iff a ¢ P iff P ¢ h(a) iff PEP - h(a).

0

MAXIMAL FILTERS

REPRESENTATION THEOREMS

304

It is of some interest to see how we obtain a more trivial representation for finite Boolean algebras (and a certain generalization thereof), comparable to the representation we obtained for finite distributive lattices, where we mapped every element into the set of (non-zero) irreducible elements below it. We first introduce the following definition. An element a of a Boolean algebra B is an atom of B iff (1) a i= 0, and (2) x E B, x ~ a =? x = a or x = O. We say that a covers 0, meaning that a > 0 and that there is no element lying stlictly between a and

O. We have the following connection between atoms and irreducible elements of a Boolean algebra. Theorem 8.10.2 Let B be a Boolean algebra. Then an element a of B is an atom is a (non-zero) join-irreducible element.

iff a

Proof (1) If a is an atom, then clearly a is not join-reducible, for if a = b V c where b, c < a, then either 0 < b < a or 0 < c < a, and in either case a is not an atom. (2) Suppose a is non-zero join-irreducible ele.:nent and yet not an atom. The~ for some element b, a > b > O. a = (a /\ b) V (a /\ b) since a = t.:./\ 1 = a /\ (b V b) = (a /\ b) V (a /\ b). a is join-irreducible, thus a = a /\ b or a = a /\ b. However, a i= a /\ b, since a > b. Assume that a = a /\ b. Then b ~ a, and by transitivity b ~ b, which means b /\ b = b i= 0, a contradiction to B being a Boolean algebra. But then neither a = a /\ b, nor a = a /\ b, in contradiction to a being joint-irreducible, thus there is no such b, and therefore a is an atom. D

Theorem 8.10.3 If a is an atom of a Boolean algebra B, then for all x a

E

B, a ~ x

iff

ix.

x

Proof (1) If a ~ x and a ~ X, then a ~ x /\ = 0, and so a = 0 and is not an atom. (2) Suppose a and a ~ 1 = x V X. Since a is prime (by Theorem 8.10.2 and D Exercise 8.4.1 (2», a ~ x or a ~ X. Hence a ~ x.

i x

Since we already have for a finite distributive lattice that h(a) = {x: x ~ a and x is (non-zero and) irreducible} is an isomorphism onto a ling of sets, the above theorem gives us the following. Theorem 8.10.4 For a finite Boolean algebra B, if h is defined for a E B as h(a) = {a E B : a ~ a and a is an atom}, then h is an isomorphism of B onto a field of sets (which are subsets of the set of atoms). Proof By the previous theorem, a ¢ h(a) iff a ~ a iff a A - h(a) (where A is the set of atoms of B).

i

a iff a

i

h(a) iff a E D

The above theorem may be improved, once we notice that the mapping h is onto Let {al, ... , a,d ~ A. Consider al V ." V all' Clearly (al, ... , a,d ~ h(al V ... van). And heal V ... V all) ~ {al, ... , an}, for suppose a E h(al V ... Van), i.e., a ~ al V ... Vall' But then since a is prime, a ~ aI, or ... ,or a ~ an. But then a = aI, or ... , or a = a'b since if a < ai, then either a or ai are not atoms. So h is onto ~(A). This is captured by the following theorem. ~(A).

305

Theorem 8.10.5 (Finite representation theorem) Let B be a finite Boolean algebra. Then B is isomorphic to the field of all subsets of the set of atoms ofB. The question arises as to whether the general representation theorem for Boolean algebras can be improved so the field of sets in question is the field of all subsets of some set. Clearly if it can, then every Boolean algebra has a lot of atoms, since any unit set {x} would be an atom in the field. Indeed, every Boolean algebra B would be atomic in the sense that for every x E B, if x i= 0, then for some atom a E B, a ~ x. Also, clearly if the general representation theorem can be so improved, then every Boolean algebra is complete in the sense that for every X ~ B, the greatest lower bound of X is an element of B, and the least upper bound of X is an element of B. This is because arbitrary (not just finite) intersections performed on a power set give results in the power set, and similarly for arbitrary unions. In analogy with the finite operations, the greatest lower bound of X is denoted by "/\ X," and the least upper bound of X is denoted by "V x." But not every Boolean algebra is atomic, as may be seen by considering the same example (see Section 8.4) we used to show that not every distlibutive lattice satisfies the prime element separation principle. Remember Z was an infinite set, and for X, Y ~ Z we defined X == Y iff X and Y differ in at most a finite number of elements. Then where [X] = (Y ~ Z : Y == X} we defined [X] /\ [Y] = [X n Y], and [X] V [Y] = [X U Y]. We can also define [X] = [Z - X], and so obtain a Boolean algebra. (Verify that - is a well-defined operation and has the complementation properties.) Zj== has not a single atom, since it had no non-zero irreducible elements (such a Boolean algebra is called atomless). And not every Boolean algebra is complete. Let X be an infinite set, for the sake of concreteness the set of positive integers. Call a subset Y of X cofinite iff X - Y is finite. Let C be the collection of finite and cofinite subsets of X. C is a field of sets and hence a Boolean algebra. But C is not complete. For consider the unit sets of the odd positive integers: {I}, {3}, (5}, .... The union of these is the set of odd positive integers, which is not in C. 8.11

Maximal Filters and Two-Valued Homomorphisms

We let 2 be the two-element Boolean algebra. It is unique (except for isomorphic copies) and has the diagram depicted in Figure 8.8. We have the following connection between maximal filters of Boolean algebras, and homomorphisms onto 2. 1

I o

FIG. 8.8. 2

MAXIMAL FILTERS

REPRESENTATION THEOREMS

306

Theorem 8.11.1 Let M be a maximal filter of a Boolean algebra B. Then h : B -+ 2, defined so that for a E B, h(a) = 1 if a E M, is a homomorphism ofB onto 2. Proof (1) This may be verified directly: h(a 1\ b) = 1 iff a 1\ b E M, iff a, b E M, iff h(a) = 1 and h(b) = 1, iff h(a)l\h(b) = 1. From this it follows that h(al\b) = h(a)l\h(b). Similarly, h(a V b) = 1 iff a V b E M, iff a E M or b E M, iff h(a) = 1 or h(b) = 1, iff h(a) V h(b) = 1. Lastly, h( -a) = 1 iff -a E M iff a ¢. Miff h(a) = 0 iff h(a) = 1. (2) Or more elegantly, the result may be inferred from Exercise 8.8.1. Consider BIM, which is a homomorphic image of B under the natural homomorphism, h(a) = [a]. We next notice that if a E M, then a ~ 1 E M, i.e., Ca V 1) 1\ (0 V a) E M. And if a ¢. M, i.e., Ii E M, i.e., -a E M, then a ~ 0 E M, i.e., (Ii V 0) 1\ (1 Va) E M. Since either [a] = [1] or [a] = [0], BIM is isomorphic to 2, and if a E M, h(a) = [1], and if a¢. M, h(a) = [0]. D

Example 8.11.5 The elements of finite direct products of 2 may be regarded as finite sequences of 0 and 1. IIi<22i is shown in Figure 8.9, and IIi<32i is shown in Figure 8.10.

(1,1)

(1,0)

(0,1)

(0,0) FIG. 8.9. IIi<22i

Exercise 8.11.2 Prove, conversely to the theorem above, that if h is a homomorphism of a Boolean algebra B onto 2, then h- 1(1), i.e., {x E B : hex) = 1}, is a maximal filter ofB. The theorem and the exercise give us a natural one-one correspondence between maximal filters of a Boolean algebra and homomorphisms of the algebra onto 2. We may thus restate our prime filter separation principle in telms of separation by homomorphisms as follows.

(1,1,1)

(1,1,0)

(0, 1, 1)

(1,0,0)

(0,0,1)

Theorem 8.11.3 (Fundamental homomorphism) Let B be a Boolean algebra with a, bE B such that a b. Then there is a hOlll011101phism h ofB onto 2 so that h(a) = 1 and h(b) = 0, i.e., so that h(a) h(b).

i

307

i

We first recall some concepts of universal algebra from Chapter 2, but specialized to the case of Boolean algebras. A direct product of an indexed set of Boolean algebras (Bi )iEI is that algebra whose elements are the indexed sets (ai)iEI, ai E Bi, and whose operations are defined "componentwise" as follows: (ai)iEII\ (bi)iE[ (ai)iEI

V

= (ai 1\ bi)iEI,

(bi)iEI= (ai

(ai)iEI

V

bi)iEI,

= (ai)iEI.

Evidently, by the componentwise definitions of the operations, the direct product of Boolean algebras is again a Boolean algebra. The direct product of (Bi) iEI is written IIiEI Bi. The class of all Boolean algebras is thus closed under the processes of taking subalgebras, homomorphic images, and direct products. Any "equationally definable" class of algebras has this property. Birkhoff showed that conversely, any class of algebras closed under these three processes is equationally definable. Exercise 8.11.4 Verify that the direct product of an indexed set of Boolean algebras is a Boolean algebra.

(0,0,0) FIG. 8.10. Ili<32i

Exercise 8.11.6 Construct a diagram of IIi<4 2i. We may represent every finite Boolean algebra except the so-called degenerate, oneelement Boolean algebra, via a direct product of 2. And more generally, we have the following: Theorem 8.11.7 (Fundamental Embedding) Let B be a (non-degenerate) Boolean algebra. Then there is some direct product of 2, IIiEi 2 i, and some function h : B -+ IIiEi 2i, such that h is an isomorphism ofB into IIiEI 2 i.

REPRESENTATION THEOREMS

308

MAXIMAL FILTERS

Proof Let (hi )iEI be an indexed set of all homomorphisms ofB onto 2. This set is nonempty because of the fundamental homomorphism theorem. Now consider IIiEI2i. Define h : B -+ IIiEI2i so h(a) = (hi(a)iEI. h is obviously a function of B into IIiEI 2i , since each hi is a function ofB into 2. That h is a homomorphism follows from the facts that operations on IIiEI 2i were defined componentwise, and that each hi is a homomorphism onto 2. Thus:

a

o

h(a /\ b) = (hi(a /\ b)iEI = (Ma) /\ Mb)iEI = (Ma)iEI /\ (hi(b)iEI = h(a) /\ h(b); h(a V b)

309

FIG. 8.11. A Boolean algebra

= (hi (a V b»iEI = (hi (a) V Mb)iEI = (Ma»iEI V (hi(b)iEI = h(a) V h(b);

= (hi(a)iEI = (hi(a)iEI = (hi(a)iEI = h(a). h is one-one. For suppose a i b. Then assume without loss of generality h(a)

(1,1)

Finally, that a i b. Then by the fundamental homomorphism theorem there is a homomorphism hi of B onto 2 such that hi (a) i hi(b). But then h(a) differs from h(b) at the "ith" components. 0 We have thus "embedded" B into IIiEI2i, mapping B in an isomorphic fashion onto a certain sub algebra of IIiEI 2i, h(B). Notice that h(B) has stronger properties than a mere subalgebra, namely each of the elements 0 and 1 of 2 is such that Vi E I,3(ai)iEI E h(B) such that ai = 0 and 3(bi)iEI E h(B) such that bi = 1. This is because each hi maps onto 2. So for each i E I, 3a E B such that hi(a) = 0 and 3b E B such that hi(b) = 1. Then h(a) = (Ma)iEI E h(B) and h(b) = (Mb)iEJ E h(B). To put things roughly, neither of the elements 0 and 1 of 2 is "wasted" in the embedding. Each shows up someplace in h(B) at each component i. A subalgebra S of a direct product IIiEI Bi such that Vi E I, Vs E S, 3 (ai) iEI E S such that ai = s is called a subdirect product of the algebras (Bi)iEI (cf. Chapter 2). This gives the following theorem. Theorem 8.11.8 Every (non-degenerate) distributive lattice (in particulm; Boolean algebra) is isomorphic to a subdirect product of 2. Example 8.11.9 Consider the Boolean algebra in Figure 8.11. It has the following two homomorphisms onto 2 (since only [a) and La) are maximal filters): hi : 1 -+ 1 a -+ 1 a-+O 0-+0

h2 : 1 -+ 1 a-+O a-+I 0-+0

Then h(1) h(a) h(a) h(O)

= (hl(1),h2(1) = (hl(a),h2(a) = (hl(7i),h2(a) = (hI (0), h2(0)

= (1,1), = (1,0), = (0,1), = (0,0).

a

(1,0)

o

(0,0)

FIG. 8.12. h applied to the Boolean algebra in Figure 8.11 This situation is pictured in Figure 8.12. Notice, irrelevantly, that the contents of Figure 8.13 are subdirect products of the algebras in Figure 8.12. Exercise 8.11.10 Find all of the two-valued homomorphisms of the Boolean algebra diagrammed in Figure 8.14, and similarly embed it in a direct product of 2 according to the construction contained in the proof of the embedding theorem. Solutiou (1) [a), [b), and [c) are the maximal filters. Thus the three homomorphisms hi, h2 and

h3 are as follows: hl:I-+I a-+O b-+I c-+I a-+I b-+O c-+O 0-+0 h: 1-+ (1,1,1)

h2:1-+1 a -+ 1 b-+O c -+ 1 a-+O b-+l c-+O 0-+0 a-+(1,O,O)

h3: 1 -+1 a-+I b-+I c-+O a-+O b-+O c-+l 0-+0

310

MAXIMAL FILTERS

REPRESENTATION THEOREMS

(1,1)

1

MFP

I

I o

311

*4~

~ ET 3 FIG. 8.15. Implications Between (MFP), (RT), and (ET)

RT

(0,0)

E

FIG. 8.13. Subdirect products use of any other of the implications which have already been established. Thus establishing implication 2 by showing that implications 3 and 6 hold and thus getting 2 by transitivity of implication would not count as a "direct" proof. This requirement makes the project very redundant, but there is valuable conceptual information to be gained by each "direct" connection. Hints:

1

(2) Establish that if F is a field of sets, a X is a maximal filter of F.

U F, and X

= {x E F : a EX}, then

(3), (4) Remember characteristic functions. A characteristic function for a set Y ~ X may be regarded as a function f defined on X and taking values in the twoelement Boolean algebra 2, so that for x E X, if x E Y, then f(x) = 1, and if x ¢ Y, then f(x) = O.

o FIG.8.l4. A Boolean Algebra a -+ (0, 1, 1)

b-+(O,I,O)

b -+ (1,0,1)

C -+ (0,0,1) 0-+(0,0,0)

c-+(I,I,O)

E

(6) Let ITiEI2i be a direct product of 2. Show that if i 1 E J, then the set of elements of the direct product such that the i 1th component is 1 is a maximal filter of the direct product, that is, show that the set of elements (ai)iEI such that for i = iI, ai = 1, is a maximal filter of ITiEI 2i. Solution

Exercise 8.11.11 Consider the following three propositions: (I) Maximal filter principle (MFP): For any (non-degenerate) Boolean algebra B, for any a, b E B, if a i b, then there is a maximal filter M of B such that a E M and b ¢ M. (II) Representation theorem (RT): Any (non-degenerate) Boolean algebra is isomorphic to a field of sets. (III) Emhedding theorem (ET): Any (non-degenerate) Boolean algebra is isomorphic to a subdirect product of the two element Boolean algebra 2. Consider the implications between these propositions as depicted in Figure 8.15. In proving (RT) and (ET), we used (MFP), and have thus effectively established that implications 1 and 5 hold (that is, we have proven implications 1 and 5 without using any equivalent of the Axiom of Choice). The exercise is to establish directly that the remaining implications hold effectively (that is, hold in standard set theory without the Axiom of Choice). By "directly" we mean that each implication is to be established without the

(2) By (RT), B is isomorphic to some field of sets F. Identifying B with F, we see that fora,b E F, a ~ b =? 3x(x E aandx ¢ b). LetP = {c E F: x E c}. Pisafilter, for x E c and xEd iff x E c n d. Further, for every c E F, exactly one of x E c and x E c hold, so P is maximal. (6) Let B be identified by the isomorphism with a subdirect product of ITiEI 2i. For (ai)iEI, (bi)iEJ E B, (ai)iEJ i (bi)iEI then for some i1 E J, ail i bil' i.e., ail = 1 and bil = O. Let P = {(Ci )iEI E B : Cil = 1). P is a filter, for Cil 1\ dil = 1 iff Cil = 1 and dil = 1. Further, for every (q)iEI, either (Ci)iEJ E P or (Ci)iEI E P, for either Cil = 1 or ~ = 1. And not both (Ci)iEJ, (Ci)iEJ E P,for then both Cil = 1 and~ = 1. (3) Let B be identified with a field of sets F. Let J = U F. We map F into ITiEI 2i as follows: for a E F, let hi(a) = I if i E a, and otherwise let hi (a) = O. (Each hi is thus a characteristic function for each a E F.) We then let h(a) = (hi(a»iEJ. Then h is a homomorphism since h(a 1\ b) = (hi(a 1\ b»iEI = (hi(a) 1\ hi(b»iEJ = (hi(a»iEJ 1\ (hi(b»iEI = h(a) 1\ h(b), and similarly for V and -. h is one-one, for

DISTRIBUTIVE LATTICES WITH OPERATORS

REPRESENTATION THEOREMS

312

if a =J b, then either 3i such that i E a and i ¢ b, or vice versa. In the first case hi(a) = 1 and hi(b) = 0, and in the other case it is the other way around. Finally, h(b) is a subdirect product of IIiEI 2i, since each hi is onto 2. This is because if i E a, then i ¢ li. (4) Let B be identified with a subdirect product of IIiEI 2 i. For (ailiEI E B, let h( (ai liEI) = {i E I : ai = I}. h is the desired isomorphism. The equivalences above give us another route to Stone's representation for Boolean algebras. We can show the embedding theorem as an instance of Birkhoff's prime factorization theorem by showing:

Lemma 8.11.12 Every (non-degenerate) irreducible Boolean algebra is isomOlphic to the two-element Boolean algebra 2. Proof We modify the two congruences used in the similar result for distributive lattices (Lemma 8.6.6) by tacking on a clause guaranteed to see that complement is respected: x~Ay x~Vy

iff x /\ a = y /\ a & - x /\ a = -y /\ a; iff xVa=yVa & -xVa=-yVa.

Thus, for example, if x ~A y, then x /\ a = y /\ a & - x /\ a = -y /\ a, and so (using strong double negation), -x /\ a = -y /\ a & - -x /\ a = - - Y /\ a, i.e., -x ~A -yo We leave to the reader the verification of the corresponding fact for x ~v y, as well as the verification that the modified relations ~A and ~v are still equivalence relations and still respect meet and join. The proof is otherwise just as for Lemma 8.6.6. 0

Remark 8.11.13 As the reader can easily see by just dropping the extra parts of the solution to Exercise 8.11.11, there are corresponding "direct" equivalences among the prime filter separation principle, the representation of distributive lattices as rings of sets, and the embedding of distributive lattices as subdirect products of a direct product of the two-element distributive lattice 2. For each positive integer n, it is customary to denote the finite direct product IIi<1l 2i as simply 211 . We extend this to the trivial one-element Boolean algebra by letting n = O. This is motivated by the following considerations. Note that 2° is the set of all mappings from the empty set 0 into 2. But the only mapping whose domain is the empty set is itself the empty map, i.e., 0. Thus 2° = {0} has just one element. It is easy to see that any "componentwise" operation involving 0 leads again to 0.

Theorem 8.11.14 For each positive integer n, 211 is isomorphic to a subalgebra of 211+1. Hencefor i ~ j, 2i is a subalgebra of 2J. Proof For motivation of the following, please consult Figures 8.9-8.11 diagramming 2 1, 2 2 , and 2 3. It should be reasonably clear that the mapping h(al' ... , all I = (a 1, ... , all, all I is the desired embedding, and of course an induction (using the fact that the composition of two embeddings is an embedding) gives the more general result. We leave the details to the reader, but we note that the proof has nothing to do with the

313

algebra being 2 or the operations being Boolean, and in fact represents a law of universal algebra. 0

Exercise 8.11.15 Prove that for every algebra A, the finite direct product All is a subalgebra of A 11+ 1. We close this section by observing: Proposition 8.11.16 Every finite Boolean algebra B is isomorphic to a subdirect product 211 for some integer n. Proof Every finite Boolean algebra B is isomorphic to a subdirect product (2i/iEI. Because the projections are required to be onto, this means that I must be finite. Taking I to be the indices i is just in effect a relabelling, and does not change the isomorphism. There is still the niggling question as to what happens if B is a one-element algebra.

o 8.12

Distributive Lattices with Operators

This section is based on what would appear to be a wrongly neglected paper: "Boolean algebras with operators," by B. Jonsson and A. Tarski. 1951, 1952. Without too much exaggeration it may be said that this remarkable paper contains the Kripke semantics for modal logic (with the dyadic accessibility relation-cf. esp. Kripke 1963). Even more surprisingly, it anticipates the Routley-Meyer generalization of Kripke's semantics so as to provide a semantics for the relevance logics (with the triadic accessibility relationcf. esp. Routley and Meyer 1973). It goes without saying, though we say it anyway to make our views unmistakable, that there is incredibly much that is original and unique in Kripke and Routley-Meyer (and that goes beyond mere "labeling"). But still the extent of the J onsson-Tarski anticipation is not to be understated either. What Jonsson and Tarski do is show how to take any Boolean algebra with additional operations on it of any degree (there are simple conditions on these "operators," mainly that they be "additive"-a condition we shall define in a moment), and represent that Boolean algebra as a field of sets upon which are defined certain extra operations corresponding to each extra operator on the given Boolean algebra. These extra operations are defined as generalized image-forming operations in a certain natural sense we shall make clear, an n-place operation being defined as forming the image of a corresponding (n + 1)-place relation. The connection with the KripkelRoutley-Meyer semantics will eventually be made clear. But in anticipation, if we adopt the familiar idea (already in Boole) that a proposition be identified with a set-the set of cases (possible worlds) in which it is true-then additional (non-classical) connectives can be interpreted as extra operations on sets. And when these operations are "operators" (as they are in many cases), they can be defined as the "image operators" of J onsson-Tarski. This explanation does not work itself out quite so smoothly for the relevance logic semantics of Routley-Meyer, for the Lindenbaum algebra of relevance logics does not straightforwardly yield a Boolean algebra. That is one reason we prefer to develop the Jonsson-Tarski ideas in the more general

DISTRIBUTIVE LATTICES WITH OPERATORS

REPRESENTATION THEOREMS

314

+ I)-place relation

315

R ~ Un+l

setting of distributive lattices. Goldblatt (1989) also considers distributive lattices with operators, calling them "complex algebras.,,4 We now begin to set the framework. An operation 0i is additive just when it distributes over V (in each of its places), that is,

Indeed, following J6nsson and Tarski, given any (n we can define an n-ary operator:

(1) Oi(Xl, ... ,Y V Z, ... ,xn) = Oi(Xl,··· ,y, ... ,xn) VOi(Xl,··· ,Z, .. · ,xn).

This is the natural generalization of the image of a set under a function to the notion of the image of an n-tuple of sets under a relation. We leave to the reader the verification that R* is additive in the field of all subsets of U. This motivates the following definitions. Let (U, (Ri) iEl)' be any relational structure. By the associated distributive lattice with image operators we mean the ring of all subsets of U with the operators (Rt)iEI. The associated Boolean algebra with image operators is the same but with "field" in place of "ring." By a ring (field) of subsets ofU with image operators we mean any ring (field) of subsets of U closed under all the image operators An operation 0i on a lattice L is normal just in the case that if the lattice has a least element 0, and for some i (1 :S: i :S: n) Xi = 0, then

Thus, where

@

is a unary operation we require that

(2) @(Y V z) = @y V @z,

and where (3) x

0

is a binary operation, both

0

= (x x = (y

(y V z)

(4) (y V z)

0

0

y) V (x

0

z) and

0

x) V (z

0

x).

By a distributive lattice with operators we shall mean a structure {L, (Oi)iE[}, where each 0i is an additive operation on the distributive lattice L. A Boolean algebra with operators is the same except a Boolean algebra B replaces L. It is easy to see that an additive operation (henceforth, just operator) is isotonic (in each of its places), i.e., (5) y:S: z

-+

Oi(Xl, ... ,y,···,xn):S: Oi(Xl, .. ·,Z, ... ,xn).

Suppose that Y :S: z. Then z = Y V z. But then Oi(Xl, ... , z,···, xn) = Oi(Xl, ... , Y V Z, . .. , xn) = Oi(Xl, ... , y, . .. , x/J) V Oi(Xl, ... , z, ... , x/J) (the last step is by (1)). So we have the consequent of (5) as desired. A distributive lattice with a single binary operator 0 is called in the literature (cf. Fuchs 1963) a lattice-ordered groupoid, or an I-groupoid. There seems to be no familiar name for the analogous case with a single unary operator @, but this too shall be a very important case for us. Examples of binary operators are furnished by the Cartesian product on sets and relative product on relations (intersection and union are the lattice operations)-also by ordinary multiplication on the natural numbers (greatest common divisor is 1\, least common multiple is v). Examples of unary operators include the converse on relations and closure on sets of points in a topological space (again with intersection and union the lattice operations). Where U is any non-empty set and f : U -+ U, a particularly important motivating example is the f-image operator f*. As is familiar, for any X ~ U, f*(X) = {y : 3x E X such that y = f(x)}, and f*(X U Y)

pi : Un

(IfL has no 0, then every operation on L is trivially normal.) Note that any Rt is normal. (Also all of the examples of operators given above are normal.) Theorem 8.12.1 (Jonsson-Tarski) Every distributive lattice (L, (Oi )iEI) with normal operators is isomOlphic to a field of sets with image operators. Proof. We know by the proof of Stone's representation for distributive lattices that the mapping h(a) = {P : P is a prime filter and a E P}

is an isomorphism between L and a ring R of sets of prime filters of L. We let from now on the Ps range over the set of all prime filters of L. We show how to define a relation Ri for each operator 0i so that (#) h(Oi(Xj, ... , x/J))

= Rt(hxI, ... , hX/J).

Thus define Ri(PI, ... ,Pn,Pn+l) ifandonlyif'v'xI, ... ,Xn(XI E PI & '" & xn E Pn =? Oi(Xi, ... , Xn) E ~l+r). We express this last relationship compactly as Oi(PI, . .. , Pn) ~ Pn+I.

To show (#), it clearly suffices (upon removing definitions) to show (##) Oi(XI, ... ,X n) E ~z+1 P n) ~ Pn+r).

= f*(X) U f*(Y)·

This can clearly be generalized to any n-ary operation n-ary operator:

R7.

-+

U, giving rise to an

4The submission date makes it clear Goldblatt had this representation since at least 1985. We have presented the substance of this section to seminars since the early 1980s.

¢>

3Pl'''''~I(Xl E

PI & ... & xn E Pn & Oi(PI, .. ·,

The implication {= is immediate. The implication =? will, however, occupy us a while. Thus hypothesize O;(XI, . .. , xn) E Pn+l. Consider the principal filters [xr), ... , [xn)' We verify that Oi([Xr), ... ,[xn)) ~ ~1+I. Thus suppose Yl E [XI), ... ,Yn E [xn), i.e., Xl :S: YI,··., xn :S: Yn. By monotonicity, it follows that Oi(XI, ... , xn) :S: Oi(Yl, ... , Yn). Thus by our hypothesis and the fact that Pn+I is a filter, it follows that Oi(Yl, ... , Yn) E Pn+l, which completes the verification.

REPRESENTATION THEOREMS

316

The point is that we now know there exists an n-tuple of filters FI, ... , Fn like the PI, ... , Pn that we want except that they are most likely not prime. So we know that the following set is non-empty (where the Fs are all filters):

LATTICES WITH OPERATORS

and a'j E [Pj, y), i.e., 3p E Pj such that p 1\ x ::; a'j and 3p' E Pj such that p' Further standardizing, set p" = p 1\ p'. It is clear that

317

1\ Y ::;

ai.

(3) p" E p.J' p" 1\ x -< a"J' p" 1\ Y -< a'~j '

But then by the distributive law and lattice properties, Note that as always, we take filters to be proper filters. The careful reader may then worry about what happens if some Xi = O. The answer is that since 0i is normal, then o(X!, ... , x n ) = 0, and so the hypothesis that Oi(XI, ... , x n) E Pn+l is contradictory (Pn+l being proper). Define a partial order on E by componentwise inclusion, i.e., define (Fl, ... , Fn) ::; (GI, ... , G n ) iff Fi ~ Gi for 1 ::; i ::; n. It is easy to verify that each chain in this ordering has an upper bound in E formed by componentwise union of the chain's members. Thus, being more explicit, where e is a chain, let ei be the set of ith components of members of e. Then define Ve = (Uq, ... , U en), which clearly is an upper bound. The question is whether VeE E. It is easy to see that each Uei is a filter, the argument being precisely the same as in the analogous step in Stone's theorem when we needed that the union of a chain of filters is a filter. That Xi E Uei is clear, since Xi is an element of some Fi E ei. What remains to be shown, then, is that 0i(Uel, ... , Ue n ) ~ Pn+l. Suppose that ai E Uei. So there are (Fll, ... ,Fl;), ... ,(F{Z, ... ,F,~) E E such that

a! E FI!"'" an E F,;l. Let (Fl, ... , Fn) be the greatest (in the sense of ::;) of these n-tuples. Consider the n-tuple (Fl, ... , F/z1 ). It need not be in E. However, it is easy to see that (Fl, . .. , F/z ) ::; (F{Il, ... , F/z ), i.e., that Fj ~ FF(1 ::; j ::; n). Then each aj EFt Since (F{n, ... , Fin E E, then we know that oi(F{Il, ... , F,;n) ~ Pn+l, and so oi(al, ... , an) E P'l+l, as desired. Since we have now shown that each chain has an upper bound, we can conclude by Zorn's lemma (in a technically more general form than has been previously introduced since the relation::; is not just ~) that E has a maximal element (PI, . .. , Pn ).5 We now show that each of the PjS is prime, completing the proof of (##). So suppose that X V Y E Pj and yet X ¢ Pj and y ¢ Pj. Let [Pj, x) be the filter generated by Pj U {x}, and similarly for [Pj, y). Since [Pj, x) :J Pj, then (PI, ... , [Pj , x), ... , P'l) > (PI, ... , Pj, ... , Pn), and similarly for [Pj, y). So each of (PI, ... , [Pj , x), ... , Pn ) and (PI, ... , [Pj , y), . .. , Pn) must fail to be in E. Checking the conditions for membership in E quickly reveals that this can only be because there exist elements al,···, aj, ... , an, a'l"'" ai, ... , a~ such that: 1

ll

(1) al E PI, ... , aj E [Pj, x), ... , an E Pn, yet Oi(al,···, aj, ... , an) ¢ Pn+l; (2) a'l E PI, ... ,ai E [Pj,y), ... ,a;l E P,z, yet Oi(a'l"" ,ai,··· ,a;l) ¢ Pn+!. We next go through a series of moves that might be confusing in detail, but the basic spirit of which is to standardize the parameters so as to allow in the end a distribution (or two). Thus for k =f. j, set a~; = ak 1\ a~, but set a'j = aj V ai. It is clear that a'j E [Pj, x) 5This more general, abstract form says that if (E, ::;) is any partially ordered set such that each chain C S;; E has an upper bound, then E has a maximal element.

(4) p" 1\ (x V y)

= (p" 1\ x) V (p" 1\ y) ::; a'j.

Since p" E Pj, and our hypothesis has been that x V y E Pj , clearly p" 1\ (x V y) E Pj. It thus follows from (4) that a'j E Pj . It is further clear that by setting a~; = ak 1\ a~ (for k. =f. j) we have also guaranteed that a% E Pk. So in general (for all k) we have a% E Pk. Smce (PI, ... , Pj, ... , P'l) E E, we have (5) Oi(Pl, ... , Pj , . .. , Pn ) ~ P'HI.

So (6) 0i (" " ... ,a") al' ... ,aj' n E Pn+l.

B ut reca11 mg ' th at a"j

= aj V a"j , th'IS means

(7) Oi(a'{, ... , aj V ai, ... , a;;) E Pn+l.

But by the fact that 0i is additive, we can distribute this element to obtain (8) 0i (" al' ... ,aj, ... ,a") n

VOi (" al,

, ... ,a") .. ·,ap n EPn+l.

Since Pn+ 1 is prime, (9) oi(a'{, ... , aj, ... , a~) E Pn+l, or

(10) oi(a7, ... , ai, ... , a;;) E P'l+!'

Assume (9). Since a% ::; ak for k =f. j, we have by repeated applications of monotoniCity that (11) oi(a7, ... , aj, . .. , a;;) ::; Oi(al, ... , aj, . .. , an). From (9) and (1]) it follows by the filterhood of Pn+l that

(12) oi(al, ... ,aj, ... ,an )

E P'l+l,

contradicting our hypothesis (]). The other case (assuming (10)) can be argued symmetrically to give

(13) oi(a~, ... ,ai, ... ,a~)

E P'l+l,

contradicting our hypothesis (2).

8.13

D

Lattices with Operators

This section gives a representation for lattices with operators, where there is no assumption as to the distributivity of the lattice. Obviously, the representation cannot be a set of sets with the usual set-theoretical operations nand u, since these operations distribute over each other. As we mentioned above in Section 8.3, there are various representations for lattices; two of them were considered in detail there, and some others are mentioned

REPRESENTATION THEOREMS

318

in Chapter 13. All these representations can be extended by operators (cf. Ono 1993; Hartonas 1993b; Allwein and Dunn 1993). We present one of the extension here. This is based on the principal pair lattice representation, which is a modification of the Hartonas-Dunn representation, and looks somewhat as if it is a dual to the Urquhart representation. 6 The basic idea of adding an operator to the lattice representation is essentially the same as in the J 6nsson-Tarski representation, although the frame has to be augmented with extra conditions due to the lack of disttibutivity. First we define the structure we are to represent:

Definition 8.13.1 A lattice with operators is a structure (L, /\, V, (Oi )iEI) where (L, /\, V) is a lattice and each 0i is a normal additive operation.

Just as before, it does follow that 0i is isotonic in each argument place: (1) y:S

z :::}

Oi(XI, ... ,y,···,xn):S Oi(Xl,···,Z, ... ,xn).

As an example of such a structure one might consider the positive additive fragment of linear logic, together with multiplicative conjunction: the additive conjunction and disjunction gives the lattice, and 0 is a lattice ordered binary operation which is isotonic in both places. Before jumping to the technicalities of the principal pair representation, let us give some intuitive motivations. The set of the principal filters (equally, the set of principal ideals) of a lattice is isomorphic to the lattice itself (cf. Birkhoff 1967). This seems to make these two sets somewhat trivial or too easy to use, but simplicity might have advantages when things start to get complicated. Notice also that this mapping (where each element is mapped into the principal filter (or ideal) it determines) can be easily "typelifted." If one takes the power set of all principal filters (or principal ideals), then the map which assigns to each element a set of principal filters (or ideals) each of which contains the particular element, then this provides and equally good representation, though the first is a join representation, and the typelifted one is meet representation. In these cases the lattice is viewed as a relational rather than an operational structure, ~ (or its converse) being the ordering. Thus, if L = (L, :S) and F is the set of all principal filters, then the two maps are: (i) hl(a) (ii) h2(a)

= {x: a:S x}, i.e., hl(a) = [a) E F; = {F : F E F & a E F }, i.e., h2(a) = [[a))

LATTICES WITH OPERATORS

maximal, and not every inconsistent pair is minimal. Maximalization might require an application of Zorn's lemma, so might minimalization. However, one does not need to minimalize on inconsistent pairs in the representation, because minimally inconsistent pairs are always made up from a principal filter and a principal ideal, as follows from the next lemma.

Lemma 8.13.2 Let L be a lattice. If (F, 1) is a principal filter-ideal pair on L, that is, (F,1) is a minimal element in the set of overlapping filter-ideal pairs, then there is an a E L such that F = [a) and I = (a].

The proof of this lemma is easy but somewhat tedious, and left to the interested reader. The converse of the lemma holds as well, that is, a pair formed from a principal filter and a principal ideal, where these two are generated by the same element, is a minimal element among the overlapping pairs. The idea is then to take advantage of the fact that for each element of the lattice there is exactly one principal filter-ideal pair generated by that element? Consider a relational structure (U, C, (Ri )iEI) where C is a transitive relation. Define two functions r and I on subsets of U as follows: reX) = {y : Vx(x EX:::} x C y)} and g(y) = {x : Vy(y E Y :::} Y ex)}. We call a subset X of U hereditary or upward closed (with respect to C) in the same sense as in Section 8.3, i.e., if x E X and x C y implies y E X. We call a subset X of U dual hereditary or downward closed if y E X and x C y implies x EX. The two functions rand g are Galois connections between the upward closed and downward closed subsets of U. A set X is called stable if X = gr X. A stable lattice is a set V of stable subsets of U closed under intersection, as well as satisfying the condition that if X, Y E V then r X n r Y = r Z for some Z E V. (Meet is defined as n, and the join of X, Y is g(r X n rY).) Furthermore, we will call a set V of stable sets fit for R7 if V is closed under Rt and additionally satisfies the condition:

* * * (7) - Ri(XI, ... , Y vZ, ... , X n) ~Ri(XI, ... ,y, ... ,Xn)VRi(XI, ... ,Z, ... ,Xn). Theorem 8.13.3 Every lattice (L, /\, V, (Oi )iEI) with operators is isomorphic to a fit stable lattice of sets with image operators. Proof The canonical map is defined as h(a) = { (F,1) : (F,1) is a principal pair and a

~ F.

To avoid distributivity Urquhart uses maximal filter-ideal pairs (Definition 3.18.37) (and join in the representation is defined from Galois connections). Instead of "maximally disjoint" one might consider "minimally overlapping," and that is what we called a principal filter-ideal pair in Definition 3.18.38. Clearly, not every consistent pair is

319

E

F }.

For the ordering relation we choose ~ on the first members of the pairs, that is, set inclusion on filters. It is straightforward to check that h(a/\b) = h(a)nh(b) and h(aV b) = g(rh(a) n rh(b)). We define the relation R on principal filters as follows: R i ( (FI,II,), ... , (Fn'!Il)' (Fn+l ,!n+l»

iff

Val, ... , an((al E FI & ... & all E F Il ) :::} Oi(al, .. ·, an) E Fn+l). 60 n the one hand, the use of two relations in the frame, as well as using filter-ideal pairs in the canonical model, resembles the Urquhart representation, though there is no translation between the two representations. On the other hand, the frame can be viewed as a special case of the frame used by Hartonas and Dunn, though canonically we use a different set. A further extension of this representation was used in Bimbo (2000) to give a semantics for certain structurally free logics.

The image operator R* which corresponds to 0i is defined as before. 7Note that we do not require in this representation the filters and ideals to be proper.

320

REPRESENTATION THEOREMS

We show that h(oJaj, ... ,an)) = R;' (h(aj), ... ,h(a n)). Going from right to left, the claim amounts to a universal instantiation in the definition of Ri. Going from left to right, let us assume that (F', I') E h(oi(aj, . .. ,an))' By the definition of h this means that oi(aj, ... , an) E F'. Consider then the principal filters determined by aj, ... , an: [aj), ... , [an). Due to the monotonicity of 0i for all a~ ~aj, ... ,a;1 ~an, oj(aj, ... ,an) :S 0i (a~ , ... ,a;I)' Then, Ri( ([aj), (all), ... , ([an), (an]), ([oi(aj, ... , an)), (Oi (aj, ... , an)]») holds, and hence by the definition of the image operator ([oi(aj, ... , an)), (oi(aj, ... , an)]) E Rt(h(aj), ... , h(a n )), as desired. Lastly, h is one-one in view of our remark that each element generates a principal ~ D

9 CLASSICAL PROPOSITIONAL LOGIC 9.1

Preliminary Notions

Classical propositional logic has associated with it a number of different algebraic logics, which we will call Frege logic, Boolean logic, and unital Boolean logic. Although these logics are characterized by different classes of logical matrices, as we will see later, they are all strongly equivalent, which is to say that they validate exactly the same asymmetric consequences. However, they do not validate all the same symmetric consequences. First of all, Frege logic is characterized by a single matrix, the Frege matrix, which is discussed in Chapter 5. Frege logic may be thought of as the algebraic representative of classical truth-functional logic; in particular, insofar as there are only two propositions in the Frege matrix ("the true" and "the false"), each formula is interpreted directly as a truth value. Closely related to Frege logic is Boolean logic, which is characterized by the class of Boolean matrices. A Boolean matrix is a matrix (B, F), where B is any Boolean algebra and F is a proper filter on B. Recall from Chapter 3 that a filter on a lattice L is any non-empty subset F of L satisfying the following conditions for all a, bEL: (fl) a E F and a :S b only if bE F;

(f2) a E F and b E F only if a /\ b E F. Equivalent to (fl) and (f2) is the following single condition: (f3) a E F and bE F if and only if a /\ bE F.

A filter on L is said to be a proper filter if it is a proper subset of L. Recall that in a matrix, F is intended to be the set of true propositions. Now, the partial ordering among propositions is intended to represent the relation of entailment (not to be confused with a -+ b, which represents the conditional connective); in particular, "a :S b" may be read "a entails b." (Note that, where "a :S b" is a statement about the lattice elements, a -+ b is itself a lattice element; this conesponds to the difference between the metalanguage entailment and the object language conditional.) This allows us to interpret the conditions above that define filters: (fl') if a is true and a entails b, then b is true;

(f2') if a is true and b is true, then a /\ b is true. The class F(L) of all filters on a lattice L is partially ordered by set inclusion, with respect to which F(L) forms a complete lattice. In particular, if C is any collection of

CLASSICAL PROPOSITIONAL LOGIC

322

n

filters on L, then C is also a filter. Since F(L) is closed under arbitrary intersection, we can speak of the filter F(S) generated by a subset S of L, sometimes written as [S). Specifically, F(S) is defined as follows: (1) F(S) = n{F E F(L): S ~ F}.

Note also the following general theorem concerning F(S): (T) F(S)

= {x E L : x 2: glb(T) for some finite T ~ S} = {x E L : x 2: SI A S2 A ... A Sn for some SI, S2, ... , Sn E S}.

BOOLEAN LOGIC AND FREGE LOGIC

all Y E r. On the other hand, h(l(a)) not K-valid.

E

A' - D', so vl!o/(a) :f. T. It follows that r I- a is D

Our next few lemmas concern the theory of Boolean lattices. Lemma 9.2.2 Let B be any Boolean lattice, and let F be any jilter on B. Dejine a relation 8 on B as follows: a8b (If a '"7 b E F. Then 8 is a congruence relation on B.

Proof. The reader can easily "calculate" this from the properties of a Boolean algebra. D

In the special case that S is finite, F(S) is given as follows: (T*) F(S)

= {x E L: x 2: glb(S)}.

In the special case that S is a singleton {a}, F( {a}) is called the principal jilter generated by a, and is given as follows: (PF) F ( {a}) = {x E L : x 2: a}

We sometimes write [a) for F ( {a} ). A special subclass of Boolean matrices are the unital Boolean matrices. A Boolean matrix (B, F) is said to be unital if F = {I}. Unital Boolean logic is characterized by the class of unital Boolean matrices. This logic corresponds to the supervaluational logic of van Fraassen (1971). This has been a quick reprisal of the properties of a filter. We will assume additional notions and results from Chapter 8, in particular, those relating to maximal filters and the Stone representation theorem. 9.2

The Equivalence of (Unital) Boolean Logic and Frege Logic

In order to show that Frege logic and Boolean logic are strongly equivalent, that they validate exactly the same asymmetric consequences, we proceed in steps, first showing that Boolean logic is strongly equivalent to unital Boolean logic, and then showing that unital Boolean logic is strongly equivalent to Frege logic. The key general lemma is the following. Lemma 9.2.1 Let J and K be two classes of similar matrices, all appropriate for a given algebra of sentences. Then the following is a s£flficient condition that every Kvalid argument is also J-valid (here M = (A, D), M' = (A', D')). For every M E J, and every a E A - D, there exists an M' E K and a homomorphism h from A into A' such that h(D) ~ D', and such that h(a) E A' - D'.

Proof. Suppose the condition, and suppose that r I- a is not J-valid. Then there is a valuation v E V(J) such that v(y) = T for all Y E r but v(a) :f. T. From this it follows that there is a J-matrix M and an interpretation I such that I(Y) E D for all y E rand I(a) E A-D. According to the hypothesized condition, there is a K-matrix M' and a homomorphism h such that h(D) ~ D', and such that h(l(a)) E A' - D'. Since h is a homomorphism, the composition of h with I yields an M-interpretation, which in tum yields a valuation Vho/. In particular Vho/(Y) = Tiff h(I(Y)) ED', but I(Y) E D, and h maps every true proposition into a true proposition, so h(I(Y)) ED', so Vho/(Y) = T for

323

Corollary 9.2.3 EvelY jilter on a Boolean lattice B induces a quotient algebra which is itself a Boolean lattice, denoted as B I F.

Proof. Follows from Chapter 2, noting that Boolean algebras form a variety.

D

Lemma 9.2.4 Let B be a Boolean lattice, let F be any jilter on B, and let 8 be the congruence relation determined by F. Then we have the following:

a

E

F

¢:=?

Proof. a81 iff a '"7 I E F iff a E F (since a

a8l.

= (a A 1) V (-a A-I)).

D

Lemma 9.2.5 Let B be a Boolean lattice, let F be a maximal jilter on B, and let 8 be the associated congruence relation on B. Then for every element a E B, either a8I or a80.

Proof. Suppose F is a maximal filter on B. Then F is complete, so for all a E B, either a E F or -a E F. So by Lemma 9.2.4, for all a E B, either a81 or -a81. If -a81, then - - a8 - 1, that is, a80. D Corollary 9.2.6 IfF is a maximaljilter on a Boolean lattice B, then the quotient algebra BI F has exactly two-elements, so it is the two-element Boolean lattice, or the Frege algebra.

Proof. Immediate from Lemma 9.2.4 and facts about universal algebra.

D

Lemma 9.2.7 Let B be a Boolean lattice, and let F be any jilter on B. Then there exists a Boolean lattice B* and a homomorphism hfrom B into B* such that F

= h- 1({1}) = {x E B: hex) = l}.

Proof. Let F be a filter on the Boolean lattice B. According to Corollary 9.2.3, BI F is a Boolean algebra, which is a quotient of B. Consider the canonical homomorphism h from B into BI F, defined so that h(a) = [a]F, where [a]F = {x E B : a '"7 x E F}. Therefore, a E h- 1({ I}) iff h(a) = 1, that is, iff [a]F = 1. Now, [a]F consists of all those elements congruent to a (modulo F), and 1 consists of all those elements congruent to 1 (modulo F). If these two congruence classes are the same, then a is congruent to 1 (modulo F), from which it follows that a E F. Conversely, if a E F, then a81, so h(a) = [a]F = [1]F = 1, so a E h- 1( {I}). D

324

SYMMETRICAL ENTAILMENT

CLASSICAL PROPOSITIONAL LOGIC

Lemma 9.2.8 Let B be any Boolean lattice, and let a be any element different from l. Then there exists a homol1101phism hJrom B into the two-element Boolean lattice such that h(a) = o. Proof Let a be a non-unit element of Boolean lattice B. Since a :I I, {I} is a proper filter. So in virtue of Stone's maximal filter separation theorem, I is contained in a maximal filter, call it F*, with a ¢ F*. According to Lemma 9.2.7, there is a homomorphism h into the two-element Boolean lattice such that h- I ( {I}) = F*. Therefore, D since a ¢ F*, h(a) = O.

First we recall some notions we introduced earlier. Remember that the customary relation of consequence holds between sets of formulas and individual formulas; specifically, we say that r entails a relative to a class V of valuations if every valuation v E V that satisfies r also satisfies a. Since the two sides of the relation are categorically different, we might call this consequence asymmetrical consequence. By contrast, a symmetrical consequence relation is a relation holding among sets. Symmetrical entailment is defined as follows (perhaps not in the manner one might expect!): (1) Let V be a class of valuations on language L, with formulas W, and let rand li

Theorem 9.2.9 Boolean logic and unital Boolean logic are strongly equivalent. Proof Since the class of Boolean matrices includes the class of unital Boolean matlices, every Boolean valid argument is automatically unital Boolean-valid. Concerning the converse, we appeal to Lemma 9.2.1. Consider any Boolean matlix M = (B, F). According to Lemma 9.2.7, there is a Boolean lattice B* and a homomorphism h from B into B* such that F = h- I ({ID. It follows that there is a unital Boolean matrixthe one based on B*-and a homomorphism (namely, h) such that h(F) ~ {I}, and such that for any a ¢ F, h(a) ¢ {l}. We thus have the antecedent condition to apply Lemma 9.2.1, so every unital Boolean valid argument is also Boolean valid. (Note that Lemma 9.2.1 says that if for any a ¢ D, there is a homomorphism of the required nature, then J-Iogic subsumes K-Iogic. In this proof, we have shown something stronger-that there is a single homomorphism which does the job for every a ¢ D.) D

be subsets of W; then r is said to entailli relative to V if for every v satisfies r, then there exists an a E li such that v satisfies a.

Proof Since the Frege matrix is a unital Boolean matrix, every unital Boolean valid argument is automatically Frege valid. In order to prove the converse, we appeal to Lemma 9.2.1. Consider a unital Boolean matrix M = (B, D), where D = {I}, and consider any element a different from 1. According to Lemma 9.2.8, there is a homomorphism h from B into the Frege algebra such that h(a) = O. We accordingly have the antecedent condition of Lemma 9.2.1, from which it follows that every Frege valid argument is also unital Boolean valid. D Corollary 9.2.11 Boolean logic and Frege logic are strongly equivalent. D

Corollary 9.2.12 Boolean logic has a finite characteristic matrix. Proof Immediate from Corollary 9.2.6 together with the fact that the Frege matrix is finite (two elements). D

V, if v

We say that two logics LI and L2 are strictly equivalent, or equivalent with respect to symmetrical entailment, if for any sets r, li, r LI-entails li iff r L2-entails li. Note that ifLI and L2 are sttictly equivalent, then they are automatically strongly equivalent, and hence weakly equivalent; for strong equivalence is a special case of strict equivalence obtained by restricting one's attention to pairs of sets r, li where li is a singleton set. In the following theorems, we show that although Boolean logic is strictly equivalent to unital Boolean logic, it is not strictly equivalent to Frege logic. We begin by proving an important lemma that states that, from the point of view of truth-value semantics, Boolean logic is identical to unital Boolean logic.

Proof Since every unital Boolean matrix is a Boolean matrix, every unital Boolean valuation is a Boolean valuation: V(Ku) ~ V(KB). Concerning the converse inclusion, suppose v E V(KB). Then there is a Boolean algebra B, a filter F, and an interpretation I : W -+ B, from the word algebra into B, such that v is the valuation induced by I; that is, v(a) = T if I(a) E F, v(a) = F if otherwise. In virtue of Lemma 9.2.7, there is a homomorphism h from B into some Boolean algebra B* such that F = h- I ( {I D. We can form a unital Boolean matrix simply by designating 1 as the true proposition. Now, the composition c of h and I is an interpretation of L into a unital Boolean mattix, and the associated valuation is defined so that vc(a) = T if c(a) = 1, vc(a) = F otherwise. Claim: Vc = VI(= v); forvl(a) = Tiffz(a) E F iff h(z(a)) = 1 iff c(a) = I iffvc(a) = T. Thus, v = VI E V(Ku). D Theorem 9.3.2 Boolean logic and unital Boolean logic are strictly equivalent. Proof Immediate from Lemma 9.3.1.

9.3

E

Lemma 9.3.1 Where KB is the class oJBoolean matrices, and Ku is the class oJunital Boolean matrices, V(KB) = V(Ku).

Theorem 9.2.10 Unital Boolean logic and Frege logic are strongly equivalent.

Proof Immediate from Theorems 9.2.9 and 9.2.10.

325

Symmetrical Entailment

Having shown that Boolean logic and Frege logic are equivalent in the sense that they validate exactly the same arguments, in the present section we show how tlley are different.

D

Theorem 9.3.3 Boolean logic and Frege logic are not strictly equivalent. Proof It is sufficient to produce two sets r, li such that r F -entails li but r does not Bentailli. The following is a simple example: r = 0, li = {p, ~ p}. To say that 0 entails

326

CLASSICAL PROPOSITIONAL LOGIC

COMPACTNESS THEOREMS

./:::,. is to say that ./:::,. is unassailable, which is to say that every valuation satisfies some formula or other in ./:::,... Thus, the claim is that although {p, ~ p} is unassailable relative to Frege logic, it is not unassailable relative to Boolean logic. Suppose v E V(F), and v does not satisfy p. It follows that there is an interpretation I into the Frege algebra such that l(p) = O. But l(~p) = -l(p) = -0 = 1, so l(~p) = 1, so v(~p) = T. Thus, {p, ~ p} is Frege unassailable. On the other hand, consider the four-element unital Boolean matrix, where the propositions are 1, a, b, 0, and consider any interpretation I such that l(p) = a, so that l(~p) = b. Since neither a nor b is designated true or false, the corresponding valuation VI satisfies neither p nor ~p. Thus, {p, ~ p} is not unassailable relative to Boolean logic. D

9.4

Compactness Theorems for Classical Propositional Logic

Earlier we introduced three different notions of compactness, defined as follows. (1) (L, V) is I-compact iff for every subset r of W, r is unsatisfiable (relative to V) only if X is un satisfiable for some finite subset X of r. (2) (L, V) is U-compact iff for every subset r of W, r is unassailable relative to V only if X is unassailable for some finite subset X of r. (3) (L, V) has finitary entailment iff for every subset ru {a} of W, r V-entails a only if X V -entails a for some finite subset X of r. We begin by noting that the notions of unsatisfiability and unassailability can be succinctly expressed in terms of symmetrical entailment as follows: (a) r is unsatisfiable iff r entails 0 (relative to V). (b) r is unassailable iff 0 entails r (relative to V).

(2) Suppose r is unassailable. Then by (b) above, 0 entails r. So by strong compactness, there are finite subsets X of 0 and Y of r such that X entails Y. But the only subset of 0 is 0, so we have that 0 entails Y for some finite subset Y of r, which is to say that Y is unassailable. Thus, (L, V) is U-compact. (3) Suppose r entails a. Then by (c) above, r entails {a}. So by strong compactness, there are finite subsets X of rand Y of {a} such that X entails Y. Since there are two subsets of {a}, we have two cases to consider. Case 1: Y = {a}; in this case X entails {a}, so X entails a. Case 2: Y = 0; in this case, X entails 0, so X entails every formula, so in particular, X entails a. In either case, we have that X entails a for some finite subset X of r. Thus, (L, V) has finitary entailment. D

Lemma 9.4.2 Suppose (L, V) is I -compact, and suppose that (L, V) has exclusion negation. Then (L, V) is strongly compact. (Recall that (L, V) has exclllsion negation ifffor every a E W, there is a 13 E W such that yea) = V - V(f3).) Proof Suppose (L, V) is I -compact and has exclusion negation. Suppose r entails ./:::,... Then (V (a) : a E r} ~ U (V (a) : a E ./:::,..} . So by set theory X ~ Y ¢=} X - Y = 0,

n

n

and (Yea) : a E r} - U (Yea) : a E ./:::,..} = 0; also by set theory (generalized De Morgan laws), (yea) : a E r} n (V - Yea) : a E'/:::"'} = 0. At this point, we appeal to the exclusion negation property, which tells us that for every a E ./:::,.., there is an a* such that V - yea) = V(a*). (a* is a particular formula 13 such that V(f3) = V - yea); we are now appealing to the Axiom of Choice.) Thus, (V (a) : a E r} n (V (a*) : a E'/:::"'} = 0. Now, let'/:::"'* be the image of./:::,.. under the star function (which is a choice function). This allows us to write n{V(a) : a E r} n n{V(a) : a E ./:::,..*} = 0, from which it follows that (yea) : a E r u./:::,..*} = 0, which is to say that r u./:::,..* is unsatisfiable. So by I -compactness, there is a finite subset of r u./:::,.. * that is unsatisfiable, which may be written X U Y, where X ~ r, and Y ~ ./:::,. *. Since X U Y is unsatisfiable, we have that n{V(a) : a EX U Y} = 0, from which it follows that n{V(a) : a E X}nn{V(a): a E Y} = 0,sobyDeMorganlaws, n{V(a): a E X}-U{V-V(a): a E Y} = 0, so by set theory, n{V(a) : a E X} ~ U (V - yea) : a E Y}. Recall that every a E ./:::,..* is an exclusion negation of some formula 13 or other in ./:::,..; so for every a E Y, there is a 13 E ./:::,.. such that V - yea) = V(f3). For each a E Y, let ail be any such 13. Thus, (V - yea) : a E Y} = (Yea) : a E yiI}, where yil is the image ofYunder the U function. So we have that (yea) : a E X} ~ U (yea) : a E yiI}, which is to say that X entails yiI; here, X is a finite subset of rand yil is a finite subset of ./:::,... Thus, (L, V) is strongly compact. D

n

n

n

n

n

As noted earlier, asymmetrical entailment can be thought of as a special case of symmetrical entailment, where the consequent set is a singleton: (c) r entails a iff r entails {a}. In addition to the above three notions of compactness, there is a stronger notion of compactness which we call strong compactness or finitmy symmetric entailment: (4) (L, V) is strongly compact iff for every pair r,'/:::'" of subsets of W, r entails ./:::,. (relative to V) only if X entails Y for some finite subset X of r and some finite subset Y of'/:::"'. The following lemma states that strong compactness is in fact strong.

Lemma 9.4.1 If (L, V) is strongly compact. then it is I -compact. U -compact. and has finitary entailment. Proof Suppose (L, V) is strongly compact, and suppose r

327

~

W, and a

E

W.

(1) Suppose r is unsatisfiable. Then by (a) above, r entails 0. So by strong compactness, there are finite subsets X of rand Y of 0 such that X entails Y, but the only subset of 0 is 0, so we have that X entails 0 for some finite subset X of r, which is to say that X is unsatisfiable. Thus, (L, V) is I-compact.

n

Theorem 9.4.3 If (L, V) has exclusion negation, then all of the following conditions are equivalent: (1) (2) (3) (4)

(L, V) (L, V) (L, V) (L, V)

is I -compact; is U-compact; has finitary entailment; is strongly compact.

328

CLASSICAL PROPOSITIONAL LOGIC

COMPACTNESS THEOREMS

Proof Strong compactness implies each of the other three, by Lemma 9.4.1. Also, by Lemma 9.4.2, if (L, V) has exclusion negation, then I-compactness implies strong compactness. We leave the interested reader to prove, as an exercise, that with exclusion negation U -compactness implies finitary entailment, and that (with exclusion negation again) finitary entailment implies strong compactness. D

As noted earlier, Frege logic (but not Boolean logic) has exclusion negation, so in order to show that Frege logic is strongly compact, it is sufficient to show, for example, that it is I -compact. In order to do this we first prove a number of lemmas. Lemma 9.4.4 Let K

U {A} be a class of similar algebras; let H (A, K) be the class of homomorphisms that map A into B for some algebra or other B E K. Define a relation e on A as follows:

aeb iff h(a)

= h(b) for all h E H(A, K).

polynomial function, we have P(h(W1), . .. , h(wm)) =f. If/(h (x]), ... , h(xn)). This contradicts our earlier claim that p( ai, ... , am) = If/( bl, ... , bn ) for all Boolean algebras B, and for all elements a1, ... , am, bl, ... , bn E B. Thus, W Ie is a Boolean algebra. D

Remark 9.4.7 Polynomials on an algebra A are the set-theoretic counterparts of terms in the associated first-order language. Let A be any algebra. Then the polynomials on A-denoted poly(A)-are partial functions from UnEOJ An into A. Each polynomial has a degree: P has degree njust when dom(p) = An. poly(A) is defined inductively as follows: (1) The identity function on A is an element of poly(A): i(x)

E

1 1 2 2 n n] 0(P1, ... , Pn)[ a I' ... , ad I ' ai' ... , ad2' ... , a I' ... , ad ll

Proof e is obviously an equivalence relation. Concerning the replacement property, suppose (0) is an n-place operation on A, and suppose alebl, a2e b2, ... , aneb n . Then forallh E H(A,K),h(ai) = h(bi)(i E n).Considerany hE H(A,K). Thenh(0(a1,a2, ... , an)) = o*(h(a]), h(a2),"" h(a n )), and h(o(bl, b2,··., bn )) o*(h(b]), h(b2), ... , h(b n )). Since h(ai) = h(bi) for all i E n, it follows that 0*(h(a1), h(a2), ... , h(a n )) = o*(h(bl), h(b2), ... h(b n )), so h(0(a1, a2,··· ,an)) = h(o(bl, b2, ... , bn )), for all h E H(A, K). Thus, o(al, a2, ... , an )(}0(b1, b2, ... , bn ), and e satisfies the replaceD

Corollary 9.4.5 Let L be the standard sentential language, and let W be the associated algebra offormulas; let B be the class of Boolean algebras. Then the relation e, defined so that aeb iff h(a) = h(b) for all h E H(W, B), is a congruence relation. Proof Immediate.

= x for all x

A. (2) If PI, P2, ... , Pn are polynomials on A, where d(Pi) = d i , and 0 is an n-place operation on A, then 0(P1, P2, . .. , Pn) is a polynomial on A of degree dl + d2 + ... + d n , and is defined as follows:

Then e is a congruence relation on A.

ment property.

329

D

Lemma 9.4.6 Let e be the congruence relation cited in the above corollary. Then the quotient algebra W Ie is a Boolean algebra. Proof Since Boolean algebras form a variety (equational class), if WI e is not a Boolean algebra, then there is at least one equation E satisfied by every Boolean algebra but not satisfied by W Ie. It follows that if W Ie is not a Boolean algebra, then there are two polynomials p and If/ such that p(al, ... , am) = If/(bl, ... , bn ) for all Boolean algebras B, for all elements al, ... , am, bl, ... , bn E B, and such that p([wIJ, ... , [w m]) =f. If/([x]], ... , [x,d) for some [wIJ, ... , [w m], [x]], ... , [xn] E W Ie. It can be shown by induction that every polynomial preserves every homomorphism: h(p(a1, . .. , an)) = p(h(a1), ... , h(an)). Then, it follows that p([w]], ... , [wm]) = [P(W1, ... , wm)], and If/([x]], ... , [xn]) = [If/(XI, ... , xn)]. Thus, [P(W1, .. ·, wm)] =f. [If/(X1, ... , xn)], which is to say that p(WI, ... , wm) is not congruent to If/(XI, ... , xn). So there is a Boolean homomorphism such that h(P(WI, ... , wm)) =f. h(lf/(XI, ... , xn)). Since h preserves every

= 0(p1(ai,· .. ,a~I)'''·'Pn(a7,···,a~)). (3) Nothing is a polynomial on A except in virtue of clauses (1) and (2) . Lemma 9.4.8 There is a natural correspondence between Boolean interpretations on Wand homomorphisms on W Ie. In particular, every interpretation I on W determines a unique homolllOlphism hi on WI e, and every homomorphism h on WI e determines a unique intelpretation Ih on W. The functions h 1-+ Ih and I 1-+ hi are inverses of each othel: Proof Suppose I is an interpretation of W into a Boolean algebra B. Define the function hi : wle -7 B so that h,([a]) = I(a). First of all, hi is well-defined; for suppose [a] = [b]; then aeb, so I(a) = I(b), so h,([a]) = h,([b]). Next, hi is a homomorphism; for h[(oc([a]], ... , [an])) = h[([cal, ... ,an]) = I(cal, ... , an) = fc(l(a1», ... , I (an)) = fc(h([a]]), ... , h([al/])); here, Oc is the operation on W Ie associated with connective c, and fc is the operation on B associated with c. Thus, hi is a homomorphism on W Ie. Now, suppose h is a homomorphism from W Ie into Boolean algebra B. Define Ih : W -7 B so that lh(a) = h[a]. In other words, Ih is simply the composition of h with the canonical homomorphism from W into W Ie; it follows immediately that Ih is a homomorphism from W into B. To see that the map h -7 Ih is the inverse of the map Ih -7 h, consider the interpretation Ih(I); by definition Ih(I)(a) = h,[a], and by definition h[[a] = I(a); thus Ih(l) = I; by similar reasoning, we can show that h,(h) = h.

D

Lemma 9.4.9 Let B alld B* be any Boolean algebras (lattices), let F be any filter on B*, and let h be any homomOlphism from B into B*. Then the pre-image h- I (F) of F under h is a filter on B.

COMPACTNESS THEOREMS

CLASSICAL PROPOSITIONAL LOGIC

330

Proof In order to show that h- I (F) is a filter on B, given the hypotheses, it is sufficient to show: a E h-I(F) and bE h- I (F) only if a /\ bE h- I (F), and a E h-I(F) and a :S b only if bE h-I(F). Suppose a E h-I(F) and bE h-I(F). Then h(a) E F and h(b) E F, so h(a /\ b) = h(a) /\ h(b) E F, so a /\ bE h-I(F). Suppose a E h-I(F) and a :S b; then h(a) E F, and a /\ b = a, so h(a /\ b) E F, so h(a) /\ h(b) E F, so h(b) E F, so bE h-I(F). D

Lemma 9.4.10

h- I (F)

is a proper filter provided F is a proper filter.

331

Proof Suppose F(S) is proper. Then 0 fj. F(S), since 0 :S x for all x E B. In virtue of the definition of F(S), we have that it is not the case that al /\ ... /\ an :S 0 for some al, ... , an E S, which implies that for all al, ... , all E S, al /\ ... /\ an ::f. O. Conversely, suppose F(S) is not proper. Then F(S) = B, so 0 E F(S). Therefore, for some aI, ... , all E S, al /\ ... /\ an :S 0; but 0 :S x for all x E B, so al /\ ... /\ an = O. D

Theorem 9.4.15 Frege logic is strongly compact.

Proof Suppose F is a proper filter on B*. Then F ::f. B*, so 0 fj. F. Since h is a homomorphism, h(O) = 0, from which it follows that 0 fj. 11- 1(F), for 0 E h-I(F) iff 11(0) E F iff 0 E F. D

Corollary 9.4.11 Let h be any homomorphism from W 18 into the Frege algebra, denoted by 2; then 11- 1( {I} ) is a proper filter on WI 8. Proof Immediate from Lemmas 9.4.9 and 9.4.10, noting that {I} is a proper filter on 2. D

Lemma 9.4.12 A subset r of formulas of W is satisfiable (by a Frege intelpretation) iff the corresponding family r I 8 of equivalence classes is contained in a proper filter on W 18; here r/8 = {[a] : a

En.

Proof Suppose r is satisfiable. Then there is an interpretation I : W -+ 2 such that I( a) = I for all a E r. Accordingly, the associated homomorphism h/ : WI 8 -+ 2 has the property that h/([a]) = 1 for all [a] E r/8 or, equivalently, for all a E r. It follows that r 18 is contained in h- I ({1 D, which is a proper filter on W 18, according to Lemma 9.4.10. Conversely, suppose r I 8 is contained in a proper filter F on WI 8. It follows that r 18 is contained in a maximal filter F* on W 18. According to Lemma 9.4.6, the quotient algebra on WI 8 induced by F* is a two-element Boolean lattice, that is, the Frege algebra. Consider the canonical homomorphism c from WI 8 into WI 8 I F*; c maps every element of F* into 1, and every element of WI8 - F* into O. The associated interpretation Ie maps every formula a whose equivalence class [a] is in F* into 1. Therefore, since r I 8 S;;; F S;;; F*, le( a) = 1 for every a E r, which is to say that r is satisfiable. D

Lemma 9.4.13 Let B be any Boolean lattice, and let S be any subset of B. Define F(S) = {x E B: for some al, ... , an E S, al /\ ... /\ an :S x}. Claim: F(S) is a filter on B. Proof Suppose p E F(S) and suppose p :S q. Then al /\ ... /\ an :S p for some aI, ... , an E S; but then al /\ ... /\ all :S q, so q E F(S). Suppose p, q E F(S). Then aI/\ .. . /\ am :S p and bl/\ ... /\ bn :S q for some aI,···, am, bI, ... , b ll E S. So by lattice theory, al /\ ... /\ am /\ bi /\ ... /\ bn :S p /\ q, from which it follows that p /\ q E F(S). D

Lemma 9.4.14 Let B be any Boolean lattice, let S be any subset of B, and let F(S) be the filter described in Lemma 9.4.13. Then F (S) is a proper filter (f and only if for all aI, ... , all E S, al /\ ... /\ all ::f. O.

Proof Since Frege logic has exclusion negation, in virtue of Theorem 9.4.3, it is sufficient to show that Frege logic is I -compact. We proceed contrapositively. Suppose X is satisfiable for every finite subset X of r. Then by Lemma 9.4.12, the corresponding family XI 8 of equivalence classes is contained in a proper filter on WI 8. Since X is finite, we write X = {XI,X2, ... ,xm }, and we write XI8 = {[xIJ, [X2], ... , [x m]}. Since XI8 is contained in a proper filter, it follows that [xIJ /\ [X2] /\ ... /\ [x m] ::f. O. Recall that X is an arbitrary finite subset of r, so we have that for all [Xl], [X2], ... , [Xm] E r 18, [xIJ /\ [X2] /\ ... /\ [xm] ::f. O. So by Lemma 9.4.14, F(r 18) is a proper filter on W 18. Since r 18 S;;; F(r 18), we have that r 18 is contained in a proper filter on W 18. So by Lemma 9.4.12, r is satisfiable. D We have now shown that Frege logic is strongly compact. Since Boolean logic is strongly equivalent to Frege logic, it follows that Boolean logic has finitary entailment. On the other hand, since Boolean logic and Frege logic are not strictly equivalent, we do not automatically know whether Boolean logic is compact in any of the other senses of compactness. As it turns out, Boolean logic is strongly compact, but we proceed to this result in small steps.

Lemma 9.4.16 Boolean logic is I -compact. Proof Suppose that r is Boolean-unsatisfiable. Then there is no valuation v E V(B) such that vCr) = {T}. It immediately follows that r entails every formula, so in particular, r entails (relative to B) the constant falsity f (f can be defined as p/\ '" p). Since Boolean logic is strongly equivalent to Frege logic, we have that r F-entails f. But Frege logic has finitary entailment, so there is a finite subset X of r such that X F-entails f. Appealing to strong equivalence again, we have that X B-entails f, which is to say that every valuation v E V(B) that satisfies X also satisfies f. But no valuation satisfies f, so no valuation satisfies X. Thus, there is a finite subset X of r that is D unsatisfiable.

Lemma 9.4.17 Let S be any set offormulas that is satisfiable. Then there is a Boolean valuation Vs that satisfies S, and moreover has the property that every valuation v that satisfies S is an extension ofvs. Proof Suppose S is satisfiable. Then there is a Boolean interpretation I from W into B such that I(a) = 1 for every a E S; equivalently, the associated homomorphism h/ is such that h/ ([ a]) = 1 for all [a] E S 18. It follows that S 18 is contained in a proper filter on WI 8, so that the filter F (S18) generated by S 18 is proper; call this filter F (S). Let 8

A THIRD LOGIC

CLASSICAL PROPOSITIONAL LOGIC

332

be the congruence relation determined by F(S), and let s be the associated (canonical) homomorphism from W into W lB. Then sex) = 1 iff x E F(S), by Lemma 9.2.4. Consider the valuation Vs determined by s. Consider any valuation v that satisfies S; v is induced by I for some homomorphism I, where v(a) = Tiff I(a) = 1, v(a) = F iff I(a) = O. In particular, v satisfies S only if SIB c h- 1({I}). Suppose a E dom(vs); then either [a] E F(S) or [a] ¢ F(S); in the former case vs(a) = T, in the latter case vs(a) = F. In the first case, since F(S) ~ h;-1 ({ I}), [a] E h;-1 ( {I}), so I(a) = 1, so v(a) = T. (F(S) ~ h;-\ {I}) because h;-1 ({ I}) is a filter containing SIB, and F(S) is the smallest filter containing SIB.) In the latter case, [a] ¢ h;-I({I}), so I(a) = 0, so v(a) = F. In either case, v extends Vs. 0 Lemma 9.4.18 There is a Boolean valuation v such that a Boolean valid or Boolean contra-valid.

E

dom(v) only

if a is

Proof Consider the unital Boolean matrix whose underlying algebra is W IB, and consider the canonical homomorphism c from W into W lB. The associated valuation Vc has the property that a E dom(v c) only if c(a) = [a] = 1, or c(a) = [a] = O. Suppose a E dom(vc). In virtue of Lemma 9.2.5, every Boolean interpretation maps a into 1, since it maps [a] = 1 into 1, or every Boolean interpretation maps a into 0, since it maps [a] = 0 into O. It follows that a is either Boolean valid or Boolean contra-valid.

o Lemma 9.4.19 valid.

r

is Boolean unassailable only

if there is an a

E

r

that is Boolean

Proof Suppose r is unassailable. Then for every v E V(B), there is an a E r such that v(a) = T. Consider the minimal valuation defined in Lemma 9.4.17, Vc; there is an a E r such that vc(a) = T, but vc(a) = T only if a is Boolean valid. Thus, there is an a E r that is Boolean valid. 0

Corollary 9.4.20 Boolean logic is U -compact. Proof Suppose r is unassailable. Then by Lemma 9.4.19, there is an a E r that is Boolean valid. It follows that {a} is unassailable, but {a} is a finite subset of r. 0

Lemma 9.4.21 r Boolean entails A only entails a, provided that A is non-empty.

if there is an a

E

A such that

r

Boolean

Proof Suppose r Boolean entails A. Then for every v E V(B), if v satisfies r, then v satisfies a for some a E A. There are two cases to consider, according to whether r is or is not satisfiable. In the latter case, there is no valuation that satisfies r, so r entails every fOlmula, so it entails at least one formula a E A (assumed to be non-empty). In the former case, we can speak of the valuation Vi as described in Lemma 9.4.17; since Vi satisfies r, there is an a E A such that Vi satisfies a; call it ao. Now, consider any valuation v that satisfies r. By Lemma 9.4.17, v extends Vi, so in particular v(ao) = T. It follows that r entails ao. Thus, in either case, there is an a E A such that r entails a.

o Theorem 9.4.22 Boolean logic is strongly compact.

333

Proof Suppose r entails A. There are two cases to consider, according to whether A is or is not empty. In the former case, r entails the empty set, so r is unsatisfiable. According to Lemma 9.4.16, there is a finite subset X ofr such that X is unsatisfiable, so X entails 0 (= A). In the latter case, since A is non-empty, by Lemma 9.4.21, there is an a E A such that r entails a. Since Boolean logic is strongly equivalent to Frege logic, we have that r Frege entails a. But Frege logic is compact, so there is a finite subset X of r such that X Frege entails a, so by strong equivalence, X Boolean entails a. Thus there are finite subsets X of rand {a} of A such that X entails {a}. 0

Thus, we see that although Frege logic and Boolean logic are not strictly equivalent (so they provide different characterizations of symmetrical entailment), they are nevertheless both strongly compact. 9.5

A Third Logic

We now have two logics, Frege logic and (unital) Boolean logic, that are strongly equivalent, but not strictly equivalent. We now give a simple example of a logic that is weakly equivalent, but not strongly equivalent, to Frege logic. The logic we have in mind is characterized by a single matrix M = (A, F), where A is the four-element Boolean lattice (as in Boolean logic), and F = {l}. Here, a and b complement each other. In order to show that M-Iogic is weakly equivalent to Frege logic, we appeal to the following general lemma, which is similar to Lemma 9.2.1. Lemma 9.5.1 Let J and K be two classes of similar logical matrices, all appropriate for a given algebra of sentences. Then the following is a sufficient condition that every K-validformula is also J-valid. (Here, M = (A, F), N = (B, G).) For every M E J, and every a E A - F, there exists an N E K and a homomorphism hfrom A into B such that h(a) E B - G. (For every non-true J-proposition b, there is a homomorphism that maps b into a non-true K -proposition.) Proof Suppose a is not J-valid. Then there is an interpretation I into some J-mauix M = (A, F) such that I(a) E A-F. According to the hypothesized condition, there is a K-matrixN = (B, G), and a homomorphism h from A intoB, such that h(l(a)) E B-G. It follows that there is an interpretation-namely, the composition c of h and I-into a 0 K-matrix such that c(a) is not a true proposition, so a is not K-valid.

Theorem 9.5.2 Frege logic and M-logic are weakly equivalent, which is to say that for every formula a, a is Frege valid if and only if a is M-valid. Proof Consider the map h : A -+ 2 such that h(l) = h(a) = 1, h(b) = h(O) = O. One can readily show that h is a homomorphism. Moreover, every non-true proposition (there is only one, 0) is mapped into a non-true proposition (namely, 0). Thus, we have the antecedent condition of Lemma 9.5.1. It follows that every Frege valid formula is M-valid. Concerning the converse inclusion, consider the map h : 2 -+ A such that h(l) = 1, h(O) = 0; h is readily shown to be a homomorphism; it also maps each nontrue proposition into a non-u·ue proposition. Thus, applying Lemma 9.5.1, we have that every M-valid formula is also Frege valid. 0

CLASSICAL PROPOSITIONAL LOGIC

334

PRIMITIVE VOCABULARY AND DEFINITIONAL COMPLETENESS

Theorem 9.5.3 Frege logic is not strongly equivalent to M-logic. Proof It is sufficient to provide an argument r I- a that is Frege valid but not M-valid. The following is a Frege valid argument that is not M-valid: {p, q} I- p /\ q; for let /(p) = a, /(q) = b; then /(p /\ q) = 0; thus, v,(p) = v,(q) = T, but v,(p /\ q) = F. On the other hand, this is a standard valid argument of classical (Frege) logic. D

9.6

Axiomatic Calculi for Classical Propositional Logic

We have now provided formal semantic characterizations of Boolean logic and Frege logic, and we have shown that although they are not strictly equivalent, they are strongly equivalent. In other words, although they validate exactly the same asymmetrical arguments, they do not validate exactly the same symmetrical arguments. In addition to formal semantic characterizations, logicians have traditionally been interested in axiomatic characterizations of various logics, and have accordingly constructed various axiomatic calculi. For our purposes, an axiomatic calculus is an inductive definition of a particular subset of syntactic objects, where there are at least four sorts of objects that one might deal with, and hence at least four sorts of axiomatic calculi. These are listed as follows: (1) A unary calculus is an inductive definition of a subset of individual formulas.

(2) A binary calculus is an inductive definition of a subset of ordered pairs of formulas. (3) An asymmetrical sequent calculus is an inductive definition of a subset of ordered

pairs, whose first components are sets of formulas, and whose second components are individual formulas. (4) A symmetrical sequent calculus is an inductive definition of a subset of ordered pairs of sets of formulas. In each case, the set of objects defined by a calculus C are called the theses of C; this set is denoted by T(C). Thus, the theses of a unary calculus are individual formulas, the theses of a binary calculus are ordered pairs of formulas, etc. For the moment, we concentrate on unary calculi. Consider the set W of formulas of some particular sentential language L. An inductively defined subset S of W is specified by giving the base elements of S, and by giving the inductive generating clauses by which "new" elements are "added" to S. In the case of an axiomatic calculus, the base elements are called axioms, and the inductive generating clauses are called transformation (or derivation) rules. However, before discussing axiomatic calculi, it might be useful first to look at an already familiar inductively defined set, namely, the set W of formulas of a sentential language L. Specifically, W is an inductively defined subset of the set S of all strings of symbols in the vocabulary of L. The base elements are the atoms, and the inductive generating clauses are the clauses pertaining to the construction of molecules, the general clause being given as follows: (c) If

C

is an n-place connective, and if is a formula.

CWI W2 ... Wn

WI, W2, ... ,Wn

are formulas, then

335

Now, the relation between this sort of definition and ordinary mathematical induction may be seen as follows. The following is a theorem of set theory: (T) Let X be any set, let a be any element of X, and let g be any function from X into X; then there is a function II : (i) -+ X having the following properties: u(O) = a; lI(n

+ 1) =

g(u(n)).

In order to convert the earlier definition into an inductive definition in the strict sense, we proceed as follows. We begin with the power set \p(S) of the set of strings of symbols of L; this plays the role of the set X. Next, the base element a is the set A of atoms. Finally, our function g is defined as follows. First, for each connective Ci, define a function fi on \p(S) as follows: fi(X) = {CiXjX2 ... Xn : Xj,X2, ... ,Xn E X}; juxtaposition is multiplication (concatenation) in the semi-group of strings; thus, fi takes each n-tuple of strings of X and makes an (n + I)-tuple by prefixing the connective Ci. Next define the function g on \p(S) as follows: g(X) = UiEI fi(X) U X. Now, we have a set \p(S), a base element A, and a generator function g; the induced recursive function II : (i) -+ \p(S) is defined as usual. u(O) consists of just atoms; u(1) consists of atoms, as well as what one can obtain from atoms by applying connectives; lI(2) consists of all of these strings, as well as all those strings obtainable from these by applying the connectives; etc. The set W of formulas is the union of all of these sets: W = U{lI(i) : i E (i)}. So if we want to prove something about every formula in W, it is sufficient to prove it for u(O), and to prove it holds for lI(n + 1) on the condition it holds for lI(n); for then it holds for u(i) for all i E (i). Also, W E W iff W E u(i) for some i E (i).

9.7

Primitive Vocabulary and Definitional Completeness

Just as the well-formed formulas of a sentential language are an inductively defined subset of the set S of all shings of symbols, the theses of a (unary) calculus formulated in a given language form an inductively defined subset of the set W of all formulas. Now, we could construct an axiomatic calculus for classical logic on the basis of the standard sentential language, but since there are seven connectives in this language, this would prove to be too cumbersome. What we do instead is take a pmticular fragment of this language as the basis for our calculus. Afragment of a language L with connectives C is basically the set of formulas of L that are expressed in terms of a proper subset C* of C; that is, where C = {Ci : i E I}, C* = {Ci : i E J}, with J ~ I, but J =1= I. There are two sorts offragments that are considered. On the one hand, there are proper fragments; on the other hand, there are definitionally complete fragments, or simply complete fragments. Whether a particular fragment is complete depends upon the semantics one is considering. Intuitively, a fragment is complete if every connective of the parent language is "definable" in terms of the fragment language, where "definable" is relative to a particular semantics. In order to clarify this intuitive notion, we begin with the algebra W of formulas, and we construct a reduced algebra of formulas W*. The carrier set of Wand W* is the same, the set of all formulas of L; on the other hand, whereas the set F of operations on W is the family {oc : C E C} or {Oi : i E I}, the family F* of operations on W* is the family

CLASSICAL PROPOSITIONAL LOGIC

336

{oe : C E C*} or {Oi : i E J}. Note that although we have removed a number of connective operations, we have not correspondingly removed the connectors themselves. Now, we can define the polynomials on the respective algebras, denoted poly(W) and poly(W*), and we can define a congruence relation e on Wand W* relative to a given semantics K (a class of logical matrices appropriate for L); in particular, we say that two formulas a and p are congruent modulo e if for every K-interpretation h, h(a) = h(P), that is, a and Pare always interpreted as the same proposition. We are now in a position to define definitional completeness:

(DC) Let L be a sententiallanguage, with connectives {Ci : i E J}, let {Ci : i E J} (J c I) be a proper subset of connectives of {Ci : i E J}, and let K be a class of logical matrices appropriate for L. Then {Ci : i E J} forms a definitionally complete set of connectives for L relative to K if for every polynomial ¢ on W (of degree n), there is an n-place polynomial Iff on W*, such that for all a1, a2, ... , an E W, ¢(al, a2,···, an)

e Iff(al, a2,·· ., an).

THE CALCULUS BC

337

where Ie is the Boolean operation corresponding to c. Let us consider Ov as an example. Since ID is join v, condition (a) reduces to the following condition: (aD) h(ov(a, fJ)) = h(a) V h(fJ). But ov(a, fJ) is by definition equal to (a --+ fJ) --+ fJ, so we have (1) h(ov(a, fJ)) = h((a --+ fJ) --+ fJ);

but by hypothesis, h preserves the conditional operation, so (2) h((a --+ fJ) --+ fJ) = (h(a) --+ h(fJ)) --+ h(fJ),

where the arrow operation on a Boolean algebra is defined so that a --+ b = -a the question whether (aD) is true boils down to whether the following is true:

V

b. So

(aD*) a V b = (a --+ b) --+ b for all Boolean algebras A, for all a, bE A.

The particular logic that concerns us is Boolean logic. In this case, there are a number of different definition ally complete subsets of connectives, including the following: {--+,f}, {--+,~}, {V,~}, {/\,~}, {+-+, v,f}, {+-+, V,~}, {+-+,/\,f}, {+-+,/\,~}. The particular fragment we are going to employ uses --+ and I as its connectives; the associated set W of formulas is defined as follows:

9.8 The Calculus BC

(wI) Every atom is a wff. (w2) I is a wff. (w3) If a and pare wffs, then so is (a --+ P). (w4) Nothing is a wffexcept in virtue of clauses (wl)-(w3).

Having specified the language L, with its associated algebra of formulas and with its defined operations, we now describe the axiomatic calculus BC. Specifically, the class T(BC) of theses of BC is the smallest subset of formulas satisfying the following clauses:

The remaining connectives do not appear explicitly in our primitive language. In place of the explicit syntactic symbols, we have their set-theoretic surrogates, defined not in the language itself, but in the associated algebra of formulas. These operations on W are defined as follows:

(Al) (A2) (A3) (A4) (RI)

(D) ov(a, P) (K) oi\(a, fJ)

= (a --+ fJ) --+ fJ· = (a --+ (fJ --+ I)) --+ I·

(N) o~(a) = a --+ I. (B) o_(a, fJ) = [(a --+ fJ) --+ ((fJ --+ a) --+ I)] --+ (t) t

=I

--+

I·

I.

Note carefully the difference between Ov, which is a defined operation, and 0_", which is a primitive operation. Whereas 0-. is a map that sends each ordered pair of formulas (a, fJ) into the formula (a --+ fJ), Ov is a map that sends each ordered pair of formulas (a, fJ) into the formula ((a --+ fJ) --+ fJ). Thus, although we have a set-theoretic counterpart of disjunction, we do not have disjunction itself. In order to show that the above definitions are adequate for Boolean logic, one must show for each connective c, and for each Boolean interpretation h, that the interpretation preserves the corresponding operation 0e, which is to say

But this can be shown routinely in the theory of Boolean algebras. That the remaining definitions are adequate for Boolean logic is shown in a completely similar manner.

For all For all For all For all For all

a, PEW, a --+ (fJ --+ a) E T(BC). a, fJ, yEW, (a --+ (fJ --+ y)) --+ ((a --+ fJ) --+ (a --+ y)) E T(BC). a, fJ E W, ((a --+ fJ) --+ a) --+ a E T(BC).

a

E W,

I

--+

a

E T(BC).

a, fJ E W, if a E T(BC) and a --+ fJ E T(BC), then fJ E T(BC).

Adopting the traditional notation for axiomatic calculi, we write "BC f- a" in place of "a E T(BC)," or, when BC is understood, we simply write "f- a." If we also agree to drop universal quantifiers, we can rewrite the above clauses as follows: (AI) (A2) (A3) (A4) (RI)

f- a --+ (fJ --+ a). f- (a --+ (fJ --+ y)) --+ ((a --+ fJ) --+ (a --+ y)). f- ((a --+ fJ) --+ a) --+ a. f- I --+ a. If f- a and f- a --+ fJ, then f- p.

We now consider the adequacy of the proposed calculus with respect to Boolean Frege logic, which divides into two parts, soundness and completeness. To say that calculus BC is sound for Boolean logic is to say that every thesis ofBC is Boolean valid;

CLASSICAL PROPOSITIONAL LOGIC

338

to say that BC is complete for Boolean logic is to say that every Boolean valid formula is either a thesis of BC or is definitionally equivalent to a thesis of BC. Since every formula is definition ally equivalent to a formula written exclusively in the primitive vocabulary, this amounts to saying that every Boolean valid formula expressed in the primitive vocabulary is a thesis of Be. The following theorem concerns soundness.

Theorem 9.8.1 For all a E W,

if a

E T(BC), then a is Boolean valid.

Proof We proceed by induction, showing that every axiom of BC is Boolean valid, and

that the single rule (Rl) preserves Boolean validity. That each axiom is Boolean valid amounts to the fact that the cOlTesponding polynomial is identically equal to the unit element for every Boolean lattice (thus, for example, a -7 (b -7 a) = 1 is a theorem of Boolean lattices); this is shown routinely. That the rule of generation preserves Boolean validity may be seen as follows. Suppose a and a -7 fl are both Boolean valid. Then for every Boolean interpretation h, h(a) = h(a -7 fl) = 1, but since h is a homomorphism h(a -7 fl) = h(a) -7 h(fl), so h(a) -7 h(fl) = 1. Therefore 1 -7 h(fl) = 1, from which it follows that h(fl) = 1. Thus, fl is Boolean valid. 0 Our strategy in proving the completeness of BC is to construct the Lindenbaum algebra LA(BC) of the calculus BC, and show that it is a Boolean algebra the unit element of which is the equivalence class of theses of BC. LA(BC) is a quotient algebra of the algebra W of formulas, where the congruence relation e is defined as follows: (DI) aefl iff I- a

-7

fl and I- fl

-7

a.

Since e is a congruence relation on W, it sets up a homomorphism c, the canonical homomorphism, from W into W Ie, where cCa) = [a] for all a E W. Since W /e is a Boolean algebra, c is a Boolean interpretation; we can also form the unital Boolean matrix based on W Ie. Also, since the unit element of W /e is the equivalence class [t] of theses of BC, c has the following property: c(a) = 1 iff a E T(BC) (i.e., x E [tD. Thus, if any formula a is not a thesis ofBC, then c(a) f: 1, from which it follows that a is not Boolean valid, which is to say that every Boolean valid formula (in the primitive vocabulary) is a thesis of BC. The details of the completeness proof of BC are somewhat involved. We begin by proving various lemmas that demonstrate that e is indeed a congruence relation on W. (We use dots and colons to mark the (relatively) principal OCCUlTence of the alTOW connective. Thus, for example, "a -7 (a -7 a. -7 a)" is short for "a -7 ((a -7 a) -7 a)." A dot to the left of an alTOW substitutes for a right parenthesis, with the mated right parenthesis to be inserted at the nearest point where it is needed to make the formula well-formed. Similarly, a dot to the right of an anow substitutes for a left parenthesis. Colons are stylistic vmiants used when the alTOW is the main connective.)

Lemma 9.8.2 I- a

-7

a.

Proof

Suppose I- fl. By (AI), I- fl

Lemma 9.8.4 If I- a

-7

fl.

-7

fl and I- fl

(a

-7

fl), so by (RI), I- a

-7

y, then I- a

-7

339

-7

-7

Suppose I- a -7 fl and I- fl -7 y. Then by Lemma 9.8.3, I- a -7 (fl (A2), I- a -7 (fl -7 y). -7 (a -7 fl. -7 a -7 y), so by RI (twice), I- a -7 y.

Lemma 9.8.5 I- (fl

-7

y)

-7

(a

-7

fl.

-7

a

o

fl.

y.

Proof

-7

y), but by

Corollary 9.8.6

If I-

fl

-7

y, then I- (a

fl)

-7

0

y).

-7

Proof By (A2), I- a -7 (fl -7 y). -7 (a -7 fl. -7 a -7 y), but by (AI), I- (fl (a -7 .fl -7 y), so by Lemma 9.8.4, (fl -7 y) -7 (a -7 fl. -7 a -7 y). -7

(a

-7

-7

y)

Lemma 9.8.7 I- (a

-7

fl)

-7

(fl

-7

y.

-7

a

-7

-7

0

y).

Proof Immediate from Lemma 8.5 and (RI).

0

y).

Proof By Lemma 9.8.5, I- (fl -7 y) -7 (a -7 fl. -7 a -7 y), so by (A2) and (RI), I(fl -7 y. -7 a -7 fl) -7 (fl -7 y. -7 a -7 y). By (AI), I- (a -7 fl) -7 (fl -7 y. -7 a -7 fl), so by Lemma 9.8.4, I- (a -7 fl) -7 (fl -7 y. -7 a -7 y). 0

Corollary 9.8.8

If I- a

-7

fl, thell I- (fl

-7

y)

-7

(a

-7

y).

o

Proof Immediate from Lemma 9.8.7 and (RI).

Lemma 9.8.9 e is a congruence relation on W, which is to say for all a, fl, y, 8

E

W:

(1) aea. (2) If aefl, then flea.

(3) If aefl and fley, then aey. (4) If ae fl and ye8, then a -7 ye fl

-7

8.

(1) follows from Lemma 9.8.2, (2) from the definition of e, (3) from Lemma 9.8.4. Concerning (4), suppose aefl and ye8. Then I- a -7 fl, I- fl -7 a, I- y -7 8, I8 -7 y. So by Lemma 9.8.7, I- (fl -7 y) -7 (a -7 y) and I- (a -7 y) -7 (fl -7 y), and by Lemma 9.8.5, I- (fl -7 y) -+ (fl -7 8) and I- (fl -7 8) -+ (fl -+ y). So by Lemma 9.8.4, I- (a -+ y) -7 (fl -+ 8) and I- (fl -7 8) -+ (a -+ y), which is to say a -7 yefl -+ 8. Proof

o Lemma 9.8.10 If I- a, then aefl (ff I- fl. Proof Suppose I- a and aefl. Then I- a -+ fl, so by (RI), I- fl. Suppose I- a and I- fl. Then by Lemma 9.8.3, I- fl -+ a and I- a -+ fl, so aefl. 0

Corollary 9.8.11 The set T(BC) of theses ofBCform a congruence class modulo e.

o

Proof Immediate from Lemma 9.8.10.

Lemma 9.8.12 I- (a

Proof By (A2), I- a -7 (a -7 a. -7 a) -7: (a -7 .a -7 a) -7 (a -7 a), but by (AI), I- a -7 (a -7 a. -7 a) and I- a -7 (a -7 a), so by RI (twice), I- a -7 a. 0

Lemma 9.8.3 Ifl- fl, then I- a

THE CALCULUS BC

-7

.fl

-7

y) -+ (fl

-7

.a

-7

y).

Proof By (Al), I- fl -+ (a -7 fl), so by Lemma 9.8.7, I- (a -+ fl. -7 a -+ y) -+ (fl -+ .a -+ y). By (A2), I- (a -+ .fl -7 y) -+ (a -+ fl. -+ a -+ y), so by Lemma 9.8.4, I- (a -+ .fl -+ y) -+ (fl -+ .a -7 y). 0

THE CALCULUS D(BC)

CLASSICAL PROPOSITIONAL LOGIC

340

Lemma 9.8.13 I- (a -+

/3.

-+

/3)

-+

(/3

-+ a. -+ a).

Proof By Lemma 9.8.5, I- (/3 -+ a) -+: (a -+ /3. -+ /3) -+ (a -+ /3. -+ a), so by Lemma 9.8.12 and (R1), I- (a -+ /3. -+ /3) -+: (/3 -+ a) -+ (a -+ /3. -+ a). By (A3), I- (a -+ /3. -+ a) -+ a, so by Lemma 9.8.5 I- (/3 -+ a) -+ (a -+ /3. -+ a). -+ (/3 -+ a. -+ a), so by Lemma 9.8.4, I- (a -+ /3. -+ /3) -+ (/3 -+ a. -+ a). 0

Lemma 9.8.14 For every a, /3, r E W:

/3) -+ a B a; (/3 -+ r) B /3 -+ (a -+ r); (3) (a -+ /3) -+ /3 B (/3 -+ a) -+ a;

(2) a -+

Lemma 9.8.16 The Lindenbaum algebra LA(BC) of the calculus BCforms a Boolean ortholattice.

One lemma remains to be proved before we can state (and prove) the completeness theorem for calculus BC. We have already shown (Lemma 9.8.10) that the theses T(BC) form an equivalence class modulo B; we next show that T(BC) is in fact the unit element of the Boolean ortholattice cited in Lemma 9.8.16.

Proof

(AI) and (A3). Lemma 9.8.12. Lemma 9.8.13. (A4) and Lemmas 9.8.2 and 9.8.3.

As a consequence of Lemma 9.8.15 and Abbott's results, we have the following lemma.

o

(4) f-+aBa-+a.

Immediate from Immediate from Immediate from Immediate from

(d4) a A b = (a -+ (b -+ 0)) -+ 0; (d5) -a = a -+ O.

Proof Immediate from Lemma 9.8.15 and Abbott's results, noting that the non-primitive operations (conjunction, disjunction, etc.) correspond to the non-primitive operations (meet, join, etc.) on a bounded implication algebra; compare the definitions.

(1) (a -+

(1) (2) (3) (4)

341

Lemma 9.8.17 T(BC) is the unit element 1 ofLA(BC).

(11) (a -+ b) -+ a = a; (12) a -+ (b -+ c) = b -+ (a -+ c);

Proof By Lemma 9.8.10, T(BC) forms an equivalence class, which we denote [t]. To show that [t] is the unit element of LA(BC), in light of (dl) and (d2) above, it is sufficientto show that [a] -+ [t] = [a] -+ [a], for then [a] :s; [t] for all [a] E W lB. Here, t is an arbitrary thesis (for example, f -+ I), so by Lemma 9.8.3 (twice), I- (a -+ a) -+ (a -+ t). Also, by Lemma 9.8.2, I- a -+ a, so by Lemma 9.8.3, I- (a -+ t) -+ (a -+ a). Therefore, a -+ tBa -+ a, from which it follows that [a] -+ [t] = [a] -+ [a]. 0

(I3) (a -+ b) -+ b = (b -+ a) -+ a; (14) 0 -+ a = a -+ a.

Theorem 9.8.18 BC is complete for the class of Boolean valid formulas; that is, every Boolean valid formula (in primitive notation) is a thesis of Be.

o

Lemma 9.8.15 For every a, b, c E WI B:

Proof Each part follows from the corresponding part of Lemma 9.8.14, together with the following facts: for all a E WIB, there is an a E W such that a = [a]; [a] -+ [/3] = [a -+ /3]; 0 = [f]; [a] = [/3] iff aB/3. 0

At this point, we appeal to the work of J. C. Abbott (1967,1969,1976), who defines an implication algebra (IA) to be an algebra of type (2) such that the single binary operation -+ (implication) satisfies conditions (11)-(I3) above. He shows that every IA induces an upper-bounded join-semi-Iattice under the following definitions: (dl) 1 = a -+ a (for any a); (d2) a:S; b iff a -+ b = 1; (d3) a V b = (a -+ b) -+ b. A bounded IA is, by definition, an IA which has a lower bound as well as an upper bound, with respect to the partial ordering given by (d2). Equivalently, a bounded IA (BIA) is an algebra of type (2,0), where the two-place operation -+ and the zeroplace operation 0 satisfy conditions (11)-(14) above. Abbott shows that every BIA induces a Boolean ortholattice under the following definitions, together with (dl)(d3):

Proof We argue contrapositively: suppose a is not a thesis of BC, to show that a is not Boolean valid. It is sufficient to construct a Boqlean matrix M = (A, D) and an M-interpretation h such that h(a) E A-D. Define M so that A is LA(BC), D = ([t]}, and define h to be the canonical homomorphism from W into W IB, which is LA(BC). By Lemma 9.8.16, LA(BC) is a Boolean ortholattice; by Lemma 9.8.17, [t] = T(BC). Thus, M is a Boolean matrix, and h is a Boolean interpretation. Now, suppose a is not a thesis ofBC. Then a is not congruent to t, in virtue of Lemma 9.8.10, so [a] f= [t]. But h(a) = [a], so h(a) f= [t], so h(a) E A - D, from which it follows that a is not Boolean 0 valid.

9.9

The Calculus D(BC)

The calculus BC is a unary calculus, and so provides an axiomatization of the class of Boolean valid formulas. We next turn to the class of Boolean valid arguments (of the asymmetrical sort), and construct two diverse axiomatic calculi for this class. In axiomatizing the class of Boolean valid arguments, we can proceed either directly or indirectly. The direct method specifies an asymmetrical sequent calculus in a completely straightforward manner-by specifying the base elements, and by specifying the inductive generating schemes. This method is completely analogous to the unary calculus

THE CALCULUS D(Be)

CLASSICAL PROPOSITIONAL LOGIC

342

BC, the chief difference being that, whereas the theses of a unary calculus are single for-

mulas, the theses of an asymmetrical sequent calculus are ordered pairs, where the first components are sets of formulas and the second components are individual formulas. On the other hand, the indirect method of constructing the theses is based on the more traditional method offormal derivations (or formal proofs). This technique is indirect because it detours through a unary calculus in order to construct an asymmetrical sequent calculus. This is the method we examine in this section. Let C be a unary calculus based on language L with formulas W, and let r u {a} be any subset of W. Then a (formal) derivation of a from r in C is, by definition, any finite sequence (s 1, s2, ... , S n) of formulas of W having the following properties: (1) Sn = a. (2) Every element (line) of (Sl, S2, ... , sn) satisfies one of the following:

(i)

Si E

r;

(ii) (iii)

Si E

A(C);

Si

follows from (s j) j
Here, A(C) is the set of axioms of C. Recall that the transformation rules of a unary calculus provide the basis for the inductive generating function g; each transformation rule Ri is a function from \p(W) into \p(W); g is defined so that g(X) = U {Ri(X) : i E I} U X. Then the class of these of C is the set T( C) = U {u(i) : i E co}, where u is the inductively defined function from co into \p(W) such that u(O) = A(C), u(n + 1) = g(u(n)). To say that a formula a follows from formulas {Pj : j E J} according to a transformation rule of C is to say that a is an element of the set g( {Pj : j E J}). If there is a derivation d in C of a from r, then we say that a is deducible from r in C, and we write "C : r f- a" or simply 'T f- a" if the calculus C is understood. Given a unary calculus C, the associated calculus of derivations (or derivational calculus, or deduction calculus) is D(C), which may be given a strict inductive definition as follows. D( C) = U{d(i) : i E co}, where d(i) consists of all the ordered pairs (r, a) such that there is a derivation (Sl, S2, ... , sm) of a from r in C, and such that m ~ i; in other words, (r, a) E d(m) iff there is a derivation of a from r in C that has a length less than or equal to m. Also, (r, a) E D(C) iff there is a derivation of a from r in C of any (finite) length. Our notation so far is as follows: C f- a iff a E T(C); C : r f- a iff (r, a) E D(C). This notation is consistent since one can show that a E T(C) iff (0, a) E D(C), which is the content of the following general lemma. Lemma 9.9.1 Let C be any unmy calculus with theses T(C), and let D(C) be the associated derivational calculus. Then a formula a is a thesis of C (C f- a) iff there is a derivation of a from the empty set 0 in C (C : 0 f- a). That is, a E T(C) (ff (0, a) E D(C).

Proof Since T( C) is inductively defined as the set U {u(i) : i E co}, we proceed by induction to show that for all a E T(C), (0, a) E D(C). Base case: to say that a E u(O) is to say that a E A(C); then the sequence (a) consisting simply of a is a derivation of a

343

from 0; thus, (0, a) E D(C). Assume the inductive hypothesis: for all a E u(n), (0, a) E D(C). We wish to show that for all a E u(n + 1), (0, a) E D(C). Suppose a E u(n + l); then by the definition of u, a E g(u(n)), where g(X) = U {Ri(X) : i E I} U X. Thus, either a E u(n) or a E Ri(u(n)) for some i E I. In the former case, (0, a) E D(C) by the inductive hypothesis. In the latter case, there is an i E I such that a is obtained from formulas PI, P2, ... , Pm E u(n) by rule R i . Since the various Pj are in u(n), by the inductive hypothesis, (0, Pj) E D(C), so there are derivations dj of Pj from 0 in C. We now form a new sequence juxtaposing all the dj and then appending the formula a at the end; we claim that dl, d2, ... , dill> a is a derivation of a from 0 in C-every line is either an axiom of C or follows by some rule R i . Concerning the converse direction, we note that D( C) is inductively defined as the set U {d(i) : i E co}, so we proceed by induction. Base case: (0, a) E d(l), which is to say that there is a derivation of length less than or equal to 1 of a from 0; this can be only if a E A(C), from which it immediately follows that a E T(C). Assume the inductive hypothesis: for all formulas a, if (0, a) E den), then a E T(C). We wish to show that for all formulas a, if (0, a) E den + 1), then a E T(C). Suppose (0, a) E den + 1). Then there is a derivation of length m less than or equal to n + 1 of a from 0 in C. In the case that m < n + 1, the inductive hypothesis applies, so we have a E T(C); in the case m = n+ l, there is a derivation (Sl, S2, ... , sm) of a from 0 in C. Now, for all j < m, the sequence (Sl, S2, ... , sj) is a derivation of formula Sj from 0 of length strictly less than n + 1, so for all j < m, (0,sj) E den), so by the inductive hypothesis, for all j < m, Sj E T(C). Next, a (= sm) is either an axiom of C, in which case it is a thesis of C, or it follows from Sl, S2, ... , Sj(j < m) by a transformation rule Ri. In the latter case, since each S j(j < m) is a thesis of C, it follows that a is also a thesis of C. D In addition to Lemma 9.9.1, there are three more general lemmas that we appeal to in proving the completeness of D(BC). Lemma 9.9.2 If (r, a)

E D(C) and

r

~

fl, then (fl, a) E D(C).

Proof Immediate from the definition of formal derivation in calculus C.

Lemma 9.9.3

fr a E T(C), then for all r

~

W,

(r, a)

D

E D(C).

Proof Suppose a E T(C). Then by Lemma 9.9.1, (0, a) E D(C), so by Lemma 9.9.2, (r, a) E D(C). D

Lemma9.9.4 If(r,a;) E D(C) for i D(C), then (r, P) E D(C).

= 1,2, ... ,m, and (ru

{aI,a2, ... ,am}'P) E

Proof Suppose (r, a;) E D( C) for i E 111, and (r u {aI, a2, ... , am}, P) E D( C). Then for each i E m, there is a derivation di = (si, s~, . .. , S!l) of ai from r in C, and there is a derivation d = (Sl, S2, ... , Sk) of P from r u {aI, a2, ... , am} in C. FOlm the sequence d* = (01,02, ... , OJ) obtained from d by replacing each formula ai by the corresponding derivation di. Claim: d* is a derivation of P from r in C. D

The remaining lemmas leading up to the completeness theorem are peculiar to the calculus C; once again, we use a, P, r to range over formulas of W.

THE CALCULUS D(BC)

CLASSICAL PROPOSITIONAL LOGIC

344

Lemma 9.9.5 ({a

(1) if[a] E [deer)]' and [a] ~ [fJ], then [fJ] E [deer)];

-+ fJ, fJ -+ y}, a -+ y) E D(C).

Proof By Lemma 9.8.7, I- (a -+ fJ) -+ (fJ -+ y. -+ a -+ y), so by Lemma 9.9.3, {a -+ fJ,fJ -+ y} I- (a -+ fJ) -+ (fJ -+ y. -+ a -+ y). Also, in virtue of (Rl), {a -+ fJ,fJ -+ y} U {(a -+ fJ) -+ (fJ -+ y. -+ a -+ y)} I- a -+ y, so by Lemma 9.9.4, {a -+ fJ,fJ -+ y} I- a -+ y.

Lemma 9.9.6 I- a Proof

0

-+ (a -+ fJ· -+ fJ).

By (AI), I- a

-+ (fJ -+ a. -+ a),

and by Lemma 9.8.12, I- (fJ

-+ a. -+ a) -+

(a -+ fJ. -+ fJ), so by Lemma 9.8.4, I- a -+ (a -+ fJ· -+ fJ).

Lemma9.9.7 {a,fJ} I- (a -+.fJ

0

-+ y) -+ y.

(2) if[a], [fJ] E [deer)], then [a] A [fJ] E [deer)]. Proof (1) Suppose [a] E [de(r)] and [a] ~ [fJ]. Then by (D2), [a] -+ [fJ] = 1, so [a -+ fJ] [t], which is to say that a -+ fJ E T(BC). Therefore, by Lemma 9.9.9,

=

fJ E deer), so [fJ] E [de(r)]. (2) Suppose [a], [fJ] E [de(r)]. Then by Lemma 9.9.9, a, fJ E deer), which is to say that r I- a and r I- fJ. By Lemma 9.9.7, {a, fJ} I- (a -+ .fJ -+ f) -+ f, so by Lemma 9.9.2, r U {a, fJ} I- (a -+ .fJ -+ f) -+ f, so by Lemma 9.9.4, r I- (a -+ .fJ -+

f) -+ f, that is, (a -+ ·fJ -+ f) -+ f E deer)· Now, by definition, [a] /\ [fJ] [(a -+ .fJ -+ f) -+ fl· Thus, [a] A [fJ] E [de(r)].

Theorem 9.9.13 D(BC) is complete for Boolean logic, i.e., for every r Boolean entails a, then (r, a) E D(BC).

Proof By Lemma 9.9.6, I- a -+: (a -+ ·fJ -+ y) -+ (fJ -+ y), and I- fJ -+ (fJ -+ y. -+ y), so by Lemma 9.9.3, {a, fJ} I- a -+: (a -+ ·fJ -+ y) -+ (fJ -+ y), and

r

{a, fJ} I- fJ -+ (fJ -+ y. -+ y). Also, {a, fJ} U {a -+: (a -+ ·fJ -+ y) -+ (fJ -+ y), fJ -+ (fJ -+ y. -+ y)} I- (a -+·fJ -+ y) -+ (fJ -+ y), and {a,fJ} U {a -+: (a -+·fJ -+ y)-+ (fJ -+ y), fJ -+ (fJ -+ y. -+ y)} I- (fJ -+ y) -+ y, by (Rl) twice. But by Lemma 9.9.5, {(a -+ .fJ -+ y) -+ (fJ -+ y), (fJ -+ y) -+ y} I- (a -+ .fJ -+ y) -+ y, so by Lemma 9.9.2

Proof

{a,fJ} U {(a -+ .fJ -+ y) -+ (fJ -+ y), (fJ -+ y) -+ y} I- (a -+ ·fJ -+ y) -+ y, so by -+ .fJ -+ y) -+ y. 0

Lemma 9.9.4, {a, fJ} I- (a

We next define the set deer) of deductive consequences of the set r, and we prove a number of lemmas pertaining to it; intuitively, deer) consists of precisely those formulas that are deducible from r in BC: (D!) Let r be any subset of formulas in W; then deer) = {a E W :

(D2) Let deer) be defined as in (Dl); then [de(r)] = {[a] Lemma 9.9.8

r

(r, a)

E

U

D(BC)}.

E W/B: a E de(r)}.

~ deer).

Evidently, {a} I- a, so r U {a} I- a, by Lemma 9.9.2, therefore, if a {a} = r, and so r I- a, which is to say a E deer).

Proof

r

Lemma 9.9.9 If a

E deer), and a -+ fJ E

T(BC), then fJ

E

r, 0

E deer)·

Suppose a E deer) and a -+ fJ E T(BC). Then r I- a and I- a -+ fJ, so by Lemma 9.9.3 r I- a -+ fJ. By (RI), {a, a -+ fJ} I- fJ, so by Lemma 9.9.4, r I- fJ, so

Proof

fJ E deer).

Corollary 9.9.10 If a Proof

0 E deer), and aBfJ, then fJ E deer).

o

Immediate from Lemma 9.9.9 and definition of B.

Corollary 9.9.11 [de(r)] is de(r) factored by B. That is, [deer)] Proof

= de(r)/B.

Immediate from Corollary 9.9.10 and the definition of [de(r)].

0

Lemma 9.9.12 For any subset r of W, [de(r)] is a filter on LA(BC) (= W /B); that is,for all [a], [fJ] E W / B,

345

= [a A fJ] = 0

U {a} ~ W, if

We argue contrapositively. We suppose that (r, a) ¢ D(BC) to show that r does not Boolean entail a. It is sufficient to produce a Boolean matrix M = (A, D) and interpretation h such that h(y) ED for all y E r, but h(a) ¢ D (for then the associated valuation viz satisfies r but not a). Define M so that A is LA(BC), and so that D = [de(r)]. Define h to be the canonical homomorphism from W into W /B(= LA(BC» such that for all a E W, h(a) = [a]. As noted in the proof of Theorem 9.8.18, LA(BC) is a Boolean lattice, and h is a homomorphism; also, according to Lemma 9.9.9, [de(r)] is a filter on LA(BC), so the definitions are adequate. Now, suppose that (r, a) ¢ D(BC). Then a ¢ de(r) , so [a] ¢ [de(r)], so h(a) ¢ D. On the other hand, for all y E r, y E deer), so h(y) = [y] E [de(r)] = D. Thus, r does not Boolean entail a. 0 We have now shown that the derivational calculus D(BC) is complete for Boolean logic. That D(BC) is also sound for Boolean logic is the content of the following theorem. Theorem 9.9.14 For every ru {a}

~ W, if

(r, a)

E

D(BC), then r Boolean entails a.

Since D(BC) is inductively defined as the set U{d(i) : i E ill}, we proceed by induction. Base case: (r, a) E d(1); in this case, there is a derivation of a from r that has length less than or equal to 1; accordingly, a E T(BC) or a E r. But every thesis of BC is Boolean valid, and any Boolean valid formula is entailed by any set; also r entails a if a E r. Thus, r Boolean entails a. Inductive hypothesis: for any r U {a} ~ W, if there is a derivation of a from r of length less than or equal to n, then r Boolean entails a. Suppose there is a derivation of length less than or equal to n + 1 of a from r. If the length is strictly less than n + I, then the inductive hypothesis applies and r Boolean entails a. So consider the case in which the length is exactly n + 1. We have a derivation (d 1, d2, ... , d n+ 1) where d n+ 1 = a, and each di is either an axiom of BC, an element of r, or follows from (dj) j
ASYMMETRICAL SEQUENT CALCULUS

CLASSICAL PROPOSITIONAL LOGIC

346

it is entailed by r, or it follows from (dj) j5,/l by the sole transformation rule (RI). This means that, for some i,j, n, di = p, and d j = P -* a, for some formula p. As already noted, r Boolean entails both p and p -* a. So every Boolean filter that contains her) contains both h(P) and h(P -* a) = h(P) -* h(a). But every filter that contains band b -* a contains b A (b -* a), and b A (b -* a) = b A (-b V a) = (b A -b) V (b A a) = bAa. Since b A a ::; a, b A (b -* a) ::; a. It follows that every filter that contains her) also contains h(a), which is to say that r Boolean entails a. D 9.10

Asymmetrical Sequent Calculus for Classical Propositional Logic

Earlier we noted that one can axiomatize the class of valid arguments in one of two ways, directly or indirectly. Having shown an example of the indirect approach, we now consider an example of the direct approach. According to this approach, the theses of any particular axiomatic calculus are not individual formulas, as they are in unary calculi, but are ordered pairs of the form (r, a), where r is a set offormulas, and a is a formula. Because of the asymmetry of the components of the ordered pair, we call such calculi asymmetrical sequent calculi. The calculus we consider is the calculus ASC. This has the following base elements:

(RI) If (r, a) E T(ASC), and r ~ il, then (il, a) E T(ASC). (R2) If (r u {a}, P) E T(ASC), then (r, a -* P) E T(ASC). (R3) If (r,a) E T(ASC), and (ru (aLP) E T(ASC), then (r,P) E T(ASC).

iff a

E

T(BC).

Proof Conceming the "if" direction, since T(BC) is inductively defined, we proceed by induction, showing that every axiom has the property, and that the transformation rule preserves this property. The base case is trivial in virtue of (AI). That the transformation rule preserves this property follows immediately as a special case of Lemma 9.10.2. Thus, every thesis of BC has this property. Conceming the "only if" direction, we first note that ASC is sound, so if 0 f- a, then oBoolean entails a, which is to say that a is Boolean valid. As we have already shown, BC is complete, so a is a thesis ofBC. Thus, if0 f- a, then a E T(BC). D

Lemma 9.10.4 For any formula a, {a} f- a. Proof By Lemma 9.8.2, a -* a E T(BC), so by Lemma 9.10.3, 0 f- a -* a, so by (RI), {a} f-a-*a.Also,by(A2), {a,a-*a} f-a,soby(R3), {a} f-a. D

Lemma 9.10.6 Ijr f- a, and a -* P E T(BC), then r f- p.

Henceforth, in order to render our notation similar to traditional axiomatic notation, we write 'T f- a" instead of "(r, a) E T(ASC)." Thus, we can write the above clauses as follows: 0 f- a for every axiom a ofBC. {a, a -* P} f- p. If r f- a, and r ~ il, then il f- a. If r u {a} f- p, then r f- a -* p. Ifrf-a,andru{a} f-p,thenrf-p.

Proof Suppose r f- a and a -* P E T(BC). Then by Lemma 9.10.3,0 f- a -* p, so by

(RI), r f- a -* p, so by Lemma 9.10.2, r f- p.

D

Corollary 9.10.7 ffr f- a, and aep, then r f- p. Proof Immediate from Lemma 9.10.6 and the definition of e (i.e., aep iff a -* p E T(BC) and p -* a E T(BC)). D

Lemma 9.10.8 (a, P} f- a A p.

We begin by stating the soundness theorem for calculus ASe.

if r f-

a, then r Boolean en-

tails a. Proof This is routinely shown by induction, by showing that the base elements correspond to Boolean valid arguments, and showing that the rules preserve this property. D

Lemma 9.10.2 ffr f- a, and r f- a -* p, then r f- p.

Lemma 9.10.3 For any formula a, 0 f- a

Proof Suppose r f- a -* p and r f- p -* 8. Then by (RI), r u {a} f- a -* p and ru {a} f- p -* 8. Also, by Lemma 9.10.4, {a} f- a, so by (RI), ru {a} f- a. Therefore, by Lemma 9.10.2, r u {a} f- p, so by Lemma 9.10.2 again, r u {a} f- 8, so by (R2), r f- a -* 8. D

Its inductive clauses are as follows:

Theorem 9.10.1 For evelY subset r u {a} of formulas,

Proof Suppose r f- a and r f- a -* p. Then by (RI), ru {a} f- a -* p. Also, by (A2), {a,a -* P} f- p, so by (RI), ru {a,a -* P} f- p, that is, ru {a} U {a -* P} f- p. So by D (R3), r u {a} f- p, and by (R3) again, r f- p.

Lemma 9.10.5 Ijr f- a -* p, and r f- p -* 8, then r f- a -* 8.

(Al) (e, a) E T(ASC) for every axiom a of calculus Be. (A2) ({a, a -* P}, P) E T(ASC).

(Al) (A2) (Rl) (R2) (R3)

347

Proof By Lemma 9.9.6, p -* (P -* y. -* y) E T(BC), and a -* (a -* (P -* y). -* (P -* y)) E T(BC), so by Lemma 9.10.3, 0 f- P -* (P -* y. -* y) and f- a -* (a -* (P -* y). -* (P -* y)), so (P} f- P -* (P -* y. -* y), and {a} f- a -*

o

(a -*

(P -*

y). -*

(P -*

y)), by (RI), so by Lemmas 9.10.4 and 9.10.2, and finally (RI),

{a,p} f- (P -* y) -* y, and {a,p} f- a -* (P -* y). -* (P -* y), so by Lemma 9.10.5, (a, P} f- (a -* .p -* y) -* y, so in particular {a, P} f- (a -* .p -* f) -* f, which is to say {a,p} f- a A p. D

Definition 9.10.9 Let r be any subset offormulas; then deer) is the set offormulas a such that r f- a, i.e., deer) = {a : r f- a}.

348

SEMI-BOOLEAN ALGEBRAS

CLASSICAL PROPOSITIONAL LOGIC

Lemma 9.10.10 Let Wiebe the Lindenbaum algebra of unary calculus BC, alld let [de(n] be the set of equivalence classes of formulas in de(n, for some subset r of formulas. Then [de(n] is afilter on W Ie. Proof Suppose [a] E [de(n] and [a] ::; [Pl. Then by Lemma 9.10.6, r I- a, and by the theorems about W Ie proved previously, a -+ P E T(BC). So by Lemma 9.10.3 and (R1), r I- a -+ P, so by Lemma 9.10.2, r I- P, so [P] E [de(n]. Next, suppose [a], [P] E [de(n]. Then by Lemma 9.10.7, r I- a, and r I- p. By Lemma 9.10.8, {a,p} I- a /\ P, so r U {a,p} I- a /\ P and r U {a} I- a, by (R1), so by (R3) twice, r I- a /\ P, from which it follows that [a] /\ [P] = [a /\ P] E [de(n]. Thus, [deer)] is a ~oo~. D Theorem 9.10.11 ASC is complete for Boolean logic. That is, for any subset r U {a} offonnulas, r Boolean entails a only if (r, a) is a thesis of ASC. Proof Completely analogous to the proof of Theorem 9.9.13.

9.11

D

Fragments of Classical Propositional Logic

We have now considered the class of Boolean valid formulas and the class of Boolean valid arguments. An additional interesting logical question concerns various special subclasses of these, where each subclass is determined according to a proper subset C* of the set C of connectives of the standard sentential language. For example, we might be interested in those Boolean valid formulas (arguments) that exclusively involve the conditional connective, or exclusively involve conjunction and disjunction, or exclusively involve the non-negative connectives (those besides ~ and f). The subclass of valid formulas (arguments) of a logic that involve a proper subset of connectives is sometimes referred to as afragment of the logic. We have, in fact, already discussed a particular fragment of Boolean logic in our discussion of axiomatic calculi, the fragment involving just the conditional (-+) and the falsity constant (f). Also, as we noted, -+ and f form a definitionally complete set of connectives for classical logic, so the (-+, f) fragment of classical logic is, in some sense, not a proper fragment. Where C is a proper subset of connectives, we say that the C-fragment of a logic L is a proper fragment if C does not form a definitionally complete set of connectives for L. Thus, for example, although the (-+, f) fragment of classical propositional logic is not a proper fragment, the (-+, f) fragment of intuitionistic propositional logic is a proper fragment. There are basically two interesting proper fragments of classical logic, respectively called the implicative fragment, which concerns the conditional connective, and the positive fragment, which concerns all connectives except ~ and f. We could be through simply by saying that the implicative fragment of Boolean logic consists of all the Boolean valid formulas (arguments) that exclusively involve implication, and that the positive fragment of Boolean logic consists of all the Boolean valid formulas (arguments) that do not involve negation or falsity. This is not very satisfying, however. Ideally, we would like an independent specification of each of these in algebraic terms, and an independent specification of each of these in axiomatic terms. We do just that.

9.12

349

The Implicative Fragment of Classical Propositional Logic: Semi-Boolean Algebras

Just as the full classical propositional logic is algebraically characterizable in terms of Boolean lattices (Boolean algebras), its implicative fragment is characterizable in terms of semi-Boolean algebras, first studied in detail by Abbott (1967, 1969). Semi-Boolean algebras come in two forms, which are dual to one another. An upper semi-Boolean algebra (USBA) is, by definition, a join-semi-Iattice with a greatest element, which is such that every principal filter is a Boolean lattice (in this context, a distributed complemented lattice). By contrast, a lower semi-Boolean algebra (LSBA) is, by definition, a meet-semi-Iattice with a least element, which is such that every principal ideal is a Boolean lattice. Semi-Boolean algebras (SBAs) are equationally definable, where the characteristic equations are given as follows: (el) (ab)a = a; (e2) a(be) = b(ae); (e3) (ab)b = (ba)a. Just as a semi-group can be written both additively and multiplicatively, axioms (e1)(e3) can be written (or interpreted) in two different ways, either implicatively or subtractively. Briefly, an implication algebra (IA) is an algebra (A, F) of type 2, where the sole binary operation (denoted -+) satisfies the following conditions: (i1) (a -+ b) -+ a = a; (i2) a -+ (b -+ e) = b -+ (a -+ e); (i3) (a -+ b) -+ b = (b -+ a) -+ a. On the other hand, a subtraction algebra (SA) is an algebra (A, F) of type 2, where the sole binary operation (denoted -) satisfies the following conditions: (s1) a - (b - a) = a; (s2) (e - b) - a = (e - a) - b; (s3) b - (b - a) = a - (a - b). Notice that, whereas (i1)-(i3) are written in the same manner as (e1)-(e3), simply by infixing an arrow, (s1)-(s3) are written in reverse order, following the usual convention concerning how to read subtraction. Note that whereas (i1)-(i3) characterize classical implication, (s1)-(s3) characterize set difference (A - B consists of precisely those elements of A not in B). Having noted that set difference is dual to material implication, we leave subtraction algebras aside and concentrate on implication algebras. We begin by connecting lAs with upper semi-Boolean algebras. First of all, every USBA induces an lA, where the implication operation -+ is defined as follows: (dO) a -+ b

= (a V b)lb.

Here, alb is, by definition, the complement of a in the principal filter generated by b. Since by the definition of a USBA, this filter is a Boolean lattice, a Ibis well-defined. On the other hand, every IA induces a USBA according to the following definitions:

350

AXIOMATIZING THE IMPLICATIVE FRAGMENT

CLASSICAL PROPOSITIONAL LOGIC

(dl) 1 = a -+ a (for any a); (d2) a ~ b iff a -+ b = 1; (d3) a V b = (a -+ b) -+ b. Now, every Boolean lattice is a semi-Boolean algebra (both a USBA and an LSBA), so every Boolean lattice is an IA (and an SA). It can be shown in fact that the arrow operation is simply the Boolean arrow (a -+ b = -a V b). Next, since IAs are equationally definable (they are a variety), every implication subalgebra of a Boolean lattice is an IA; an implication subalgebra is a subset closed under the arrow operation. An IA that is a sub algebra of a Boolean lattice is called a Boolean IA. According to the basic structure theorem for IAs, the class of Boolean IAs is canonical, which is to say that every IA is isomorphic to a Boolean IA. To state matters equivalently, every IA can be embedded into a Boolean lattice preserving the arrow operation. This structure theorem for IAs (which can be dualized to SAs) has the following immediate result: every equational theorem of the theory of Boolean ortholattices that is expressible purely in terms of the Boolean arrow is a theorem of the theory of IAs. This can be argued as follows. Suppose P = Q is not a theorem of IA theory, where P and Q are terms in the corresponding language. Then by the completeness of first-order logic, there is an IA which fails to satisfy P = Q; call it A. According to the embedding theorem cited above, A can be embedded into a Boolean ortholattice; call it B, and call the embedding e. Thus, for every a, b E A, e(a -+ b) = e(a) -+ e(b) (= -e(a) V e(b», and e(a) = e(b) only if a = b. Let p,q be the polynomials on A that correspond to P, Q respectively. Since A does not satisfy P = Q, it follows that there are sequences (al, ... , am), (bl, ... , bn ) of elements of A such that p(al, ... , am) :j:: q(bl, ... , bn ). Since every polynomial is recursively defined, e(p(al,"" am» = p*(e(ar), ... , e(am», and e(q(bl,···, bn » = q*(e(br), ... , e(bn », where p*, q* are the polynomials on B corresponding to p, q, respectively. Since e is one-one, and since p(al, ... , am) :j:: q(bl, ... , bn ), we have p*(e(ar), ... , e(am» :j:: q*(e(br), ... , e(bn ». Since p*, q* correspond to P, Q, we have that B does not satisfy P = Q, so by the soundness of first-order logic, P = Q is not a theorem of Boolean lattice theory. We thus see that the theory of IAs provides a complete algebraic characterization of Boolean implication. This is a useful result if we wish to construct an axiomatic calculus for the implicative fragment of Boolean logic, as we see presently.

9.13 Axiomatizing the Implicative Fragment of Classical Propositional Logic We begin by constructing a unary axiomatic calculus BC*, which is based on a sententiallanguage L in which there is exactly one connective, the conditional. In order to show that BC* is adequate, we must show that every thesis of BC* is Boolean valid, and we must show that every Boolean valid implication formula is a thesis of BC*. The calculus BC* is specified as follows: (AI) I- a -+ (P -+ a); (A2) I- [a -+ (P -+ y)] -+ [(a -+ P) -+ (a -+ y)];

351

(A3) I- [(a -+ p) -+ a] -+ a;

(Rl) if I- a and I- a -+

p, then I- p.

Here I- a means that a is a thesis of BC", i.e., a E T(BC"). Notice that BC" is very similar to BC, the only difference being the absence in BC* of axiom schema (A4): (A4) I-

f

-+

a.

That BC* is sound follows from the fact that BC is sound, since every thesis of BC* is a thesis of Be. In order to show that BC* is complete for the implicative fragment of Boolean logic, we appeal to results 9.8.2-9.8.15, none of which appeals to (A4), except for the fourth parts of Lemmas 9.8.14 and 9.8.15, and we appeal to Lemma 9.8.17. As a result of these lemmas, we have that the Lindenbaum algebra of BC* is an implication algebra, the unit element of which is the equivalence class of theses of BC*. In virtue of the embedding theorem for lAs, LA(BC*) can be embedded into a Boolean ortholattice preserving implication. Now, suppose that a is not a thesis ofBC*. Then [a] :j:: [t] = 1. Consider the function h, which is the composition of the canonical homomorphism of the algebra of formulas into LA(BC*) with the embedding function e. Since h is the composition of two homomorphisms, it is a homomorphism. Since [a] :j:: [t], and since e is one-one, we have h(a) = e[a] :j:: e[t] = eel) = 1. Since h(a) :j:: 1, a is not Boolean valid. In order to discuss whether the associated deduction calculus D(BC*) is sound and complete for the class of Boolean valid implicative arguments, we need to discuss the implicational analog of Boolean filters. Specifically, we define an implicational filter on an IA, A, to be any subset F of A satisfying the following conditions: (iF!) 1 E F; (iF2) a E F and a -+ b E F only if b E F. In other words, an implicational filter is a set of propositions containing the tautological proposition 1, and closed under modus ponens. Note that if A is in fact a Boolean lattice, then every implicational filter on A is a lattice filter on A, and conversely. Recall that every Boolean filter F is the inverse image h- l ( {I} ) of a Boolean homomorphism h, since for any Boolean algebra B and any filter F on B, there is a Boolean algebra B* and a homomorphism h from B into B* such that F = h- l ({ I}). As shown by Abbott (1967), the same is true for implicational filters; specifically, he shows that, given any implicational filter F on A, the relation B on A, defined so that aBb iff a -+ b, b -+ a E F, is a congruence relation on A, and moreover, a E F iff aB1. This allows us to prove the completeness of D(BC*), as follows. (The surrounding details are exactly as they are for D(BC).) We begin with the Lindenbaum algebra of BC*, which we have previously shown to be an implication algebra, the unit element of which is the congruence class of theses of BC*. We next consider any subset r of formulas, and we consider its deductive closure deer), as before, and we consider the corresponding equivalence class [de(n], as before. The key lemma is that [de(n] is an implicational filter on LA(BC*). In virtue of the above mentioned theorem, there is an lA, A, and a homomorphism h from LA(BC*) into A such that [de(n] = h- 1 ({ I}). By the embedding theorem for IAs,

we know that A can be embedded into a Boolean ortholattice; call it B, and call the embedding e. Now, suppose that a is not a deductive consequence ofr. Then a ¢ deer), so [a] ¢ [de(r)], so h([a]) t= 1, so e(h[a]) t= 1. Consider the composition of e and h, and the canonical homomorphism, call it I, i.e., I(a) = e(h[a]). Then I is a Boolean interpretation such that I(Y) = 1 for all y E r, and such that I(a) t= 1. It follows that r does not Boolean entail a. The details of this proof are left as an exercise for the reader. Also left as an exercise for the reader is the proof that (A4) may not be derived from (Al)-(A3), (Rl), which simply amounts to showing that not every implication algebra is a Boolean algebra. 9.14

The Positive Fragment of Classical Propositional Logic

By the positive fragment of a propositional logic, we mean the subset of valid formulas (arguments) that involve the so-called positive connectives, which include conjunction, disjunction, conditional, biconditional, and the truth constant. By contrast, the negative connectives are negation and the falsity constant. As we have seen, whereas the full Boolean logic is characterized by Boolean lattices, the implicative fragment of Boolean logic is characterized by semi-Boolean algebras or implication algebras. In the case of the positive fragment of Boolean logic, the associated mathematical structures are Boolean rings, or, more specifically, their duals. This identification constitutes one of those remarkable coincidences in logic and mathematics. Recall that a ring is an algebra having an addition operation +, a multiplication operation x, an additive inverse operation *, and an additive identity element O. A ring must satisfy the following postulates (we write ab in place of ax b): (Rl) (R2) (R3) (R4) (R5) (R6)

THE POSITIVE FRAGMENT

CLASSICAL PROPOSITIONAL LOGIC

352

a + (b

+ e) = (a + b) + e; a + b = b + a; a + 0 = a, 0 + a = a; a + a* = 0, a* + a = 0; a(b + e) = ab + ae; (a + b)e = ae + be; a(be) = (ab)e.

An element b of a ring A is said to be idempotent if b2 = b, where b2 = bb. A Boolean ring is, by definition, a ring in which every element is idempotent. Equivalently, it is a structure satisfying (Rl)-(R6) as well as (R7) aa = a. Given these axioms, one can show first of all that every Boolean ring is a commutative ring, which is to say that for all a, b E A, ab = ba. One can also show that every element b E A is its own additive inverse: b* = b. Perhaps the most common example of a ring is a ring of subsets of some set X. Where X is any non-empty set, a collection S of subsets of X is a ring of subsets of X just when S is closed under the formation of unions and set differences. That is, (RS) if A E S, and B E S, then A - B E S and A U B E S.

353

S is afield of subsets of X if S is a ring of subsets of X and YES for all YES. A ring of sets is an algebraic ring under the following definitions:

(FSl) A + B = (A - B) U (B - A) [= (A U B) - (A n B)]; (FS2) A x B = An B = A - (A - B). It is easy to verify that every ring of sets is an algebraic ling, indeed a Boolean ring. What is the relation between Boolean rings and the positive fragment of Boolean logic? First of all, according to the customary reading of Boolean rings, every such ring induces a lattice with a least element, according to the following definitions: (dl) least element = 0; (d2) a V b = a + b + ab; (d3) a A b = abo Every lattice has a dual, so we can just as easily say that every Boolean ring induces a lattice with a greatest element, according to the following definitions: (dl') greatest element = 0 (the lattice unit equals the ring zero); (d2') a V b = ab; (d3') a A b = a + b + abo Observe that in the former case, the ring + is interpreted as binary exclusive disjunction (aVb = (a V b) A -(a A b) = (a A -b) V (b A -a». On the other hand, in the latter case, things are turned upside down, so that V becomes A and A becomes V; a + b = (a A b) V -(a V b) = a ~ b; that is, + is interpreted as the biconditional. The remaining positive connective operations may be defined as follows: (d4) a -+ b = b + ab; (d5) 1 = a + a (for any a). Recall that in a Boolean ring, every element is its own additive inverse: a + a = O. Just as every implication algebra can be embedded into a Boolean lattice preserving the implication operation, every Boolean ring can be embedded into a Boolean lattice preserving the ring operations (and hence preserving all the positive connective operations). In order to show this, one first shows that every Boolean ring induces an implication algebra under definition (d4). One next shows that this implication algebra is a lattice, not merely a join-semi-Iattice, where the meet operation is given by definition (d3). One next appeals to Abbott's (1967, 1969) work on implication algebras, according to which every IA can be embedded into a Boolean lattice preserving the implication operation, and preserving all existing infima (meets) in the IA. Since every IA that is a lattice is a Boolean ring, under the following definitions, the Abbott embedding preserves the Boolean ring operations: (d6) a + b = (a (d7) a x b = (a

-+ -+

b) b)

A

(b -+ a); b.

-+

In virtue of the embedding theorem for Boolean rings, we have that the theory of Boolean rings provides a complete algebraic characterization of the positive fragment of the theory of Boolean lattices: every equational theorem of Boolean lattice theory that

354

CLASSICAL PROPOSITIONAL LOGIC

is expressible in terms of the positive operations is a theorem of Boolean ring theory. The argument is completely analogous to the argument for implication algebras given earlier. Concerning the axiomatic characterization of the positive fragment of Boolean logic, the unary calculus BC+ is obtained from BC* by adding the conjunction symbol to the language, and by adding the following axiom schemata: (AS) I- (a /\ fJ) ~ a;

(A6) I- (a /\ fJ) (A7) I- (a

~

~

.fJ

fJ;

~ y) ~

(a

~

(fJ

~

.fJ /\ y)).

An alternative to (AS) and (A6) is the following: (AS+6) I- (a

~

·fJ

~ y) ~

(a /\ fJ. ~ y).

That BC+ is sound follows from the fact that BC* is sound (which follows from the fact that BC is sound), together with the fact that axiom schemata (AS)-(A 7) are Boolean valid. The completeness of BC+ is shown by showing that the Lindenbaum algebra of BC+ is a lattice ordered implication algebra, and hence a Boolean ring. That LA(BC+) is an IA has already been shown in connection with BC*. In order to show that it is lattice ordered, it is sufficient to show that [a] /\ [fJ] is the greatest lower bound of [a] and [fJ]. That it is a lower bound follows from (AS) and (A6), noting that [a] /\ [fJ] ~ [a] (for example) if and only if [a /\ fJ] ~ [a] iff [a /\ fJ] ~ [a] = 1 = [t] iff [(a /\ fJ) ~ a] = [t] iff (a /\ fJ) ~ a E T(BC+). That [a] /\ [fJ] is the greatest lower bound follows from (A7). For suppose [a] ~ [fJ] and [a] ~ [y]. Then I- a ~ fJ and I- a ~ y, so by (AI) and (R1), I- fJ ~ (a ~ y), so by Lemma 9.8.12 and (R1), I- a ~ (fJ ~ y), so by (A7) and (R1), I- a ~ (fJ ~ .fJ /\ y), so by (A2) and (Rl), I- (a ~ fJ) ~ (a ~ .fJ /\ y). But I- a ~ fJ, so by (R1), I- a ~ (fJ /\ y), from which it follows that [a] ~ [fJ] /\ [y]. We thus have that LA(BC+) is a Boolean ring. The completeness of BC+ for the positive fragment follows from this together with the fact that every Boolean ring can be embedded into a Boolean lattice. For suppose a is not a thesis of BC+. Then [a] =1= [t], so e([a]) =1= e([t]) = 1. It follows that there is a Boolean interpretation I, defined so that I(a) = e([a]), such that I(a) is not designated, which is to say that a is not Boolean valid (details may be filled in by the reader). Next, we may show that the deductive calculus C(BC+) associated with unary calculus BC+ is both sound and complete as follows. Soundness is routine and may be proved by induction on the length of the derivation. Completeness is shown in three steps. First of all, one shows that the deductive closure deer) of any subset r of formulas, when factored by the congruence relation B, is a lattice filter on LA(BC+). As the second step, one shows that every filter on a Boolean ring (thought of as a lattice ordered implication algebra) is the inverse image h- 1 ( { 1}) of a Boolean ring homomorphism h. This is shown as follows. Consider any filter F on a Boolean ring A. Define a relation B on A so that aBb iff a +-+ b E F. Next, we show that B is a congruence on A (the result is a general ring-theoretic result, when F is a ring ideal), and we also show that a E F

THE POSITIVE FRAGMENT

355

iff aBl. One then defines the quotient algebra AlB and the associated canonical homomorphism from A into AlB; this is the desired Boolean ring homomorphism. In the last step, one appeals to the embedding theorem for Boolean rings. The completeness proof for D(BC+) is then completely analogous to the completeness proof for D(BC+). Once again, the details are left as an exercise for the reader.

MODAL LOGICS

357

At the same time it would be natural to switch from D to 0 as primitive, and we officially do so.

10 MODAL LOGIC AND CLOSURE ALGEBRAS 10.1

Exercise 10.1.1 Show that the formulations K, K' and K" are all equivalent in the sense that they have the same theorems. (Of course, one has to use the translations given by the dual definitions of D into 0 and vice versa in going back and forth to K". (Hint: cf. Hughes and Cresswell 1968, Ch. VII.)

Modal Logics

Other standard modal logics which we will want to discuss are:

The reader who has no previous familiarity with modal logic may wish to consult Hughes and Cresswell (1968, 1996). The classic, but now dated reference, is Lewis and Langford (1932). The modal logics which we shall consider are only a few among many. The reader wanting to know more about the many should consult Chellas (1980), Segerberg (1971) or Gabbay (1976). Incidentally, this section is very much influenced by Lemmon (1966) and Lemmon with Scott (1977). The weakest modal logic which we shall consider is the system K (for Kripke). Its morphology is like that of the system TV of classical sentential calculus except that it has an additional unary connective D (Dp is read as "necessarily p"). In terms of this one can define Op (read "possibly p") as ~D ~p (or alternatively 0 could have been taken as primitive, setting Dp = ~O ~p). As axioms we take all instances of the following:

T (sometimes called M): K + Dp:J P (or T + Dp:J DDp (or OOp:J Op); B: T + p:J DOp (or ODp :J p); S5: S4 + Op:J DOp (or ODp:J Dp).

These logics are related to one another as shown in Figure 10.1 (where -+ means subsystem), as can be easily seen. S5

B

(1) tautologous schemes of TV; (2) D(p:J l/f) :J (Dp :J Dl/f) (Axiom of Kripke) ((p:J l/f) is to be understood as always abbreviating (~

As rules we have: (4) If P and p :J l/f, then l/f (MP). p, then I- Dp (Necessitation).

== (Dp 1\ Dl/f) (in this section, p == l/f is to be understood as abbreviating (p :J l/f) 1\ (l/f :J p», and the

following is to be added as a rule (additional to MP and necessitation): (6') If P == l/f, then Dp == Dl/f

(Necessitation of provable equivalents).

Alternatively we could replace necessitation with necessitation of provable equivalents, but we would have to add the axiom scheme D (p V ~ p) or some such-cf. Hughes and Cresswell (1968, Ch. VII). We formulate K" as a minor and somewhat dual variant of K'. Thus replace (2') with (2") O(p V l/f)

== (Op V Ol/f),

and replace (6') with (6") if

p == l/f, then Op == Ol/f.

"'/ t

S4

K

(5) If I-

(2') D(p 1\ l/f)

/'" T

p V l/f».

There are two other equivalent formulations that are often preferred in algebraic treatments. Thus let K' be formulated just like K except that (2) is replaced with

p :J Op);

S4:

FIG. 10.1. Relations between K, T, B, S4, and S5 In defining the notion of a deduction in K or any of the other modal logics, we must be careful with the rule of necessitation. It is best viewed as a rule that produces theorems from theorems, rather than consequences from premises. The inference (7) pI- Dp

is just plainly invalid. Thus, let p be any contingent truth, e.g., that you are reading this book. Although this is true, it is hardly necessary (close the cover and go get a cup of coffee if you do not believe us). On the other hand, the rule of necessitation is perfectly in order when applied to theorems, which are presumably logically true and hence necessary. Thus letting S be one of the systems K, T, B, S4, S5, we define a proof in S as a finite sequence of sentences of S, each one of which is either an axiom of S or follows from preceding sentences by either MP or necessitation. A theorem of S is then the last line of a proof in S (we write I-s p). A deduction in S of the sentence p from the set of sentences r will then be a finite sequence of sentences of S, each one of which is either

MODAL LOGIC AND CLOSURE ALGEBRAS

358

BOOLEAN ALGEBRAS

a member of r, or a theorem of S, or follows from preceding sentences by MP. We then say that cjJ is deducible (is a deductive consequence) from r in S when cjJ is the last line of a deduction from r in S (we write r I-s cjJ). In the sequel we shall assume that any "modal logic" is an extension of K, shares the same rules (MP and necessitation), and that the class of theorems is closed under substitution. In the literature these are often called "normal" or "classical" modal logics.

10.2

Boolean Algebras with a Normal Unitary Operator

Recalling definitions, a Boolean algebra with a normal unary operator is a structure (B, /\, v, -, e), where (B, /\, v, -) is a Boolean algebra, and (Ic) eO S 0 (2c) e(x V y)

(can be written as eO = 0); = e(x) Ve(y).

Anticipating applications, let us call such a structure a modal algebra or a "K-algebra." If a K-algebra satisfies additionally

359

1 s il (or il = 1); i(x /\ y) = ix /\ iy; ix s x; ix s iix; (Si) cix six; (6i) cix s x.

(Ii) (2i) (3i) (4i)

Because of this duality, and duality for the underlying Boolean algebras, for any equation or inequation that is derivable for a given system, its dual is also derivable (where its dual is defined as switching /\ and V, i and e, and I and 0). Exercise 10.2.3 Prove that each of (li)-(6i) is equivalent to its dual among (Ic)-(6c). (Hint: Use the fact that ix = -e - x. Thus, for example, 1 S il iff -il s -1 iff -i-O s 0 iff eO sO.) Proposition 10.2.4 Both e and i are isotonic, i.e., x S y implies both ex S ey and ix S iy.

(3c) x sex, we call it a "T-algebra" (in the literature such a thing has been called an "extension algebra"). And ifit satisfies (lc)-(3c) and (4c) eex S ex

(can be written as eex

Proof. We argue the first, the second being dual. If x S y then x ex V ey = e(x V y) = ey, and so ex S ey.

it is called an "S4-algebra" or "closure algebra" (which is why we have chosen to denote the operator by "e"). Closure algebras have important connections to topology, which we shall be describing in a moment. But first note that an operator i (the "interior" operator) can be defined as the dual of e: ix = -e-x. This allows us to define neatly the notion of an "SS-algebra" by adding on to (I c )-( 4c) the requirement

(Sc) ex S iex.

= y. But then 0

Proposition 10.2.5 For any modal algebra, i(a ::) b) S ia ::) ib.

Proof. (a::) b) /\ as b, so by isotonicity i[(a ::) b) /\ aJ sib. But using (2i), we obtain i[(a ::) b) /\ aJ = i(a ::) b) /\ ia. By substitution of identicals, we obtain i(a ::) b) /\ ia sib, and by Boolean algebra moves we obtain the proposition. 0 Exercise 10.2.6 Fill in the "Boolean algebra moves" in the above proof. Also, show that in any modal algebra i(a ::) b) S ea ::) eb. Note that the following corresponds to the rule of necessitation (if a proposition is provable, then so is its necessity).

While we are at it we shall define a "B-algebra" as satisfying (lc)-(4c) and the weaker requirement

Proposition 10.2.7 For any modal algebra, a

(6c) x S iex.

Proof.

The definition of the interior operator from the closure operator can be reversed, as the following shows.

Proof. -i-x = - - e - -x = ex.

y

The modal algebraic equivalent of the Kripke axiom is i(a ::) b) S ia ::) ib.

= ex by (3c)),

Proposition 10.2.1 In a modal algebra, ex

V

= -i-x.

= 1 implies ia = 1. Assume that a = 1. Then by isotonicity ia = ii, and by (il) ia =

1 as desired.

o For T we have the converse:

= 1 implies a = 1. a by (3i). However, then a = 1 since always a S

Proposition 10.2.8 In aT-algebra, ia

o

Remark 10.2.2 The axioms for a modal algebra favor "possibility" in their frequent use of the operator e. Of course one can write dual and equivalent axioms favoring "necessity," and we shall make implicit use of this fact by citing whichever form seems easiest for the purpose at hand. The dual axioms are:

Proof. If ia = 1, then 1 S

Define the operation of "strict implication" -< as

a -< b = i(a ::) b),

1.

o

360

MODAL LOGIC AND CLOSURE ALGEBRAS

FREE BOOLEAN ALGEBRAS

where a :J b is the usual "material implication" defined as -a V b. Define the operation of strict equivalence in the obvious way: a>< b = (a -< b)

1\

Proposition 10.2.13 In any S5-algebra the following hold: i(a 1\ cb)

(b -< a).

c(a

Proposition 10.2.9 We can derive as a theorem in any S4-algebra (and in any algebra "above," i.e., satisfying at least (Ji)-( 4i)): (a>< b) ::; (ia >< ib). Proof The key to this is of course to derive

(a -< b) ::; (ia -< ib).

361

V

= ia 1\ cb,

i(a V ib) = ia V ib;

ib) = ca V ib,

c(a 1\ cb)

= ca 1\ cb.

Proof We prove the two equations on the first line. (The two others are duals to the ones above them.) (1) i(a 1\ cb) = ia 1\ icb by (2i), and ia 1\ icb = ia 1\ cb by (3i) and (Sc). (2) Notice that ia ::; a ::; a V ib. So by isotonicity, iia ::; i(a V ib). Since ix ::; iix, this means that ia ::; i(a V ib). We can similarly show that ib ::; i(a V ib) starting with the lattice fact that ib ::; a V ib. Using lattice properties, we see that ia V ib ::; i(a V ib). We next show the inequality in the other direction. Using Proposition 10.2.12, we obtain i(aVib) ::; iaVcib. Since cib = ib, we obtain i(aVib) ::; iaVib as

D

desrred.

The main steps of the derivation are as follows: 1. 2. 3. 4.

i(a:J b) ::; (ia :J ib) (by Proposition 1O.2.S) ii(a:J b) ::; i(ia :J ib) (1, by Proposition 10.2.4) i(a:J b) ::; ii(a:J b) ((4i)) i(a:J b) ::; i(ia:J ib) (2,3, by transitivity of ::;).

10.3 Free Boolean Algebras with a Normal Unitary Operator and Modal Logic

D

Let us form the Lindenbaum algebra of the modal logic K (in symbols (LA(K))) as follows (actually, we will work with K"). For sentences if; and \fI, define if;;::;j \fI iff f- K " if; == \fl. It is easy to see that ;::;j is a congruence on the algebra of sentences, and that given the definitions

Proposition 10.2.10 The following holds in all S5-algebras: (ia >< 1)

V

- [if;] = [if;]

(ia >< 0) = 1.

1\ [\fI]

[~if;],

= [if; 1\ \fI],

[if;] V [\fI] = [if; V \fI],

Proof Note that the above expression is equivalent in effect to saying that a necessary

proposition is either itself necessary or else impossible. In S5 there is no room for contingency when it comes to modal propositions. We leave to the reader the details of the "translation." The heart of the proof is to demonstrate: iia

V

-cia

= 1.

we obtain that the Lindenbaum algebra of K" is a Boolean algebra with operator c. Indeed, it is easy to check that it is free in the class of Boolean algebras with a unary operator by an "induction on proofs" (soundness). Lindenbaum algebras for T, B, S4, and S5 can be constructed analogously. Theorem 10.3.1 Let S be anyone of the systems K, T, B, S4, or S5. valid in all S-algebras.

By Boolean algebra, this is equivalent to cia ::; iia.

The latter follows from the axioms cia::; ia (Si) and ia ::; iia (4i) by transitivity.

c[if;] = [Oif;],

If f-s if; then if; is

Corollary 10.3.2 For each of the logics S listed above, LA(S) is afree S-algebra. D

10.4 Exercise 10.2.11 Fill in the missing steps in the proof above. Proposition 10.2.12 In any modal algebra the following hold: i(x V y) ::; ix V cy; ix 1\ cy ::; c(x 1\ y). Proof We prove only the first, the second having a dual proof: i(x V y) = i( -y :J x) ::; i-y:Jix=-i-yVix=cyVix=ixVcy. D

The Kripke Semantics for Modal Logic

A good topic for gossip is "Who first invented the 'Kripke semantics' for modal logic?" We have already contributed enough of a gossipy nature with our discussion of the anticipations in the J onsson-Tarski representations of Boolean algebras with operators. Thus, here we go on to record as fully convinced that the Kripke semantics really should be called the Kripke semantics by vrrtue of the philosophically vivid and mathematically developed form in which Kripke first expressed these ideas. It rightly caught the imagination. The reader might especially want to consult Kripke (19S9, 1963a, 1963b, 1965) for first-hand knowledge.

For us, a (Kripke) frame shall be a structure (U, R), with U a non-empty set and R a binary relation on U. The elements of U are thought of as "possible worlds" and the relation R is thought of as "relative possibility." The reader should be warned that for reasons having to do with the Jonsson-Tarski representation using image-forming operators, we shall read aR/3 as "a is possible relative to /3," whereas Kripke (and most everyone else, following Kripke) would read it the other way around. This is merely a matter of convention. The reader should also be cautioned that the vivid terminology need not be taken too seriously. One beauty of the Klipke idea is that frames may be variously interpreted. Thus the elements of U may be thought of as moments in time with R the relation of temporal priority, to give just one important example (which is central to "tense logic"). This abstractness is nicely captured by (following Montague) calling the elements of U "reference points" or "indices," perhaps while also calling R by the neutral name "accessibility relation." But for the most part we shall not shrink from the words "possible world" (although in dealing later with the Kripke semantics for intuitionistic logic, we shall follow Kripke in regarding the elements of U as "evidential situations"). Subsets of U are quite naturally thought of as "propositions" (sometimes called "UCLA propositions") on the idea of "identifying" a proposition with the set of worlds in which it is true. In the end this may be a bad thing to do (since, for one thing, all contradictory propositions tum out to be the empty set, and hence identical-cf. Dunn 1976). But there seems to be not too much wrong with it as a heuristic in the context of modal logic, since whether a proposition is necessary or possible depends not upon any "fine-grained content" of the proposition, but merely upon which possible worlds have it true. (It would be a terrible heuristic in the context of relevance logic-again cf. Dunn (1976, 1986).) Accordingly, it makes good sense to view a modelfor (an interpretation of) a modal sentential language as a triple (U, R, I), (taking "', /\, 0 as the primitive connectives): (1) for each sentential variable p, I(p) ~ U; (2) 1(",rjJ) = U - l(rjJ);

(3) l(rjJ /\ lfI) = l(rjJ) (4) l(OrjJ)

n l(lfI);

= R*I(rjJ),

where R* X = {y : 3x E X such that y Rx }. Alternatively, and more standardly, a (world relativized) valuation function v can be taken instead of the interpretation function l. Then, v is a function from pairs (rjJ, a), where rjJ is a sentence and a E U, to truth values 2 = {O, I}. So then (1') v(p, a) E 2;

= 1 ¢;> v( rjJ, a) = 0; v(rjJ /\ lfI, a) = 1 ¢;> v(rjJ, a) = 1 and V(lfI, a) = 1; v(OrjJ, a) = 1 ¢;> 3x E U(xRa and v(rjJ, x) = 1).

(2') v(",rjJ, a) (3') (4')

COMPLETENESS

MODAL LOGIC AND CLOSURE ALGEBRAS

362

(Once again, note that we write xRa, where Kripke uses aRx.) On the first alternative, a sentence rjJ is said to be valid in a Kripke frame (U, R) iff for all interpretations I on (U, R) (in the first sense, of course), I( rjJ) = U. On the second

363

alternative, rjJ is valid in (U, R) iff for all valuations v on (U, R) (in the second sense) and for all x E U, v( rjJ, x) = 1. On either alternative, then, a sentence is said to be valid in a class of Kripke frames iff it is valid on each member of the class. The reader should convince himself that these two alternatives really boil down to the same ideas. Which alternative one prefers depends upon whether one wants to write "a E I( rjJ)" or "v( rjJ, a) = 1," whether one wants to work with a set or its characteristic function. In the present context we have a reason for wanting to work with sets, to wit, it is more "algebraic," i.e., by working with l(rjJ) ~ U, we are evaluating rjJ in the Boolean algebra which is the power set of U. Indeed by the material on Jonsson-Tarski, we are evaluating rjJ in a corresponding Boolean algebra with an operator (\(.I(U), n, -, R*). R* being additive and normal, this is in fact a K-algebra. In the sequel we shall examine various classes of frames appropriate to the various modal logics. (The class of K-frames will simply be the class of all frames.) The frames are characterized via properties of the accessibility relation as follows. T-frames: R reflexive. B-frames: R reflexive, symmetric. S4-frames: R reflexive, transitive. S5-frames: R reflexive, symmetric, transitive (= an equivalence relation). 10.5

Completeness

Theorem 10.5.1 (Weak completeness for Kripke semantics) Let S be anyone of the modal logics K, T, B, S4, or S5. Then I-s rjJ iff rjJ is valid in the class of S-frames. Proof. We work through the case of K by way of example. We leave it to the reader to work through the other cases. If not I- rjJ, then in LA(K), [rjJ] I- 1. But since LA(K) is a K-algebra, and hence a Boolean algebra with an operator, we know by the Jonsson-Tarski representation that each element of LA(K) can be identified with a set of "worlds" in some underlying universe U. It is then easy to see that l(rjJ)

= h[rjJ]

is an interpretation where I( rjJ) I- U. The proof for the other cases is analogous, except it must be verified that the way that R is defined in the proof of the J onsson-Tarski theorem gives rise to the appropriate properties in each case (reflexivity for T, etc.). D We can actually prove a stronger theorem equating deducibility and validity, but we have to be careful about how we define deducibility (r I-s rjJ). In constructing a derivation we must not apply the rule of necessitation to any step which has not been justified as being a theorem of S, for otherwise we get rjJ I-s DrjJ, which we know to be invalid. We must also define r !=s rjJ, which means for all S-frames (U, R), and for all a E U, if for all y E r, v(y, a) = 1, then v(y, rjJ) = 1.

TOPOLOGICAL REPRESENTATION

MODAL LOGIC AND CLOSURE ALGEBRAS

364

365

Theorem 10.5.2 (Strong completeness for Kripke semantics) Let S be anyone of the modal logics K, T, B, S4, or S5. Then r I-s rjJ ijjT Fs cp.

(11) I(X) = X; (12) I(Y) ~ Y; (13) I(I(Y)) = I(Y);

Exercise 10.5.3 Prove the above theorem.

(14) I(Y

10.6

We shall say that Y is a closed set iff Y = CCY), and that Y is an open set iff Y = I(Y). If Y is simultaneously closed and open, we say it is clopen. The following

Topological Representation of Closnre Algebras

In this section we shall show how every closure algebra (S4-algebra) can be represented so that the closure operator is represented as a closure operator on a topological space. McKinsey and Tarski (1944) were the first to make this connection, though we think our own proof here, using the Kripke semantics, is more intuitive for present day readers. We first review some relevant topological notions. A quasi-metric on a set X is a function d defined on the Cartesian product X x X taking non-negative real numbers as values, satisfying (MI) dCa, a) = 0, (M2) dCa, y) ::; dCa, p) + d([3, y)

(Triangle inequality).

A pseudo-metric additionally satisfies

(M3) dCa, [3)

= d([3, a)

(Symmetry).

A full-fledged metric satisfies also

(M4) if dCa, [3)

= 0, then a = [3.

(M4) may be regarded as a converse of (MI), since (MI) says that if a = [3, then dCa, [3) = o. By a (quasi- or pseudo-) metric space we mean an ordered pair (X, d), where X is a non-empty set and d is a (quasi- or pseudo-) metric on X. By a two-valued (quasi- or pseudo-) metric we mean one where dCa, [3) = 0 or 1. We shall also call this a discrete metric. Observe that there is only one two-valued metric on a set X, although there may be many two-valued quasi- or pseudo-metrics on X. Informally, dCa, [3) is to be thought of as the distance from a to [3. By a topological space we mean an ordered pair (X, C) where X is a non-empty set and C is a unary operation on the power set of X satisfying the closure axioms due to Kuratowski (1958): (Cl) (C2) (C3) (C4)

C(0) = 0; Y ~ C(Y); C(CCY)) = CCY);

CCY U Z) = CCY) U CCZ). We read CCY) as "the closure of Y," and think of it as the result of adding to Y all the points in X that are arbitrarily close to points of Y. There is a notion dual to a closure operator, called an interior operator. I(Y) may be defined as X - CCX - Y). We might have taken I as a primitive notion, defining C(Y) = X - I(X - Y), and characterizing I by the following interior axioms, the duals of (Cl)-(C4):

n Z) = I(Y) n I(Z).

exercises should be worked at seriously by anyone previously unfamiliar with topology, as they contain well-known facts of topology that we shall appeal to in the sequel. Exercise 10.6.1 Verify that Y is closed iff its complement (relative to X) is open. Hence Y is open iff its complement is closed. Exercise 10.6.2 Verify that the collection of clopen subsets of X is a field of sets. Exercise 10.6.3 Verify that the closed sets of a topological space are closed under finite unions and arbitrary intersections, and hence dually for the open sets. Exercise 10.6.4 The fact in Exercise 10.6.3 is the basis for an alternative way of characterizing a topological space. Let be a topology on S iff is a collection of subsets of X closed under finite intersections and arbitrary unions. We call the members of the open sets of the topology. (There is obviously a dual approach aimed toward the closed sets.) Define I(Y) to be the largest open set included in Y, i.e., as the union of all so there is at least one member members of that are included in Y. (Note that 0 E of included in Y.) Verify that I satisfies the interior axioms.

r

r

r

r

r

r,

Exercise 10.6.5 By the discrete topology on X is meant the set of all subsets of X. Show that the discrete topology on X is determined by the discrete (i.e., two-valued) metric on X. Given a quasi-metric d on a set X, we can define the closure operator C determined by d:

CCY)

= {a EX:

for every r E jR+, there exists [3 E Y such that dCa, [3) < r}.

We now verify that C so defined satisfies (CI) through (C4). (Note that (MI) is used for (C2), (M2) for (C3), and that (Cl) and (C4) come more or less for free.) (CI) Obvious, since there exists no [3 E 0. (C2) Obvious, since if a E X, then, since d(a,a) = 0 (MI), for every positive real number r there exists a [3, namely a itself, such that dCa, [3) < r. (C3) That CCY) ~ CCCCY)) is actually a special case of (C2), so it needs only to be argued that CCCCY)) ~ CCY). Suppose a E CCCCY)) and yet a (j. CCY). Then by the second there is some positive real number r so that for all [3 E Y, dCa, [3) ~ r. But by the first it must be the case that for some y E CCY), dCa, y) < ~. Further, for any such y E CCY), it must be that there is some [3 E Y so dey, [3) < ~. But by the triangle inequality (M2), dCa, [3) ::; dCa, y) + dey, [3) < ~ + ~ = r. So dCa, [3) < r, but remember r was chosen so that dCa, [3) ~ r for all [3 E Y. Contradiction!

366

MODAL LOGIC AND CLOSURE ALGEBRAS

THE ABSOLUTE SEMANTICS FOR SS

(C4) It is reasonably straightforward that C(Y) ~ C(YU Z) and similarly that C(Z) ~ C(Y U Z). Thus, for example, if a E C(Y) then for each positive r there must be some fJ E Y so dCa, fJ) < r. This same fJ is in Y U Z and so will serve for the same choice of r to show a E C(Y U Z). We argue the more interesting C(Y U Z) ~ C(Y) U C(Z). Suppose a E C(Y U Z) and yet a ~ C(Y) U C(Z). Since a ~ C(Y) there is some positive real r so that for all fJ E Y, dCa, fJ) ~ r. Similarly, there is some positive real s so that for all y E Z, dCa, y) ~ s. Let t = miner, s). Then obviously for all u E Y U Z, dCa, u) ~ t. This contradicts our supposition that a E C(Y U Z). The reason for our special interest in two-valued quasi-metrics stems from our use of quasi-ordered sets (U, <) in representing closure algebras. Define a function don U x U so dCa, fJ) = 0 if a < fJ, and otherwise dCa, fJ) = 1. This is the characteristic function for <. Note that a < a iff dCa, a) = O. Also transitivity of < just amounts to the triangle inequality. Thus if a < fJ and fJ < y, then dCa, y) ~ dCa, fJ) + d(fJ, y) = 0 + 0 = 0, so a < y. Conversely, if dCa, fJ) = lor d(fJ, y) = 1, then clearly dCa, y) ~ dCa, fJ)+d(fJ, y). So suppose dCa, fJ) = d(fJ, y) = 0, i.e., a < fJ and fJ < y. But then by transitivity a < y, i.e., dCa, y) = O. So again dCa, y) ~ dCa, fJ) + d(fJ, r). Thus we can freely interchange the notions of a quasi-ordered set (U, <) and a quasi-metric space (U, d). We shall exploit this fact so as to give a topological representation of closure algebras. Definition 10.6.6 Given a topological space (X, C), we call the structure (\9(X), n, u, -, C) (where - is complement relative to X) the full closure algebra on the space. By the J6nsson-Tarski representation we know that a closure algebra can be represented using a frame (U, R), where U is a non-empty set and R is a binary relation on U. In the canonical frame the elements of U are prime filters and aRfJ is defined as \ta(if a E a then Oa E fJ). It is easy to see in the case of a closure algebra that R so defined is a quasi-ordering. Thus, as the reader can easily verify, reflexivity follows from a ~ Oa, and transitivity follows from OOa ~ Oa. Thus the J6nsson-Tarski theorem provides the following.

Exercise 10.6.9 Show that every S5-algebra can be represented on a pseudo-metric space.

10.7

The Absolute Semantics for S5

Completeness of S5 is relative to a Kripke semantics with an accessibility relation that is an equivalence relation. Let us call this the "relative" semantics because it uses the notion of one world being possible relative to another, and defines necessity in terms of being true at all worlds that are relatively possible. There is a simpler semantics, as established in Kripke (1959), which we shall call the "absolute" semantics. This defines necessity as truth in all worlds period. Of course, what is meant here is in all the worlds of the frame. Thus a relative frame is of the form (U, whereas an absolute frame is just a set U. It is easy to see how an absolute frame can be turned into an equivalent relative frame. Just let the equivalence relation be the universal relation on U. Going the other way requires a bit more subtlety. Suppose ¢ is refuted in a relative frame (U, ==). Then for some w E U, w ¢ [¢]. Consider the equivalence class [w] of worlds w' such that w == w'. An easy induction shows that in evaluating ¢ at w we never need look outside of [w]. The only thing that forces us to look at worlds other than w is when we evaluate necessity (or possibility-but let us assume it has been defined in terms of necessity using negation). The key then is that in evaluating a subsentence DlJf of ¢ we need examine the truth lJf only at worlds w' such that w == w'. SO it metaphorically is as if the partition into the equivalence classes divides the worlds up into isolated universes that have no effect on each other. 10.8

Henle Matrices

Let us abstract out the algebraic structure of the absolute semantics. A Henle algebra is a structure (B, /\, v, -, i), where (B, /\, v, -) is a Boolean algebra with top element 1, and i ("necessity") is defined as follows: 1

=1 ia = 0

ia

Corollary 10.6.7 Every closure algebra can be represented as afield of sets with the closure operator interpreted as an image operator with respect to a quasi-ordering <:

CX

= {fJ : 3a E X, a < fJ}.

367

if a

= 1,

if a =f. l.

A Henle matrix is a Henle algebra with 1 the only designated element. We shall not always bother to distinguish these.

PIVof From Theorem 8.12.1 (cf. also Section 10.2 above to see how closure algebras fall under the J 6nsson-Tarski conditions). D This corollary can be reinterpreted given the discussion above about the relation between quasi-orderings and quasi-metrics. Theorem 10.6.8 Every closure algebra can be represented as a closure subalgebra of the full closure algebra on a topological space, indeed a quasi-metric space.

Proof The only trick is to switch the relation < with the corresponding quasi-metric d. D

Exercise 10.8.1 (Soundness) Show that every Henle algebra is an S5-algebra. Hence each theorem ¢ of S5 is valid in every Henle matrix. A good way to think of Henle matrices is in terms of the (absolute) Kripke semantics for S5, where each proposition is a set of possible worlds. The definition then says, in a two-valued on/off way, that a proposition is necessary if it is true in all possible worlds, and is not necessary otherwise. J Of course,

0 can be defined as Oa = -0 - a, in which case Oa = 1 if a

i' 0, and Oa =

°

if a = 0.

368

ALTERNATION PROPERTY FOR 84

MODAL LOGIC AND CLOSURE ALGEBRAS

The above definitions of "Henle algebra" and "Henle matrix" are all "up to isomorphism." Because every Boolean algebra is isomorphic to a subdirect product of 2, we can always let the Boolean part of a Henle matrix be some product of 2. We shall be particularly interested in the finite Henle matrices, and so (given Proposition 8.11.16) each can be nicely presented as a Boolean algebra of the form 2/l (where n is a positive integer), with the additional operator i. We shall denote this concrete Henle matrix as Hn. For convenience, we include the case of the one element Henle matrix as Ho. The following is an immediate consequence of the similar fact about Boolean algebras (Proposition 8.11.16): Proposition 10.8.2 EVelY finite Henle matrix is isomorphic to some Hn. Lemma 10.8.3 Let A be an S5-algebra and let a t= b. Then there exists a Henle matrix H and a homomorphism h of A onto H such that /1(a) t= h(b).

i

Proof Suppose without loss of generality that a t= b because a b (the argument is symmetric). We know from the prime filter separation principle (Lemma 8.6.2), together with the fact that in a Boolean algebra, prime filters are just maximal filters (Theorem 8.9.1), that there exists a maximal filter M with a E M and b ¢ M. We define a congruence on A as follows: a ~ b iff a

>< b E M.

Note that we use strict equivalence instead of material equivalence because the former supports replacement in necessity contexts (a ~ b implies Da ~ Db) because of Proposition 10.2.9. We now consider the quotient algebra A' = A/~. This is a homomorphic image of A under the canonical homomorphism (see Section 2.6), and it is easy to argue that a 'f!. b, and so [a] t= [b]. Otherwise a >< b E M. It is easy to see that a >< b ~ a :J b, and so a :J b E M. Since we have also that a E M, we conclude that (a :J b) /\ a E M. But (a :J b) /\ a ~ b, and so bE M, contrary to our selection of M as separating a and b. Since S5-algebras are equationally definable, we know from Theorem 2.10.3 that they are also S5-varieties. So we know that A' is an S5-algebra. But is it a Henle matrix? We need to prove for x E A/~ that ix = 1 if x = 1. Proposition 10.2.7 says that this holds of any modal algebra. We also need to prove that ix = 0 if x t= 1. This does not hold of all modal algebras and in fact is the distinguishing characteristic of a Henle matrix. From Proposition 10.2.10 and the fact that 1 EM, it follows that (ia>< 1) V (ia >< 0) EM.

Since M is prime, this means that (ia >< 1)

E

M or (ia >< 0)

E

M, i.e., ia ~ 1 or ia ~ O.

This means that for x E A/~, ix = 1 or ix = O. Now suppose that x t= 1. Then by Proposition 10.2.8, ix t= 1. Thus, ix = 0 is the only alternative remaining. D

369

Theorem 10.8.4 EvelY S5-algebra is isomorphic to a subdirect product of Henle algebras. Proof This is immediate from Lemma 10.8.3 and Theorem 2.8.12.

D

Remark 10.8.5 There are at least two other methods of proving the above theorem. The first is to show that only Henle matrices are irreducible and then apply Birkhoff's prime factorization theorem. If A is irreducible and not a Henle matrix, there is an element of the form ca t= 0,1. We define congruences ';::jfl and ';::jv just as for Lemma 8.11.12, but putting ca in place of a. The only thing then that needs doing is to show that these respect the modal operator, say c. If x ';::jfl y, then x /\ ea = y /\ ea & -x /\ ea = -y /\ ca. From the first we can conclude e(x /\ ea) = e(y /\ ea), and from this we can obtain ex /\ ea = ey /\ ea, using Proposition 10.2.13. This is just half of showing that ex ';::jfl ey. We must also show that -ex /\ ea = -ey /\ ca. We have that -x /\ ea = -y /\ ea, and so i( -x /\ ea) = i( -y /\ ea). But again using Proposition 10.2.13 we obtain i-x /\ ea = i - Y /\ ea, and hence -ex /\ ea = -ey /\ ea as needed. All that really remains to be shown is that ';::jv respects e, where the remaining two modal distribution principles from Proposition 10.2.13 play their role. The rest of the proof is as for Lemma 8.6.6. Another way to prove the embedding theorem is given implicitly by the following exercise. Exercise 10.8.6 We know from Exercise 10.6.8 (cf. also Section 10.7) that every S5algebra A is representable using a frame (U, ==), where == is an equivalence relation on U. We also know that the U is partitioned by == into a number of disjoint equiValence classes {[ w] : w E U}. Look at each power set \p([ w]) as a field of sets. Show that it becomes a Henle matrix, given the definition for X ~ [w], C(X) = {P : 3a E X, a == P}.1t is obvious that if X is empty then C(X) = 0. Show that when X is not empty, then C(X) = [w]. Go on to show that the S5-algebra A is isomorphic to a subdirect product of these Henle matrices. (This proof is somewhat analogous to showing implication 3 in Exercise 8.11.11 (Figure 8.15), and also has antecedents in the informal proof given above of the relative and absolute semantics for S5.) 10.9

Alternation Property for S4 and Compactness

A topological space (U', C') is a subspace of a topological space (U, C) iff (1) U' ~ U and (2) C' is just C restIicted to U'. (The latter is equivalent to the more usual requirement that each open set 0' of the first topology is the intersection 0 n U' for some open set 0 of the second topology.) A topological space U is compact iff for every indexed family of open subsets {Oi} iEi, wh~n~ver ~iEi Oi = U then ~here exists some finite J ~ I such that UjEJ OJ = U. ThIS IS easIly seen to be eqUIvalent to the dual condition that for every indexed family of closed subsets {C} iEi, if for every finite J ~ I, njEJ Cj t= 0, then niEi C:i t= 0. The antecedent of this conditional, i.e., that n'EJ Cj is non-empty for all fimte J, is often called the "finite intersection property," ~nd the consequent, i.e., that niEi C is non-empty is often called the "intersection property." Thus compactness

MODAL LOGIC AND CLOSURE ALGEBRAS

370

ALGEBRAIC DECISION PROCEDURES

371

may be neatly stated as requiring that every family of closed sets which has the finite intersection property also has the intersection property. We shall be interested in a somewhat stronger property, to wit, a topological space U is strongly compact iff for every indexed family of open subsets {Oi} iEI, whenever UiEIOi = U, then::li E I such that Oi = U. This is again easily seen to be equivalent to the dual condition that for every indexed family of closed subsets {Ci } iEI, if for every i E I, C =f. 0, then niEI Ci =f. 0. We want now to prove the one-point compactification theorem.

algebra (where [pJJ ... [Pk] are the generators, and PI, ... , Pk are all the subformulas of lfI). This finitely generated Boolean algebra B is finite (because of normal forms) but may not be closed under c[p] = [Op]. We define a new operation c' on B' that agrees with the old c when it exists in B' (all the theorems are about this). Thus we can now falsify lfI in a finite K-algebra. Indeed I-K lfI ifflfl is valid in all K-algebras of size:::; 22", where 11 is the number of subformulas of lfI. For other modal logics, e.g., K, T, and S4, we can obtain similar results. The details are to be found in the following theorems.

Theorem 10.9.1 Every topological space (U, C) is a subspace of a strongly compact topological space (U+, C+) such that U+ - U is a one-element set and is closed.

Theorem 10.10.1 (McKinsey 1941) Let (B, 1\, V, -, c) be a K-algebra and let (B', 1\',

Proof Let aD be anything not in U, and let U+ = U U {aD}. Define C+(X) = C(X) U {aD} (think of aD as "close" to all points). Clearly, by the definition of C+, (U, C) is a subspace of (U+, C+). 0

Let us say that a closure algebra (B, 1\, v, -, c) is finitely compact iff whenever ia V ib = 1 then ia = 1 or ib = 1 (or dually, ca =f. 0 and cb =f. 0 imply ca 1\ cb =f. 0). This terminology is frankly invented and is the vestigial form of strong compactness that remains when one is working with closure algebras that are not necessarily complete. Thus, it is easily seen to be equivalent to the condition that for every finite indexed family of open elements {Oi} iEI, if AiEl 0i = 1, then there is an i E I such that 0i = 1. Theorem 10.9.2 Every closure algebra (B, 1\, V, -, c) is a subalgebra of afinitely compact closure algebra. Proof By the representation theorem for closure algebras we know that B is isomorphic to a topological field of sets of some topological space (U, C). By the one-point compactification theorem, we know that (U, C) is a subspace of a strongly compact topological space (U+, C+). 0

Theorem 10.9.3 (Alternation property for S4) If 1-84 Dp V DlfI, then 1-s4 Dp or 1-84 DlfI· Proof Consider the Lindenbaum algebra LA(S4). By Theorem 10.9.2 it can be embedded in a finitely compact closure algebra. Let us assume the hypothesis I- Dp V DlfI. Then i[p] V i[lfI] = 1 in LA(S4) and so h(i[p] V i[lfI]) = ih([p]) V ih([lfI]) = 1 as well, where h is the embedding. But then by the definition of finite compactness, ih([p]) = 1 or ih([lfI]) = 1, i.e., h(i[p]) = 1 or h(i[lfI]) = 1. So i[p] = 1 or i[lfI] = 1 back in 0 LA(S4), i.e., 1-s4 p or 1-84 lfI.

10.10

Algehraic Decision Procedures for Modal Logic

We shall present a proof of McKinsey's (1941) theorem that S4 is decidable, which shows that S4 has the finite model property. We shall show the same for K and T. Decidability will then follow using Harrop's theorem (Theorem 6.16.5). The rough idea is that if, for example, 11K lfI then lfI is falsifiable in a K-algebra under the canonical valuation [lfI]. The K-algebra is "almost" a finitely generated Boolean

V', -') be an infinitely distributive complete Boolean subalgebra ofB, i.e.,

(i) (B', 1\', V', -') is a complete Boolean algebra;

(ii) (B', 1\', V', -') is a subalgebra of(B, 1\, V, -) where 1\' and V' are defined for 0, bE

B' as a 1\' b = A' (a, b} and a V' b = V' (a, b} ; (iii) B' is infinitely distributive, that is, a V' A' X = X ~ B'; and (iv) c1 E B'.

A' {a v' x

: x E X} for a E B',

Then there exists a unalY operation c' on B' such that (B', 1\', V', -', c') is a K-algebra, and whenever a, ca E B', then c' 0 = ca. Proof Before we begin, we remark that the principal application of the theorem (as in Corollaries 10.10.2 and 10.10.3) is to the case when B' is finite, in which case conditions

(i)-(iii) collapse to the simple requirement that B' be a Boolean subalgebra of B. The reader may want to keep this case in mind as he goes through the proof to "fix ideas." Also, to simplify notation, we shall drop the primes from the symbols denoting the operations of the sub algebra with the exception of the "new" operation c' which we wish to focus attention on. Now to begin the proof, define for a E B', (1) c'(a)

= A{cx: ex E B' and a :::; x},

which is non-empty due to (iv). We first verify that c' a = co for a, ca E B'. Thus, suppose a, ca E B'; then since 0 :::; a, ca is itself a component in the meet that makes up c' a. So clearly c' a :::; ca. But ca :::; c' a as well, since we can argue that ca :::; any component ex in the meet displayed above. For this, suppose ex E B' and a :::; x. Then by monotonicity of c, ca :::; ex, which is what is wanted. We will now occupy ourselves with showing that B' is a K-algebra. We need to show (2) c'o = 0 and (3) c'(aVb)=c'aVc'b.

It is trivial that (2) holds given (1). Thus, cO = 0 E B', and so c'o = cO = O. The calculations for (3) are somewhat complex. We first obtain a useful form of the right-hand side of (3). Thus, (4) c' a V c' b

= A {ex: ex E B' &

a :::; x} V A {cy : cy E B' & b :::; y}.

But by infinite distribution, this equals

(5)

!\ U\ {cx : cx E B' &

a S x} V cy : cy E B' & b s y}.

And by a further infinite distribution, this equals (6) !\{!\{cxVCy:cxEB'&asx} :cyEB'&bSy}·

!\ {cx V cy : cx, cy E B' & a S x

(9) c'a V c'b

Recalling the definition of c' a as the meet of cxs satisfying a certain condition, it suffices to assume (2) cx E B' and a S x

(3)

& b s y}.

These computations are straightforward but confusing (because of the nested braces, etc.). Asserting that our principal application of the theorem will be to the case where B' is finite, we invite the reader to work through all our computations with c' a = CXj, V ... V cx n . Now going on to show (3), it obviously suffices to show both (8) c'(a V b)

S c'a V c'b and S c'(a V b).

as cx.

But (3) clearly follows from the second conjunct of (2) and x S cx (which is the Talgebra postulate (3c)). 0 Corollary 10.10.3 Let B andB' be as in Corollary 10.10.2, except that B is required to be an S4-algebra. Then there exists a unmy operation c' on B' such that (B', /\', V', -', c') is an S4-algebra and whenever a, ca E B', then c' a = ca. Proof Tn virtue of the proof of Corollary 10.10.2, we need only check that the c' as defined in the proof of the theorem satisfies the characteristic S4-postulate:

To prove (8), given the definition of c'(a V b) and (7), it suffices to show

(1) c'c'a

s c'a.

(10) !\{cz:cZEB'&avbsz}scxvcy

It obviously suffices to suppose that

upon the assumption of

(2) cx E B' and a S x

(11) cx, cy E B' & a S x & b

s

y.

to and show

This we do if we show cx V cy to be a component of the meet that constitutes the lefthand side of (10). The trick is to show that c(x V y) is such a cz. From (10) it follows that c(x V y) = cx V cy E B'. It only remains to show then that a V b S x V y, but this follows by lattice properties from (11). To prove (9), given (7) and the definition of c'(a V b), it suffices to show (12)

!\ {cx V cy : cx, cy E B' &

a S x & b s y}

s cz

upon the assumption of

S z. By moves familiar by now, it suffices to show that such a cz is in fact a component of the meet on the left-hand side of (12). The trick is note that cz = cz V cz. By (13), cz E B' and a S z and b S z, which is all that is needed to put cz in the set being meeted. 0 (13) cz E B' & a vb

Corollary 10.10.2 Let Band B' be as in the theorem except that condition (iv) is dropped and in its place it is required that B be aT-algebra. Then there exists a unary operation c' on B' such that (B', /\', V', -', c') is aT-algebra and whenever a, ca E B', then c' a = ca. Proof Condition (iv) can be dropped since it is trivially satisfied by every T-algebra that c1 = 1 E B'. Since a T-algebra is a K-algebra, we need then only check the definition of c' used in the proof of the theorem to see that it satisfies the additional postulate for a T-algebra, namely, (1)

as c' a.

373

and to show

But by "infinite association," this equals (7)

ALGEBRAIC DECISION PROCEDURES

MODAL LOGIC AND CLOSURE ALGEBRAS

372

(3) c' c' a S cx, i.e.,

(4) !\{cy: cy E B' & c'a S y} S cx.

To show (4) it obviously suffices to show that cx is a component of the left-hand meet, i.e., to find some y such that cx = cy and cy E B' and c' as y. Since ccx = cx (the (4c) condition of an S4-algebra), cx is a possible choice for y (x itself is a very poor choice for y) provided only that the other two conditions on yare met. But c(cx) = cx E B' (from the S4-condition (4c) and (2)). So it remains to show only (5) c'a

s cx.

But this is immediate from (2) and the definition of c'.

o

Remark 10.10.4 For certain modal systems the definition of c' can be varied somewhat. Thus, for example, for S4-algebras (closure algebras) one may use the definition

c~4(a)= /\'{X:XEB' and cx=x/\aSx}. This definition accords with the familiar topological intuition that the closure of a set is the intersection of all its closed supersets (x is "closed" if cx = x). For other modal systems the definition of c' must be varied somewhat. Shukla (1970) gives an ingenious variation needed for some of the weaker modal systems in the neighborhood of SI. Exercise 10.10.5 Show for an S4-algebra that cS4(a)

= c'(a).

374

85 AND PRETABULARITY

MODAL LOGIC AND CLOSURE ALGEBRAS

Exercise 10.10.6 For certain systems stronger than K the definition of e' must be varied somewhat. Thus consider the logic D = K + O(¢ V ~¢). (O(¢ V ~¢) may be replaced by D¢ :J O¢, which has been thought particularly appropriate for a deontic logic with D read as "it is obligatory that" and 0 read as "it is pemlissible that.") The condition on the accessibility relation is that every world have a world possible relative to it. (a) Show that the theorem can be proven for D-algebras (K-algebras with the added postulate that e 1 = 1) with the definition

e~(a) =

I\. {ex: ex

E B' & a:S ex}.

(b) Does e~(a) = e'(a) in aD-algebra? Exercise 10.10.7 Can some analog of the theorem be proved for B and S5? It would seem that the definition of e' must be varied. (Strangely, there seems to be no discussion of this problem in the literature.)

375

VI(OIff) = e'[Iff], the result cannot be guaranteed to be [Olff]' But it can be when Olff is a sub sentence of ¢, and this is good enough. Thus one can prove the following by an easy induction on sentences:

Lemma 10.10.9 Let VI be as defined above. Then VI(Iff) = [Iff].

if Iff

is a subsentence of ¢, then

The finite model property now follows directly. Since we have been assuming that IIH ¢, then [¢] -:j:. 1, and so v I (¢) = [¢] is undesignated. The corresponding results for T and S4 follow using Corollaries 10.10.2 and 10.10.3 in place of Theorem 10.1 0.1. D Corollary 10.10.10 The set of theorems for each of the modal logics K, T, and S4 is decidable. Proof From Theorem 10.10.8 using Harrop's Theorem (Theorem 6.16.5).

D

Theorem 10.10.8 The modal logics K, T, and S4 all have the finite model property. Proof We suppose that the logic in question is K. The proofs for T and S4 follow by obvious modifications. Suppose then that 11K ¢. Then form the Lindenbaum algebra of K, which is a K-algebra. Let Iff], ... , Iffn be all of the subsentences of ¢. Consider the Boolean algebra B' generated by [Iff]], ... , [lffn]. (For D we also need to add [O(lffl V ~Iffr)] to the generators.) We will show that B' is finite. Using De Morgan's laws and double negation, every element can be put into "meetnormal" form (Xl,] V ... V Xl, 1111) /\ ... /\ (X n ,] V ... V x n , 1111/)' where the Xi} are the generators or their complements. Drive complement signs inside meets or joins (switching meets withjoins), removing double complements as they arise, so complement signs flank generators. Then use distribution to drive meets in past joins. Because of associativity, commutativity, and idempotence this form can be defined to be unique (up to an ordering of the generators). So clearly the sublattice L' is finite, since it is easy to see that there are at most 22211 such forms. Indeed, a little fiddling so as to not allow joins in which both a generator and its complement occurs reduces this bound to 221/.2 There is not yet a closure operation defined on B'. We cannot simply use c[1ff] = [Olff], since the result might not be defined when Olff is not a subsentence of ¢ (or even a conjunction of disjunctions of negations of such). And we cannot simply expand B', closing it under e, since the result need not be finite (as it was when we did the corresponding construction with meet, join, and complement). But we do know by Theorem 10.10.1 that a new closure operation e' can be defined on L' that agrees with the original closure operation e when a and ea are both in B'. This turns out to be good enough, and (B', /\, v, -, e') will be our desired finite model. Let us consider the interpretation l(p) = [p], for each atomic sentence p which is a sub sentence of ¢, and otherwise l(p) is arbitrarily defined. This is very much like the canonical valuation in the Lindenbaum algebra except that when one comes to compute 2 Alternatively, the reader can use the fact that Boolean tenns can be identified when they define the same operations in 2, and note that there are only 22" such n-p]ace functions.

10.11

S5 and Pretabularity

The reader may have noticed that in the previous section we did not bother to show that S5 is decidable. This is because S5 has the finite model property in a very strong sense which we shall call "uniform." It is not just that each non-theorem ¢ can be refuted in a finite model, but the size of the model can be calculated in a very simple manner from the size of ¢, indeed from the number of distinct atomic sentences occurring in ¢. Theorem 10.11.1 (Uniform finite model property for S5) If there are exactly n atomic sentences in a sentence ¢, then ¢ is a theorem of S5 iff ¢ is valid in the Henle matrixHIl • Corollary 10.11.2 (Soundness and completeness for S5) A sentence ¢ is a theorem of S5 (fifor all positive integers n, ¢ is valid in Hn. We shall eventually get around to proving the theorem. The corollary clearly follows from the fact (Exercise 10.8.1) that a Henle-matrix is an S5-matrix (provided Theorem 10.11.1 holds). We will prove the even more surprising theorem (Theorem 10.11.7 below) due to Scroggs (1951) that all modal logics extending S5 have a finite characteristic matrix, namely, one of the finite Henle matrices. This, together with the fact that S5 itself has no finite charactelistic matrix (Theorem 10.11.9 below), establishes that S5 is "pretabular" in the sense that while it does not itself have a finite characteristic matrix, every modal extension of it does. Lemma 10.11.3 Let X be a modal extension of S5. is not valid in some Henle matrix H.

If ¢

is not a theorem of X then ¢

Proof We leave to the reader the verification that Henle matrices satisfy the axioms and rules of S5. Consider the Lindenbaum algebra LA(X) fonned using the equivalence relation of provable material equivalences (Iff == X iff (Iff :J X) /\ (X :J Iff) is a theorem

376

MODAL LOGIC AND CLOSURE ALGEBRAS

85 AND PRETABULARITY

of S5), or alternatively, of provable strict equivalences. This is an S5-algebra in which [p] i= 1, since P is not a theorem of X. We can now apply Lemma 10.8.3 to say that there is a Henle matrix H and a homomorphism h from LA(X) onto H so that h([ p]) i= h(l) = 1. In the interpretation z(p) = h([p]), then z(p) i= 1. D Notice that there is no reason to think that H is finite, so we work towards replacing the Henle matrix H with some finite Henle matrix Hn. The following will be of use in that task: Proposition 10.11.4 For natural numbers i and j, i

~ j

= c(h(al,

Theorem 10.11.7 (Scroggs 1951) Let X be a modal extension of S5. Some finite Henle matrix H/1 is characteristic for X in the sense that p is a theorem of X (ff P is valid in H/1'

Proof We then let I be the set of indices such that Hi validates all of the theorems of X. Either I is infinite or finite. If infinite, then because of Proposition 10.11.4 we can take I to be all of :£;+, and so (by Theorem 10.11.5) X = S5. If finite, then (assuming non-emptiness) there is a largest index k. Because of Proposition 10.11.4 and

Theorem 10.11.5, it is easy to see than Hk is the desired characteristic matlix.

iff Hi is a subalgebra of Hj.

Proof The "if" part is obvious on size considerations. For the "only if" part we show that Hn is a subalgebra of Hn+ I, and then the proposition follows by an obvious induction. We know from the similar Theorem 8.11.14 that the Boolean part of Hn is isomorphic to a sub algebra of the Boolean part of Hn+l under the mapping h(al, ... , an) = (aI, ... , all, an). It remains to show that h(c(al, ... ,all))

Remark 10.11.8 Note that in the above proof, if I extension in which all sentences are theorems.

Note that we cannot simply apply the general law of universal algebra stated in Exercise 8.11.15 because (unlike the Boolean operations) c "thinks globally" and is not computed componentwise. Rather c( a I, ... , all) = (1, ... , 1) if some component ai = 1, and c(al, ... ,all) = (0, ... ,0) if every component ai = O. The key trick is that some component of (aI, ... , all) is 1 iff some component of its "expansion" (aI, ... , an, an) is 1, and similarly every component of (aI, ... , an) is 0 iff every component of its expansion (aI, ... , an, all) is O. Note that either h(c(al, ... , an)) is all Is, or else it is all Os. But h(c(al, ... ,all)) is allIs iff c(al, ... ,an ) is allIs iff (aI, ... ,an ) contains a 1 iff (aI, ... , an, an) contains a 1 iff c( (aI, ... , an, an)) is allIs. And h(c(al, ... ,an)) is all Os iff c(al, ... , an) is alIOs iff (aI, ... , an) is alIOs iff (aI, ... , an, an) is all Os iff c(al, ... ,an,an )) is allIs. In either case, h(c(al, ... ,an )) = c(h(al, ... ,an )).) D

Theorem 10.11.5 Let X be a modal extension of S5. If P is not a theorem of X then P is not valid in some finite Henle matrix Hn (where n is the number of atomic sentences with occurrences in p).

P of X is rejectable

in some Henle matrix H. We observe that the Henle submatrix H' of H generated by the elements assigned to atomic sentences in p is finite and no larger than 2n. This is because of normal forms for Boolean algebras, together with the fact that in a Henle matrix the modal operators do not lead to new elements. Because of Proposition 10.8.2 we can take H' to be of the form 2k, with k ~ n. Because of PropositionlO.11.4 this means that 2k is a subalgebra of 2n. Since validity is preserved under subalgebra, this D means that p is rejected in 2/1. Proposition 10.11.6 Setting X to be S5 in the above theorem gives the "completeness" half of Theorem /0.11.1, the discussion of which began the present section. The "soundness" half (that each H/1 validates all of the theorems of S5) is given by Exercise /0.8.1.

= 0 then

D

X is the inconsistent

Theorem 10.11.9 S5 has no finite characteristic matrix. Before we present the proof proper we discuss some background. Using == for material equivalence, the following is a well-know tautology of classical logic:

... ,all))'

Proof We know from the previous lemma that any non-theorem

377

(p ==

lfI) V (p

== X)

V (lfl

== X)·

The reason is that with only two truth values one ends up having to always assign the same truth value to two of the three sentences p, lfI, X. This can be generalized to Henle matrices. By the finitizing equality n we mean a formula of the form: (Xl

>< X2)

>< X3) V ... V (Xl >< Xn+t) (X2 >< X3) V ... V (X2 >< X/1+I)

V (Xl

V V

The displayed formula is a bit opaque, but the idea is that n says of n + 1 variables that some two of them stand for the same proposition. By the corresponding finitizing sentence L/1 we mean a sentence in the modal language of the following form, with p >< lfI (strict equivalence) defined as necessary material equivalence (D[(-.p V lfI) A (-'lfI V p))):

(PI >< P2) V (PI >< P3) V ... V (PI >< p/1+d V (P2 >< P3) V ... V (P2 >< Pn+l) V

V

(Pn >< P/1+I) = 1.

Proof Suppose that some finite matrix M is characteristic for S5 and that M has n elements. Just on size considerations it will thus validate the finitizing sentence <1'>/1. But there is a bigger Henle matrix Hk, and in it we can assign distinct objects to distinct variables. It is easy to show that this refutes n, contradicting that M is characteristic for S5. D

378

MODAL LOGIC AND CLOSURE ALGEBRAS

Exercise 10.11.10 Show that any SS matrix with more than n elements refutes < Xl) is designated, and (2) a disjunction is designated if any disjunct is designated.) We have the following corollary, which shows that modal extensions of SS have particularly simple axiomatizations. Corollary 10.11.11 Every proper modal extension of SS can be axiomatized by adding one of the jinitizing sentences L,n to the axioms of SS. Proof Let X be a proper modal extension, by which we mean that X c SS. We know that X has a characteristic Henle matrix Hn. It is easy to see that adding the finitizing sentence L,2" to the axioms of SS (call the resulting system SSn) forces Hn to be also the characteristic matrix of SSn. Hence X = SSn. 0 It turns out that "SS can be approximated from above" by its finitary counterparts SSn (defined as in the proof just above):

Corollary 10.11.12 The extensions of SSline up as a chain asfollows: SSo :J SSl :J ... :J SSn :J SSn+1 :J ... :J SS. Proof The weak version of these inclusions (replace :J with 2) follows from Proposition 10.11.4 and that fact that submatrices preserve validity. All that remains is to show that the inclusions are proper by showing distinctness of the displayed systems. That no SSn = SS amounts to Theorem 10.11.9. The finitizing sentence L,2" is a sentence valid in SSn but not valid in SSn+ 1. 0

Corollary 10.11.13 SS

= nnEw SS/l.

Classical logic can be viewed as the limiting case of modal logic. Viewed in truthfunctional terms, 0 is simply the identity function, and can be read in English as "it is true that." If one considers the two element Boolean algebra as a Henle matrix, one obtains the same result. We shall say that a modal logic "collapses to classical logic" when DcjJ >< cjJ is a theorem. A logic is said to be Post complete if every proper normal extension of it is Post inconsistent in the sense that every sentence is a theorem. Sometimes writers call these notions absolute completeness and absolute inconsistency, and we sometimes stray into this way of talking. Note that in logics where any contradiction implies every sentence, Post consistency (absolute consistency) and ordinary consistency (not both cjJ and ..,cjJ are theorems) coincide. This last is sometimes called "negation consistency" for emphasis. Corollary 10.11.14 The only consistent and Post complete modal extension of SS collapses to classical logic. Proof The proof can be more or less read off of the approximation of SS given by Corollary 10.11.12. First note that SSl is consistent, for there is a sentence provable in SSo which is not provable in SSl. Further, SSI is Post complete, since its only extension is SSo, which can be easily seen to be the absolutely inconsistent modal logic. And

S5 AND PRETABULARITY

379

for n > 1, SSn is not Post complete, since it can always be properly extended to the 0 consistent system SSn-l. Remark 10.11.1S The above results can be given a "purely algebraic" formulation. For example, the fundamental Theorem 10.11.7 can be restated to say that if any equations are added to those axiomatizing SS-algebras, then the resulting set of equations axiomatizes some finite Henle matrix H/l. Remark 10.11.16 An amazing fact is that SS is one of exactly five pretabular modal extensions of S4, as was shown independently by Maksimova (1975), and Esakia and Meskhi (1977).

IMPLICATIVE LATTICES

381

It is worth remarking that (H5) is given in an exported form, but it can be given in an imported form

11 INTUITIONISTIC LOGIC AND HEYTING ALGEBRAS

(H5') ((cjJ -- If!) /\ (cjJ -- X)) -- (cjJ -- (If! /\ X)) at the cost of adding either the axiom (H12) cjJ -- (If! -- (cjJ /\ If!)) or the adjunction rule:

11.1

Intuitionistic Logic

We here present a Hilbert-style formalism for the sentential calculus H due to Heyting (1930). We assume an infinite set of atomic sentences, binary connectives __ , /\, V for implication, conjunction, and disjunction respectively, and a unary connective ..., for negation. The axioms consist of all sentences of the following forms: (HO) (HI) (H2) (H3)

cjJ -- cjJ; cjJ -- (If!-- cjJ); (cjJ -- (If! -- X)) -- ((cjJ -- If!) -- (cjJ -- X)); (cjJ /\ If!) -- cjJ;

(H4) (cjJ /\ If!) -- If!;

(H5) (H6) (H7) (HS) (H9) (HlO)

(ADJ) If cjJ and If!, then cjJ /\ If!. One can also replace (HS) with its imported form (HS') ((cjJ -- If!) /\ (X -- If!)) -- ((cjJ V X) -- If!). The point of this is to make the axioms more obviously give /\ and V the properties of lattice meet and join. Incidentally, from (HI) and (H2), one can prove the following principle of transitivity: (H13) (cjJ -- If!) -- ((If! -- X) -- (cjJ -- X))·

This, with (HO), shows that f-H cjJ -- If! establishes a pre-order, indeed a pre-lattice. We shall see that it is distributive.

(cjJ -- If!) -- ((cjJ -- X) -- (cjJ -- (If! /\ X))); cjJ -- (cjJ V If!);

11.2 Implicative Lattices

If! -- (cjJ V If!);

We shall call a structure (L, /\, V, :::}) an implicative lattice if (L, /\, V) is a lattice and for all a, b, x E L,

(cjJ -- If!) -- ((X -- If!) -- ((cjJ V X) -- If!)); (cjJ -- ""X) -- (X -- ...,cjJ); cjJ--(...,cjJ--lf!).

As sole rule we take modus ponens: (MP) If cjJ and cjJ -- If!, then If!. We now make a few remarks of an axiom-chopping sort. Axiom (HO) is redundant. (Show this as an exercise if you have never done so before.) Axioms (HI) and (H2) completely characterize the "pure implicational fragment" (those theorems whose only connective is --). Axioms (HI) through (H5) completely characterize the "implicationconjunction fragment" (those theorems whose connectives are only -- and /\). And axioms (HI) through (HS) characterize so-called "positive logic" (those theorems that are negation-free). We do not prove these "separation results," but we mention them for the sake of calling attention to the importance of the various fragments. It turns out that if one has a primitive constant false proposition j, one can define ...,cjJ = cjJ -- j (in the style of Johansson 1936), thus dispensing with the need for a primitive negation connective. One must, though, add the axiom scheme (Hll) j -- cjJ

to get the effect of (HlO). But (H9) follows even without this addition, and so (HI) through (H9) amount to the axioms of what Johansson called "minimal logic" (positive logic supplemented with Johansson's definition of..., but no special axioms about f).

(*) x /\ a ~ b iff x ~ a :::} b.

(The reader may want to verify, as an exercise, that this condition implies that:::} is antitonic in the first position and monotonic in the second.) The motive for the name is obvious once it is remarked that the postulate (*) from left to right amounts to the "deduction theorem," and from light to left amounts to modus ponens. Nonetheless, it should be remarked that not all "implications" discussed in the literature satisfy the deduction theorem as it is usually understood. Thus, in particular, the strict implication of C. I. Lewis (191S), the counterfactual implications of Stalnaker and Thomason (1970) and D. Lewis (1973), the Sasaki implication of quantum logic, and the relevant implication of Anderson and Belnap (1975) all reject the deduction theorem in the form (**) r,cjJf-lf!onlyifrf-cjJ--lf!,

where f- stands for ordinary deducibility. The quick reason for this rejection is that by setting paradox of implication

r

to be {If!}, one obtains the

(***) If!f-cjJ--lf!,

which is anathema for strict, counterfactual, and relevant implication. But the deduction theorem in its form (**) is central to the classical and intuitionistic systems, particularly to the intuitionistic system wherein all the pure implicational theorems may be deduced from (**) and its converse (modus ponens). Accordingly, (*) is central to the algebraic

INTUITIONISTIC LOGIC

382

HEYTING ALGEBRAS

treatment of the intuitionistic system. Note that (*) is just a special case of residuation (see, in particular, Sections 3.10, 3.16, and 3.17), which does allow for more general ways of combining premises than simply the ordinary conjunction in (*). The careful reader may have noticed that our definition of an implicative lattice does not explicitly postulate distributivity. This is because to do so would be redundant, as the following shows. Theorem 11.2.1 (Skolem-Birkhoff) Every implicative lattice is distributive.

(1) b/\as(a/\b)V(a/\c), (2) c /\ a S (a /\ b) V (a /\ c).

Corollary 11.2.4 Let (L, /\, v) be a finite distributive lattice. Then there is a unique implicative lattice (L, /\, V, =?). 11.3

Heyting Algebras

(1) ..,a

From these we obtain by "exportation," using (*), -+

but this is transparent since each component a /\ x is postulated to be less than or equal D to b. Finally (*) trivially uniquely characterizes =?

A Heyting algebra (sometimes called a Heyting lattice) is an implicative lattice (L, /\, V, =?,O) with least element 0. The Heyting complement, also called pseudo-complement, can be defined as

Proof By lattice properties, we have

(3) b S a

383

= a =? 0,

and can be easily seen to have the properties ascribed to it in Chapter 3.

[(a /\ b) V (a /\ c)],

(4) c S a -+ [(a /\ b) V (a /\ c)].

11.4

And from (3), (4) we obtain by lattice properties

Let (U, t:) be a quasi-ordered set, i.e., t: is a reflexive and transitive relation on U. By a proposition p we mean a subset of U "closed upward," i.e., for a, p E U, if a E p and a t: P, then PEp. By a full Heyting algebra of propositions we mean a structure (A, /\, V, =?, 0), where A is the set of all propositions on some partially ordered set (U, t:), /\ and V are intersection and union, p =? q = {a E U: for all P E U such that a t: P, P jt p or P E q}, and 0 is the empty set. By a Heyting algebra of propositions we mean a subalgebra of a full one. We shall call (U, t:) an evidential frame. The idea comes from Kripke (1965). An item a E U is thought of as an "evidential state," and a t: P means that the information at a is contained in that of p. The requirement that p be closed upward corresponds to the idea that if p is established by a piece of information, it is also established by any piece of information that extends it. "Established" is intended in a very strong sense then, i.e., it means "proven." Such an assumption might well be given up for some weaker sense, such as "shown to be highly probable." So-called "non-monotonic logics" might well reject this assumption.

(5) b V c S a

-+

[(a /\ b) V (a /\ c)].

But from (5), we obtain by "importation," using (*) (with commutation), (6) a /\ (b V c) S (a /\ b) V (a /\ c),

which inequality suffices for distribution (cf. Chapter 2).

D

Remark 11.2.2 The above theorem has been thought important by many writers (including Birkhoff) for showing that "quantum logic" can have no decent implication, since distribution fails. In light of our remarks above about the dangers in the nomenclature "implicative lattice" in the light of many "non-exporting" implications, this moral must be regarded as questionable. Indeed, many recent workers on quantum logic have questioned this once orthodox moral; see especially Hardegree (1975), who relates a quantum implication to the Stalnaker conditional. Theorem 11.2.3 Let (L, /\, v) be a complete lattice which is infinitely distributive, i.e., a /\ VX = V{a /\ x : x E X}. Then there is a unique operation =? on L such that (L, /\, V, =?) is an implicative lattice (with a /\ b = !\ {a, b}, a V b = V {a, b D. Proof Define a =? b = V{x : x immediate, since if x /\ a S b, then x is a =? b. From right to left is only x /\ a S (a =? b) /\ a. So it suffices to

/\ a S b}. Then (*) from left to right is all but is in fact a component of the "infinite" join which slightly harder. Thus, assume x sa=? b. Then show that

a /\ (a =? b)

S b,

i.e., modus ponens holds. By virtue of the definition of a =? b as "infinite join" and "infinite distribution" it then suffices to show that

v

{a /\ x : x /\ a S b}

s b,

Representation of Heyting Algebras using Quasi-ordered Sets

Theorem 11.4.1 A full Heyting algebra ofpropositions is a Heyting algebra. Corollary 11.4.2 A Heyting algebra ofpropositions is a Heyting algebra. Proof Despite appearances, neither the theorem nor the corollary is of the form "an unmarried male is unmarried." We chose the name "full Heyting algebra of propositions" in anticipation of the theorem, but calling it a "Heyting algebra" does not make it a Heyting algebra. Such patterns of nomenclature are ubiquitous in logic and mathematics. The corollary follows by virtue of the fact that Heyting algebras are equationally definable, and hence, by Birkhoff's varieties theorem, are closed under subalgebras. As for the theorem proper, we first need to verify that the set of propositions A is closed under all of the operations. Because of our requirement that propositions be closed upward, A need not be the power set of U and so it is not immediate. But if a E p n q and a t: P, then since a E p and a E q and since both p and q are closed upward we have PEp, q, i.e., PEp n q. The case for V is argued similarly. As for =?,

384

INTUITIONISTIC LOGIC

suppose a E p =} q and a b {3. To show that {3 E P =} q we must argue that for arbitrary y such that {3 b y, y rt p or y E q. The trick is that since a b {3 and {3 b y, it follows that a b y, and it was required for a E p =} q that for all y such that a b y, y rt p or r E q. Finally, note that the empty set is vacuously closed upward. In verifying postulates, the only thing that is not obvious is that =} is a relative pseudo-complement. Suppose p n q ~ r. We show p ~ q =} r. Let a E p. To show a E q =} r it suffices to consider arbitrary {3 such that a b {3 and show {3 rt q or {3 E r. If {3 E q, then, since p is closed upward and a E p, {3 E P as well, and so {3 E P n q ~ r, i.e., {3 E r. Conversely, suppose p ~ q =} r and a E pn q, i.e., a E p and a E q. Then, obviously a E q =} r and since a b a, by the definition of p =} q it follows that a rt q or a E r. But since a E q, a E r, and thus p n q ~ r. 0

TOPOLOGICAL REPRESENTATION OF HEYTING ALGEBRAS

385

That AnY ~ B iff Y ~ (X - A) U B may be easily verified. But Y ~ (X - A) U B implies I(Y) ~ I((X - A) U B). This follows from the more general fact that Y ~ Z implies I(Y) ~ CCZ), which fact comes rather trivially from (14). But, since Y is open, Y = I(Y), so Y ~ A =} B. Arguing the converse direction, suppose Y ~ A =} B, i.e., Y ~ I((X - A) U B). Then by (12), Y ~ (X - A) U B, and so by the "iff" that starts off this paragraph, AnY ~ B. This gives the following.

Theorem 11.5.1 A full Heyting algebra of open sets is a Heyting algebra. By a Heyting algebra of open sets we shall mean a subalgebra of a full Heyting algebra of open sets. We then have the following corollary, which follows in the same way the corollary to Theorem 11.4.1 followed.

Corollary 11.5.2 A Heyting algebra of open sets is a Heyting algebra.

Theorem 11.4.3 EvelY Heyting algebra is isomorphic to a Heyting algebra ofpropoWe are next going to provide a topological representation for Heyting algebras.

sitions. Proof Let A = (A, /\, v, =}, 0) be an arbitrary Heyting algebra. Let U be the set of prime proper filters on A, and for a, {3 E U define a b {3 iff a ~ {3. We shall embed A into the full Heyting algebra of propositions on (U, b), using the mapping h(a) = {a E U : a E a}. By Stone's representation for distributive lattices, we know h preserves

Theorem 11.5.3 Every Heyting algebra is isomorphic to a Heyting algebra of open sets. Proof Instead of giving a direct proof of this theorem, we shall obtain it as a special

/\ and V and is one-one. Also, since it is evident that the only filter containing 0 is the whole lattice, h(O) = 0. We need only argue then that h(a) is a proposition, i.e., is closed upward, and that h preserves =}. As to the first, if a E h(a), i.e., a E a, and a ~ {3 E U, then a E {3, i.e., {3 E h(a). As to h preserving =}, suppose a E h(a =} b), i.e., a =} b E a E U, and suppose ab{3 E U. Ifwe can show {3 rt h(a) or {3 E h(b), i.e., art {3 or bE {3, we will have shown a E h(a) =} h(b). Since a b {3 means a ~ {3, we have a =} b E {3. So if a E {3, then a /\ (a =} b) E {3 and, since a /\ (a =} b) ~ b, b E {3. Arguing the other direction, we assume contrapositively that a rt h(a =} b), i.e., a E U but a =} b rt a. We argue that the latter means b rt [x, a). Recall that if b E [a, a), then 3c E a so c /\ a ~ b. But then c ~ a =} b and so a=} bE a. Since b rt [a, a), [a, a) may be extended to a prime filter {3 with b rt {3. So since a b {3 and a E {3 yet b rt {3, it follows that art h(a) =} h(b). 0

case of the representation in terms of "propositions" given in Theorem 11.4.3, exploiting the connection we have discovered between quasi-ordered sets and quasi-metrics in the previous chapter. The idea is to take some arbitrary quasi-ordered set (K, b) and then to consider the topological space (K, C) determined by the quasi-metric d which is the characteristic function of Con K x K. We shall show that the full Heyting algebra of propositions on (K, b) is the same as the full Heyting algebra of open sets of (K, C). Before proceeding we observe that for Y ~ K, I(Y) = K - CCK - Y). So for x E K,x E I(Y) iff x ~ CCK-Y)iff3r E JR+,Vy E K-Y,d(x,y) ~ r.But since d is two-valued, this is true iff Vy E K - Y, d(x, y) = 1, i.e. (since d is the characteristic function for b), iff Vy E K(y rt Y =} x g y), i.e. (by contraposition), iff Vy E K(x bY=} Y E Y). The latter gives us a workable characterization of the members x of I(Y). We first argue that the "propositions," i.e., the subsets of K that are closed upward, are precisely the open sets. Recall that a subset Y of K is a proposition iff

11.5

(1) Vx, y E K(x E Y & x bY=} Y E Y),

Topological Representation of Heyting Algebras

Given a topological space (X, C), we can construct a Heyting algebra which we shall call the full Heyting algebra of open sets of the space. Let it be (0, n, u, =}, 0) where 0 is the set of all open sets of X and where for A, B E 0, A =} B = I((X - A) U B). That o is closed under nand U follows from the well-known topological fact asked for in Exercise 10.6.3. That 0 is closed under =} follows from the clause (I3) of Section 10.6 together with our definition of an open set as one that equals its own interior. That 0 is open follows also from that definition, but using (12) (from the same section). That (0, n, U) is a distributive lattice with least element 0 is obvious. We do need, however, to verify that =} satisfies the residual law: AnY

~

B iff Y

~

A =} B.

and a subset Y of K is open iff (2) Y ~ I(Y).

But, by our characterization of I at the end of the last paragraph, (2) is equivalent to (3) Vx(x E Y =} Vy E K(x bY=} Y E Y)).

But (1) and (3) are essentially just stylistic variants of one another and are trivially equivalent. Since /\, V, and 0 are n, u, and 0 in both Heyting algebras of propositions and Heyting algebras of open sets, it remains only to argue that p =} q = I(K - p) U q, where =} is the implication operation defined on propositions. By definition, x E p =} q iff

INTUITIONISTIC LOGIC

386

ALTERNATION PROPERTY FOR H

Vy E K, x !: y implies x ¢ p or x E q. But by our characterization ofl, x E I(K - p).u q iff Vy E K, x !: y implies x E (K - p) U q, i.e., x ¢ p or x E q. So x E P ~ q iff

x E I((K - p) U q).

D

11.6 Embedding Heyting Algebras into Closure Algebras We have already established that a Heyting algebra of open sets is indeed a Heyting algebra in Theorem 11.5.1 and its Corollary 11.5.2. This result can be cast a little more generally. Given a closure algebra B, define an element a of B to be open iff ia = a (where i is defined in terms of the given closure operator by ia = -c-a). For a given closure algebra, we define a Heyting algebra of open elements as an abstract version of the notion of a Heyting algebra of open sets. Thus, where (B, II, v, -, c) is a closure algebra, the Heyting algebra of open elements of B is a structure (A, II, V, ~,O) where A is the set of open elements of B, II and V are the corresponding operations of the closure algebra restricted to A, a ~ b = i( -a V b), and 0 is the least element of B. We leave it as an exercise for the reader to verify that A is closed under the operations and that ~ is indeed relative pseudo-complement (the proof being precisely analogous to the corresponding proof for Heyting algebras of open sets).

Theorem 11.6.1 The Heyting algebra of open elements of a closure algebra is a Heyting algebra. We can also recast our Theorem 11.5.3 that represented Heyting algebras as Heyting algebras of open sets more abstractly as follows.

Theorem 11.6.2 Every Heyting algebra is isomO/phic to a Heyting algebra of open elements in some closure algebra. Notice that no proof is needed since (unlike the situation with Theorem 11.6.1) Theorem 11.6.2 is actually a weakening of the Oliginal, more concrete theorem.

11.7 Translation of H into S4 McKinsey and Tarski (1948) demonstrated that in a certain sense the intuitionist sentential calculus H may be translated into the modal logic S4. We define the translation * inductively:

= Dp; (-.cjJ)* = D-cjJ*; (cjJ IIIJI)* = cjJ* IIIJI*; (cjJ V IJI)* = cjJ* V IJI*; (cjJ -> IJI)* = D(cjJ* J IJI*)·

(1) p*

(2) (3) (4) (5)

Lemma 11.7.2 Let (B,II,V,-,O) be a closure algebra and let (A,II,V,~,O) be the Heyting algebra of open elements of B. Let / be an interpretation of S4-formulas in B and let t' be an interpretation of H-formulas in A which is such that for all atomic sentences pJ(p) = /(p). Thenfor all H-formulas cjJJ(cjJ) = /(cjJ*). Proof Basically the reader should be able to gestalt the lemma by reason of the fact that

definitions of the operations in a Heyting algebra of open elements and the "definitions" of the connectives of H given by the translation * parallel one another, but technically we need an induction on the length of the formula cjJ. For the base case, where cjJ is an elementary sentence p, we note that /(p) is an open element of B, hence t'(p) = i/'(p) = i/(p) = t(p*). We leave the tlivial cases when cjJ is a conjunction or disjunction to the reader and jump to the case when cjJ is IJI -> x. t'(cjJ -> X) = l(lJI) ~ lex), and, by definition of ~, this equals i(/(IJI) J z'(X)). Then, by inductive hypothesis i(/(IJI*) J z(X*)), and further /(D(IJI* J X*)) = z((1JI -> X)*). The case of negation is handled similarly, but is complicated slightly by the fact that we did not take the pseudo-complement operation -. as primitive in Heyting algebras but instead defined -.a = a ~ O. We argue that z'(-.cjJ) = -.z'(cjJ) = z'(cjJ) ~ 0 = i(/(cjJ) J 0) = i(-z(cjJ*)) = iz(-cjJ*) = /(D-cjJ*) = z((-.cjJ)*). D Now we tum to the proof of the preceding theorem. Proof From left to right is rather trivial. It may be proven relatively mechanically by induction on the length of proof of cjJ in H, producing for each axiom cjJ of H a proof of cjJ* in S4, and observing that if cjJ*, D(cjJ* J X*) are theorems of S4, then so is X*. Alternatively, we can take our Theorem 11.6.1 as establishing the faithfulness of the translation from left to right, for it really just amounts to a statement of that faithfulness in algebraic language. Thus, if f-H cjJ then by the generalized soundness theorem for H we have that cjJ is valid in all Heyting algebras. Hence, by Theorem 11.6.1, cjJ is valid particularly in the class of all Heyting algebras of open elements in closure algebras. Hence, by Lemma 11.7.2, cjJ* is valid in all closure algebras and thus, by the generalized completeness theorem for S4, we have f-S4 cjJ*. From right to left we proceed contrapositively, showing that if not f-H cjJ then not f-S4 cjJ. Supposing not f-H cjJ, we have by the general completeness theorem for H that there is a Heyting algebra (A, II, V,~, 0) and an interpretation z ofH sentences in A such that z(cjJ) =/: 1. By our Theorem 11.6.2 we know that A is isomorphic to a Heyting algebra of open elements in some closure algebra. Let the closure algebra be (B, II, v, -, c) and let the Heyting algebra of open elements be (B', II, V,~, 0). It is clear that cjJ can be rejected in an isomorphic image of B just as well as in B, so there is an interpretation I' in B' so that t' (cjJ) =/: 1. We now define an interpretation /' of S4 sentences in a so that zl/(cjJ*) =/: 1. Set II/(p) = z'(p) for each elementary sentence p. We know by the previous lemma that /(cjJ) = t"(cjJ*). Thus since /(cjJ) =/: 1, t"(cjJ*) =/: 1. Hence, by the general D soundness theorem for S4, not f-S4 cjJ.

Theorem 11.7.1 For each sentence cjJ of H, f-H cjJ Uff-s4 cjJ*. Before we prove the theorem we state the following, which may be established by a straightforward induction on formulas.

387

11.8 Alternation Property for H Theorem 11.8.1 Iff-H cjJ

V

IJI, then f-H cjJ or f-H IJI.

INTUITIONISTIC LOGIC

388

ALGEBRAIC DECISION PROCEDURES

Proof We could give a direct proof analogous to the proof given of the alternation property for S4, but instead we shall actually use the alternation property for S4 by way of the GOdel-McKinsey-Tarski translation of H into S4. We shall argue as follows. Suppose I-H ¢ V lfI. Then, by the translation, I-S4 ¢* V lfI*. But then, by the lemma that we will prove below, I-S4 O¢* V olfl*. Then, by the alternation property for S4, 1-s4 O¢* or I-S4 Olfl*. So by the translation, I-H ¢ or I-H lfI, as desired. 0

So we only need the following lemma. Lemma 11.8.2 Let ¢* be the Godel-McKinsey-Tarski translation of a sentence ¢ of H. Then 1-s4 ¢* +-+ o¢*. Proof We induct on the complexity of ¢. (1) Base case. ¢

= p, where p is a sentential variable. Then p* = op, and obviously

I-S4 0 p +-+ DO P by the Axiom of Necessity and the characteristic S4 axiom. (2) ¢ = "'lfI. Then ¢* = 0 -lfI, and the reasoning proceeds as in the base case. (3) ¢ = lfI A X. Then (lfl A X)* = lfI* A X*· By inductive hypothesis, I-S4 lfI* +-+ olfl* and I-S4 x* +-+ 0 X*. So by the replacement theorem for S4, 1-s4 (lfI* A X*) +-+ (Olfl* A 0 X*). But since it is a well-known fact that 1-s4 O(a A p) +-+ (Oa A OfJ) (distribution of necessity over conjunction), we have 1-s4 (lfI* A X*) +-+ O(lfI* A X*) as desired. (4) ¢ = lfI V X. The proof proceeds as in case (3), but, of course, cannot appeal to simple distribution of necessity over disjunction, since that is a well-known modal fallacy. Instead the trick is to proceed from the step I-S4 lfI* V X* +-+ olfl* V 0 X* by the S4 axiom (with Axiom of Necessity) to 1-s4 lfI* V X* +-+ OOlfl* V ooX*. Then use I-S4 OOa V OOfJ +-+ O(Oa V OfJ) to obtain I-S4 lfI* V X* +-+ O(Olfl* V 0X*). Inductive hypothesis then strips the inner necessity signs off, giving I-S4 lfI* V X* +-+ O(lfI* V X*), as desired. (5) ¢ = lfI -+ X. Then ¢* = O(lfI* :J X*), and the proof proceeds as in the base case.

o 11.9

Algebraic Decision Procedures for Intuitionistic Logic

It is possible to show that the theorems of H are decidable by using the translation of H

into S4 and the fact that S4 has the finite model property. Indeed, by fussing with this it is possible to show that H itself has the finite model property, but instead we shall sketch a more direct proof. The reader is advised to compare the presentation with that of the corresponding theorems for S4 of Section 10.10, Since we shall be more brief here. Theorem 11.9.1 Let (L, A, V, =}, 0) be a Heyting lattice and let (L', A, V, 0) be a complete infinitely distributive sublattice ofL (with same lower bound 0). Then there exists a binary operation =}' on L' such that (L', A, V, =}', 0) is a Heyting algebra and when a, b, a =} bEL', then a =}' b = a =} b. Proof We use the same symbols to denote the meet and join operations in the original

lattice and in the sublattice. We define

389

(1) a=}'b=V{XEL':xAa~b}.

We know from the proof of Theorem 11.2.3 (which uses infinite distributivity) that (2) a=}b=V{xEL:xAa~b},

and so clearly a =}' b ~ a =} b. We show that the inequality holds as well in the other direction when a, b, a =} b E L'. It suffices to recall (again from the proof of Theorem 11.2.3) that (a =} b) A a ~ b (modus ponens). Since a =} bEL', then a =} b is actually one of the components of the join that defines a =}' b; pictorially a =}' b is ... Va=} b V .... But then clearly a =} b ~ a =}' b.

0

Remark 11.9.2 When L' is finite the conditions above collapse to the condition that L' is a sublattice of L (with the same lower bound). Theorem 11.9.3 The intuitionistic propositional calculus H has the finite model property. Proof Suppose that IIH ¢. Then form the Lindenbaum algebra of H, which is a Heyting lattice L. Let lfII, ... , lfIn be all of the subsentences of ¢. Consider the sublattice L' generated by [lfII], ... , [lfIn], [f]. We will show that L' is finite.

Using distribution, every element can be put into "meet-normal" form (Xl, I V ... V Xl, III I ) A ... A (x n, I V ... V xn, Ill,), where the Xi} are the generators. Because of associativity, commutativity, and idempotence this form can be defined to be unique (up to an ordering of the generators). So clearly the sublattice L' is finite, since it is easy to see that there are at most 22"+1 such forms. There is not yet an implication operation defined on L'. We cannot simply define [lfI] =} [X] = [lfl -+ X], since the result might not be defined when lfI -+ X is not a subsentence of ¢ (or even a conjunction of disjunction of such, even throwing in f). And we cannot simply expand L', closing it under =}, since the result need not be finite (as it was when we did the corresponding construction with meet and join). But we do know by Theorem 11.9.1 that a new implication operation =}' can be defined on L' that agrees with the original implication operation =} when a, b, and a =} b are all in L'. This turns out to be good enough, and (L', A, V, =}', [f]) will be our desired finite model. 0 Let us consider the interpretation l(p) = [p], for each atomic sentence p which is a subsentence of ¢, and otherwise l(p) is arbitrarily defined. This is very much like the canonical valuation in the Lindenbaum algebra except that when one comes to compute V/(lfII -+ lfI2) = [lfII1 =}' [lfI2], the result cannot be guaranteed to be [lfII -+ lfI2]. But it can be when lfII -+ lfI2 is a subsentence of ¢, and this is good enough. Thus one can prove the following by an easy induction on sentences: Lemma 11.9.4 Given VI as defined above, iflfl is a subsentence of ¢, then VI(lfI)

= [lfI].

Proof The finite model property now follows directly. Since we have been assuming that IIH ¢, then [¢] f: 1, and v I (¢) = [¢] is undesignated. 0

390

11.10

INTUITIONISTIC LOGIC

LC and Pretabularity

Dummett (1959) presented the sentential calculus LC which is obtained from the intuitionist sentential calculus H by the addition of all sentences of the form (1) (¢ -+ 1Jf) V (1Jf -+ ¢). Dummett then gave a completeness proof for LC with respect to the sequence of matrices that Godel (1933) used in showing that LC has no finite characteristic matrix. Dummett proved that although LC too has no finite characteristic matrix, still each (n + 2)-valued GOdel matrix is characteristic for those LC sentences containing but n distinct sentential variables. Ulrich (1970) proved that every extension of LC that is closed under substitution and modus ponens (we call these normal extensions) has the finite model property. In this section we report results of Dunn and Meyer (1971), giving an alternative proof of Dummett's completeness theorem by algebraic means, but more importantly strengthening Ulrich's result by showing that every normal extension of LC has a finite characteristic matrix. Similar results have been obtained for S5 by Scroggs (1951) (presented in Section 10.11) and for RM by Dunn (1970). The proofs we give are exactly parallel to those of Dunn (1970). Maksimova (1972) has set these results about LC into the context of a very pleasant general result, to wit that there are only three normal extensions of the intuitionistic sentential logic H that have the property of pretabularity (that all their normal extensions have a finite characteristic matrix). Where X is an extension of LC (perhaps LC itself), by an X-algebra we mean a pseudo-Boolean algebra in which all of the theorems of X are valid. In pseudo-Boolean algebras generally, ...,a = a =? O. So in considering LC-algebras, we need only concern ourselves with A, V, =?, and O. Certain LC-algebras are especially important. By Goo we mean that algebra whose elements are the negative integers and 0 together with -(j) (where -(j) is the least element), and whose operations are defined as follows: (i) a A b = min(a, b); (ii) a V b = max (a, b); and

(iii) a =? b =

{Ib

if a ::; b if a> b. By Gil we shall mean that subalgebra of Goo whose elements are the negative integers -n to -1 inclusive, together with 0 and -(j). We take Go to consist of just -(j) and O. Incidentally, our definitions of these matrices are dualized from those of Godel (1933) and other references, where conjunction is interpreted as max(a, b), etc. Generalizing, by a Godel algebra we shall mean any algebra whose elements form a chain with least and greatest elements, and whose operations are defined in an analogous way. All GOdel algebras are LC-algebras. Where A is a pseudo-Boolean algebra and F is a filter of A, we define the quotient algebra AI F. The elements of AI F are the equivalence classes of [a] of all elements b of A such that a=? b, b =? a E F.

LC AND PRETABULARITY

391

Theorem 11.10.1 If X is an extension of LC, A is an X-algebra, and F is a filter of A, then AI F is an X-algebra and is a homomOlphic image of A under the natural homomOlphism, h(a) = [a]. Proof. AI F is a pseudo-Boolean algebra and is a homomorphic image of A. We need then only observe that every theorem of X is valid in AI F. Since AI F is a homomorphic D image of A and every theorem of X is valid in A, this follows.

Theorem 11.10.2 Let X, A, and F be as in Theorem n.IO.I, but let F be prime, i.e., a V b E F only if a E F or b E F. Theil AI F is a Godel algebra. Proof. That AI F is a chain is immediate given (1) and the primeness of F. Further, AI F must have least and greatest elements since every pseudo-Boolean algebra does.

We need then only check that the operations are defined as on a GOdel algebra. This is obvious for A and V, and the following theorem of LC, proved in Dummett (1959), ensures that =? is all right:

Thus since F is prime, either a =? b E F or (a =? b) =? b E F. But a =? b E F iff [a] ::; [b]. So if [a] ::; [b], then a=? b E F. But in general for a pseudo-Boolean algebra, x E F only if [x] is the greatest element in the quotient algebra. So [a] =? [b] = [a =? b], which is the greatest element of F, as it should be. On the other hand, if [a] i [b], then a =? b ¢. F, and then (a =? b) =? b E F. So then [a =? b] ::; [b]. And in any pseudoBoolean algebra, since b =? (a =? b) is the greatest element and hence in F, then [b] ::; [a =? b]. So [a] =? [b] = [a =? b] = [b], as it should. D Exercise 11.10.3 Prove that (¢ -+ 1Jf) V «¢ -+ 1Jf) -+ 1Jf) is a theorem ofLC. Theorem 11.10.4 Let X and A be as in Theorem n.lO.l, and let a E A be such that a :j:. 1. Then there is a homomorphism h of A onto a Godel algebra which is an Xalgebra, such that h(a) :j:. 1. Proof. Immediate from Theorems 11.10.1 and 11.10.2 once we invoke Stone's prime D filter separation theorem.

We remark that it easily follows from Theorem 11.10.4, by a familiar construction used by Stone, that every LC-algebra is isomorphic to a subdirect product of Godel algebras. Since the only GOdel algebra which is a Boolean algebra (excluding the degenerate one-element algebra) is Go, this result may be regarded as a generalization of the embedding theorem of Stone's for Boolean algebras. Theorem 11.10.5 Consider the sequence of Godel algebras Go, G], G2, .... tence ¢ is valid in G i , then ¢ is valid in G j for all j ::; i. Proof. This is immediate since each G j is a subalgebra of G i .

If a senD

Where X is a sentential calculus and A is a set of atomic sentences, let XI A be that sentential calculus like X except that its sentences contain no atomic sentences other than those in A. The following theorem is then obvious.

392

INTUITIONISTIC LOGIC

Theorem 11.10.6 If X is a normal extension of LC, then A(X/A) is an X-algebra, and in fact is characteristic for X/A, since any non-theorem may be falsified under the canonical valuation that sends every sentence ¢ to 1i¢11. The hard part of Dummett's completeness result for LC is showing that if a sentence ¢ is not a theorem, then there is some GOdel algebra G n such that ¢ is not valid in G n . This is contained in the following theorem, though generalized to arbitrary normal extensions of LC. Theorem 11.10.7 Let X be a normal extension of LC. Then if a sentence ¢ is not a theorem of X, then there is some Godel algebra G n such that G n is an X-algebra and ¢ is not valid in G n . Proof. It follows quickly from Theorems 11.10.6 and l1.lOA. Thus if ¢ is not a theorem of X, then by Theorem 11.10.6, ¢ is falsifiable in the X-algebra A(X/ A) where A is the set of atomic sentences occurring in ¢. But since 1i¢11 f:. IllfI:J lfIll, the greatest element, then by Theorem 11.10.4, there is a homomorphism h of A(X/A) onto a GOdel algebra G such that G is an X-algebra and h(II¢II) f:. 1. We may then falsify ¢ in G by the interpretation l(¢) = h(II¢I\). Note that G is an X-algebra and h(lllfIll) f:. 1. We may then falsify lfI in G by the interpretation l(lfI) = h(lilfll\). Note that G is finitely generated since it is the homomorphic image of A(X/A), which itself is finitely generated by the elements Ilpli such that pEA. Thus G is finitely generated by the elements h(llpll) such that pEA. It is obvious that every finitely generated GOdel algebra is finite, and it is further obvious that every finite Godel algebra containing at least two elements is isomorphic to some Gil' Thus G is isomorphic to some Gil' which completes the 0 theorem.

We now tum to the proof of our principal result. Theorem 11.10.8 Every consistent proper normal extension of LC has a finite characteristic matrix, namely, some Godel algebra Gil' Proof. The reasoning mimics that of Scroggs (1951). Let I be the set of indices of those GOdel algebras Gil that are X-algebras, where X is the given consistent proper normal extension of LC. By Theorem 11.10.7, since X is consistent, I is non-empty. If I contains infinitely many indices, then I contains every index because of Theorem 11.10.5. But then it follows from Dummett's completeness result that X is identical to LC. But if I contains only finitely many indices, then by Theorem 11.10.5, there must be some index i such that I contains exactly those indices less than or equal to i. By construction, G; is an X-algebra. Now suppose that a sentence ¢ is not a theorem of X. Then by Theorem 11.10.7, ¢ is not valid in some X-algebra G;, and by our choice of i, k ::; i. But then by Theorem 11.10.5, ¢ is not valid in G;. So G; is the desired finite characteristic matrix. 0

We remark that Theorem 11.10.8 has as a corollary that every proper normal extension of LC may be axiomatized by adding as an axiom to LC one of the sentences

LC AND PRETABULARITY

393

Godel (1933) used in showing that H has no finite characteristic matrix, and that from this it easily follows that the only consistent and complete normal extension ofLC is the classical sentential logic. (Compare the proof of similar corollaries at the end of Section 10.11.) It should also be remarked that Thomas (1962) contains another interesting way of ax iomati zing all of the GOdel matrices G,z, in which each of them is axiomatized by the addition of some appropriate pure implicational sentence as an axiom to LC. We finally allude to the fact that strong completeness results for LC are readily obtainable from Theorem 11.10.7, which are along the lines of strong completeness results for RM in Dunn (1970).

RESIDUATION AND GALOIS CONNECTIONS

12 GAGGLES: GENERAL GALOIS LOGICS 12.1

Introduction

The aim of this chapter is to provide a uniform semantical approach to a variety of non-classical logics, including intuitionistic logic and modal logic, so as to recover the representation theorems of Chapters 10 and 11 as special cases. The strategy is to adopt the basic framework of the Kripke-sty Ie semantics for modal and intuitionistic logic (cf. Chapters 10 and 11), using accessibility relations to give truth conditions for the connectives. We generalize this in line with Jonsson and Tarski (1951, 1952) so that in general an n-place connective will be interpreted using an (n+ 1)place accessibility relation (cf. Section 8.12). Besides the Kripke semantics for modal logic, there are motivating precedents with the Routley and Meyer (1973) semantics for relevant implication and the Goldblatt (1974) semantics for orthonegation. The problem with the Jonsson Tarsld result is that while it shows how Boolean algebras with n-place "operators" can be realized using (n + 1 place)-relations, the context is more restrictive than one would like. For example, the underlying structure must be a Boolean algebra, and the "operators" must distribute over Boolean disjunction in each of their places. We have already shown in Section 8.12 that Boolean algebras can be replaced with distributive lattices. But in this chapter we shall examine structures that we call "distributoids," and which relax the constraints of Jonsson Tarski. Distributoids are not the full abstraction we are seeking, because there need be no interaction between the various operators. We have noticed that many important logical principles can be seen as involving relationships between pairs of logical operators that may be seen under the algebraic abstractions of residuation and Galois connections. I We shall abstract these relationships into an algebraic structure called a "gaggle." Incidentally, we owe the name "gaggle" to Paul Eisenberg (a historian of philosophy, not a logician), who supplied it at our request for a name like a "group," but which suggested a certain amount of complexity and disorder. It is a euphonious accident that "gaggle" is the pronunciation of the acronym for "general galois logics.,,2 The general approach here is algebraic, thus we will think of a logic in terms of its "Lindenbaum algebra," formed by dividing the sentences into classes of provable equivalents, defining operators on these equivalence classes by means of the connectives applied to representatives. We shall represent the algebras in a way pioneered by Stone

395

(1936, 1937), and extended by Jonsson and Tarski, so that elements are mapped into sets (thought of as "propositions," or sets of states where the sentences are true). This gives completeness results for the various logics. See Chapter 1 for a discussion of the general relation between representation results and completeness theorems. In their original incarnation (Dunn 1991), gaggles were required to have underlying distributive lattices so that meet ("and") is represented as intersection, and join ("or") is union. Canonically then states are prime filters. However, this condition can be weakened to where the underlying structure is just a partial order (as with the Lambek calculus). Then states can just be principal cones and the complements of principal dual cones, and one does not need Zorn's lemma. In certain cases (as with orthologic and at least the non-exponential part of linear logic) where the logic is a meet-semilattice in any case, for consistency and join can be defined from meet using a negation that is a lattice involution, the methods can be extended so both meet and join are given reasonable interpretations. At the end of this chapter we shall give some applications. We should caution that in most cases, the general representation theorem leads to something other than the usual semantics known in the literature. Thus, for example, the usual semantics for intuitionistic implication (cf. Chapter 11) uses a two-place accessibility relation, whereas the gaggle approach yields a three-place accessibility relation. It then becomes necessary to examine the details of the general representation, applying specific algebraic properties of the logic in question to see that the usual result falls out as a special case after "fiddling with the representation." A toy example of this is given, showing how the usual Stone representation for Boolean algebras (where Boolean complement becomes set complement) can be obtained from the gaggle representation (where Boolean complement is represented using a two-place accessibility relation). 12.2

Residuation and Galois Connections

Consider two po sets A

= (A,::;) and B = (B, ::;') with functions f :A

1---+

B,

g: B

1---+

A.

The pair (f, g) is called residuated iff (rp) fa::;' b iff a::; gb.

The pair (f, g) is called a Galois connection iff (gc) b::;' fa iff a::; gb. A dual Galois connection is a pair (f, g), where (dgc) fa::;' b iff gb::; a. A dual residua ted pair (f, g) is a pair (f, g), where (rp) b::;' fa iff gb::; a.

I There

has been an anticipation of this in Sections 3.10, 3.17, and 8.1, but where the underlying order structure was only a partial order and not a (distributive) lattice. 2We do not ourselves endorse the alternative pronunciation "giggle."

Remark 12.2.1 We have already defined a Galois connection in Section 3.13 for the special case where A = B. The definitions above differ from one another only in the

GAGGLES: GENERAL GALOIS LOGICS

396

direction of an inequality here and there. Thus turn around the left inequality in the definition of a residuated pair, and we obtain a Galois connection, and turning around the right inequality gives a dual Galois connection. If both the left and right inequalities are turned around, of course a residuated pair becomes a dual residuated pair, and similarly for a Galois connection and its dual. Incidentally, observe that a dual residuated pair (f, g) is just a residuated pair (g, J), and so for the most part we shall not bother to look separately at dual residuated pairs. The moral is that as long as the two posets A and B are treated as distinct, one is of course free to turn the inequalities around, since the converse of a partial ordering is again a partial ordering. As someone in Australia can testify, one person's "up" is another person's "down." But these are abstractly all the same, and yet can be distinguished if we assume that A and B are the same, as we shall do henceforth. Our next theorem is easy to prove.

Theorem 12.2.2 (1) For a residuated pair, the following is an equivalent definition: (a) both f and g are monotonic, and fgx S x, x S gfx.

Moreover

if the poset is a lattice,

then

(b) f(x V y) = f(x) V fey) and (c) g(x 1\ y) = g(x) 1\ g(y). (a) both f and g are antitonic, and x S fgx, x S gfx.

if the poset is a lattice,

then

(b) f(x V y) = f(x) 1\ fey) and (c) g(x V y) = g(x) 1\ g(y).

(3) For a dual Galois connection, the following is an equivalent definition: (a) both f and g are antitonic, and fgx S x, gfx S x.

Moreover

if the poset is a lattice,

then

(b) f(x 1\ y) = f(x) V fey) and (c) g(x 1\ y) = g(x) V g(y).

Terminological note. The definition of a Galois connection was first introduced by Birkhoff (1940), and was defined in terms of condition (2a) above. (Birkhoff attributes to 1. Schmidt the equivalence stated as (gc) and which constitutes our definition.) Galois connections were extensively studied by Ore (1944) and Everett (1944). The notion of a Galois connection of course abstracts out the correspondence between the sub fields of a given separable field extension and the subgroups of the Galois group of transformations that leave that given subfield unmoved. Note the relation Fl ~ F2 iff H(F2) ~ H(Fd, which gives a clear motivation for turning the partial order around on the right-hand side of a Galois connection. There is an issue about this, as we shall see. The notion of residuation has most often been discussed in the case of binary operations, in the context of "residuated partially ordered groupoids" (see below), but the

397

definition of a residuated pair is a natural extension of that concept to the unary case (with the further understanding that the function need not be an operation taking values in its domain). Our unary form of a residuated pair can be found (after some decoding) in Blyth and Janowitz (1972), where it is explicitly contrasted with a Galois connection (pp. 18-19). It is also somewhat confusingly to be found in Gierz et al. (1980), where it is called a "Galois connection," with the ironic remark "notice that we have to keep the order straight." Gratzer (1979, p. 51) also follows this usage in an exercise. These usages reinforce the moral above about the essential abstract equivalence of these notions. Finally, let us note that in the language of category theory, a residuated pair can be understood as a pair of "adjoint functors" (cf. MacLane 1971). We believe that the theorem in MacLane "Galois connections are adjoint pairs" (p. 93) has been the driving force in identifying what we are distinguishing as Galois connections and residuated pairs (cf. also Lambek 1981), though MacLane himself is explicit about the fact that Galois connections are anti tonic, and that one must take the "opposite" of the righthand category (the dual of the po set) in order to obtain the desired result. Given a binary relation R on a set U (a frame), it is easy to construct examples of residuated pairs and Galois connections (also their duals) defined on subsets of U as follows.

Example 12.2.3 Residuated pair (0 tA

(2) For a Galois connection, the following is an equivalent definition: Moreover

RESIDUATION AND GALOIS CONNECTIONS

~

B

A

¢

~

DB):

0tA = {X: 3a(aRx & a E A)}, DA = {X: Va(xRa =? a E A)}.

Example 12.2.4 Dual residuated pair (0 A

~

B

¢

A

~ D t B):

OA = {X: 3a(xRa & a E A)}, DtA = {X: Va(aRx =? a E A)}.

Example 12.2.5 Galois connection (A ~ B.l.. .l..A = {X: Va(a E A A.l.. = {X: Va(a E A

=? =?

¢

B ~ .l..A):

XRa)}, aRX)}.

Example 12.2.6 Dual Galois connection (There is no customary notation-we use ? for "possibly false."?A ~ B ¢?tB ~ A): ?A = {X: 3a(xRa & a ¢ A)}, ?tA = {X: 3a(aR x & a ¢ A)}.

All of the above examples, except for the last, have occurred prominently in the literature, but not "on a single page." Thus, for example, thinking of U as a set of "possible worlds," or "states," and thus A and B as "propositions," DA and 0 A are just the usual definitions of the necessity and possibility operators in the Kripke semantics for modal logic. The relation R is called an "accessibility" relation, and aRfJ is read as "fJ is possible relative to a." Of course DtA and 0 tA are just the duals ("backward possibility and necessity"). It is not common to have these together with their forward version in the same modal logic, but this does happen in temporal logic, where aRfJ is read "fJ is an alternative future to a." Then DA becomes the usual tense operator GA for

398

GAGGLES: GENERAL GALOIS LOGICS

"it will always be the case," OA is FA for "it will (sometimes) be the case," and D~A and 0 ~A become the past tense versions H A and P A, respectively. Note also that in standard modal logic, when the accessibility relation is symmetric, (as for the logics Band S5), the "backward" operators are indistinguishable from the "forward" operators, and we can thus strike out the 'T' in the residuated pair and dual residuated pair laws above. The interesting thing is that these laws always hold if we distinguish backwards from forwards. Finally, we discuss Galois connections. It is interesting to note that these definitions can be found in Birkhoff (1940, 1948, 1967) under the heading of "polarities," and Everett (1944) showed that all Galois connections defined on power sets can be obtained from polarities. (Our results concerning gaggles will obtain, as a very special case, the more general result for Galois connections defined on distributive lattices, or, by trivial modification, Galois connections defined on posets). In interpreting R it is best to think of it as an "inaccessibility relation," or better as "incompatibility." It is customary to denote this relation as..L, and in Birkhoff (1940, p. 125) it is connected to Hilbert spaces as the "orthogonal" or "perp" relation. Goldblatt (1974) gives a completeness theorem for ortholattices (but not orthomodular lattices) in effect using A..L as the definition of negation. In many cases, including Goldblatt's, it is natural to require that the relation be symmetric, in which case A..L = ..LA. But this is not forced. It is easy to establish from our representation results below, that the above examples are "canonical," i.e., all residuated pairs, Galois connections, and their duals are isomorphic to those defined as above on a collection of subsets of some frame. Our representations assume that the po set is a distributive lattice, since we want representations that carry meet and join into intersection and union, respectively. But if one does not care about that one can have an arbitrary poset, and it is left as an exercise for the reader to see how to rework our results so that they apply to this "more general" case. Section 8.1 is a good model.

12.3

Definitions of Distributoid and Tonoid

As a first approximation a distributoid is a structure D = (A, 1\, v, (Oi) iEI), where (A, 1\, V) is a distributive lattice, and each I E (Oi) iEI is a (finitary) operation on A that "distributes" in each of its places over at least one of 1\ and V, leaving the lattice operation unchanged or switching it with its dual. Note that it is easy to see that for each I E (Oi) iEI, I is in each of its argument positions either isotonic or antitonic: '<:Jb,eEA: b::;e::} l(a1, ... ,b, ... ,a n)::;/(a1, ... ,e, ... ,an), '<:Jb,eEA: b::;e::} l(a1, ... ,b, ... ,an)?/(a1, ... ,e, ... ,an ).

A tonoid is roughly just a poset (A,::;, (Oi)iEI) where each I E (Oi)iEI is either isotonic or antitonic. We shall first consider distributoids. More explicitly, for a distributoid let T = {I\, v}, and let T (with subscripts) range over T. We associate with each I E (Oi )iEI a distribution type:

DEFINITIONS OF DISTRIBUTOID AND TONOID

Then, where * is V if Tj way on the value of T,

399

= V, and is 1\ if Ti = 1\, and # is V or 1\ depending in the same

l(a1, ... , b * e, ... , an) = l(a1, ... , b, ... , an )#/(a1, ... , e, ... , an).

There is one more requirement (which generalizes a requirement of Jonsson and Tarski). For convenience we require that the lattice be bounded, i.e., there is a least element and a greatest element 1 (a lattice can always be trivially extended with two additional elements to serve as the bounds). These bounds are useful, since with them, all finite subsets B of A have both greatest lower bounds (/\ B) and least upper bounds (V B). For a non-empty set B, just form the pairwise meets and joins, and for the empty set 0, /\ 0 = 1, and V0 = 0, as the reader can easily see. It is natural, and for technical reasons important, to have our operators "distribute" not just over finite nonempty meets or joins, but over the empty ones as well. Let us calculate the result for an operator 0 that distributes over join, leaving it as join. Then 00 = 0 V 0 = V 00 = V0 = 0. (Here we naturally understand 00 as the result of applying 0 to each member of 0, i.e., as 0.) Similar arguments can be produced for an operator D that distributes over 1\, leaving it as 1\, to show that D 1 = 1; for an operator ..L that distributes over v, changing it to 1\, to show that ..L = 1; and for an operator? that distributes over 1\, changing it to v, to show that ?1 = 0. More explicitly, and generalizing to operators of arbitrary finite degree, if the operator I is of distribution type (T1, ... ,Tj, ... ,Tn) ~ T, then I(C], ... ,ei, ... ,en) = e, where ej is when Tj = v, and is 1 otherwise, and the same for e in relation to T. We shall refer to this as "I respects the bounds."

°

°

°

Remark 12.3.1 Since, for a given operation, one is free to assign to each place as "input type" either 1\ or V, there are lots of degrees of freedom in terms of whether the operation I distributes over meet or join in that place. However, the fact that the "output type" T is always the same for each place forces uniformity in telms of whether a meet or ajoin results. Remark 12.3.2 The requirement that the operator respects the bounds can be conditionalized to remove the assumption that the lattice is bounded, but it is convenient to retain it throughout this chapter. Example 12.3.3 Boolean algebras with operators as in Jonsson-Tarski, and in particular closure algebras and the Lindenbaum algebras of normal modal logics with possibility as the operator, provide examples, as well as the dual of Jonsson-Tarski with necessity as the operator. Example 12.3.4 Further examples are: Boolean algebras, with complement as the (unary) operator; and various other distributive lattices with "complements" that arise in

the algebraic study of logic, including pseudo-complement (intuitionistic logic), De Morgan or quasi-complement (relevance logic, Lukasiewicz logic), impossibility and possibly false (modal logic).

400

GAGGLES: GENERAL GALOIS LOGICS

Example 12.3.5 Various (binary) implication operators, including relative pseudocomplement (intuitionistic logic), the implication of relevance logic, modal logic, and Lukasiewicz logic are examples too in virtue of: (a V b) -+ c

= (a -+ c) /\ (b -+ c),

a -+ (b /\ c) = (a -+ b) /\ (a -+ c). It is possible to generalize the notion of a distributoid so as not to require the existence of meets and joins. We call these structures tonoids (cf. Section 3.10). Recall that a tonoid is a poset (A,::;, (0; );EI) where each n-ary operation f E (0; );EI is in each of its argument positions either isotonic or antitonic. "Mix and match" is allowed, so that it can be isotonic in one position and anti tonic in another. For convenience, though, we assume that there always exist least and greatest bounds 0, 1, and each f will have to "respect the bounds" in the sense that for each of its argument positions there is a way of getting a uniform output of one of 0 or 1 by plugging in either 0 or 1 (this time it need not be uniform, one can "mix and match" in the argument positions), independent of the values of the other arguments. Putting all this formally, as for a gaggle, we assume that each n-ary operation f E (0;) ;EI has a "distribution type," but since this time preserving the bounds is the only issue, it is more appropriate to formalize this as simply an (n + I)-tuple of Os and Is and to call it a trace. 3 We use the notation (c I, ... , c 11) 1-+ c. The first n signs indicate at each position which of 0, 1 the function is preserving or co-preserving, and the value of the (17 + l)th indicates the output value. From these we can then determine for each position i whether the function is isotonic or antitonic, simply by seeing whether the value C; = c or not. Let us consider the two modal operators 0 and O. Both are isotonic, so how can we distinguish them? We do it by virtue of the two equations 01 = 1 and 00 = O. This is recorded by saying that the distIibution type of 0 is 1 1-+ 1, whereas that of 0 is o 1-+ O. Both of these types indicate isotonicity since the bounds are preserved, rather than being inverted.

12.4

Representation of Distributoids

We start with a definition of the "target" structures that will be used in the representation.

Definition 12.4.1 Given a distributoid D = (A, /\, V, (0;) ;EI), by a frame for D is meant a structure (U, (R;);EI), where there is a one-one correspondence between (0;) and (R;), and for each f E (0; );E[, if the degree of f is n, then the corresponding R; ~UI1+I. 3In Dunn (l993a) a somewhat reversed notion was called a "trace" and denoted by ±I, ... , ±Il f-7 ±. The idea was that a + in an argument position indicated isotonicity, whereas a - indicated anti tonicity. A + in the ouput position indicated that 1 was the value, whereas a - indicated O. From the bound which was output plus the information as to whether the function preserved or inverted the order, one could deduce the information about which bound was preserved or co-preserved.

REPRESENTATION OF DISTRIBUTOIDS

401

We state a rough theorem, the exact statement of which will not be given until after we have some motivating examples. Showing how various examples are treated and how this treatment is to be generalized will constitute all of the proof which we will give here of the theorem. A rigorous version of the proof would run along the lines of our proof Of. the Jo~sson-Tarski representation (Theorem 8.12.1), but with an appropriate schematIc notatIOn so as to "hide from the user" which operation of meet or join is bein a distributed or co-distributed over in which places. With such a notation the proof can b: made to look just like the proof of Jonsson and Tarski, where they had only to consider distribution over join.

Theorem 12.4.2 (Representation of distributoids) Every distributoid can be represented as a lattice of sets, with /\ as n, V as U, and each operator f E (0; );EJ intelpreted as an operation F defined on the power set of U, using R; according to a pattern to be described after motivation by the two examples below. The first examples which we give are those that relate directly to the Jonsson-Tarski representation. (1) Possibility (Kripke):

F(A) = OA = {p: 3a E A, R(p,a)}. (2) Generalized image operator (Jonsson and Tarski):

F(AI, ... ,An)=R;"(AI, ... ,An)=A, where A = {a: 3al E AI, ... , 3an E An, R;(al, ... , an, a)}.

Notice that it is tempting to say that the Kripke example is just the unary case of the Jonsson-Tarski example. This is right in spirit, but wrong in detail since the direction of the relation R is reversed in the two examples. Clearly nothing essential hangs on this (one person's relation is another person's converse), but it will be a constant theme of this chapter that one must pay as much attention to the "backwards" representations as to the "forwards" ones (witness the relation between menu A and menu C below), and ind.eed t~is is cru~ial to representing interactions between operators that are related by reslduatIOn, GalOIS connection, or their duals. Before setting out the general pattern for choosing the relation R, we first exhaustively analyze the case where R is binary. Thus let f be a unary operator. There are then only four distlibution types, and we list the condition in each case for p E f(A). Each "menu" represents alternative ways of realizing the operators. These examples show that one obtains various patterns of distribution and co-distribution depending on how one defines the operation. We have written in the vicinity some standard or non-standard notations fo!.:,.some operators. We use A for the set theoretic complement of A (relative to U). MENU A

P E +tA (/\ 1-+ /\)

Va(a E A or aRp)

PE (V

1-+

0tA E A and aRp)

v) 3a(a

REPRESENTATION OF DISTRIBUTOIDS

GAGGLES: GENERAL GALOIS LOGICS

402

13 (V

1-+

1\)

E A.L

= ..itA

Va(a E A or aRfJ)

13 E ?t A (1\ 1-+

V)

3a(a E A and aRfJ)

MENU B: R in place of R (1\ 1-+

(V

1-+

13 E 0t A 1\) Va(a E A or aRfJ) 1\)

Va(a E A or aRfJ)

(V

1-+

(1\ 1-+

V) V)

(1\ 1-+

13 (V

1-+

1\)

E .LA

(V

1-+

v)

= ..iA

Va(a E A or fJRa)

E OA 3a(a E A and fJRa)

(1\ 1-+

V)

3a(a E A and fJRa)

(1\ 1-+ 1\)

13 E OA Va(a E A or fJRa) Va(fJRa :::} a E A)

(V

V)

3a(a E A and fJRa)

(V

Va(a E A or fJRa)

(1\ 1-+ v)

3a(a E A and fJRa)

1\)

1-+

Remark 12.4.3 The above tables show that any menu will do as well as any other menu in realizing operators of various distribution types. One can pick and choose among the sets for different operators as in a Chinese restaurant, one from menu A for f, one from menu B for g. However, if one wants to preserve certain relationships between operators, the choice may not be so free. Thus observe that the customary d~fi~ition of possibility comes from menu C, while that for necessity comes from D. ThIS IS not arbitrary, but reflects the fact that the distributoid in question is a Boolean algebra, that possibility and necessity are related by the equation Oa = -0 - a, a~d th~t Bo~lean complement is to be realized by set complement. More profound relatIOnshIps wIll be explored when we come to residuation and Galois connection type phenomena. We finally set out the general rule, first with reference to menu A, which the reader should consult for motivation from time to time. We shall be treating V as a "positive sign" and 1\ as a "negative sign" and the reader is invited to think of them, respectively, as 1 and O. Accordingly Ev is just E, whereas E/\ is ¢. Similarly Rv is R, and R/\ is R. We also adopt the convention that 7\ = V and V = 1\. Given an n-ary operator f of distribution type

(i) ifr = v, then

E

Fi(AI,· .. , An) iff

13)).

We next notice that the second condition can be transformed into a semblance of the first by negating both sides (contraposing):

13 ¢ Fi(AI,· .. ,An) iff 3al, ... , an(al ErJ Al 1\ ... 1\ an Erll An 1\ Ri(al, ... , alb 13))·

13 E?A

MENU D: (R)-l in place of R

1-+

13

Val, ... , an(al ETJ Al V ... Van ETII An V Ri(al,· .. , an,

3a(a E A and aRfJ)

13

13

(ii) if T = 1\, then the realization condition for t is appropriately dual, i.e., the realization condition is a universally closed disjunction and the rules reverse for determining whether a component ai E Ai is negated:

3a(a E A and aRfJ)

MENU C: R- l in place of R E +A 1\) Va(a E A or fJRa)

403

The final trick is to stick in a couple of more subscripts to indicate "sign," and we thus obtain one general rule that covers both of the cases (when T = V and T = 1\): (1)

13

Er Fi(AI, ... ,An) iff

3al, ... , an(al ErJ Al 1\ ... 1\ an Erll An 1\ R i, r(al, ... , an, 13))·

We get the other three menus by switching R with R (= U - R), R- l , and (R)-l. We shall not need to consider these other menus until Section 12.7, and so the following theorem is stated explicitly only for menu A. However, the reader should recognize that it, of course, applies to the complement of R, or to any relation which is obtained from R by a simple permutation of its terms (as with R- 1 in the binary case), since these are also relations of the same degree. Theorem 12.4.4 (Realizing distribution types) Let (U, (Ri )iEI) be aframe. Associate with each Ri of degree n + 1 a distribution type ti : (Tl, ... ,Tn) 1-+ T. Define the n-my operation Fi on subsets of U as in (l). Then the operation Fi is of the distribution type ti· Proof We content ourselves with observing that the reason why the realizing condition is required to be an existentially closed conjunction when T = V is that both existential quantifiers and conjunction distribute over disjunction, and dually for when T = 1\. The reason why some atomic sentences ai E Ai are negated is that we at that place want to tum a conjunction into a disjunction, or vice versa, by De Morgan's laws. 0

Theorem 12.4.5 (Representation of distributoids) Every distributoid (A, 1\, V, (Oi)iEI) is representable as a distributoid defined as above on aframe (U, (Ri)iEl). Proof The set U is the set of prime filters on A. Using the method of Stone, for each element a E A, we set h(a) to be the set of prime filters P such that a E P. It is well known that h carries meet into intersection, and join into union. So all that remains is

404

GAGGLES: GENERAL GALOIS LOGICS

to provide for each n-ary operation defined on prime filters, such that

I in (Oi) (of type t) an

(n

+ 1)-place relation

REPRESENTATION OF DISTRIBUTOIDS

Rf

405

maximalization argument). The obvious definitions for the relations are then as follows (with the first two immediately guaranteeing their canonical equivalences from left to right, and the last two guaranteeing theirs in the other direction): P R+Q iff 3x(x

where for any sets of prime filters X 1, ... , X n , and a prime filter Q,

¢ P and +X

E Q);

PR.J..Q iff :lx(x E P and ..Lx E Q); P R?Q iff Vx(x ¢ P implies ?x E Q) iff Vx(x E P or ?x E Q);

f

:lPj, ... , Pn(Pj ErJ X 1 /\ ... /\ P n Erll Xn /\ Rr (Pj, ... , Pn, Q)).

Before jumping straight to the definition of the relation R{, we exhaustively analyze the cases where Ii is a unary operation, in hopes of shedding some light on the final definition. We shall focus on menu A. First let us suppose that we have an operation of type V l-+ V. For mnemonic reasons we will denote it as 0 (strictly speaking it should be subscripted with t since we are working with menu A, but we omit the subscript). We need then to show for some appropriate relation R, that h(Oa) = Fo(h(a)),

where Fo is the corresponding operation on sets of prime filters defined using R. This amounts to showing that for each prime filter Q: Q E h(Oa) iff Q E Fo(h(a)), i.e., Oa E Q iff :lP(P ErJ h(a) /\ PRQ), i.e. (using menu A), iff :lP(P E h(a) & PRQ), i.e., iff :lP(a E P & PRQ). We need then to define the relation PRQ so that a E P & PRQ implies Oa E Q. The obvious definition is to simply define PRQ to hold whenever Vx(x E P implies Ox E Q). Of course, this does not guarantee so immediately the other direction of the above "iff," i.e., if Oa E Q, then :lP(a E P & PRQ), but it turns out that such a prime filter P can be produced by a standard Stone-style argument using Zorn's lemma, maximalizing on the condition "pi is a proper filter, such that a E pi, and pi RQ." It is easy to show that the principal filter determined by a, [a) = {x : a ~ x}, satisfies this condition. (That [a) is proper is the whole point of the technicality that operations must respect the bounds.) Now let +, ..L, and? be unary operations of types /\ l-+ /\, V l-+ /\, /\ l-+ V, respectively. Again, we omit the subscript t, and work through what is required for these operations to be preserved, we obtain the following "canonical equivalences" (we list the one for 0 again for the record): (+) +a E Q iff VP(a E P or PR+Q);

(..L) ..La E Q iff VP(a ¢ P or PR.J..Q); (?) ?a E Q iff :lP(a ¢ P and P R?Q); (0) Oa E Q iff :lP(a E P and P RoQ). Adopting the strategy used above for 0, we define the relations so as to immediately guarantee half of each of the canonical equivalences (again the other half will come by a

P RoQ iff Vx(x E P implies Ox E Q) iff Vx(x

¢ P or Ox

E Q).

Briefly examining the halves of the canonical equivalences that are not an immediate consequence of the definition of the relation, we observe that the proof for ..L is similar to the proof sketched above for 0, but we start with the assumption that ..L a ¢ Q. We maximalize on the condition "pi is a proper filter, such that a E pi, and pi R.J..Q." The stories for + and ? are more interesting. For ?, we must show that if ?a E Q, then for some prime filter P, a ¢ P and P R?Q. This time we consider the ideal generated by a, (a] = {x_: x ~ a}, and maximalize on the condition "I is a proper ideal, such that a E I, ~d ~ R?Q," ~oting that (a] is such an ideal. The ideal we obtain can be argued to be pnme m a routme manner, and the complement of the prime ideal is our desired prime filter. The argument for +a similarly maximalizes on ideals. We at last try to figure out the general rule for defining an accessibility relation to use in representing an operation I of type t: Cr], ... , Tn) l-+ T.4 If the output type is V (think of 0 as an example), Rf (PI, ... , Pn, Q) iff VXI ... xn(Xj ETJ PI V ... V Xn ETII P n V I(XI, ... , xn) Er Q).

And if the output type is /\ (think of + as an example), Rf (PI, . .. , Pn, Q) iff

:lxI ... Xn(XI

ErJ PI /\ ... /\ Xn Ern P n /\ I(XI, ... , Xn) E:r Q).

~gain we can tum this into a form similar to the preceding condition by negating both SIdes: -f

R (PI, ... , Pn, Q) iff

VXj ... xn(Xj ETJ PI V '"

V Xn ETII P n V I(X1, ... , xn) Er Q).

Sticking in one more sign variable gives us our general rule:

(2)

R{ (Pj, ... , Pn, Q) iff VXj '" Xn(X1 ETJ Pj V '"

V Xn ETII P n V I(XI, ... , xn) Er Q).

4This notation is essentially due to Gerard Allwein.

406

GAGGLES: GENERAL GALOIS LOGICS

PARTIALLY ORDERED RESIDUATED GROUPOIDS

We need to show that h(f(a}, ... , an»

= F(h(aJ}, ... , h(an»,

i.e.,

Q E h(f(a}, ... ,an» iff Q E F(h(a}), ... ,h(an», i.e., I(a}, ... ,an) Er Q iff

::IPj, ... , Pn(aj ErJ Pj /\ ... /\ an Erll P n /\ Rr(Pj, ... , P n , Q», i.e., I(aj, ... , an) Er Q

iff ::IP}, ... , Pn(a} ErJ p} /\ ... /\ an Erll Pn/\ \;fXj ... xll(Xj E r ] Pj V ... V Xn Erll P n V I(xj, ... , x l1 ) Er Q)).

It is easily straightforward to see that the right-to-Ieft half of the last "iff" is guaranteed by the definition of R. The conditional the other way can be argued for by a maximalization argument (cf. J 6nsson and Tarski 1951), the details of which are routine but confusing given the level of generality. Note that the right-hand side is equivalent to an existentially quantified conjunction, and we are set up for a maximalization argument, much like the one in the proof of Theorem 8.12.1. But one must then be careful to maximalize on the principal ideal (ai], rather than on the filter [ai), when Ti = /\. D

Exercise 12.4.6 Prove the representation theorem for distributoids by modifying the proof of Theorem 8.12.1. With the right notation the modified proof should look to the eye much like the original, if one is not focusing on the "signs." For example, one useful notation is [ai)r;.

There are two familiar examples of residuation. First, think of 0 as multiplication, think of S as "divides," and think of say a -+ R c as the quotient cia. A more central example to our project is to think of 0 as some way of "conjoining" propositions into "premises," then think of S as logical deducibility, and think of (say) a -+ R c as "if a then c." (Conjoining premises need not be done by the logical operation usually called "conjunction," but instead by some other operation, as for relevance logic in Dunn (1966).) In both of these examples it is natural to think that 0 is commutative, and so there is no reason to distinguish left and right residuals, but in a general setting we wish to do so. Incidentally, given the quotient interpretation, it is common, especially in older literature, to use notations such as alc for one of the residuals and (say) a\c or allc for the other. These are not memorable notations for keeping track which is the left and which is the right residual, and they have frequently been reversed. Given the logical interpretation, and also the felt need for a notation that is more memorable as to its "sidedness," we introduce the following definitions following Pratt (1991).

Definition 12.5.1 x

-+ R Y

=x

-+ y

and x -+ L Y

T(*) = (Tj, T2)

where

T

0

b S c iff a S b -+ L c

(left residual).

Right residuation can be defined symmetrically: (rr) a

1-+

T,

(with or without subscripts) ranges over /\ and V. Thus, (V, V)

1-+

V.

Partially Ordered Residuated Groupoids

We have already discussed the notion of residuation for unary operations. We now want to examine it in the case of binary operations, where the standard jargon has to do with "partially ordered residuated groupoids." These were already introduced in Section 3.7. Good extemal sources are Birkhoff (1967) and Fuchs (1963). The general notion of "residuation," although it has its origins in such nineteenth-century figures as De Morgan and Peirce, seems first to have been abstracted in its modem form by Ward and Dilworth (1939). A partially ordered groupoid is just a structure (A, 0, S), where 0 is a binary operation on A, S is a partial order on A, and 0 is monotonic in each of its places with respect to s. When the partial order is a lattice ordering, we speak of a lattice-ordered groupoid when 0 distributes over V from both directions (in which case monotonicity becomes redundant). Such a groupoid is said to be left residuated when there always exists an element denoted by b -+ L c. This element can be shown to be unique, satisfying (lr) a

= Y +- x.

We want a more abstract way of stating the residuation properties, and these will ultimately help motivate our definition of a gaggle. Thus to each binary operation * of a residuated groupoid (0 and the two residuals) we assign a "type"

T(o):

12.5

407

0

b S c iff b S a -+ R c

(right residual).

The type of +-, a "contrapositive" of T( 0), is T( +-):

(/\, V)

1-+ /\,

and the type of -+, another "contrapositive" of T( 0), is T( -+):

(V, /\)

1-+ /\.

Note that we chose to view the left residual as composed from its arguments in the opposite order from the right residual. This is important, as we shall soon see. We next introduce some abbreviations that are useful for our abstract statement of residuation. · { a *b< c .if T S( *, a, b) ,c abb revlates c S a * b If T

= V' = /\.

The laws of the left residual and of the light residual can now be stated as follows: (SIr)

S(o,av,bv,cv)

iff

S(+- ,cA,bv,aI'J

bS c

iff

a

a

0

S c +- b;

REPRESENTATION OF GAGGLES

GAGGLES: GENERAL GALOIS LOGICS

408

(SIT)

S(o,av,bv,cv)

iff

S(-+,av,clI,brJ

b$ c

iff

b $ a -+ c.

a

0

The join and meet signs as subscripts are intended to aid the reader in discerning the patterns. What happens in each case is that the sign of the operation (which we attach to the third, or "output" variable) determines whether the operation applied to the first two terms (the "input" variables) is put on the left- or the right-hand side of the inequality. And the term c switches with the term with which it "contraposes," i.e., it is switched with the term under which there is indicated a change in input type.

Example 12.6.3 The following are examples of gaggles: 1. A residuated pair, Galois connection, or their duals, on a distributive lattice, with (Oi) chosen to be any of the following families of operations (either I or g can be the head of the family consisting of the two): {f}, {g}, {f, g} . 2. A distributive lattice-ordered residuated groupoid, with (OJ) chosen to be any of the following families of operations (0 is the head of the families of which it is a member, otherwise -+ or +- is the head): {o}, {o, +-}, {o, -+}, {o, +-, -+}, {+-}, {-+}, {-+, +-}.5 The operations are realized on a frame as follows (cf. Section 8.1.2): A

Exercise 12.5.2 Analyze the case of unary operations in a similar manner, assigning types to the operations in residuated pairs, Galois connections, and their duals so that the appropriate laws fall under the general scheme

Definition of a Gaggle

After this motivation, we are at last in a position to define a gaggle. First we define some useful terms. Definition 12.6.1 (1) Let D = (A, /\, V, (Oi )iEI) be a distributoid. Recall that evelY I E (0; );EI has a distribution type T(f). As a notational convention, let -/\ = V and -v = /\.

T(f) = (''1, ... , Ti, ... , Tn)

1-+ 1',

I and g that g is a contrapositive of I (with then T(g) = (1'1, ... , -

1', ... ,

Tn)

1-+

-Ti·

(3) If l' = V we write S(f, al, ... , a'b b) for tal ... an $ b, and if l' = /\ we write S(f, al,···, an, b) for b $ tal .. , an· (4) We say that two operations I and g satisfy the abstract law of residuation (in their ith place) when I and g are contrapositives (with respect to their ith place) and

(5) Two operations I, g E (Oi) are relatives when they satisfy the abstract law ofresiduation in some position. (6) The family of operations (0;) is founded when there is a distinguished operator f E (0;) (the head) such that any other operation g E (Oi) is a relative of I· Definition 12.6.2 A gaggle is a distributoid D = (A, /\, V, a founded family.

B = {y: 3a E A,3jJ E B, RajJy},

A+- B = {y : Va, VjJ( if RyjJa and jJ E B, then a E A)}, A -+ B = {y: Va, VjJ( if RayjJ and a E A, then jJ E B)}.

12.7

where [a, bY has the terms rearranged according to the contraposition of types.

(2) We shall say oltwo n-ary operations respect to the ith place) when if

0

A partial gaggle is defined quite analogously, replacing distributoids with tonoids. 6

S(f, a, b) iff S(g, [a, b]'),

12.6

409

(0; );EI),

in which (Oi )iEI is

Representation of Gaggles

We now have to figure out the realization conditions for gaggles. Concentrating on residuated pairs for concreteness, we know from our results on distributoids how each operation separately can be realized on a frame of prime filters, but now the question is how to "glue the operations together." A clue is gotten by looking at the example given in Section 12.4 of a concrete residuated pair defined using a single binary relation R. Both were realized in the standard distributoid way in terms of a relation, but one realization used the relation R, and the other used its converse R- l . The definitions were chosen from different menus. Before we can generalize we must define the "converse" of a relation with more than two terms. Of course, there is no such thing really; instead there are just all of the different ways of permuting the terms. But for our purposes here it fortunately turns out that we do not need this full generality. The following suffices:

This has to do with the fact that we look at operations whose types are "contrapositives" of each other. We cannot forbear remarking that in the case where R is binary, by a kind of universal harmony, the usual notation R- l results. If we now look at the concrete example of a residuated groupoid realized on a frame with a three-place relation (see Example 12.6.3), we see that the definition of 0 used RajJy, that of +- used R-lajJy = RyjJa, and that of -+ used R- 2 ajJy = RayjJ. 5Note that [->, <- J is literally not a gaggle, but it is very much in spirit. Either adding 0 or allowing the transitive closure of the relation "contrapositive" on operations would make it officially a gaggle. 6The reader should be warned that a subtle mistake was made in this definition in Dunn (l993a), wherein the notion of "contrapositive" was adapted from the definition of a "gaggle" without modification, even though the equivalent but somewhat reversed notion of "trace" was being used in place of "tonic type." This mistake does not really affect any results of the paper.

410

GAGGLES: GENERAL GALOIS LOGICS

The use of R- i does not by itself completely solve the problem of determining the realizing conditions. To make this clear, observe that with the case of a residuated pair, once we have decided to represent fusing O! from menu A, we of course must choose to represent its residual gas D. But 0 appears not on menu C (where R- 1 has been substituted for R), but rather on menu D (which uses (R)-l). We must address the question then as to why R- 1 gets negated. To make matters even more complicated, notice that it does not always get negated. Witness a Galois connection and the definition of A..L (menu A) and the definition of ..LA (menu C), which are precisely the same except for the direction of R. The same holds of dual Galois connections: ?!A and? A. But with a dual residuated pair we have the same phenomenon as with a residuated pair. The relation between D! (menu B) and 0 (menu C) has not simply to do with the direction of the relation, but negation again mysteriously raises its head. The moral is that we cannot determine an appropriate selection of the representing conditions by just knowing the distribution patterns, making an arbitrary choice of one menu for the one operation, and then simply going to "the dual" menu for the other. There are really two "dual" menus, one with the simple converse of the relation, and the other with it complemented. Under what conditions are we determined to go to one or the other? The answer is simple, once we observe that a Galois connection (and its dual) put the operations on the same side of the inequality, whereas residuation (and its dual) put them on opposite sides. It is this that determines whether we choose from the simple dual menu, or rather its negated version. Let us now put this more formally. We assume that we have a founded family of operations (Oi 1, where f is the head of the family. Using the distribution type t of f, let us decide to realize f by an operation F on subsets of U, arbitrarily choosing its representation condition to use R "directly" (as on menu A, rather than some inverse, complement, or inverse complement). So far everything is in accord with the way of realizing distributoids, and so we know that this operation F will have the same distribution pattern as f. Now we consider any other operation g of the family. The only question with respect to g is which "menu" to use in realizing it as an operation G on subsets of U. We know from our earlier work on distributoids how to choose a definition from a given "menu" so that the distribution pattern of G will realize that of g. But now we must choose this "menu" so that f and g satisfy the abstract law of residuation. We have used quotes here for "menu" since with the introduction of R- i , things appear more complicated than with our simple menus A through D. However, recall g is a relative of f. This requires, besides satisfying the abstract law of residuation with respect to f, that g is a contrapositive of f with respect to some position i. Now we observe that if the output type of g matches that of f, then we choose the definition of G using R- i , whereas if the output types clash, we use (R)-i. Thus once we pick our position, there are always four menus after all, and so we can continue to call them menu A, B, C, D, though we must realize that menus C and D depend on the choice of i.

REPRESENTATION OF GAGGLES

411

We leave to the reader to velify that the abstract law of residuation holds with respect to F and G defined as indicated, and record the above discussion as follows: Definition 12.7.1 Given two types t and t' of the same degree, i.e., having the same number of input types, we say that they agree if they have the same output type, and that they clash if they have different output types. Theorem 12.7.2 (Realizing contraposed distribution types) Let T be. a family of distribution types (T], ... , Ti, ... , Tn) 1-+ T (all the same degree), where to E T is the head ofT. Let (U, R) be aframe, where R is a relation on U of degree n + 1. Define the n-ary operation FO on subsets of U as follows:

P Er Fo(A], ... ,An)

iff

3al, ... , an(al ErJ Al /\ ... /\ an Erll An /\ Rr(al, ... , a'b P)). For a given type t' that is a contrapositive of t in the ith place, define the n-ary operation G on subsets of U as follows: PEr G(AI, ... , An)

iff

3a], ... ,an(al ErJ Al /\ ... /\ an Erll An /\ R~(al, ... ,an, P)),

where R' is R- i or (R)-i, depending on whether the output types oft and t' agree or clash. Then the operation FO is of distribution type to, G is of distribution type t', and Fo and G satisfy the abstract law of residuation with respect to each othe!: In short, we have a gaggle of operations with types assigned appropriately.

Theorem 12.7.3 (Representation of gaggles) Every gaggle (A, /\, V, (Oi liEf) can be represented using a frame (U, (Ri liEf), as for a distributoid, but choosing relations for different operations from different (dual) menus. Proof Actually in some ways this is more organized than the distributoid representation. There a different definition of R on the prime filters was needed for each operation. Here we use the same definition throughout, the definition coming from the type of the head of the family. The representation otherwise proceeds as for distributoids, but of course we must have a certain interaction among the relg.tions, which is ensured by the fact that they are defined using operations that all satisfy the abstract law of residuation with respect to some given operation ("the head"). We illustrate this with respect to distributive lattice residuated groupoids, leaving the general case to the reader. We have then the following ways of defining the relation R for each operation:

PL., Q) iff VxVy(x E PI & Y E PL. =? x 0 Y E Q); R-+ (p], PL., Q) iff VxVy(x E PI & x -+ Y E PL. =? Y E Q); R+-(PI, PL., Q) iff VxVy(x +-- y E PI & Y E PL. =? x E Q). Ro(PI,

We shall show only the equivalence of the definitions of Ro and R-+, since the argument for the equivalence of Ro and R+- is symmetric. Let us first suppose, then, that

GAGGLES: GENERAL GALOIS LOGICS

412

Ro(PI, PJ" Q). In order to show R .... (PI, P2, Q), let us suppose that x E PI, (x -+ y) E PJ" and show y E Q. From the definition of Ro(PI, PJ" Q), it follows that x 0 (x -+ y) E

Q. But it is easy to show that x 0 (x -+ y) s y, and since it is part of the definition of a filter that it is closed upward, then y E Q. So R .... (PI, Q, PJ,), as desired. For the converse, let us suppose that R .... (PI, PJ" Q) and that x E PI and y E PJ,. Since it is easy to show that y s x -+ (x 0 y), by upward closure we have that (x -+ (x 0 y)) E PJ,. But now using our hypothesis that R .... (PI, PJ" Q), we obtain (x 0 y) E Q, as desired. 0 Exercise 12.7.4 Prove the general case of the representation theorem for distributoids. We close this section by answering a question V. R. Pratt raised in discussion. Theorem 12.7.5 Distributoids and gaggles are equationally definable. Proof It is clear that the postulates for distributoids can be cast in the form of equations, since it is well-known that distributive lattices can be given such a formulation, and the distribution postulates for each place of 0 are all equations. It is further well known that in a (right) residuated groupoid, the inequalities a 0 (a -+ b) S band

suffice along with isotonicity for

0

b S a -+ (a

0

b),

and right-isotonicity for the residual:

if b s b' then a -+ b S a -+ b' . In a lattice (indeed, in a semi-lattice) the inequalities become equations given the equivalence a S b iff a 1\ b = a. Further, isotonicity follows from the fact that it is postulated that 0 distributes over V from both directions. Right isotonicity for -+ can be established by the inequality a -+ (b 1\ b') S a -+ b'. All these can be generalized to gaggles. It is just a matter of postulating the appropriate inequalities for each place, and seeing that the right form of isotonicity or antitonicity results. 0

GAGGLES WITH IDENTITIES

we need some way to say that a proposition a is "true," and this may be expressed as e S a (cf. Section 8.1.2). Note, for example, e S a -+ a, which follows from a 0 e S a. Of course, in classical logic e can be taken to be the greatest element of the Boolean algebra, but in general, as for relevance logic, this is too strong. Still e can be introduced as the "infinite conjunction" of all of the truths (cf. Dunn 1966). In representing certain distributoids or gaggles with identities (ej)jEJ (J ~ I), we add to a frame for each identity a partial order!: and a set Z ~ U. We require a !: p iff R(~l, ... , ~i-l, a, ~i+l, ... , ~/l' P) for some ~l, ... , ~i-l, ~i+l, ... , ~Il E Z. We also require that the representing sets (the "propositions") be "closed" as follows (where R is the (n + I)-place relation used to represent the n-ary operation fj). Hereditary condition. For each i thus closed we call it hereditary.

E J,

and a, PEA, if a !: p, then pEA. When A is

As a technical complication, we must also assume that each Zj is itself closed with respect to itself under the hereditary condition, but such sets can be "constructed" in the usual way by starting with a set of states and taking the intersection of all sets that include it and are so closed. The point of the hereditary condition is to ensure that the sets Zj function as identity elements. We illustrate this with respect to a left-residuated groupoid defined an a frame using a three-place accessibility relation. In order to verify that Z 0 A ~ A, we assume that X E Z 0 A. Then there exist ~ E Z and a E A such that R~ax. But by the hereditary condition, X E A as needed. One can also verify that A ~ Z -+ A. There is one problem that still must be dealt with. We want a guarantee that the hereditary sets are closed under the gaggle operations. The lattice operations of intersection and union are straightforward, but the only way to ensure closure under the other gaggle operations is to require that the following "monotonicity conditions" on the relations R hold: if R(al, ... , ai, ... , an, p) and a' !:i a, then R(al, ... , a;, ... , a'l> P); if R(al, ... , ai, ... , an, P) and P !:i

12.8

413

p',

then R(al, ... , ai,···, an, p').

Modifications for Distribntoids and Gaggles with Identities and Constants

Consider an n-ary operator f(XI, ... , Xi, .. , Xll) whose associated type is (-£1, ... , 7:i, ... ,7:n) H 7:. We shall say that an element e is an identity element for f with respect to the ith place if 7:i = 7: = V and f(e, ... ,Xi, ... ,e) S xi. or else 7:i = 7: = 1\ and i 7 Xi S fee, ... , Xi, ... , e). We shall sometimes denote such an element as e . In practice it is very common for gaggles to have identity elements. For example, residuated groupoids often have an element e such that eo x = x 0 e = x. In this case the single element e serves as an identity element for both places (and note further that we have =, and not merely s). In logical applications identity elements are important, since

Remark 12.8.1 Although one is accustomed to have closure requirements (such as the hereditary condition) on propositions in the case of intuitionist logic, relevance logic, and orthologic, it may strike one as strange for modal logic, where "propositions" are arbitrary subsets of a frame. And yet, where -+ is strict implication, it is clearly needed in order to have A -+ A tum out to be a logical truth. It turns out that by "fiddling with the representation" it can be discovered that the canonical accessibility relation between prime filters reduces to 0 PI ~ Q & PJ, ~ Q, but since modal logic contains classical logic, we are in a Boolean algebra and all prime filters are maximal, and so P2 = Q. SO closure is just closure under the identity relation, and so all sets are closed.

7This definition was misstated in Dunn (1991), where the requirement was omitted that the types of the input positions and the output position be identical. One could define "identity elements" where the types clash, if one postulates an underlying involution ~ on the lattice. Then if, say, Ti = V and T = II, one would require f(~e, ... , Xi, ... , ~e)::; ~Xi' Thus, for example, TI = V, T = II, and Xl .... , ~e::;, ~XI·

Usually in the Lindenbaum algebra of a logic one has few constants, maybe none. The identity, t, is the most likely constant to appear, perhaps, with its dual, f. However, in a combinatory algebra constants are abundant. The somewhat remote connection

414

MONADIC MODAL OPERATORS

GAGGLES: GENERAL GALOIS LOGICS

between implicational fragments of some logics and combinators is captured by the socalled Curry-Howard isomorphism. A more direct relation has been introduced in Dunn and Meyer (1997) which also shows how to represent such constants. Without going into the details, we give a short example referring the reader to the paper mentioned. Consider the combinator C. Taking the binary operation 0 to be application, C has a combinatory axiom

«C

0

x)

0

y)

0

z

==

(x

0

z)

0

y.

Just as for identity we required a set Z to be present in the representation, for such constants as C we require that an appropriate set, say, C exists in the representation satisfying the condition «C 0 X) 0 Y) 0 Z ~ (X 0 Z) 0 Y. Such a condition can be shown to be satisfied by the set into which a combinatory constant of the algebra is mapped. The same condition can be specified in terms of R (the accessibility relation) on the frame (3c E C)(3x E X)(3y E Y)(3z E Z)(3uj, u2)(Ruj zv & RCXU2 & RU2YUt) :::} (3x E X)(3y E Y)(3z E Z)(3u)(Ruyv & Rxzu).

(This latter is not as obvious as the condition was for identity. Examples of similar combinatory conditions can be found in Dunn and Meyer (1997), and an algorithm generating the appropriate condition for any combinator can be found in Bimbo and Dunn (1998).) We note here only that this observation seems to generalize easily whenever an operation (with respect to which the constant is a constant) is well represented.

12.9 Applications There are several topics that can be further developed. One has to do with applying distributoids or gaggles to various logics. This has already been carried out in some detail for modal, intuitionistic, and relevance logics, as we shall describe. This involves "fiddling with the canonical representation" to get the usual results, which do not always invol ve an accessibility relation of the same degree as that employed in the gaggle representation. We quickly illustrate this with the case of Boolean algebras (classical logic) viewing Boolean complement as a distributoid operation since it distributes over join, changing it to meet (it also does the dual, but one is enough). One can obtain the properties of Boolean complement by postulating that the relation R is irreflexive (A n A.l = 0) and symmetric (so A.l = .lA, and the properties of the negation of minimal logic result), and that the propositions A are "closed" in the sense that if a E A and it is not the case that a..Lx, then X E A (A U A.l = 1), and verify that all of these properties hold in the canonical representation in terms of prime filters. One finally observes that the use of the relation R is otiose, prime (proper) filters in Boolean algebras are maximal filters, and one can show that they are both consistent and complete with respect to Boolean complement, i.e., x E P iff x.l ¢ P. Since PRQ is just 3x(x E P & x.l E Q), it follows that PRQ iff P :f. Q. But then Q E A.l iff PRQ for all PEA, iff P :f. Q for all PEA, iff Q ¢ A. But then A.l becomes ordinary set-theoretic complement (relative to the class of prime filters), and so we obtain the ordinary (Stone)

415

representation for Boolean algebras out of the one delivered by the representation for distributoids.

12.10 Monadic Modal Operators In Chapter 10, we introduced Boolean algebras with normal modal operators as the algebraic way of looking at normal modal logics, and defined various classes of algebras corresponding to various well-known modal logics (K, T, B, S4, S5). We also showed how these algebras can be represented so as to get completeness results using Kripke frames. Also in Chapter 10, we introduced the operator i defined as -c-x corresponding to the necessity operator in modal logic. It is clear that we might just as well have had i as primitive and introduced c by the equation c = -i-x. By being even-handed to both sides, we can consider a normal modal Boolean algebra (or K-algebra) to be a structure (B,::;, 1\, v, -, i, c, 0,1), where (B,::;, 1\, v, -, 0,1) is a Boolean algebra, and (il\) i(a 1\ b)

= ia 1\ ib,

(il) il = 1,

(cv) c(a V b) (cO) cO

= ca V cb,

= 0,

(-i) ca = -i - a, (-c) ia = -c - a.

An interesting question arises, though, if we drop the underlying Boolean algebra in favor of a distributive lattice. Then we can no longer state (-i), and so there is no obvious relationship of duality between i and c. Indeed, in terms of a Kripke model, i could be modeled using one accessibility relation R, and c could be modeled using an entirely different one R,.8 But with the modal logic B, one has a natural connection between the operators c and i that can be expressed as the following (as the reader can easily verify): (b)

ca ::; b iff a::; i b,

which simply says that (c, i) is a "residuated pair." The Kripke semantics for modal logic is customarily presented in terms of a frame (U, R), where U is thought of as a set of possible worlds, and aRfJ is thought of as fJ is possible ("accessible") from a. A sentence is interpreted as expressing a "proposition" A, i.e., a set of worlds (those in which it is true), with the modal operators interpreted as follows: CA

= {X:

3a(xRa & a E A)},

8But Dunn (l995a) shows that if one adds the axioms i(a V b) ::::; ca V ib and ia 1\ cb ::::; c(a 1\ b), then one can (and must) use the same accessibility relation. An interesting question arises about how one might cover the axioms above and the residuation axioms below under a common abstraction.

416

GAGGLES: GENERAL GALOIS LOGICS

IA

= {X: Va(xRa =? a E A)}.

Conjunction, disjunction, and negation are just intersection, union, and complement (relative to U), and I is of course U and is the empty set. This can be viewed algebraically as the fact that a frame (U, R) determines a modal Boolean algebra, whose elements are the subsets of U, and whose operations are defined as above. We shall call this the modal algebra determined by the frame (U, R). As is well known, various conditions can be put on the accessibility relation R that give a sound and complete semantics for the corresponding logic; e.g., for T it is required that R be reflexive. From an algebraic point of view, soundness means that the modal algebra determined by the frame is of the appropriate kind, such as aT-algebra, and completeness means that all algebras of the appropriate kind, such as all T -algebras, may be embedded into the Boolean algebra determined by the frame. 9 Some of the well-1m own conditions on R include reflexivity and transitivity for S4, reflexivity and symmetry for B, and the various combinations of reflexivity, symmetry, and transitivity that correspond to various combinations of postulates on modal algebras. The condition that most interests us is the symmetry condition for B, which forces the interaction between the two modal operators in the condition (b). But the requirement of the symmetry of R is a kind of "red herring." The residuation condition CA ~ B iff A ~ IB

DYADIC MODAL OPERATORS

each of (B,::;;, /\, v, -, i, c, 0,1) and (B,::;;, /\, v, -, i', c' , 0,1) are modal Boolean algebras, and the following residuation properties are satisfied:

°

holds just as well if the direction of R is reversed in the definition of C. Of course, instead the definition of I might be changed to look backward (keeping the forwardlooking C). The point is that the only role the symmetry of R plays in verifying the residuation properties is to make the forward-looking operators and the backwardlooking ones indistinguishable. What really requires the symmetry of R is not the fact that I and C are duals in the sense of the residuation properties, but rather that they are duals in the sense of the De Morgan properties. Let us denote the backward-looking operators with a prime:

= {x:3a(aRx&aEA)}, I' A = {X: Va(aRx =? a E A)}.

C'A

The reader can readily verify the following residuation properties: C' A ~ B iff A ~ IB, CA ~ B iff A ~ l'B.

It gives a smoother theory to assume that all modal Boolean algebras are outfitted with suitable "backward" operators to match their forward operators. Thus by a symmetric modal Boolean algebra we mean a structure (B,::;;, /\, v, -, i, c, i', c' , 0,1), where 9Implicit in the above discussion is that the class of algebras satisfying given formulas is closed under subalgebra (which it always is).

417

c' a ::;; b iff a::;; ib, ca ::;; b iff a::;;

i' b.

Symmetric modal Boolean algebras can be viewed as gaggles in the sense of Dunn (1993a). The only subtlety is that they have an underlying Boolean algebra instead of a distributive lattice, but the representation of gaggles, based as it is fundamentally on Stone's (1937) representation of distributive lattices, includes as a special case Stone's (1936) representation of Boolean lattices wherein lattice complement is carried into set-theoretic relative complement. 10

12.11 Dyadic Modal Operators There are well-known connections (due fundamentally to GOdel) between Heyting algebras and S4 algebras (cf. Chapter 11). The idea is that the "open" elements x = ix of an S4 algebra form a Heyting algebra, with a ::J b = i( -a V b). But since the implication operator in a Heyting algebra is two-place, whereas the modal operators above are one-place, there is a certain lack of parallelism with the standard approach to modal logic. We fix this by first considering a modal algebra in which strict implication (-<) is a primitive, together with its adjoint o. Note that at this point the choice of the right arrow is not supposed to prejudge any relationship to the right residual, but the reader is wise to anticipate. It is standard in the Kripke semantics that ¢ -< lfI is true at a world w iff at every world w' accessible from w, either ¢ is not true at w' or lfI is true at w'. Less standardly, ¢ 0 lfI is true at a world w iff ¢ is true at wand lfI is true at some world w' from which w is accessible.!1 It is easy to verify that (1)

(¢ 0 lfI)

1= X

iff lfI

1= (¢ -<

X),

which motivates our adopting the algebraic analog, right residuation: (rr)

a

0

b ::;; c iff b::;; a -< c,

which is indeed expressed in algebraic jargon by saying that -< is the (right) residual with respect to o. We of course have a precisely parallel law for Heyting algebras, but with the Heyting implication ::J substituted for -< and the lattice meet /\ substituted for 0. 12 Put this JO Another way to look at this is that lattice complement itself is a gaggle operation being a Galois connection. While the initial gaggle representation is then in terms of a two-place relation, it can be fiddled with as in Dunn (1993a) so as to obtain the usual representation in terms of relative complement. I I Thus a 0 b is in effect a II c'b. It forms the basis for Wansing's (1994) Gentzenization of modal logics, and it is also implicit in Belnap's (1982) display logic.

12This corresponds to the well-known fact that the deduction theorem and its converse hold for intuitionistic logic.

1 I

418

GAGGLES: GENERAL GALOIS LOGICS

way, the question arises as to what distinguishes strict implication from intuitionistic implication. It is not the above law, but maybe it is the properties of 07 When 0 is just meet, as it is in a Heyting algebra, it is of course commutative, associative, and idempotent. But when it is the modal operator described above, it does not need to have all of those properties. Let us first examine commutativity. It is clearly a modality-destroying principle, since commutation on the left-hand side of (1) leads to permutation of "antecedents" on the right-hand side of (1). Thus, we get from the modally harmless ¢ 1= (If/ -< If/) to the modally noxious If/ 1= (¢ -< If/). The distinction between these two theses can be viewed as the central difference between strict implication on the one hand, and the classical and intuitionistic implications on the other. In the Kripke semantics the only natural ways to make 0 commutative are to either have just one world (classical logic) or to require the "hereditary condition" that if ¢ is true at w then ¢ is true at all worlds w' accessible from w (as in intuitionistic logic). The connective 0 is clearly always "square-decreasing," ¢ 0 ¢ I- ¢, since indeed it satisfies "left lower bound," ¢ 0 If/ 1= ¢. But the converse (¢ 1= ¢ 0 ¢) does not always hold. Indeed 0 is fully idempotent exactly for alethic modal logics (those where the accessibility relation is reflexive). Thus given a reflexive frame (U, R), it is easy to see that cjJ 1= cjJ 0 ¢ is valid, for if w 1= cjJ then w 1= cjJ and 3w' (namely w) such that w' Rw and w' 1= ¢. So w 1= ¢ 0 ¢. And conversely, if the frame (U, R) is not reflexive, then for some w, wRw. Pick an atomic sentence p and define a model where w 1= p, but for any other state w " w' ~ p. Clearly there then is no state w' such that w ' Rw and w' 1= p, and so w 1= p. We leave to the reader the verification of other such correspondences that we shall describe between properties of 0 and properties of R. Thus the reader can easily verify that 0 is associative exactly when the accessibility relation is dual Euclidean 13 as with K5. S5, of course, results when 0 is idempotent, commutative, and associative, and so these properties alone cannot distinguish intuitionistic from modal logic. Before going on to explore what does then make the difference, let us first try to identify a property of o that does express the simple transitivity of the accessibility relation. The answer can be checked to be the half of associativity that says (i)

a

(b

0

d) :S (a

0

0

b)

0

d,

while the other half, (ii)

(a

0

b)

0

d :S a

0

(b

0

d),

says that the accessibility relation is dual Euclidean. We will not go through the whole proof, but we note that it is useful in checking these correspondences to think of a 0 b as being defined as a 1\ c' b. 14 Then (i) can be expressed as 13That is, aRX and fJRX imply aRfJ, (and fJRa as we11, by the commutativity of conjunction). 14This is a convenience to guide "calculations," but they do not depend on it.

DYADIC MODAL OPERATORS

419

a 1\ c'(b 1\ c'd) :S (a 1\ c'b) 1\ c'd.

It is easy to see that the a term can be dropped, so it reduces to

c' (b 1\ c'd) :S c' b 1\ c'd.

Similarly, (ii) reduces to the converse of the expression above: c' b 1\ c'd :S c' (b 1\ c'd).

We have thus sorted out some of the distinctions between various modal logics in this framework, but still have not solved the mystery of what distinguishes modal from intuitionistic logic in terms of properties of o. There are two ways of looking at this, depending on whether we think of the intuitionistic implication :J as a left or a right residual. Both ways focus on the key intuitionistic fact that the following paradox of material implication holds: (pmi) b:S a :J b. Let us start by assuming that :J is a right residual (we shall accordingly denote it by --+) in order to facilitate its comparison with -<. Then (pmi') b:S a --+ b clearly follows from a 1\ b :S b by the law of the right residual. What distinguishes modal from intuitionistic logic then is the inequality that says that a 0 b is a lower bound of the right argument b: (lbr) a

0

b :S b.

When 0 is given the modal interpretation, this of course fails, but we still have that a a b is a lower bound of the left argument a: (lbl) a 0 b :S a. From this it follows by the law of the right residual that (psi) b:S a -< a, which is a version of the paradox of strict implication, familiar from modal logic. When o is meet, of course, both inequalities hold (because 0 is then commutative, and one can get from one to the other as indicated above), but in the present context it is interesting to ask what happens when only (lbr) holds. Then one would have a logic in which the paradox of material implication holds, but not the paradox of strict implication. There is another, and maybe prettier way to look at the distinction between modal and intuitionistic logic. Let us accept the fact that (lbl) holds, but not (lbr), i.e., let us keep as fixed the modal interpretation of o. And let us continue to regard -< as the right residual. But there is also a left residual :J satisfying (lr)

a

0

b :S c iff a:S b :J c.

From (lr) for :J and (lbl) we obtain the paradox of material implication for :J, but not the paradox of strict implication, and from (rr) and (lbl) for -< we obtain the paradox

GAGGLES: GENERAL GALOIS LOGICS

420

of strict implication for -<, but not the paradox of material implication. Of course, if we postulate that -+ is both a left and a right residual we obtain both paradoxes, and from these we can deduce the commutativity of 0, from which it follows that a 0 b ::; b after all. I like this way of looking at things, for it splits the properties of the intuitionistic implication into those of two different residuals.

REPRESENTATION OF POSITIVE BINARY GAGGLES

421

Let us say a word about how to define identity elements on a frame. As is made clear in Section 8.1.2 the best way is to work with articulated ternary frames (U, R, r:;;), where R ~ U 3 , and r:;; is a partial-order satisfying: Rapy & a' r:;; a imply Ra' py; Rapy & p' r:;; pimply Rap'y;

12.12

Identity Elements

Rapy & y r:;; y imply Rapy'.

While we are tidying up, let us observe that there is a more general way of looking at the inequalities (lbl) and (lbr), which is helpful in generalizing the framework to relevance logic. Let us assume that we have a distinguished element e. The following two inequalities say that e has half of the properties of a left and right identity, respectively: (lid) e 0 a ::; a; (rid) a 0 e ::; a, The postulates (lid) and (rid) are of course just special cases of (lbr) and (lbl) respectively, and they lead by the law of the right residual to

We think of r:;; as an "information order," and define a proposition A (a representing set) to be a hereditary subset of U, i.e., a subset that is closed upward under r:;; (that is, a E A & a r:;; a' imply a' E A). As we have already mentioned, this corresponds to requirements familiar from intuitionistic and relevance logics. It is easy then to see that if Z ~ U satisfies the following condition, then it is a left lower identity with respect to propositions (Z 0 A ~ A): (lli) 3(

E

Z(R(aP) iff a r:;; p.

Also Z is a right lower identity (A

0

Z

~

A) if it satisfies:

iff a r:;; p.

a::; e -+ a;

(rli) 3(

e ::; a

By a left assertionalframe we shall mean a structure (U, R, r:;;, Z), where (U, R, r:;;) is an articulated frame and Z satisfies (lli). A right assertional frame is defined similarly, but with Z satisfying (rli), and in a doubly assertionalframe Z must satisfy both.

-+ a.

Identity elements are important because they give us an algebraic way of saying that a proposition is "true," to wit, e ::; a. Thus we can say not only that a implies a (which is said in "algebraese" as a ::; a), but also with e ::; a -+ a that the proposition "if a then a" is true. When e satisfies the properties (lid) or (rid) we shall talk of "lower" (left or right) identities. But there are sometimes good reasons to require that e be a full identity. Thus, for example, if it is a full right identity (a 0 e = a) then one can derive what we called earlier (see Section 8.1.2) push and pop: (pp) a::;b iff e::;a-+b. The "if" direction of (pp) pushes the implication down into the object language, and the converse pops it out again. With e being only a lower light identity, one gets only the "push" half of this and not the "pop." But there may also be good reasons for sometimes not requiring "pop," and so we do not always require e to be a full right identity. Let us consider what happens when e is the top element in the structure, i.e., a ::; e for every element a (when it is we will often denote it by 1). For the cases of modal and intuitionistic logic we can make that assumption. Then, because of isotonicity, (lid) implies (lbr), and (rid) implies (lbl). We illustrate with the first implication. Since lob ::; b and a ::; 1, it follows (monotonicity) that ao b ::; b. Thus, (lid) and (lbr) are equivalent, when e = 1, and similarly for (rid) and (lbl). So the "fundamental" difference between modal and intuitionistic logic is whether e is a left, or both a left and right, lower identity element. IS 15 It turns out that this difference holds up in considering relevance logics too. amounting roughly to the distinction between the system R of relevant implication and the system E of entailment (which combines

12.13

E Z(Ra(p)

Representation of Positive Binary Gaggles

We codify some of the observations made above in the following framework.

Definition 12.13.1 A binary gaggle (BO) is a distributive lattice-ordered residuated groupoid, i.e., a structure (G, /\, v, 0, -+, e), where (1) (G, /\, V) is a distributive lattice; 0 (y V z) = (x 0 y) V (x 0 z), and (y V z) 0 x = (y 0 x) V (z 0 x) (i.e., 0 distributes over V); (3) x 0 e = x (i.e., e is a right identity); (4) x 0 y ::; Z iffy::; x -+ Z (i.e., -+ is a right residual with respect to 0).

(2) x

In standard "algebraese" this is called a right-residuated distributive-lattice ordered groupoid with right identity. There is the threat of what Whitehead called "the fallacy of misplaced concreteness" here, in that in the abstract framework of Dunn (1991) there could be other binary gaggles. Thus, for example, we might have a left residual and leftidentity instead, or in addition. And we might have a whole different family of operators, founded (say) on + (distributing over /\) instead of o. As can be seen from the discussion above, BOs are of interest for their application to modal logic, interpreting -+ as strict implication. If -+ is not only a right residual but also a left residual (0 is commutative), then it is non-modal and the BO has application necessity with relevance), but the element e is not required to be 1.

422

GAGGLES: GENERAL GALOIS LOGICS

to both intuitionistic and relevance logics. We have no very good name for such a BG, but let us label it a material BG (MBG).16

Definition 12.13.2 Given a right assertionalframe (U,!;;;, R, Z), we define the full binary gaggle (FBG) induced on it (G, /\, v, 0, -+, e) so that G is the collection of all hereditary subsets of U, /\ is n, V is U, e is Z, and if A and Bare hereditmy subsets of U, A 0 B, A -+ B, B +- A are defined as in Example 12.6.3. Theorem 12.13.3 EvelY BG is embeddable in the full BG induced on some right assertionalframe (U,!;;;, R, Z). Proof The proof is a special case of the result for general gaggles of Dunn (1991), but we run through in this special case so that the reader can understand both why the official representation of a binary operation in gaggle theory uses a three-place accessibility relation Rapy, and also how it can be resolved that the ordinary Kripke-style representation uses only a two-place one. The representation of a BG proceeds along the path started by Stone (1936, 1937) for distributive lattices and Boolean algebras, extended by Jonsson and Tarski to Boolean algebras with operators, and uses ideas from the semantics of relevance logic due to Routley and Meyer (1973) and Meyer and Routley (1972). An element a of the BG is mapped into the set of all prime filters P of the BG such that a E P. Meet is thus carried into set intersection, join into set union (as with Stone), the operation 0 is the generalized image operator defined above (as in Jonsson and Tarski), and -+ is also as defined above (as in Routley and Meyer). Of course, those definitions above of 0 and -+ assume we are given a three-place accessibility relation on the prime filters, and we next show how this is canonically defined. Thus in the canonical frame, the set UG is the set of all prime filters of G, ZG is the set of all prime filters of G containing e, and RGPIPJ.Q holds iff for all a, bEG, if a E PI and b E PJ. then a 0 b E Q. We will not here work through the details of the proof that this representation works. The interested reader can work out the details from the sources cited above. 0

Corollary 12.13.4 EvelY MBG is embeddable in the full MBG induced on some right assertional commutative frame. Proof One simply observes that if the accessibility relation is commutative, so is the induced fusion operator defined as in Example 12.6.3, and conversely, if the fusion operator is commutative, then the accessibility relation on the canonical frame is commutative. 0

12.14

Implication

We can now see the problem of relating the representation of gaggles to the usual Kripke semantics by way of a crude "dimension analysis." The gaggle representation of a bi16 As indicated above, this is equivalent to requiring that a be commutative. It should be mentioned that in the general theory of gaggles, it would be permitted that the structure have both a left and a right residual when these might not be the same, but we have chosen to simplify here to be closer to the applications discussed.

IMPLICATION

423

nary operation such as implication uses a three-place accessibility relation, whereas the Kripke semantics for either intuitionistic or strict implication uses a two-place accessibility relation. While, as we shall see, this fits perfectly the analysis of implication in the Routley-Meyer semantics for relevance logic, there are stories to be told in the cases of intuitionistic and strict implication. 17 12.14.1

Implication in relevance logic

The positive fragment of the relevance logic R can be characterized algebraically as a BG (G, /\, V, 0, -+, e), where 0 is associative, commutative, and "square-increasing" (a ::; a 0 a). (Since 0 is commutative this makes it an MBG.) We shall call such an algebra a positive De Morgan monoid. 18 A positive Routley-Meyerframe can be taken to be a right assertional frame (U,!;;;, R, Z).19 Other properties match the various other algebraic properties. Thus for an R+ frame we require the following (labeled to show their correspondences to algebraic properties) : (1) 3x(Rap X & RXyo) iff 3x(Raxo & RPy X)

(associativity);

(2) Rapy implies Rpay (3) Raaa

(commutativity); (square-increasingness).

Indeed, one obtain various relevance logics depending on which properties one adds, but we do not need to go into that here. We simply note that it must be checked that the canonical accessibility relation defined on the space of prime filters has the corresponding properties. 20 The chief point is that Routley and Meyer define X

1= cP -+ If.!

iff Va, p, if RaXP and a

1= cP,

then p

1= If.!.

And this definition (modulo the choice of the first or second position of R as the "fulcrum") perfectly matches the definition of -+ in the representation of a BG. Also the semantical clause for the fusion connective is a perfect match for the definition of o. I7For intuitionistic logic there is a related and even more dramatic problem in that the gaggle representation of a also uses a three-place accessibility relation, whereas the Kripke one does not use an accessibility relation at all (0 is just intersection). 18In various papers Meyer has called these "Dunn monoids," but modesty forbids its use here. 19The reader familiar with the relevance logic literature will notice that we have taken some liberties. First, for Routley and Meyer, Z = {OJ. This is because (cf. Meyer and Routley 1972) they in effect represent only prime De Morgan monoids (those where e ::; a V b implies e ::; a or e ::; b), obtaining a prime De Morgan monoid by dividing out the Lindenbaum algebra of R by a prime filter that excludes the given proposition that one wants to falsify. Second, they do not have an explicit partial order, but instead define one from ROa fJ, giving them in effect a left assertional frame. We have chosen to work with right assertional frames, choosing the second position of the accessibility relation R (and not the first) as our "fulcrum," in order to get the proper match to our use of"" as the right residual. Note, finally, that we have built in certain properties that Routley and Meyer explicitly require of R by connecting it with the partial order J;;;, e.g., RaOa follows from

a J;;; a. 20 Details can be worked out by consulting one or more of Routley and Meyer (1973), Meyer and Routley (1972), Dunn (1986), Anderson et al. (1992), and Dunn (l993a).

424

GAGGLES: GENERAL GALOIS LOGICS

Indeed, a large part of the original inspiration for developing gaggle theory was to tease these out as special cases. 12.14.2

Implication in intuitionistic logic

But, as already indicated, it is not so straightforward with intuitionistic implication. Some of what we say here has already been said in Section 8.1.2, but we repeat it in somewhat different words for the sake of clarity and completeness. Let us consider the case where G is a positive Heyting algebra. Then 0 is just meet, and we shall see that RPI P2 Q iff PI ~ Q and PL. ~ Q. Let us assume then that RPI P2 Q. We show that PI ~ Q, and of course PL. ~ Q follows symmetrically. Suppose, then, that a E PI. Since 1 E PL., a A 1 = a E Q, as desired. Now for the converse let us assume PI ~ Q and PL. ~ Q, and we shall show that RPIPL.Q. To this end let us suppose that a E PI and bE PL.. Since both PI and PL. are subsets of Q, both of a and b are in Q. But since Q is a filter and filters are closed under meet, a AbE Q, as needed.

Definition 12.14.1 For intuitionistic logic R is defined as: Rapy (ff Ray & RPy. We shall ultimately justify this definition, but for the moment let us observe that the gaggle representation employing them reduces to the usual definition in terms of the two-place relation. Thus for the intuitionistic ::>, the plugging in of Definition 12.14.1 into the definition of --+ in Example 12.6.3 gives us X E A ::> B iff Va, P(if RXP & Rap & a E A, then P E B).

Here more fiddling is required. Let us assume the right-hand side, Va, P(if RXP & Rap & a E A, then P E B),

If x E e' PL., then x = e' a for some a E PL.. Again 1 E PI and so 1 A e' a = e' a = x E Q, and so e' P2 ~ Q as desired. We next show that PI ~ Q as an initial step in showing that PI = Q. Thus if bE PI, then (since 1 E PL.) e'l AbE Q. But e'l A b ::; b, and so b E Q. Now since PI ~ Q, and PI is a maximal filter, it must be the case that PI = Q. But we can define the three-place relation in terms of the two-place one as follows:

Definition 12.14.2 For modal logic R is defined as: Rapy (ff RPy & a

Va(if RXa & a E A, then a E B).

Given the reflexiveness of R, this follows immediately, instantiating Pto a in the previous formula. The converse is only slightly trickier. Let us assume (xt) and the antecedent of the previous formula, RXP & Rap & a EA. We must now show that P E B. Since a E A and Rap, it follows from the hereditary condition that pEA. But since also RXP, we can apply (Xt) to obtain P E B as desired.

Thus plugging in the above definition of the three-place R in terms of the two-place one, we obtain for the modal --+: X E A -< B iff Va, P(if RXP & a = P & a E A, then P E B)

i.e., after substituting identicals, X E A -< B iff Va (if RXa & a E A, then a E B),

as desired. We leave to the reader a similar verification for a 0 b = a A e'b).

12.15

0

(showing that in effect

Negation

There are two more puzzles about the gaggle way of representing logical operations, one having to do with intuitionistic negation, and the other having to do with the De Morgan negation of relevance logic. The reader is advised to consult Dunn (1993b) to see some of the issues sUlTounding the definitions of negation in a more systematic context. 12.15.1

x::;

(1)

The gaggle treatment of negation

rvy

iff y::; ..,x.

The first problem, as the reader must have immediately recognized, is that we have two "negations." But if we simply require that.., and rv be the same operation we obtain all the properties of what is called "minimal negation." To obtain the intuitionistic negation we must also add the property that (2) a A rva ::; b.

A

12.14.3

= y.

Let us first quickly review the gaggle treatment of negation. The idea is to treat negation as a Galois connection on an underlying (distributive) lattice. Thus we require:

and now show that (xt)

425

NEGATION

The idea is that to represent negation one uses a binary frame (U, .1),21 and for a U, one can give the following definitions:

~

Modallogic

Let us next examine the case where G is a modal algebra. To make things easy we shall suppose that we have both of the modal operators e' and -<, with a 0 b = a A e' b. It would be interesting to explore in more detail what would happen if, say, we had only -< and 0 as primitives, but it would take us longer to develop the conceptual point, which is, as we shall see, that RPI PL. Q iff PI = Q and e' PL. ~ Q. Let us assume, then, that RPI PL. Q.

= A..L = {X: A.1X} = {X: Va E A,a.1x}; ..,A = ..LA = {X: X.1A} = {X: Va E A, x.1a}.

rvA

21 Actually for intuitionistic and relevance logics, the frame should be articulated with a partial order J;;;, and we should only be considering subsets A that are hereditary with respect to J;;;. But we suppress this level of detail at this moment to facilitate exposition.

426

The reader can verify that for X, Y

~

The first thing that occurs to one is to simply contrapose the KIipke definition to make it to have the same form as the gaggle definition:

U,

X :S "" Y

iff Y:S "" X. X

To ensure that ""A = "'A, it clearly then suffices to require that ..l be a symmetric relation, and to obtain (2) one simply adds the requirement that ..l be irreftexive, which ensures that there can be no X E An .LA.22 In the canonical representation of gaggles, where U is the set of prime filters, the canonical embedding is defined by h(a) = {P E U : a E P}, and P..lQ is defined as 3x(x E P & ""x E Q). It is easy to see that h(~a) = h(a).L, i.e., ""a E P iff VQ(a E Q implies Q..lP). The "if" direction is immediate. For the converse, one proceeds contrapositively, assuming that"" a ¢ P. We now must show 3Q(a E Q & Qtp). We consider the principal filter determined by a, [a) = {x : a:S x}. It is easy to see that [a) is "compatible" with P in the sense that [a)..lP. (Since otherwise 3q E [a), ""q E P. But then a :S q, and so ""q:S ""a, and then ""a E P, contrary to our assumption.) We next consider the set of all filters that contain a and are compatible with P, order it by inclusion, and a standard argument shows that the union of any chain in this set is also a filter containing a and compatible with P. This is the hypothesis of Zorn's lemma, and so we can apply it to show that there is a maximal such filter Q, and by another standard argument (using distribution) it can be shown that Q is prime. One might have chosen instead the definition P..l' Q as 3x(x E P & ..,x E Q), but, as the reader can easily verify, ..l' is just the converse of..l because of (1), and this explains why one represents one negation using.LA and the other using A.L. It is also clear that if the two operations .., and"" are identical, then the relation ..l on filters is symmetric. Also observe that if "" satisfies (2), then the relation ..l on filters is irreftexive, for if a, ""a E F, then a 1\ ""a E F, and so every b E F, contradicting the fact that we have (implicitly) taken "filters" to be proper. 12.15.2

Negation in intuitionistic logic

Having set the general context, we now examine the intuitionistic case, where the familiar Kripke definition of negation on a binary frame (U, R) is as follows: X

I=..,P iff

Va, X Ra implies a ~

p.

The standard gaggle representation of negation also involves a binary frame (U, ..l), so this time at least the "dimension analysis" is OK. Negation is a one-place operation, and so the gaggle representation uses a binary relation in representing it. But so does the Kripke semantics. So far there is no problem. The problem is that the gaggle definition uses the binary operation in an apparently different way:23 X

I=..,P iff

427

NEGATION

GAGGLES: GENERAL GALOIS LOGICS

Va, a

1= P implies x..la.

22This is essentially the semantics of Goldblatt (1974) for orthologic. 23This definition is in line with Birkhoff's definition of a Galois connection using a binary relation, and with Goldblatt's (1974) definition of negation in his representation of ortholattices.

I=..,P iff

Va, a

1= P implies XRa.

This is right, as far as it goes, but unfortunately one cannot simply take ..l to be R. The quick reason is that the binary relation ..l used in the definition of negation for intuitionistic logic in "gaggle theory" is, as we have just seen, a symmetric relation. Its complement R would then be symmetric as well, whereas the definition in the standard Kripke semantics requires R to be a partial order. Clearly, not many partial orders are symmetric (indeed, only identity). The solution reveals itself by looking at the well-known definition of negation in terms of implication and a constant false proposition ("the absurd"): ..,p = p :J F. If we apply the gaggle definition using the three-place accessibility relation we obtain: X

I=..,P iff

Va, y(Rxay & a

1= p ='? Y 1= F).

Since F is never true, this simplifies to X

I=..,P iff

Va, y(Rxay ='? a ~ p),

i.e., in the canonical model, where a and y are prime filters,

Clearly if we define XCa ("X is compatible with a") as for some y, X ~ Y & a ~ y, i.e., they have a common extension, we obtain a symmetric relation, and it turns out to be equivalent to the standard gaggle one. Compatibility is of course the complement of "perp," so X ..l a iff not (XCa). Thus the standard gaggle definition of negation becomes: X I=..,P iff Va, a 1= p implies X ..l a. The equivalence is trivial in one direction and needs Zorn's lemma for the other. What is going on is that the Kripke definition of negation uses inclusion of the prime theory a in the prime theory p, and inclusion is a subrelation of the compatibility relation. But it gives the same results because one can prove that if aCp, then there is a theory that extends both. Thus if p were in a prime theory a, but p showed up in a prime theory X such that aC x, then extend a and X to a common extension y, and this will be an extension of a that contains p, and so ..,p would be false not only on the gaggle account, but also on Kripke's. The converse is trivial since if..,p were in a prime theory a, but p showed up in an extension (i.e., ..,p is false on Kripke's account), then afortiori p would show up in a theory compatible with a (and be false on the gaggle account). 12.15.3

Negation in relevance logic

There is also a problem about how to "gaggleize" the standard semantical treatment of De Morgan negation in relevance logic. De Morgan monoids were introduced in Dunn

428

GAGGLES: GENERAL GALOIS LOGICS

NEGATION

(1966) as a way of characterizing the relevance logic R. A De Morgan monoid is a structure (G, /\, v, 0, -+,~, e), where (G, /\, v, 0, -+, e) is a positive De Morgan monoid (in the sense as we defined in Section 12.14.1), and ~ is an antitonic operation of period 2 satisfying: a 0 b :s: c iff bo ~ c :s: ~ a iff ~c 0 a :s: ~ b. Rather than saying that ~ is antitonic and of period 2, we could have equivalently said that (~, ~) is a Galois connection on G satisfying ~ ~a :s: a, i.e., every element of G is "Galois closed." A Routley-Meyerframe can be taken to be a structure (U, J;;;, R, Z, *), where (U, J;;;, R, Z) is a right assertional frame and * is a unary operation on U such that (1) a** = a (period two) and (2) Rapy iff Ry*ap* (switching). By an R-frame we mean a RoutleyMeyer frame whose right assertional frame is an R+ frame. 24 The clause for negation in the Routley-Meyer semantics is: 25

This makes clear the problem we must address, because this is simply not the form of a gaggle-theoretic definition of negation, which would rather be the following: x

F

~¢

iff Va(a

F¢

=? x.la).

Not only do the definitions have different forms, they also have different ingredients in that the Routley-Meyer definition uses a unary function, whereas the gaggle-theoretic definition uses a binary relation (not required to be functional in character). It turns out that in relevance logic ~¢ can be defined as ¢ -+ f, where f is thought of as the disjunction of all falsehoods. Note that f =I F, which last can be thought of as the conjunction of all propositions. Unlike F, f does not imply everything. It turns out that running through the analysis of the semantic conditions for ¢ -+ f in R leads to insights. Thus: x

F¢

-+

f iff

X

F

~¢ iff

F ¢ =? P F f), Va,p(RxaP & P F f =? a F ¢).

Va,p(RxaP & a

i.e.,

But one can work out that P F f iff P < 0*, but X < y abbreviates RO Xy for Routley and Meyer, and can be thought of as our information ordering 1;;;, canonically as X ~ y. So P < 0* means that whatever is true at Pis true at 0*.0* F f = ~t, since 0 Ft. So P F f· And conversely, if P F f, then P* F ~f = t, and so 0 < p*, i.e., P < 0*. So we have: X F ~¢ iff Va,p(RxaP & P < 0* =? a F ¢). Rearranging quantifiers, we obtain: 24Then in view of commutativity we can require instead of switching the customary postulate: Rapy implies Ray' p' . 25This clause is as in Routley and Routley (1972), though it actually originates in the representation of De Morgan lattices by Bialynicki-Birula and Rasiowa (1957). Cf. Dunn (1976) for a fuller discussion.

X

F

429

~¢ iff Va(3p[RxaP & P

< 0*]

=? a

F ¢).

Read in bastard English and with the canonical model in mind, this says that if X and a are "compatible" from the viewpoint of a consistent p,26 then ¢ is false at a. So it is not surprising that by contraposition we obtain something having the form of the gaggle definition of negation: X

F

~¢ iff Va(a

F¢

=?

Vp < 0*, RXaP),

taking X.la to mean Vp < 0*, RXap. Let us then in the canonical model compare the notion of "compatibility from a consistent viewpoint" with the canonical notion of XCa, or equivalently its negation X.la. First notice that if x 0 ~x is in any filter G, then f E G (since x 0 ~x :s: f). So let us suppose X.la. Then for some x E X, ~ x E a. Suppose that RXap. Given the canonical definition of R, this means that xo ~ x E p. But then f E p, and so P is not consistent. Arguing the other way around, let us suppose canonically that it is not the case that X.la. Set Po = [X 0 a) = {b : 3x E x,3a E a, x 0 a :s: b}. It is easy to see that Po is a filter. Indeed, it is a consistent filter since if f E Po, then 3x E x,3a E a, x 0 a :s: f. But then a :s: x -+ f = ~x, and so ",x E a, and so X .la, contrary to what we have supposed. So Po is a filter not containing f. But now using Zorn's lemma, one can prove by a standard argument (due to Stone) that Po can be extended to a prime filter excluding f. 12.15.4

Negation in classical logic

We save the most familiar to last. What properties does one have to put on .l to give A 1. the properties of classical negation? As we have already seen in Section 12.15.1, if .l is made symmetric and irreflexive then the following hold: X ~ yl. iff y ~ Xl.; xnxl.=0.

If, as is also needed for relevance logic, one restricts the "propositions" X to be the Galois closed sets, i.e., X

= Xl.l.,

one gets all of the properties of orthocomplement, including all of De Morgan's laws. This is the basis of Goldblatt's (1974) semantics for orthologic. We should point out that v is not the same as u, but rather, given the De Morgan definition,

We also point out that this does not give the whole orthomodular logic used in one standard interpretation of quantum mechanics, since it is missing the orthomodular law (a kind of weak form of distribution). 26"Consistent" here simply means

p or

~ a

¢ Pl.

f ¢ p. In R this implies negation consistency in the sense that 'Va(a ¢

430

GAGGLES: GENERAL GALOIS LOGICS

How does classical logic arise in this context? Obviously we need that all subsets of U are Galois closed. The natural way to ensure this is to take ..1 to be just the relation of non-identity -::j.. Then X.L is the set of things that are distinct from everything in X, i.e., just U - X. If we examine the canonical frame we note that P ..1 Q iff P i Q. Thus if P..l Q then 3x(x E P & ~x E Q). It must be the case then that P -::j. Q, for otherwise 3x(x E P & ~x E P). Then 0 = xA ~ x E P, and so P is not proper. Conversely, if P -::j. Q then 3x(x E P & x ¢. Q), or the other way around. The two cases are symmetric so we shall just consider the first case. But then since, in a Boolean algebra, (proper) prime filters are maximal (Theorem 8.9.1), we know (Theorem 8.9.2) that ~x E Q. SO canonically P ..1 Q.

12.16

Future Directions

Besides applying distributoids and gaggles as defined in this chapter, there are also three ways to generalize these structures so that they can be applied to more logics. One way has already been suggested, to weaken the distributive lattice ordering. Such a weakening to a semi-lattice ordering or just a partial ordering has been explored in Dunn (1993a). The latter is along the lines of the representation of implicational posets in Section 8.1. The weakening to a lattice ordering has been explored in Allwein and Dunn (1993) and Allwein (1992). Although operations whose degree is 3 or greater are not explicitly considered, we believe that such an extension would be routine (but complicated). Hartonas (l993a, 1993b) does explicitly consider operations of arbitrary degree, but contains a completely different style of representation. A second way to generalize gaggles would be to expand the notion of a "family of operations" so that operations of various degree can live happily together. There is a suggestion of this in the treatment of negation in intuitionistic logic, and in relevance logic, where negation is defined using the same accessibility relation as used for implication (and fusion). A third way to generalize gaggles would be to contemplate logics where various families of operations live happily together, for example intuitionistic or relevance logic with modal operators.

13 REPRESENTATIONS AND DUALITY! 13.1

Representations and Duality

The study of representations of lattices and related structures has been considerably ~n­ riched since Stone's original work, with contributions from various authors. We restnct ourselves here to a brief report for the sake of the interested reader. We begin by bringing attention to an important aspect of the Stone representation, called "duality," that we did not previously discuss. Let us first try to give philosophical meaning to duality issues. Duality gets its ultimate expression in the terms of category theory, but we shall do our best to give the reader some intuitive grasp of duality without resorting to category theory, or at least its diagrams. "Duality" in its most general sense has to do with "reversing." One talks of the "dual" of a partial order ~ being ;:::, or of A being "dual" to V. If one is lucky, laws regarding one notion become laws regarding the other, as is indeed the case with the examples given. A somewhat different kind of example is more germane to our concerns. In geometry it is a familiar idea that a line can be taken to be a set of points. But the opposite is true as well: a point can be taken to be the set of lines that intersect it. 2 So geometry can be formulated just as well in a "dual" fashion, with lines rather than points as primitive. The actual example that concerns us is that we can think of a proposition as a set of possible worlds, or dually, we can think of a possible world as a maximally consistent set (filter) of propositions. Thus given a set U of possible worlds, one can form the set of propositions ~(U) as just the power set of U (a Boolean algebra), and given a set of propositions B (viewed as having the structure of a Boolean algebra), one can form the set of worlds WeB) as just the set of maximal filters of B.3 This forms the underlying conceptual basis of Stone's representation of Boolean algebras, wherein we start with a Boolean algebra B of abstract propositions, form "possible worlds" as maximal filters of these and so obtain WeB), and then realize the abstract propositions as sets of these I We wish to thank Gerry Allwein and Chrysafis Hartonas for help on this chapter. In describing Priestley's representation and duality results we borrow heavily from Allwein (1992), who in turn borrows from Priestley (1970, 1972), and Davey and Priestley (1990). 2Indeed some sophisticated version of this is the basis of projective geometry, wherein not only do any two distinc~ points determine a line, but equally any two distinct lines determine a point (perhaps "at infinity" for lines that would otherwise be parallel). i~ a lovely 3That \?<J(U) can be viewed as standing for both "power set of U" and "propositions of mnemonic accident. But what can be done mnemonically with WeB)? The answer of course lIes III the fact that W is an upside down M and we are forming the dual set of maximal filters.

r:"

432

SOME TOPOLOGY

REPRESENTATIONS AND DUALITY

possible worlds so as to obtain '6:J WeB). Thus we go from propositions to possible worlds and back to propositions again. We of course can also start with possible worlds, construct propositions as sets of these, and then obtain possible worlds as maximally consistent sets of these propositions. So suppose we have a set U which we think of as the set of "all" possible worlds. Let us suppose that the cardinality of U is n. Then when we form the power set '6:J(U) we get the set of "all" propositions, and its cardinality is 211. So far, so good. 4 A possible problem arises when we realize that we can now apply the dual construction to '6:J(U) so as to form all of its maximally consistent sets and so obtain a set of possible worlds. Let us call this new set W'6:J(U). How big is W'6:J(U)? We know that in the power set all maximal filters are principle, and that means that for each maximal filter M there is a world W E U so that A E M iff W E A. So there is a natural one-one correspondence between U and W'6:J(U), and thus in some appropriate sense we obtain no new worlds in going from U to W'6:J(U). To make this concrete the reader is invited to consider: U '6:J(U)

= {WI, W2}, = {{WI, W2}, fwd, {W2}, 0},

W'6:J(U) = {{ fwd, {WI, W2}}, {{W2}, {WI, W2}}}.

Things are not so easy if one starts with an abstract Boolean algebra and forms WeB), '6:JWeB), etc. The question, put roughly, is whether we get back to where we started, or whether we have a new set of propositions, and then a new new set of propositions, etc. Where does it stop? If we do not cut this off at the first iteration, there seems no hope of stopping this construction, and it completely undermines any claim we might have had about B being the set of all propositions (say, about some particular subject matter). If the reader considers only finite Boolean algebras, say the four-element Boolean algebra of Figure 8.11 for concreteness, there may be a false conclusion, for here we do get back to where we started. Thus B = p, a, G, O}, WeB) = {{ 1, a}, p, G} }, and '6:JW(B) = {{ {l, a}, {I, G}}, {{I, a}}, {{I, G}}, 0}. But the problem is caused by Boolean algebras that are not isomorphic to any power set, and for this we need to consider some appropriate infinite Boolean algebra. One that is not too exotic is the Lindenbaum algebra formed from a language with a denumerable set of atomic sentences: At = {PI, P2, ... }. Where ¢ is a sentence of the language, [¢] is the set of sentences provably equivalent (in classical logic) to ¢. We know that this Lindenbaum algebra is the same as the free Boolean algebra with denumerably many free generators, with the generators being members of the set G = 4 Actually the reader may have noticed that already there is a mathematical problem, since the collection of all possible worlds would seem to be too large to form a set, and would seem to be a proper class. But in certain special circumstances, it may be that we can consider all possibilities and collect them together in a set. Think, for example, of a phase space in Newtonian mechanics, with some fixed finite number of (point) particles, each of which is assigned a position and momentum. Each "possible world" is given by such an assignment.

433

{[pd, [P2], ... }. We know that the cardinality of the set of sentences of the language is

No, and it is easy to see that the cardinality of the set of elements of the Lindenbaum algebra is also No, since it is not any larger and is not finite since always [pd f:. [Pj] when if:. j. We know from Cantor that the size of the power set of G is non-denumerable, indeed it is of cardinality 2 No. It is also clear that each subset of G corresponds to a set that has in it just every item in the subset together with the complement of every item not in the subset. Each such set is consistent and can be extended to a maximal filter, but no two such sets can be extended to the same maximal filter since they will differ in that one has, say, [pd and the other has -[Pd = [""pd. So the size of the set of maximal filters is also 2No. Now considering this as a set of worlds, we know when we form the set of its 2NO propositions, i.e., its power set, we get a set of the size 2 , and so we do definitely get a new set of propositions, and we are off and running on a kind of infinite regress, since we can now form a new, new set of worlds, and a new, new set of propositions, etc. Perhaps it is stretching things, but Stone, in his representation of Boolean algebras, can be viewed as cutting off this regress by providing the set of maximal filters WeB) with topological structure so as to determine the field of sets isomorphic to the original Boolean algebra independently of the representation map in terms of properties of subsets of WeB) arising from this additional structure. Of course, to complete this story we should give a philosophical interpretation of this topological structure, but we do not at this point have such an interpretation and commend this project to some reader. From this point on we stop being philosophers and turn to being pure mathematicians. We know that Stone's representation cannot always represent a Boolean algebra as a full power set. Thus we seem forced to describe the target representation algebras as "fields of sets" (sometimes "concrete Boolean algebras"), where these are just any collection of subsets of some given "universe" U where these subsets are closed under intersection, union, and complement relative to U. It would be nice to have some more intrinsic characterization of these target algebras, and this is what Stone provides by way of looking at U not just as a mere set, but rather as a set with a topological structure.

13.2 Some Topology Some of the notions discussed here have appeared in other parts of the book. But we collect them all together here in a unified fashion. The reader who wishes to learn more is advised to consult some standard book on topology such as Kelley (1955) or Simmons (1963) (our own personal favorite). One of the prettiest and most useful notions of pure mathematics is that of a topological space = (U, (Od iEI), where U is a non-empty set and {Oi} iEl is a collection of subsets of U (called a topology) which is closed under finite intersections and arbitrary unions. Among the finite intersections we include the empty intersection, and so U is open. Also 0 is open since we include the empty union. Each Oi is called an open set, and its complement U - Oi is called a closed set. Alternatively the closed sets might be taken as primitive, requiring them to be closed under finite unions and arbitrary intersections, with then the open sets characterized as their complements (relative to U). Note that U and 0 are always closed as well as open.

r

434

DUALITY FOR BOOLEAN ALGEBRAS

REPRESENTATIONS AND DUALITY

A set is clop en if it is both open and closed. Note that if 0 is clopen, then so is U - O. Since clopen sets are also closed under nand u the clopen sets constitute a field of sets. Sometimes we are given a collection {Bj liEf of subsets of U, where the collection is closed under finite intersections but not under arbitrary unions. Such a collection is called a base for a topology, because we can just close it under arbitrary unions, and the resulting collection of subsets then of course can be viewed as a collection of open sets. There is an even weaker notion called a subbase, where the given collection of sets has no special closure properties. But we can close it under finite intersections (obtaining a base) and then under arbitrary unions to obtain a topology. By a clop en subbase we mean a collection of subsets {Sj j JEJ closed under relative complement: (U - Sj). Given U' ~ U, we can view it as a subspace of T (the relative topology): T' = (U', {O;liEI), where 0; = Oi n U'. The additional topological structure that is needed for duality is given by two notions, total disconnectedness and compactness. Total disconnected means that distinct points are separated by a clopen set, i.e., x f= y implies that there is a clopen set 0 such that x E 0 and y ¢ O. Since U - 0 is also clopen, this means that the pair is separated by two disjoint clopen sets. A space is Hausdorjfif any two distinct points are separated by disjoint open sets, so any totally disconnected space is Hausdorff. Hausdorff spaces playa very important role in topology, and are midway in a hierarchy of separation principles, the weakest of which defines a space to be To if for each pair of distinct points there is an open set that contains one but not the other. Compactness is a very important property of topological spaces that arises from the real line. It says that if there is a collection of open sets {Oi) iEI such that U iEI Oi = U, then for some finite J ~ I, UJEJ OJ = U. An important feature of the real line is the set of open intervals. This is already closed under intersection, and so is a base for the so-called usual topology of the real line, since all one has to do is close it under arbitrary unions and it becomes a collection of open sets. The famous Heine-Borel theorem states that every closed and bounded subspace of the real line is compact. In particular, this means that every closed interval, such as the closed unit interval [0, ... 1], is compact. This is related to the following more abstract theorem.

Theorem 13.2.1 Every closed subspace of a compact space is compact (in the relative topology). Corollary 13.2.2 Let X be a closed set and let {Oi j iEI be a collection of open sets that covers X in the sense that UiEI Oi :2 X. Then there is a finite subcollection {OJ}jEJ that covers X, again in the sense that UJEJ OJ :2 X. Proof We shall not prove Theorem 13.2.1, which can be found in any standard topological text. The reader should be warned, though, that it is sometimes found stated as our Corollary 13.2.2. This is sometimes disguised by changing the sense of cover so as to replace :2 with =, and so we introduce the terminology exact cover for the latter, i.e., when U jEJ OJ = X. It is worth noting how the corollmy is related to the theorem. The trick is to notice that we can view X as a topological space in its own right with the relative topology. Then Oi n X is an open set in the relative topology, and

435

X = UiE1(Oi n X). Since X is closed in the original topology, we can now apply Theorem 13.2.1 to conclude that X = UjEJ(Oj n X) for some finite subunion, and hence

X~UjEfOj.

D

It is often convenient to investigate compactness by focusing not on the open sets, but on the basic, or even subbasic sets. The following is therefore useful.

Theorem 13.2.3 (Alexander subbase theorem) If S is a subbase for the topology T = (U, (9) sllch that eveJY cover ofU by members of S has afinite subcover then T is compact. We need one last notion before continuing our discussion of duality.

Definition 13.2.4 Given two topological spaces T = (U, (9) and T' = (U', (9'), we call a function I from U to U' open if it preserves open sets, i.e., if 0 E (9, then the image 1*(0) = {tea) : a E OJ E (9'. A function I from U to U' is called continuous if it pulls open sets back to open sets, i.e., if 0' E (9', then the inverse image l-h(O') = {a: I(a) EO'} E (9. Continuous maps play roughly the SaIne role in topology as do homomorphisms in algebra. We need then to introduce the analog of an isomorphism, which (somewhat confusingly given the algebraic analogies) is called a homeomorphism. The reader is COlTect to think that for a homeomorphism we need to require that I be one-one and onto, as well as, of course, continuous. But it is also necessmy to require I to be open in order to meet the requirement that the inverse of a homeomorphism is also a homeomorphism.

Definition 13.2.5 A homeomorphism between two topological spaces T = (U, (9) and T' = (U', (9') is a I-I function I from U onto U' which is both open and continuous. Two topological spaces will be said to be homeomorphic images of each otherjust when there is a homeomorphism between them. 13.3

Duality for Boolean Algebras

We are now in a position to prove a version of Stone's representation that builds in the duality aspects. A compact and totally disconnected space has come to be known in the literature as a Stone space (though Stone himself, of course, spoke more modestly of Boolean spaces). We assume knowledge of the proof of the representation theorem for a Boolean algebra B, recalling that the isomorphism is defined as h(a) = {M : M is a maximal filter of B and a E M j. We will show how to add to this proof so as to obtain a topologized version of it which builds in duality considerations.

Theorem 13.3.1 (Stone) EvelY Boolean algebra B is isomorphic to the Boolean algebra of clopen subsets of a Stone space S(B). The proof of this theorem is distributed among various lemmas and comments. First we describe how S(B), called the dual space ofB, is determined. We let U be the set of maximal filters of B, and we use the collection of all sets of the form h(a) as a base, which it is because h(a) n h(b) = h(a!\ b). We thus let the open sets of the dual space

436

DUALITY FOR BOOLEAN ALGEBRAS

REPRESENTATIONS AND DUALITY

be unions of sets of the form h(a). Note that sets of the form h(a) are clopen since there complements are also open by virtue of the fact that U - h(a) = h( -a). Lemma 13.3.2 S(B) is totally disconnected. PlVO! Suppose we have M, M' E U with M -::j:. M'. Then 3a E B such that a E M and a ¢ M', or vice versa (the argument is symmetric). But then M E h(a) and yet M' ¢ h(a). And so h(a) is a clopen set that separates the point M from the point M'. D

Compactness is harder. We find it easiest to work with compactness in a well-known dual form. 5 We shall call the original form of compactness U-compactness. A collection of sets C is said to have the finite intersection property if every finite subcollection has non-empty intersection. A space U is said to be n-compact if every collection of closed sets which has the finite intersection property also has non-empty intersection. Lemma 13.3.3 U-compactness and n-compactness are equivalent. PlVO! Apply set-theoretic complement and De Morgan's laws. We need to show that if UXEX hex) = U, then for some finite Y ~ X, UYEY hey) = U. Note that this is compactness for the (sub)basic sets, but it implies compactness for the open sets by the alexander subbase theorem 13.2.3. D

Lemma 13.3.4 (Maximal Filter Principle) Let F be a plVper filteJ: Then F can be extended to a maximalftlter M'. PlVO! This is just another variation on Stone's .prime/maximal filter separation principles (see Lemma 8.6.2 and Exercise 8.11.11), and is sometimes called the maximal filter theorem. The idea is to set E = {G : G is a filter and F ~ G} and order E by inclusion. All of its chains have upper bounds (their unions), and so by Zorn's lemma E has a maximal member M. It is easy to see that this is a maximal filter simpliciter (and not just a filter maximal among those extending F), for if M c M' then F ~ M c M', and so M would not be a maximal member of E after all. D

Lemma 13.3.5 The space of maximal filters U is n-compact. PlVO! Suppose X ~ B and for no finite Y ~ X does nyEy hey) = 0. Let [X) be the filter determined by X. We first show that [X) is proper. Otherwise 3Yl, ... , YIIl E X such that Yl/\ .. . /\Ym = O. But this is impossible given that some maximal (proper!) filter ME h(Yl) n ... n h(Ym). Because then Yl, ... , YIIl EM, and so Yl/\ ... /\ Ym = 0 E M, meaning that M is not proper after all. Now that we know that [X) is proper we can extend it to a maximal filter M' using the maximal filter principle (Lemma 13.3.4). Clearly for x E X, x EM', i.e., M' E hex). Thus nXEX hex) -::j:. 0. D

Lemma 13.3.6 h is onto the class of clopen sets. SThe various fonus of compactness we discuss are analogous to various semantical notions introduced in Section 9.4.

437

PlVO! By using sets of the form h(a) as a base, we have trivially ensured that each h(a) is open. We are finally in a position to show the converse. Let 0 be a clopen set. Then 0 = UXEX hex) for some X ~ B. But 0 is also a closed set of a compact Hausdorff space, and so by Theorem 13.2.1 is thus itself compact. But then there are Yl, ... , YI1 E X such that 0 = hY1 U ... U hYI1 = h(Y1 V ... V YI1). D

Let us now assess the duality aspects. We start with a Boolean algebra B, form the Stone space S(B), and then form the Boolean algebra of the clopen sets B(S(B». Contrary to the famous novel of Thomas Wolfe, we can sometimes get home again (at least in category theory). This is shown by Theorem 13.3.1, which tells us that B is isomorphic to B(S(B». There is a symmetric question concerning duality. What if we start with a Stone space S, form the Boolean algebra B(S) (the dual algebra of S) of its clopen sets, and then form the Stone space S(B(S»? When we examined this construction with respect to the power set construction we discovered that the answer is yes: simply map an element a E U to the set of subsets of U that contain it as a member. This is a one-one mapping onto the maximal filters of the power set. Let us then try the same trick, mapping an element a to the set f(a) of clopen subsets of U that contain a. It is easy to see that f(a) is a complete and consistent filter, and hence a maximal filter. The map is clearly one-one because of total disconnectedness. Is it onto? Suppose we have a maximal filter M. We need to know that there is some a E nM. Note that M is closed under binary intersections since it is a filter, and so under all finite intersections. This means that we have the finite intersection property, and so by compactness nM is non-empty and contains some a. We show that f(a) = M. Clearly M ~ f(a). But since f(a) is a filter and M is a maximal filter, M = f(a). We are not yet through showing that f is a homeomorphism, but we leave to the reader the completion of this as an exercise. Exercise 13.3.7 Show that f is both open and continuous. The duality above is often called "object" duality and is distinguished from "functorial duality," with "full duality" including both. Functorial duality is at the level of morphisms (in this case the isomorphisms and the homeomorphisms), and not just at the level of the objects (in this case the Boolean algebras and the Stone spaces), and says that there is a natural one-one correspondence of isomorphisms between two Boolean algebras and homeomorphisms between their two dual spaces, and that there is a similar one-one correspondence of homeomorphisms between two Stone spaces and the isomorphisms between their dual algebras. Further, these correspondences between isomorphisms and homeomorphisms compose in ways so that "nothing new" arises by repeated compositions. For example, if one were to start with a isomorphism between two Boolean algebras, go to the corresponding homeomorphism between the two dual spaces, and then take the isomorphism between the two dual algebras, there is a natural sense in which one "comes home" to the isomorphism with which one started. And similarly if one were to start with a homeomorphism between two Stone spaces. In this chapter we will focus on object duality, though we shall try to keep track from time to time of functorial duality as well.

438

13.4

REPRESENTATIONS AND DUALITY

Duality for Distributive Lattices

There are two topological approaches to Stone's representation of distributive lattices. The approach taken by Stone (1937) leads to a representation in terms of spaces which Hochster (1969) calls spectral. These spaces are such that their family of compact open sets is closed under intersection and forms a base for the topology of the space. Further, they are sober in the sense that every irreducible closed subset is the closure of a unique singleton set (a "point"), where a subset C of the space is irreducible closed if C is the union of two closed sets Cl UC2 then C = Cl or C = C2. Johnstone (1982) and Vickers (1989) are good "textbook" sources on this approach, though the reader is wamed that Johnstone speaks of coherent spaces instead of spectral spaces. 6 The approach of Priestley (1970) is somewhat simpler, and also (though this is hindsight) sympathetic to the idea from philosophical logic that a "proposition" is a hereditary subset A of an information frame (U, 1:), i.e., if a E A and a I: f3 then f3 E A. Priestley introduced ordered Stone spaces T = (U,I:, {ad iEI), where I: is a partial order on U, (U, {Oi) iEI) is a compact topological space, which is totally order disconnected, i.e., every two points a ~ f3 can be separated by a hereditary clopen set, i.e., there exists a hereditary clopen set C such that a E C and f3 ¢ C. We shall call these Priestley spaces. To smooth things out Priestley considers only bounded lattices, that is, those which have a least element 0 and a greatest element 1. Theorem 13.4.1 (Priestley) Every bounded distributive lattice is isomorphic to the lattice of clopen hereditmy subsets of a Priestley space. Priestley's theorem builds on Stone's representation of distributive lattices (see Theorem 8.6.1) and is essentially the same as for Boolean algebras, except that we let U be the set of proper prime filters P, and let ~ restricted to U be the required partial order. We set h(a) = {P : a E P}. We note that this nicely generalizes the Boolean case (since for a Boolean algebra, maximal filters and proper prime filters are the same), and the proof that h is an isomorphism is even slightly easier since we do not have to check that h( -a) = U - h(a). Unfortunately, the absence of Boolean complement makes the duality part of the theorem much less obvious, since we can no longer so easily see that the set-theoretic complement of a basic open set is also a basic open set by simply observing that U - h(a) = h( -a). Accordingly we define a new class of subbasic sets: S = {h(a): a E L} U {U - h(a): a E L}. Before showing that the topology given by S is compact, we find it interesting to reformulate compactness. Definition 13.4.2 A space is symmetrically compact if given any collection of closed sets {Ci} iEI and any collection of opens sets {OJ} jEJ such that niEI Ci ~ U jEJ OJ, thenfor some finite I' ~ I, J' ~ J, niEI' C; ~ UjEJ' OJ.

DUALITY FOR DISTRIBUTIVE LATTICES

439

Proof First note that symmetric compactness includes both U-compactness and ncompactness as special cases. We examine the case of U-compactness, leaving the case of n-compactness to the reader. Suppose that UjEJ OJ = U. Then no matter which collection of closed sets we choose, niEI Ci ~ U jEJ OJ. Choose the empty collection, setting I = 0. By symmetric compactness, for some finite I' ~ 0, J' ~ J, niEI'Ci ~ U JEJ' OJ. But then I' = 0, and since the intersection of an empty collection of sets gives U, we have U ~ U JEJ' OJ, and so we have our finite subcover. To see that compactness implies symmetric compactness, we observe that if niEI Ci ~ UjEJ OJ, then (U - niEl C;) U UjEJ OJ = U, i.e., by De Morgan, UiEI(U Ci) U U 'EJ OJ = U. Since U - C; is always an open set, this means that we have a cover usIng the sets U - Ci and the sets OJ. So by compactness there must be a finite subcover, and so for some finite I' ~ I, J' ~ J, UiEl'(U - Ci) U UjEJI OJ = U. And so niEI' Ci ~ UjEJI OJ, as needed. The following generalize the prime filter separation principle (Lemma 8.6.2) in various ways. The reader may also want to consult the end of Chapter 3. Corollary 13.4.6 greatly simplifies the argument for compactness from the standard arguments using mere prime filter separation.? D

Lemma 13.4.4 (Maximal filter-ideal pair principle) Let L be a lattice (not necessarily distributive) and let F be a filteT; I an ideal, and assume that F n I = 0. Then there exists a maximal (disjoint) filter-ideal pair (P, Q) with P ;2 F and Q ;2 I. Proof Let E = {(F', I') : F' is a filter;2 F, and I' is an ideal ;2 G, and F' n I' = 0}. Pmtially order E by componentwise inclusion: (F', I') ::::; (F", I") iff F' ~ F" and I' ~ I". It is easy to apply Zom's lemma to obtain a maximal member (P, Q), noting that the union of a chain of filters is a filter, and similarly with the union of a chain of ideals. It is easy to see that this is a maximal filter-ideal pair simpliciter (and not just maximal in E). D

This pair is disjoint in the sense that P n Q = 0. When a pair is not disjoint, we shall speak of it as inconsistent. We next argue that the pair is exhaustive in the sense thatP U Q = L. Corollary 13.4.5 When L is distributive, then P U Q = L. Proof Thus suppose we have an element a that cannot be thrown into either P or Q without causing it to be the case that the resultant pair is inconsistent. Thus both ([P, a), Q) and (P, (a, Q]) are inconsistent, where [P, a) is the filter generated from P U {a} and (a, Q] is the ideal generated from Q U {a}. Using properties of filter and ideal generation, we can see that there must be inequalities of the form PI /\ ... /\Pm /\a ::::; x ::::; ql V ... V qll and of the form P'1 /\ ... /\ P;Il' ::::; Y ::::; a V q; V ... V q;l" where x and Y are elements that, respectively, make the two pairs inconsistent, and all of the P terms

Theorem 13.4.3 Symmetric compactness is equivalent to compactness. 6Vickers (1989, p. 121) says thatIohnstone's usage is common in domain theory.

7The basic construction may be found in a number of places (cf. Dunn 1995b, note 3), but the most pertinent reference is to Urquhart (1979), who used it in his extension of Priestley's results to lattices in general.

REPRESENTATIONS AND DUALITY

440

EXTENSIONS OF STONE'S AND PRIESTLEY'S RESULTS

come from the filter P, and all of the q telms come from the ideal Q. We can make this uniform by throwing all of the p terms together into one big meet, and similarly throwing all of the q terms into one big join, obtaining two inequalities:

From these we obtain p ~ p /\ (a (P, Q) is not disjoint after all.

V

P /\ a

~

q,

p

~

a V q.

q)

~

(p /\ a) V q

~

q, and so q E P. Thus the pair

0

Corollary 13.4.6 (Prime filter principle) Let L be a distributive lattice, let F be a = 0. Then there exists a prime filter P :2 F

filter, I an ideal, and assume that F n I such that P n I = 0.

Proof We show that the P from the maximal filter-ideal pair (P, Q) is a prime filter. Suppose a V b E P, but both a, b ¢ P. Then by the previous corollary, both a, b E Q, and since Q is an ideal, then a V b E Q. But then a V bE P n Q, contrary to its construction

using Lemma 13.4.4.

0

Remark 13.4.7 Corollary 13.4.6 can be understood as saying that every disjoint filterideal pair can be extended to a prime filter-ideal pair, since Q can dually be argued to be a prime ideal (or we can recall that L - P = Q is a prime ideal).

441

Lemma 13.4.10 Sets ofthefonn hex) are closed under finite unions and intersections. Similarly with sets ofthefonn U - hex). Proof hex) n hey) = hex /\ y), hex) u hey) = hex V y). (U - hex)) n (U - hey)) = U - (h(x) u hey)) = U - hex V y), (U - hex)) u (U - hey)) = U - (h(x) n hey)) = U-h(x/\y). 0

Lemma 13.4.11 The clopen sets are precisely those sets which are finite unions of sets of the form hex) n (U - hey)). Proof Let 0 be a clopen set. Then it is a union of finite intersections of sets of the form hex) or U - hey). Since 0 is a closed subset of a Hausdorff space, we can take this to be a finite union. Collect together the sets of the two separate forms, obtaining hex]) n ... n h(Xk) n (U - hey])) n ... n (U - h(y/)). By Lemma 13.4.10 this is of the form h(x]/\ ... /\ Xk) n U - hey] V .,. V y/). 0

Lemma 13.4.12 Every set of the form hex) is hereditmy, and evelY set of the form U - hey) is dual hereditary. Proof It is obvious that hex) is hereditary, for suppose that p] ~ I7. and p] E hex), i.e., x E Pl. Then x E I7., i.e., I7. E hex). It is equally obvious U - hey) is dual hereditary. Thus suppose that PI ~ I7. and I7. ¢ hey), i.e., y ¢ I7., then y ¢ I7., i.e., P2

¢ hey).

0

Lemma 13.4.8 If nXEX hex) ~ UYEY hey), then for some finite X' ~ X, yl ~ y,

Theorem 13.4.13 Every hereditmy clopen set 0 is of the form hex).

nXEX' hex) ~ UYEY' hey)·

Proof First note that if 0 is empty, then 0 = h(O), where 0 is the least element of the lattice. So pick a particular P E O. Consider U - O. For all prime filters Q E U - 0, P g; Q since 0 is hereditary. Hence for some a E L, a E P and a ¢ Q. This means that P E h(a) and Q E U - h(a). So we have shown that each Q E U - 0 is a member of U - h(a) for some a E P, a ¢ Q. SO then 0 is covered by a collection of open sets

Proof We proceed contrapositively, supposing that for every finite X' ~ X, yl ~ y, nXEX' hex) g; UYEY' hey). We show that nXEX hex) g; UYEY hey).

We first observe that our supposition means that for every finite X' ~ X, Y' ~ y, /\ X' VY'. For suppose to the contrary that /\ X' ~ Vyl. Then if P E nXEX' hex), then Vx E X', P E hex), i.e., x E P. Since P is a filter, first /\ X' E P, and then VY' E P. Since P is prime, 3y E yl such that yEP, i.e., P E hey), and so P E UyEY' hey). Thus nXEX' hex) ~ UyEY' hey), contrary to our hypothesis. The prime filter principle gives us a prime filter P such that X ~ P and Y ~ U - P. Q = U - P is a prime ideal. Since X ~ P, this means Vx E X, P E hex), and so P E nXEX hex). But P ¢ UYEY hey), for otherwise 3y E Y such that P E hey), i.e., yEP, contradicting Y ~ U - P. Thus we have shown nXEX hex) g; UyEY hey), as desired. 0

i

of this form. U - 0 is closed and hence by Corollary 13.2.2 it is covered by a finite subunion: U - 0 ~ Ui
Theorem 13.4.9 The topology generated from S as a subbase is compact. Proof By the Alexander subbase theorem, we know that it suffices to consider just the subbasic sets, i.e., sets of the form hey) or U - hex). Suppose then that UXEX(U hex)) u UYEY hey) = U. Then nXEX hex) = U - UxEX(U - hex)) ~ UyEY hey). By the lemma there exist finite X' ~ X, yl ~ Y such that nXEX' hex) ~ UYEY' hey). But

then U - nXEX' hex) u UYEY' hey)

= U,

i.e., UXEX'(U - hex)) u UYEY' hey)

= U. o

13.5

Extensions of Stone's and Priestley's Results

This is a largely bibliographic report, and we do not attempt to define all of the terminology used but hope, nonetheless, to convey something of the flavor of the results we report. Johnstone (1982) is a high-level presentation of results relating to Stone's representation in the context of category theory. Priestley's representation for distributive

442

REPRESENTATIONS AND DUALITY

lattices has been extended in several directions. Urquhart (1978) extended it to Ockham lattices, i.e., bounded distributive lattices with a dual homomorphic operator (a negation operator satisfying the De Morgan laws and switching the bounds), generalizing De Morgan lattices. A second extension was by Martinez (1988) for Wajsberg algebras, which coincide with the bounded commutative BCK-algebras (the algebraic models of BCK logic). Dosen (1989a) gave a duality result for modal algebras. Goldblatt (1989) showed how the Priestley representation of distributive lattices could be extended to distributive lattices with operators, with a duality result similar to Priestley's for distributive lattices. Hansoul (1983) had previously shown how the original Jonsson-Tarski representation of Boolean algebras with operators can be extended to include a duality result of the original type of Stone (using spectral spaces). It is worth noting that Goldblatt extended the Jonsson-Tarski representation, not just in supplying duality, but also in replacing Boolean algebras with distributive lattices, as well as allowing operators that distribute over meet in each of their places, as well as those of the J onsson-Tarski type that distribute over join in each of their places. But he did not, as with distributoids, allow operators that mix and match, distributing over meet in some places and join in others. We should also mention Sambin and Vacarro's (1988) work on duality for modal algebras. Goldblatt (1975) published a representation theorem for ortholattices, largely driven by logical motivation,S which was linked in Goldblatt (1974) to a semantic analysis of orthologic. An ortholattice is a bounded lattice with an orthocomplement operation, where the latter is an antitonic map.., with the involution property (..,..,a = a, for any element of the lattice) and satisfying a A..,a = 0 and a V..,a = 1. The representation relies on the fact that orthonegation satisfies the De Morgan laws, so that joins are definable by means of meets. Thereby, in Goldblatt's theorem, what is needed is a representation of meets, which is canonically done by intersections, and a representation of the orthocomplement operation. Since orthocomplementation cannot be interpreted as set-theoretic complementation (else joins would be interpreted as unions and the lattice would be distributive), Goldblatt's idea was to interpret the negation operator by means of an irreflexive, symmetric relation ..i~ U 2 of orthogonality on the points of the space (the filters of the lattice), defined by a..i f3 iff 3a E a(..,a E f3). Where a..i X means that for all f3 E X, a..i f3, the relation ..i induces an operation on subsets of the space, defined by ..,X = {a : a..i X}. With the usual Priestley topology imposed on the dual space, Goldblatt concludes by showing that the original ortholattice is isomorphic to the lattice of clopen, ..i-regular subsets of the space, where a subset U is ..i-regular iff ..,..,U = U. Dunn (1996) shows how Goldblatt-style representations can be extended to various other logics by playing with the conditions on ..i. This is based on a representation of Galois connections from Dunn (1993a). We give some examples. Dropping just symmetry gives two "complements": ..,X = 1. X = {a: a ..iX}, and ",X = Xl. = {a: X ..ia}, and these behave as a Galois connection. Dropping just irreflexiveness gives the De Morgan complement of relevance logic. Dunn (1993b) shows the relationship of the SGoldblatt's interest in this paper is only with representation, and not full duality, which he does not investigate.

EXTENSIONS OF STONE'S AND PRIESTLEY'S RESULTS

443

Goldblatt-style semantics using ..i to the Routley-Meyer style semantics using their *-operator. Urquhart (1979) proposed a topological representation of general lattices. The dual space, in Urquhart's representation, is a doubly ordered space (X,:Sl, :S2), consisting of maximal, disjoint filter-ideal pairs, and where :Si is coordinatewise inclusion of pairs. To represent joins, Urquhart defined a pair of maps r and fJ on sets of disjoint, maximal filter-ideal pairs increasing in the I-order and 2-order, respectively, by rU = {x : Vy(x :S2 y :::} Y ¢ U)} and fJU = {x : Vy(x :Sl y :::} Y ¢ U)}. With representation defined by a 1--+ ua = A = {x EX: a E Xl}, where a is an element of the lattice and Xl is the filter part of the filter-ideal pair x, Urquhart topologized X via the subbase

s

= {-ua : a E L} U {-rub: bEL}

and characterized the image of the representation function II as the collection of all doubly closed (both A and r A closed in the topology) stable sets (stability meaning that fJr A = A). A doubly ordered space (X, :Sl, :S2), where :Si are pre-orders on X, is called doubly disconlZected, if for any points X and y, if x $1 y, then 3U(U is doubly closed x E U, Y ¢ U) and if x $2 y, then 3U(U is doubly closed, x E rU, y ¢ rU). An L-space is then defined as a doubly ordered set (X, :Sl, :S2) with a topology such that:

r,

r

(1) X is a doubly disconnected compact space; (2) if U, V are doubly closed, then r(U n V) and fJ(rU n rV) are closed; and (3) the family {-U : U is doubly closed, stable} u {-r V : V doubly closed, stable} forms a subbase for the topology on X.

r

Urquhart's representation theorem, however, is not functorial (homomorphisms are not preserved by the representation) and cannot be extended to a full duality, as was observed by Allwein (1992), who enlarged the canonical space to include all disjoint filter-ideal pairs (not just the maximal ones) and proved full (functorial) duality in his dissertation. In an as yet unpublished manuscript, Allwein and Hartonas extended the duality to one for the lattice of congruences (strengthening the relevant result in Urquhart) and the lattice of sublattices. They also showed how to represent lattices by sheaf spaces. Allwein and Dunn (1993) and Allwein (1992) used the Urquhart representation of lattices as the basis for adding additional (binary) gaggle-type operators to a lattice that is not necessarily distlibutive, thereby providing (among other things) a "Kripke-style" semantics for linear logic. Hartonas (1993a, 1993b) does explicitly consider operations of arbitrary degree, but employs a completely different style of representation. Yet another representation is considered in BimbO (1999, 2000). Instead of maximally consistent pairs, minimally inconsistent ones are used, and implication and fusion are added. While the frame is defined more in the spirit of the Urquhart representation using a doubly ordered set, the whole representation (and especially the canonical model) is more closely related to the Hartonas and Dunn representation mentioned below. (Duality has not been investigated for this representation.)

444

REPRESENTATIONS AND DUALITY

A different kind of representation of lattices is to be found in Hartonas and Dunn (1993), together with full duality. The idea is based on an observation of Birkhoff (1948). Birkhoff defines a polarity as in effect a triple (U, U', 1..), with 1.. ~ U xU'. For X ~ U we define Xl. = {a' E U' : X1..a'} , and for X' ~ U' we define l. X = {a' E U' : X 1..a'}. Birkhoff observes that this establishes a Galois connection between the power set of U and the power set of U'. For X ~ U we call Xl. Galois closed if X = l.(Xl.), and dually for X' ~ U'.9 Birkhoff further observes then that the Galois closed subsets of U (there is a dual fact about U') form a lattice, with X f\ Y = X n Y and X v Y = l.(Xl. n yl.). Hartonas and Dunn show, generalizing the representation of ortholattices to be found in Dunn (l993a), that every lattice can be embedded in such a lattice of Galois closed subsets generated by a polarity.

REFERENCES Abbott, J. C. (1967). Semi-boolean algebra. Matematicki Vesnik, 4:177-198. Abbott,1. C. (1969). Sets, Lattices, and Boolean Algebras. Allyn and Bacon, Boston. Abbott, J. C. (1976). Orthoimplication algebras. Studia Logica, 35: 173-177. Ajdukiewicz, K. (1935). Die syntaktische Konnexitat. Studia Logica, 1: 1-27. Translated into English in McCall (1967). Allwein, G. (1992). Duality of Algebraic and Kripke Models for Linear Logic. PhD thesis, Indiana University, Bloomington. Allwein, G. and Dunn, J. M. (1993). Kripke models for linear logic. loumal of Symbolic Logic, 58:514-545. Anderson, A. R. and Belnap, Jr., N. D. (1975). Entailment. The Logic of Relevance and Necessity, Volume 1. Princeton University Press, Princeton, N1. Anderson, A. R., Belnap, Jr., N. D. and Dunn, J. M. (1992). Entailment. The Logic of Relevance and Necessity, Volume 2. Princeton University Press, Princeton, NJ. Angell, R. B. (1962). A propositional logic with subjunctive conditionals. loumal of Symbolic Logic, 27:327-343. Balbes, R. and Dwinger, P. (1974). Distributive Lattices. University of Missouri Press, Columbia, MO. Belnap, Jr., N. D. (1967). Intensional models for first degree formulas. loumal of Symbolic Logic, 32: 1-22. Belnap, Jr., N. D. (1982). Display logic. loumal of Philosophical Logic, 11:375--417. Bialnicki-Birula, A. and Rasiowa, H. (1957). On the representation of quasi-boolean algebras. Bulletin de l'Academie Polonaise des Sciences, 5:259-261. BimbO, K. (1999). Substructural Logics, Combinatory Logic and A-calculus. PhD thesis, Indiana University, Bloomington. BimbO, K. (2000). Semantics for the structurally free logic LC+. Manuscript, 14 pp. Bimb6, K. and Dunn, 1. M. (1998). Two extensions of the structurally free logic LC. Logic loumal of IGPL, 6:403--424. Birkhoff, G. (1935). On the structure of abstract algebras. Proceedings of the Cambridge Philosophical Society, 31 :433--454. Birkhoff, G. (1940). Lattice TheOlY. American Mathematical Society, Providence, RI. Birkhoff, G. (1944). Subdirect unions in universal algebras. Bulletin of the American Mathematical Society, 50:764-768.

9Urquhart's notion of "stability" is a special case of Galois closure since the pair (fl, r) constitute a Galois connection.

Birkhoff, G. (1948). Lattice TheOlY (2nd edition). American Mathematical Society, Providence, RI.

446

REFERENCES

REFERENCES

447

Birkhoff, G. (1967). Lattice Theory (3rd edition). American Mathematical Society, Providence, RI.

Davey, B. A. and Priestley, H. A. (1990). Intmduction to Lattices and Order. Cambridge University Press, Cambridge.

Birkhoff, G. and Frink, O. (1948). Representations of lattices by sets. Transactions of the American Mathematical Society, 64:299-316.

Dedekind, R. (1872). Continuity and irrational numbers. In Essays on the TheOlY of Numbers. Open Court, Chicago. W. W. Beman translation of Dedekind's "Stetigkeit und irrationale Zahlen," first published in 1901.

Birkhoff, G. and von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37:823-843. Blok, W. 1. and Pigozzi, D. (1986). Proto algebraic logics. Studia Logica, 45:337-369. Blok, W. J. and Pigozzi, D. (1989). Algebraizable Logics. Memoirs of the American Mathematical Society 396. Amelican Mathematical Society, Providence, RI.

Dipert, R. R. (1978). Development and Crisis in Late Boolean Logic: The Deductive Logics of Peirce, levons and Schroder. PhD thesis, Department of Philosophy, Indiana University, Bloomington. Dosen, K. (1988). Sequent systems and groupoid models I. Studia Logica, 47:353-385.

Blok, W. J. and Pigozzi, D. (1992). Algebraic semantics for universal Hom logic without equality. In Romanowska, A. and Smith, 1. D. H. (eds), Universal Algebra and Quasigmup Theory, pp. 1-56. Heldermann Verlag, Berlin.

Dosen, K. (1989a). Duality between modal algebras and neighborhood frames. Studia Logica,48:219-234.

Bloom, S. L. (1976). Varieties of ordered algebras. lournal of Computer and System Science, l3:200-212.

Dummett, M. (1959). A propositional calculus with denumerable matrix. Journal of Symbolic Logic, 24:97-106.

Blyth, T. S. and Janowitz, M. F. (1972). Residuation Theory. Pergamon Press, New York.

Dunn, 1. M. (1966). The Algebra of Intensional Logics. PhD thesis, University of Pittsburgh.

Boole, G. (1847). Mathematical Analysis of Logic. Being an Essay towards a Calculus of Deductive Reasoning. Macmillian, Barclay & Macmillian, London.

Dunn, 1. M. (1970). Algebraic completeness results for R-mingle and its extensions. Journal of Symbolic Logic, 35: 1-l3.

Camap, R. (1943). Formalization of Logic. Studies in Semantics. Harvard University Press, Cambridge, MA.

Dunn, J. M. (1976). Intuitive semantics for first-degree entailment and 'coupled trees'. Philosophical Studies, 29: 149-168.

Camap, R. (1947). Meaning and Necessity: A Study in Semantics and Modal Logic. University of Chicago Press, Chicago.

Dunn, 1. M. (1986). Relevance logic and entailment. In Gabbay, D. and Guenthner, F. (eds), Handbook of Philosophical Logic, Volume 3, pp. 117-224. D. Reidel, Dordrecht.

Certaine, J. (1943). Lattice-Ordered Gmupoids and Some Related Pmblems. PhD thesis, Harvard University. Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge University Press, Cambridge. Church, A. (1956). Introduction to Mathematical Logic, Volume 1. Princeton University Press, Princeton, NJ. Copi, I. M. (1979). Symbolic Logic (5th edition). Macmillian, New York. Curry, H. B. (1963). Foundations of Mathematical Logic. McGraw-Hill, New York. Czelakowski, J. (1980a). Model-Theoretic Methods in Methodology of Pmpositional Calculi. Institute of Philosophy and Sociology of the Polish Academy of Sciences, Warsaw. Czelakowski, J. (1980b). Reduced products of logical matrices. Studia Logica, 39:1943. Czelakowski,1. (1981). Equivalentiallogics I, II. Studia Logica, 40:227-236, 335-372. Czelakowski, J. (1983). Some theorems on structural entailment. Studia Logica, 42:417429.

Dosen, K. (1989b). Sequent systems and groupoid models II. Studia Logica, 48:41-65.

Dunn, J. M. (1991). Gaggle theory: An abstraction of Galois connections and residuation with applications to negation and various logical operations. In van Eijck, J. (ed.), Logics in AI. Pmceedings of the Eumpean Workshop lELIA 1990, Lecture Notes in Computer Science 478, pp. 31-51. Springer-Verlag, Berlin. Dunn, 1. M. (l993a). Partial-gaggles applied to logics with restricted structural rules. In Dosen, K. and Schroeder-Heister, P. (eds), Substructural Logics, pp. 63-108. Clarendon Press, Oxford. Dunn, 1. M. (1993b). Perp and star: two treatements of negation. Philosophical Perspectives (Philosophy of Language and Logic), 7:331-357. Dunn, J. M. (l995a). Gaggle theory applied to intuitionistic, modal and relevance logic. In Max, I. and Stelzner, W. (eds), Logik und Mathematik. Frege-Kolloquium lena 1993, pp. 335-368. W. de Gruyter, Berlin. Dunn, J. M. (1996). Generalized ortho-negation. In Wansing, H. (ed.), Negation: A Notion in Foclls, pp. 3-26. W. de Gruyter, Berlin. Dunn, J. M. (1999). A comparative study of various semantical treatments of negation: A history of formal negation. In Gabbay, D. and Wansing, H. (eds), What Is Negation?, pp. 23-61. Kluwer Academic Publishers, Dordrecht.

448

REFERENCES

Dunn, 1. M. and Meyer, R. K (1971). Algebraic completeness results for Dummett's LC and its extensions. Zeitschrift fiir Mathematische Logik und Grundlagen der Mathematik, 17:225-230. Dunn, J. M. and Meyer, R. K (1997). Combinatory logic and structurally free logic. loumal of the Interest Group in Pure and Applied Logic, 5:505-537.

REFERENCES

449

Halmos, P. R. (1963). Lectures on Boolean Algebras. Van Nostrand Mathematical Studies #1. D. Van Nostrand, Princeton, NJ. Hansoul, G. (1983). A duality for boolean algebras with operators. Algebra Uiversalis, 17:34-49.

Esakia, L. and Meskhi, V. (1977). Five critical modal systems. Theoria, 43:52-60.

Hardegree, G. M. (1975). Stalnaker conditionals and quantum logic. lournal of Philosophical Logic, 4:399-42l.

Everett, C. 1. (1944). Closure operators and Galois theory in lattices. Transactions of the American Mathematical Society, 55:514-525.

Harrop, R. (1958). On the existence of finite models and decision procedures for propositional calculi. Proceedings of the Cambridge Philosophical Society, 54: 1-13.

Fitch, F. B. (1952). Symbolic Logic. Ronald Press, New York.

Harrop, R. (1965). Some structure results for propositional calculi. Journal of Symbolic Logic, 30:271-292.

Font, 1. M. and Jansana, R. (1996). A General Algebraic Semantics for Sentential Logics, Lecture Notes in Logic 7. Springer-Verlag, Berlin. Frege, G. (1879). Begriffsschrift. In van Heijenoort, J. (ed.), From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, pp. 5-82. Harvard University Press, Cambridge, MA Trans. S. Bauer-Mengelberg.

Hartonas, C. (1993a). Algebraic and Kripke Semantics for Substructural Logics. PhD thesis, Indiana University, Bloomington. Hartonas, C. (1993b). Lattices with additional operators. Technical Report IULG-93-27, Indiana University Logic Group.

Frege, G. (1892). Uber Sinn und Bedeutung. In Feigl, H. and Sellars, W. (eds), Readings in Philosophical Analysis, pp. 85-102. Appleton Century Crofts, New York. Trans. Herbert Feigl.

Hartonas, C. and Dunn, J. M. (1993). Duality theorems for partial orders, semilattices, Galois connections and lattices. Technical Report IULG-93-26, Indiana University Logic Group.

Fuchs, L. (1963). Partially Ordered Algebraic Systems. Pergamon Press, New York.

Henkin, L. A (1954). Boolean representation through propositional calculus. Fundamenta Mathematicae, 41:89-96.

Gabbay, D. (1976). Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics. D. Reidel, Dordrecht. Gentzen, G. (1934-35). Untersuchungen tiber das logische Schliessen. Mathematische Zeitschrift, 39: 176-210, 405-43l. English translation ("Investigations into logical deduction") by M. E. Szabo in Szabo, M. E. (ed.), The Collected Papers of Gerhard Gentzen, pp. 132-213. North-Holland, Amsterdam, 1969.

Henkin, L. A, Monk, 1. D. and Tarski, A (1971). Cylindric Algebras, Part 1. Studies in Logic and the Foundations of Mathematics 64. North-Holland, Amsterdam and London.

Gericke, H. (1963). Theorie der Verbiinde. Bibliographishes Institut, Mannheim.

Hermes, H. (1938). Semiotik. Eine Theorie der Zeichengestalten als Grundlage fUr Untersuchungen von formalisierten Sprachen. Forschungen zur Logile und zur Grundlegung der exakten Wissenschaften, n.s. 5:22.

Gierz, G., Hofman, K H., Keimel, K, Lawson, J. D., Mislove, M. and Scott, D. S. (1980). A Compendium of Continuous Lattices. Springer-Verlag, Berlin.

Heyting, A (1930). Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der preussichen Akademie der Wissenschaften, pp. 42-56, 57-71, 158-169.

GOdel, K (1933). Zum intuitionistischen Aussagenkalktil. Ergebnisse eines mathematischen Kolloquiums, 4:40.

Hochster, M. (1969). Prime ideal structure in commutative rings. Transactions of the American Mathematical Society, 142:43-60.

Goldblatt, R. I. (1974). Semantic analysis of orthologic. loumal of Philosophical Logic, 3:19-35.

Hughes, G. E. and Cresswell, M. (1968). An Introduction to Modal Logic. Methuen, London.

Goldblatt, R. I. (1975). The Stone space of an ortholattice. Bulletin of the London Mathematical Society, 7:45-48.

Hughes, G. E. and Cresswell, M. (1996). A New Introduction to Modal Logic. Routledge, London and New York.

Goldblatt, R. I. (1989). Varieties of complex algebras. Annals of Pure and Applied Logic, 44: 173-242.

Huntington, E. (1904). Sets of independent postulates for the algebra of logic. Transactions of the American Mathematical Society, 5:288-309.

Gratzer, G. (1979). Universal Algebra (2nd edition). Springer-Verlag, New York.

Jaskowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1:532.

Hallden, S. (1951). On the semantic non-completeness of certain Lewis calculi. lournal of Symbolic Logic, 16: 127-129. Halmos, P. R. (1962). Algebraic Logic. Chelsea, New York.

Jevons, W. S. (1864). Pure Logic or the Logic of Quality Apartfrom Quantity. E. Stanford, London.

450

REFERENCES

REFERENCES

451

Johansson, I. (1936). Der Minimalkalkiil, ein reduzierte intuitionistischer Formalismus. Compositio Mathematica, 4: 119-136.

Lincoln, P., Mitchell, 1., Scedrov, A. and Shankar, N. (1992). Decision problems for propositional linear logic. Annals of Pure and Applied Logic, 56:239-311.

Johnstone, P. T. (1982). Stone Spaces. Cambridge University Press, Cambridge.

Los,1. (1944). On Logical Matrices, Travaux de la Societe des Sciences et des Lettres de Wroclaw, Series B vol. 19. Wroclaw. Translated into English in Ulrich (1968).

J6nsson, B. and Tarski, A. (1951). Boolean algebras with operators. American Journal of Mathematics, 73:891-939. Kalish, D. and Montague, R. (1964). Logic: Techniques of Formal Reasoning (2nd edition). Harcourt Brace Javanovich, New York. Kelley, J. L. (1955). General Topology. Van Nostrand, New York. Kiss, S. (1961). An Introduction to Algebraic Logic. Privately printed, Westport, CT. Kneale, W. C. (1956). The province of logic. In Lewis, H. (ed.), Contemporary British Philosophers, Third Series, pp. 237-261. George Allen & Unwin, London. Kripke, S. A. (1959). A completeness theorem in modal logic. Journal of Symbolic Logic, 24:1-15. Kripke, S. A. (1963a). Semantic considerations on modal logic. Acta Philosophica Fennica, 16:83-94. Kripke, S. A. (1963b). Semantical analysis of modal logic - I. Zeitschrijt filr mathematische Logik und Grundlagen der Mathematik, 9:67-96. Kripke, S. A. (1965). Semantical analysis of modal logic - II. In Addison, 1., Henkin, L. and Tarski, A. (eds), The Theory of Models, pp. 206-220. North-Holland, Amsterdam. Kuratowski, C. (1958). Topologie I (2nd edition). PWN (Polish Scientific Publishers), Warsaw. Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly, 65:154-170. Lambek, J. (1981). The influence of Heraclitus on modem mathematics. In Agassi, 1. and Cohen, R. S. (eds), Scientific Philosophy Today, pp. 111-121. D. Reidel, Dordrecht. Lemmon, E. 1. (1965). Beginning Logic. Nelson, London. Lemmon, E. 1. (1966). Algebraic semantics for modal logics, I and II. Journal of Symbolic Logic, 31:46-65,191-218. Lemmon, E. 1. (1977). An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, No. 11. Basil Blackwell, Oxford. In collaboration with D. Scott, edited by K. Segerberg. Lewis, C. I. (1918). A Survey of Symbolic Logic. University of California Press, Berkeley and Los Angeles. Lewis, c.I. and Langford, C. H. (1932). Symbolic Logic. Century, New York. Lewis, D. K. (1973). Counteifactuals. Library of Philosophy and Logic. Basil Blackwell, Oxford.

Los, J. (1957). Remarks on Henkin's papers: Boolean representation through propositional calculus. Fundamenta Mathematicae, 44:82-93. Los,1. and Suszko, R. (1957). Remarks on sentential logics. Indagationes Mathematicae, 20:177-183. Lukasiewicz, J. (1910). 0 zasadzie sprzecznosci uArystotelesa. Akademia Umiej~tnosci, Krak6w. Lukasiewicz, J. (1913). Die logischen Grundlagen der Wahrscheinlichkeitsrechnung. Akademie der Wissenschaften Viktor Oslawski'scher Fonds, Krak6w. Lukasiewicz, J. (1920). On three valued logic. Ruch Filozojiczny, 5:169-170. In Polish, translated into English in McCall (1967). Lukasiewicz, J. (1930). Philosophical remarks on many-valued systems of propositional logic. Comptes Rendus des Seances de fa Societe des Sciences et les Lettres de Varsovie, 23:51-77. In German, translated into English in McCall (1967). Lukasiewicz, J. and Tarski, A. (1930). Untersuchungen iiberder Aussagenkalktil. Comptes Rendus des Seances de la Societe des Sciences et les Lettres de Varsovie, 23:30-50. In German, translated into English in Tarski (1956). MacLane, S. (1971). Categories for the Working Mathematician. Springer-Verlag, New York. Maksimova, L. (1972). Predtablichnge superintuitionistskie logiki. Algebra i Logika, 11:558-570. (See Algebra and Logic, 1974, pp. 308-314, for English translation: Pretabular superintuitionist logic.) Maksimova, L. (1975). Predtablichnge rasshirenia logiki S4 Lewisa. Algebra i Logika, 14:28-55. (See Algebra and Logic, 1975, pp. 16-33 for English translation: Pretabular extensions of Lewis S4.) Marcev, A. I. (1973). Algebraic Systems. Springer-Verlag, Berlin. Malinowski, G. (1993). Many- Valued Logics, Oxford Logic Guides 75. Clarendon Press, Oxford. Martin, E. P. and Meyer, R. K. (1982). Solution to the P-W problem. Journal of Symbolic Logic, 47:869-886. Martinez, N. G. (1988). The Priestley duality for Wajsberg algebras. Studia Logica, 49:31-46. McCall, S. (1966). Connexive implication. Journal of Symbolic Logic, 31:415-433. McCall, S. (1967). Polish Logic: 1920-1939. Clarendon Press, Oxford. McKinsey, J. C. C. (1941). A solution ofthe decision problem for the Lewis systems S2 and S4, with an application to topology. Journal of Symbolic Logic, 6:117-134.

452

REFERENCES

REFERENCES

453

McKinsey, 1. C. C. and Tarski, A. (1944). The algebra of topology. Annals of Mathematics,45:141-191.

Schroder, E. (1890-1905). Vorlesungen iiba die Algebra del' Logik, 3 vols. Teubner, Leipzig.

McKinsey, J. C. C. and Tarski, A. (1948). Some theorems about the sentential calculi of Lewis and Heyting. Journal of Symbolic Logic, 13:1-15.

Scott, D. S. (1973). Background to formalization. In Leblanc, H. (ed.), Truth, Syntax and Modality. Proceedings of the Temple University Conference on Alternative Semantics, pp. 244-273. North-Holland, Amsterdam.

Meyer, R. K. and McRobbie, M. A. (1982). Multisets and relevant implication. Australasian Journal of Philosophy, 60:265-281. Meyer, R. K. and Routley, R. (1972). Algebraic analysis of entailment (I). Logique et Analyse, n.s. 15:407-428.

Scott, D. S. (1974). Completeness and axiomatizability in many-valued logic. In L. Henkin et al. (eds), Proceedings of the Tarski Symposium/Proceedings of Symposia in Pure Mathematics, vol. 25, pp. 411-435. American Matllematical Society.

Ono, H. (1993). Semantics for substructural logics. In Dosen, K. and Schroeder-Heister, P. (eds), Substructural Logics, pp. 259-291. Clarendon Press, Oxford.

Scroggs, S. J. (1951). Extensions of the Lewis system 85. Journal of Symbolic Logic, 16:112-120.

Ono, H. and Komori, Y. (1985). Logics without the contraction rule. Journal of Symbolic Logic, 50:169-201.

Segerberg, K. (1971). An Essay In Classical Modal Logic, 3 vols. University ofUppsala Press, Uppsala.

are, O. (1944). Galois connexions. Transactions of the American Mathematical Society, 55:493-513.

Shoesmith, D. and Smiley, T. J. (1971). Deducibility and many-valuedness. Journal of Symbolic Logic, 36(4):610-622.

Popper, K. R. (1947). Logic without assumptions. Proceedings of the Aristotelian Society,47:251-292.

Shoesmith, D. and Smiley, T. J. (1978). Multiple-Conclusion Logic. Cambridge University Press, Cambridge.

Priestley, H. A. (1970). Representation of distributive lattices by means of ordered stone spaces. Bulletin of the London Mathematical Society, 2:186-190.

Shukla, A. (1970). Decision procedures for Lewis system 81 and related modal systems. Notre Dame Journal of Formal Logic, 11(2):141-180.

Priestley, H. A. (1972). Ordered topological spaces and the representation of distributive lattices. Proceedings of the London Mathematical Society, 2:507-530.

Sikorski, R. (1964). Boolean Algebras. Springer-Verlag, Berlin.

Quine, W. V. O. (1961). From A Logical Point of View. Harvard University Press, Cambridge, MA. Quine, W. V. O. (1986). Philosophy of Logic (2nd edition). Harvard University Press, Cambridge and London. Rasiowa, H. (1974). An Algebraic Approach to Non-Classical Logics. North-Holland, Amsterdam. Rasiowa, H. and Sikorski, R. (1963). The Mathematics of Metamathematics (2nd edition). Panstwowe Wydawnictwo Naukowe, Warsaw. Routley, R. and Meyer, R. K. (1972). The semantics of entailment - II, III. Journal of Philosophical Logic, 1:53-73, 192-208.

Simmons, G. F. (1963). Introduction to Topology and Modern Analysis. McGraw-Hill, New York. Smiley, T. 1. (1959). Entailment and deducibility. Proceedings of the Aristotelian Society, n.s. 59:233-254. Stalnaker, R. C. (1968). A theory of conditionals. In Rescher, N. (ed.), Studies in Logical Theory. Blackwell, Oxford. Stalnaker, R. C. and Thomason, R. H. (1970). A semantic analysis of conditional logic. Theoria, 36:23-42. Stone, M. H. (1936). The theory of representations for boolean algebras. Transactions of the American Mathematical Society, 40:37-111. Stone, M. H. (1937). Topological representations of distributive lattices and Brouwerian logics. Casopis pro Pestowinz Matematiky a Fysiky, 67:1-25.

Routley, R. and Meyer, R. K. (1973). The semantics of entailment - I. In Leblanc, H. (ed.), Truth, Syntax and Modality. Proceedings of the Temple University Conference on Alternative Semantics, pp. 199-243. North-Holland, Amsterdam.

Strawson, P. F. (1952). Introduction to Logical The01Y. Methuen, London.

Roiltley, R. and Routley, V. (1972). Semantics of first degree entailment. Nous, 6:335359.

Szasz, G. (1963). Introduction to Lattice Theory. Academic Press, New York. Trans. B. Balkay and G. T6th.

Rutherford, D. (1965). Introduction to Lattice Theory. Hafner, New York.

Tarski, A. (1956). Logic, Semantics, Metamathematics. Clarendon Press, Oxford. Translated by 1. H. Woodger.

Sambin, G. and Vaccaro, V. (1988). Topology and duality in modal logic. Annals of Pure and Applied Logic, 37 :249-296.

Suppes, P. (1957). Introduction to Logic. Van Nostrand, Princeton, NJ.

Thomas, I. (1962). Finite limitations on Dummett's LC. Notre Dame Journal of Fonnal Logic, 3:170-174.

454

REFERENCES

Ulrich, D. E. (1968). Matrices for Sentential Calculi. PhD thesis, Wayne State University.

INDEX

Ulrich, D. E. (1970). Decidability results for some classes of propositional calculi (abstract). Journal of Symbolic Logic, 35:353-354. Ulrich, D. E. (1986). On the characterization of sentential calculi by finite matrices. Reports on Mathematical Logic, 20:63-86. Urquhart, A. (1973). An interpretation of many-valued logic. ZeitschriJt filr mathematische Logik und Grundlagen der Mathematik, 19: 111-114. Urquhart, A. (1978). Distributive lattices with a dual homomorphic operation. Studia Logica, 38:201-209. Urquhart, A. (1979). A topological representation theorem for lattices. Algebra Universalis, 8:45-58. Urquhart, A. (1985). Many-valued logic. In Gabbay, D. and Guenthner, R. (eds), Handbook of Philosophical Logic, Volume III, Synthese Library Series vol. 166, pp. 71116. Reidel, Dordrecht. Van Fraassen, B. C. (1971). Formal Semantics and Logic. Macmillan, New York. Vickers, S. (1989). Topology via Logic, Cambridge Tracts in Theoretical Computer Science 5. Cambridge University Press, Cambridge. Wansing, H. (1994). Sequent calculi for normal modal propositional logics. Journal of Logic and Computation, 4:125-142. Ward, M. and Dilworth, R. P. (1939). Residuated lattices. Transactions of the American Mathematical Society, 45:335-354. Wechler, W. (1992). Universal Algebra for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin. Whitehead, A. N. and Russell, B. (1910). Principia Mathematica to *56. Cambridge University Press, Cambridge. Wittgenstein, L. (1921). Tractatus Logico-Philosophicus. Routledge & Kegan Paul, London. Translation by D. F. Pears and B. F. McGuinness, published by Routledge & Kegan Paul in 1961. Wojcicki, R. (1970). Some remarks on the consequence operator in sentential logic. Fundamenta Mathematicae, 68:269-279. Wojcicki, R. (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations. Kluwer Academic Publishers, Dordrecht.

+,202 -,1 1,2 <,60 =,36 >,60 Cn(f"), 188 [S), 117, 120 [cP]' 4 [a), 117, 120

X(Ai),27 D.A(D),252

n,13 U,30 V3,70 n, 19 0,39,82 =,4 ~, 125 ;::,60 1;1,245 E,5

[(a/p),257 [(cP),7

&,3 <--,109 ~, 129 :S;, 1,57 :S;p,220 «,12 0,149 F,170 /,,129 .l,I03 <,359 ~, 75 j,75 II, 28 ~, 129 (Y, 137 ~, 3 1;;;,235 D,149 ~, 12 :J, 1 ;2,44 ,/, 129 0,30 ->,106

f- (V), 200 f-p,218 V, 1,402 11,67,402 h([cP]),7 v(cP, IV), 149 2,6 A/=,23 A,401 Abbott, J. C., 340, 351, 353 absolute completeness, 378 absolute frame, 367 absolute inconsistency, 260, 378 absolutely faithful, 16 absoluteness, 194, 200-202, 223 abstract algebra, I abstract law of residuation, 281, 408, 411 accessibility relation, 277, 361-363, 401-430 appropriate for operators, 401-406 for intuitionistic logic, 424 symmetry of, 416 adequacy of equations for algebras, 54 adverbs, 208 agreement of types, 411 Ajdukiewicz, K., 135 Alexander subbase theorem, 435 algebra, 10-11 and lattices, 71-74 and sets of generators, 14 definable by equations, 36, 39 free, 39-47 method of equivalence classes, 40 of propositions, 144 and algebra of sentences, 144 of sentences, 33, 130-133, 254 Leibniz congruence on, 254 of a zero-order language, 33 of strings, 125 and free semi-groups, 126 as multiple successor algebra, 129 associativity in, 126 of terms, 33 and interpretations, 34 of truth values, 146-148 of words, 34-36 on an operational syntax, 34 polynomials on, 329 simple, 25

456

algebraic compositionality, 141 algebraic logic and universal algebra, 10 algebraic prime factorization theorem, 29-33 algebraic semantics, 8, 167,273 and representation theory, 8 of modal logic, 151 algebraizability, 273-276 and the Leibniz operator, 276 partial, 276 Allwein, G., 405, 431,443 alphabet, 125 alphabetic variant of a sentence, 263 alternation property for H, 387 for S4, 370, 388 Anderson, A. R., 91, 106, 224, 227, 234, 274, 277, 287, 381 Angell, R. B., 5 anti-symmetry, 56 antitonic, 279, 397 A(S),15 appropriateness of an algebra of propositions for an algebra of sentences, 144 of atlases, 156 of matrices, 156 of medleys, 156 articulated concrete residuated partially ordered groupoid, 282 articulated frame, 281, 285 ternary, 421 ASC, 346 assertion, 80 and rule-form permutation, 80 pseudo-, 280 specialized pseudo-, 284 assertional double implicational poset, 283-285 assertional frame, 423 assertional implicational poset, 283 assignment, 34 and interpretations, 35 associated distributive lattice with image operators, 315 associativity in algebras of strings, 126 asymmetric cancellation property, 271 asymmetric consequence, 195-199,322 and explicit consequence, 197 and normal characteristic atlas, 269 completeness for, 195 Galois connection, 195 asymmetric pre-consequence relation, 188 asymmetric sequent calculus, 334, 341 for Boolean logic, 346-348

INDEX

asymmetric tautological implication, 272 asymmetry, 56 atlas, 141, 152,211-214 and logical space, 153 and possible worlds semantics, 153 Lindenbaum, 211 normal, 266-270 Scott, 260 atom of a Boolean algebra, 304 atomic definability, 253 atomic generation, 126 atomic Leibniz congruence, 253 atomic sentences, 131 automorphism, 17 axiom of necessity, 357, 388 axiom of separation, 193 axiomatic calculi for classical logic, 334 for intuitionistic logic, 380 for modal logics, 356 axiomatization, 216-224 axioms, 380 Balbes, R., 67 base for a topology, 434 BC', 350 BC,337 BCK,80 Belnap, N. D., viii, 91, 106,224,227,234,242, 262,274,277,287,381,417 BG, see binary gaggle 422 Bimb6, K., 80, 318, 414, 443 binary calculus, 334 binary consequence, 187 binary frame, 425 binary gaggle, 421 and intuitionistic logic, 421 and modal logic, 421 and positive fragment ofrelevance logic, 423 and relevance logic, 421 embeddable in full binary gaggle, 422 binary implication relation, 191-194 Birkhoff's prime factorization theorem, 30, 369 varieties theorem, 47, 103,246 for ordered algebras, see Bloom 76 Birkhoff, G., viii, 5, 10, 29, 36, 46, 48, 74, 287, 382,396,398,406,426 Blok, W. J., 246, 247, 252, 253, 274 Blok-Pigozzi matrix, 254 Bloom, S. L., 74, 83 Blyth, T. S., 397 Bochvar, D. A., 230, 245 Boole, G., 1-2

INDEX

Boolean algebra, 2, 4,88,92, 122,278,300-301, 306,328,351,371,394,402,414,434 and classical propositional calculus, 3 and completeness of classical propositional logic,

7 and consistency of classical propositional calculus,7 and homomorphisms onto 2, 306 and implication algebras, 350 and semi-Boolean algebras, 350 and subdirect product of 2, 313 and varieties, 328 as a lattice, 88 embedding theorem, 310 filter in, 351 filters in, 302 and congruences, 302 maximal, 302 fundamental embedding, 307 ideals in, 5 and deductive systems, 5 maximal filter principle, 310 representation theorem, 310 with operators, 313, 361, 399, 422, 442 Boolean interpretation, 329, 331 and homomorphisms, 329 Boolean lattice, see Boolean algebra 153 Boolean logic, 321, 322, 328, 336 and Frege logic, 322, 324 and unital Boolean logic, 325 axiomatic calculus for, 337 compactness of, 331-332 complete fragment, 336 Boolean matrix, 174,321 Boolean ring, 5, 352-354 Boolean spaces, 435 Boolean valid, 332 Boolean valuation, 331, 332 Boolos, G., 137 bundle, 152, 273 calculus of derivations, 342 cancellation property, 257-261 asymmetric version of, 260 in asymmetric logic, 261 in symmetric logic, 257 canonical homomorphism, 24 Cantor's enumeration theorem, 138 Cantor, G., 433 Carnap, R., 155 carrier set, 10 Cartesian product, 26 of a family of sets, 27 categorial grammar, 135 categoricity, 194

457

category theory, 397 Certaine, J., 74, 287 chain, 30, 58, 391 Chellas, B. F., 356 Church, A., 35, 131,262 clashing of types, 411 classical (material) implication, 106 classical logic, 2-9, 321-355 and modal logic, 356, 378 and LC, 393 classical negation, 88 classical propositional calculus, see classical logic, and Boolean logic, 2 clopen set, 179,365,434,441 subbase for a topology, 434 closed set, 179,365,433 closure algebra, 5, 8, 358, 364-367, 370-373, 386, 387,399 finite compactness of, 370 closure of a set, 365 closure operator, 358, 364 determined by a quasi-metric, 365 closure under exclusion negation, 178 closure under gaggle operations, 413 closure under substitution of a globally evaluated interpretationally constrained language, 163 of an evaluationally constrained sentential language, 163 co-consequence, 214-216 combinators algebra of, 413 and implicational fragment, 414 commutation, 72, III commutative ring, 352 compact topological space, 179,438 compactness, 176-181, 197,274,320,369,436, 438 and exclusion negation, 327 and the finite union property, 180 E-, 177 1-, 177, 326 in classical propositional logic, 326 in explicit symmetric consequence, 203 of a topological space, 179 of a topology, 440 of deductive entailment, 176 of elliptical consequence, 198 of semantic entailment, 176 relation of semantic to topological, 179 S-, 177 strong, 326 symmetrical, 438 U-, 186,326

458

compatibility, 429 complementation, 85-93 classical, see complementation in distributive lattices, 88 in distributive lattices, 88 non-classical, 88 complementation laws, 300 complemented lattice, 85 complete filter, 119 complete fragment for Boolean logic, 336 complete lattice, 67, 68, 70, 256, 322, 382 complete reduct, 205 complete ring of sets, 67 completely prime filter, 119 completely standard matrix, 272 completeness, viii, 196 and K -algebras, 53 and finite algebras, 8 and Lindenbaum algebras, 8 and modal logics, 424 and representation theorems, viii, 193 and valuations, 191, 193 definitional, 336 extensions of 85 in Henle matrices, 376 for asymmetric consequence, 195 for binary implication, 193 for compact symmetric pre-consequence, 20 I for ortholattices, 398 in formal symmetric consequence logics, 213 in formal unary assertionallogic, 210 in the sense of Hallden, 263, 287 and the asymmetric cancellation property, 264 and the symmetric cancellation property, 263 of a modal logic, 416 of asymmetric consequence logic relative to a Kripke atlas, 269 of Boolean algebras, 305 of classical logic, 7, 8, 265 of formal asymmetric consequence logics, 212 of intuitionistic logic, 8, 384 of modal logics, 8, 375 of symmetric consequence, 200 of unary assertionallogic, 266 ofLC,390-393 W.r.t. a class of algebras, 53 compositionality algebraic, 141 of a language, 168 compression, 15 concrete assertional double implicational poset, 286 implicational poset, 285 residuated partially ordered groupoid, 286 concrete residuated partially ordered groupoids, 279

INDEX

INDEX

concrete residuated pair, 409 cone, 119, 192,272,277-282 and pre-orderings, 193 principal, 120, 278 congruence, 19-25,250,254-257,323,328,336 and equivalence relations, 20 and homomorphisms, 23 and identity, 256 and lattices, 71 and quasi-congruence, 75 on a matrix, 249-252 strong and weak, 250 on algebras, 21 on relational structures, 21 congruence of indiscernibles, 252 conjunction, 64 connectivity, 57 consequence, 181, 184, 188 asymmetric and stability, 261 cardinality of, 248 asymmetric in a matrix, 244 binary, 277 elliptical, 197, 199 explicit, 197, 199 in a matrix, 234 mixed, 199 relation and operation, 188 stop-gap, 220 symmetric, 184 class of matrices satisfying, 249 symmetric in a matrix, 244 symmetric stop-gap, 244 consistency, 267 and Lindenbaum algebras, 8 of a logic, 263 of classical propositional calculus w.r.t. Boolean algebras, 7 tautological, 263 consistency of a filter, see filter, consistency, 267 continuous function, 435 contra-validity, 170 contraction, 112 contradiction, 87 contraposition, 87, 89 contrapositive, 408, 409 converse of a relation, 60 with more than two terms, 409 Copi, 1. M., 184 correspondence, 286 cover, 179 open, 179 covering relation, 62, 304 Cresswell, M., 135, 356 Cs t(M),255

Curry, H. B., 2, 130 Curry-Howard isomorphism, 414 cut, 188,267 asymmetric algebraic, 197 global, 189 in asymmetric consequence systems, 188 in explicit consequence, 197 in hemi-distributoids, 202 in symmetric consequence systems, 188 infinitary, 188 symmetric algebraic and hemi-distributoids, 203 symmetric infinitary, 189 CIV (M),255 Czelakowski, J., 237, 246, 248, 257, 273 data types, 41 Davey, B. A., 431 D(BC),343 D(C),342 De Morgan complementation, 91 Laws, 374,429,441 monoid, 428 negation, 425, 427 properties, 416 De Morgan, A., 406 decidability,370-375 of intuitionistic logic, 388 Dedekind cut, 120,296 Dedekind, R., 5, 74, 120,296 deducibility relation, 282 deduction, 218, 357 in a modal logic, 357 deduction calculus, 342 deduction theorem, 381, 417 deduction theorem property, 215, 265 deductive consequences, 344 deductive system, 5 defining equations, 274 degree of a relation, 10 of an operation, II derivation, 176, 342 calculus, 342 for classical logic (D(BC)), 341 rules of, 334, 342 designation, 226 preservation of, 237, 239-243, 243-246 designation extension, 246 dilution, 113, 188,259, 267 and elliptical consequence, 198 in explicit consequence, 197 in explicit symmetric consequence, 203 in symmetric consequence systems, 188

459

Dilworth, R. P., 74, 406 direct interpretation, 149 direct product, 26, 306, 307 and power, 28 of algebras, 27 of arbitrary family of structures, 27 of matrices, 232 of ordered algebras, 76 of relational structures, 26 disconnected family of sets of sentences, 258, 259 discrete metric, 364 discrete topology, 365 disjointness of a filter-ideal pair, 123,439 disjunction, 64, 291 display logic, 417 distribution classical, 93-98 non-classical, 98-104 distribution type, 400, 401 and trace, 400 in a gaggle, 403 of modal operators, 400 of operations in a distributoid, 408 distributive lattice, 4, 94, 207,288,293-300,314, 382,398,421,440 and distributive triple, 98 and hemi-distributoids, 207 and varieties, 97 complementation in, 96 with operators, 314, 362, 415 distributive laws, 94 distributive pre-lattice, 207 and consequence relations, 207 distributive triple, 98 and distributive lattice, 98 in modular lattices, 100 distributoid, ix, 394, 398-406, 408, 442 equational definability of, 412 with identities, 413 Dosen, K., 287, 442 double implicational poset, 280, 281 and partially ordered groupoids, 280 double negation, 87 doubly assertional frame, 421 dual Galois connection, 395, 396 dual hereditary set, 441 dual ideal, 5, 297 dual residuated pair, 395 dual space of a Boolean algebra, 435 dual-strong fidelity, 17 duality, 431, 435 and topology, 434 for Boolean algebras, 437 object and functorial, 437

460

of propositions and possible worlds, 431 Dummett, M., 233, 390 Dunn, J. M., 78, 224, 227, 234, 274, 277, 279, 286,288,318,362,390,395,409,412,414, 415,418,421,422,427,439,442,444 Dwinger, P., 67 E-compact, 177 ECSL,159 effectively decidable set, 138 of axioms and rules, 217 of sentences, 140 effectively enumerable set, 137 of strings, 139 elementary class, 167 elliptical consequence, 197, 198,206 £(M),255 embedding, 17 embedding theorem, 310, 369 and Birkhoff' s prime factorization theorem, 312 endomorphism, 17 entailment, 55, 170,321 deductive, 176 compactness in, 176 finitary, 176, 326 ordinary, 170 semantic, 176 compactness in, 176 simple, 170 symmetric, 170 epimorphism, 17 equational class, 36 equivalence of evaluationally constrained languages, 172 of matrices, 173 strict between matrices and atlases, 174 strict, of logics, 325 equivalence relation, 19, 255 and congruence relations, 20 and partitions, 20 respect for designation in, 256 Esakia, L., 379 Euclid, 184 Euclidean geometry, 216 evaluation ally constrained language, 158 Everett, C. J., 396, 398 evidential frame, 383 exact cover, 434 exclusion negation, 178,327,328 closure under, 178 exhaustive filter-ideal pair, 123,439 expansion of a matrix, 247 explicit consequence, 197-199 explicit symmetric consequence, 221

INDEX

falsifiability, 170 FBG, see full binary gaggle, 422 fidelity, 15 absolute, 16 minimal,16 field of sets, 300, 315, 353 filter, 5, 115-123,291,297,321,323,330,424, 431 and Dedekind cut, 120 and deductive systems, 5 and ideals, 5, 117 and possible worlds, 115 and theories, 115 complete, 122 consistent, 122 generated by a set, 117 join-irreducible, 304 maximal, 121,302 of the classical propositional calculus, 6 prime, 116,302 completely, 119 filter-ideal pair, 123 disjoint vs overlapping, 123 exhaustive, 123 maximal vs principal, 123 Fine, K., 277 finite algebras and completeness, 8, 371 finite direct product, 313 finite distributive lattice, 295, 304 finite intersection property, 180,369,436 finite model property, 224, 234, 375, 388 and unary assertionallogics, 224 finite union property and compactness, 180 first-order definability over a matrix, 252 Fis(~), 203 fissions, 203 Fitch, F. B., 4, 108, 184 FOLE, 38 formal derivations, 342 formality of valuations, 209 formula, 134 founded family of operations, 408 fragment of a language, 335 of a logic, 348 frame for a distributoid, 400 for a gaggle, 400 for a modal algebra, 313, 362, 416 for a tonoid, 279 free generators, 43 FK(n),44 free K -algebra, 45 free K -tonoid, 83 free semi-group, 127

INDEX

freedom, 43 and soundness, 51 model-theoretic notion, 41 of discrete word algebras in tonoids, 84 of word algebra in groupoids, 39 universal, 44 Frege algebra, 147,323,330 and Frege matrices and atlases, 153, 324 Frege interpretation, 330 Frege logic, 321, 322, 328, 333 and Boolean logic, 325-333 and M-Iogic, 333 and unital Boolean logic, 322-324 Frege semantics, 146 Frege, G., 2,141,151,184,216 Frink, 0., 288 Fuchs, L., 74,287,406 full binary gaggle, 422 full closure algebra, 366 full Heyting algebra of open sets, 384 of propositions, 383 function space, 28 fundamental embedding, 307 fundamental homomorphism theorem, 308 FusCr). 196 fusion, 109, 196,278 and its properties, 110,282 and residuation, 278 future contingent statements and multi-valued logic, 148 Gabbay, D., 356 gaggle, ix, 281, 394, 395, 408-420, 421-430 and symmetric modal Boolean algebra, 417 definition of, 408 equational definability of, 412 with identities, 413 Galois closed, 194, 429 Galois connection, 48, 89, 195, 395, 396, 401, 409,410,417,425,426,428,442,444 and residuated pairs, 396 and tonoids and inequations, 83 dual, see dual Galois connection, 395 general square-increasingness ofhemi-distributoids, 205 general sum-decreasingness ofhemi-distributoids, 205 generation, 13 generators induction from, 14 Gentzen, G., 114, 184, 188 Gericke, H., 67 Gierz, G., 397 global cut property, 189,260

461

globally evaluated interpreted language, 159 Gil' 230, 390 Godel, K., 230, 390, 393, 417 Godel algebra, 390 and LC-algebras, 390 Godel matrix, 230 Godel numbering, 138 Godel-McKinsey-Tarski translation, 388 Goldblatt's theorem, 442 Goldblatt, R., 394, 398,426,429,442 Gratzer, G., 51, 397 greatest element, 65 glb(S),66 greatest lower bound, 66, 255 groupoid, 39, 279, 406 H, 380 Hallcten completeness, 263 and the cancellation property, 271 Halmos, P., viii, 300 Hansoul, G., 442 Hardegree, G., 382 Hartonas, C., 288, 318, 431, 443 Hasse diagram, 62 Hausdorff space, 434, 441 Heine-Borel theorem, 434 hemi-distributoid, 202 and compact symmetric consequence, 205 bottom element of, 206 general square-increasingness of, 205 general sum-decreasingness of, 205 lower identity element of, 206 sufficient idempotence of, 205 and symmetric algebraic cut, 205 Henkin, L., 9 Henle algebra, 367 Henle matrix, 367-369, 377, 375 hereditary condition, 290,413,424 downward,291 upward,291 hereditary set, 421, 441 Hermes, H., 129 Heyting algebra, 5, 383-386, 417, 424 and S4-algebras, 417 of open elements, 386 of open sets, 385 Heyting complement, 91, 383 Heyting implication, 417 Heyting lattice, see Heyting algebra, 388 Heyting, A., 91,380 Hilbert spaces, 398 Hilbert, D., 184,216 Hilbert-style presentation, 216 and consequence relations, 216-222 of intuitionistic logic, 380

462

of modal logics, 356 ofLe, 390 Hochster, M., 438 Hofmann, K. H., 397 homeomorphic image, 435 homeomorphism, 435 homomorphic image, 15 homomorphism, 7,15 and congruence in matrices, 251 and congruence relations, 23, 24 and interpretations, 7, 34, 51, 329 between matrices, 237 in tonoids, 80 relational and operational, 19 Horn formulas and quasi-equations, 50 Hughes, G. E., 356 Huntington, E., 2 I-compact, 177,326 lA, 340, 349 ideal,S, 116, 115-123 and Dedekind cut, 120 ideal elements, 121, 295 idempotence, 40, 197,352,418 idempotent rings and Boolean algebras,S identity element, 282, 412, 421 and truth of propositions, 282, 412, 420 left, 282,420 right, 282, 420 identity of indiscemibles, 252 image operator, 314 implication, 55 and residuation, 109,279,287 classical, 105-109 in intuitionistic logic, see intuitionistic implication, 424 in quantum logic, 382 non-classical, 109-115, 278 push and pop, 420 implication algebra, 340, 353, 349 and Boolean implication, 350 and Boolean lattices, 350 and upper semi-Boolean algebra, 349 implication operation, 106 implication subalgebra, 350 implication tonoid, 79 and substructural logics, 79 implicational filter, 351 implicational fragment, 115,380,414 implicational po set, 279, 283 and partially ordered groupoids, 279 implicative fragment of classical logic, 348 and semi-Boolean algebras, 349 derivational calculus for, 352 unary axiomatic calculus for, 350

INDEX

implicative lattice, 38 I, 382, 383 inclusion lattice. 93 inclusion lemma, 284 inclusion poset, 58 and complete typicality, 58 inconsistency of filter-ideal pair, 123,439 toleration of, 267 indirect interpretation, 149 induction, 14 IL,75 inequational logic, 75 inequations, 75 infimum, 66 and set intersection, 66, 67 infinitary cut, see cut, infinitary, 188 infinitely distributive. 382 information order on a frame, 277, 383, 421 interior operator, 364, 358 I(L,M),157 interpretation, 6, 34, 51, 145,389 and algebra of terms, 34 and assignments, 35 and homomorphisms, 34, 51,155 and the principle of categorial compositionality, 145 and valuations, 155 in a Scott atlas, 213 into 2, 6 into a Frege algebra, 147 of a language in a matrix, 157 of a language in a medley, 157 of a language in an algebra, 155 of a language in an atlas, 157 of H formulas, 387 of S4 formulas, 387 on a language, 155 truth w.r.t. possible worlds, 149 interpretation structure, 34 interpreted language, 159 intersection of algebras, 13 intersection property, 370 finite, 181,370,436 intransitivity, 62 intuitionistic calculus (H), 380 intuitionistic implication, 395, 418, 424 and residuation, 420 intuitionistic logic,S, 8. 277, 380, 381, 418 and binary gaggles, 421. 424 and modal logic, 386, 419, 420 and non-classical complementation, 91 finite model property for, 389 fragments of, 380 Hilbert style, 380

INDEX

Kripke-Grzegorczyk semantics, 234 intuitionistic negation, 425 irreducible closed subset, 438 irreflexivity, 56 isolation lemma, 35 isomorphic, 17 isomorphism, 15, 17,292 and free K -algebra, 45 between matrices, 228 isotonic, 74, 76, 79, 196,279,282,314.359 and implication, 77 and negation, 77 Janowitz, M. E, 397 J askowski, S., 184 Jeffrey, R. C., 137 Jevons, W. S., 2 Johansson, 1., 91,380 Johnstone, P. T., 438, 441 join, 68 join-irreducible, 304 join-irreducible separation principle, 289 join-reducible, 293 join-semi-Iattice, 68, 349 J6nsson, B., viii, 313, 394, 401, 422 1SL,71 K -free algebra, 43 and free K -algebra, 45

and subdirect classes, 46 existence of, 44 Kalish, D., 184 Kelley, J. L., 433 Kleene, S. C., 35, 227 Komori, Y., 290 Kripke algebra, 150 Kripke atlas, 154, 269 Kripke matrix, 266 Kripke semantics, 290, 313, 361, 383, 417, 418 Kripke, S., 149,262,277, 356, 361, 383, 401 Kummer, E. E., 5 Kuratowski, C., 41 Kuratowski closure axioms, 364 L-space, 443 Lambek calculus, 129,287,395 of pairs (non-associative), 130 of strings (associative), 129 Lambek, J., 135,397 Langford, C. H., 356 language, 133-136 and decidabiIity, 140 and substitution, 136, 163 and unique decomposition, 133 atomic sentences of, 133

463

compositional, 168 evaluated, 162 evaluationally constrained, 158, 162, 163, 173 interpretationally constrained, 159 interpreted, 159 truth functional, 169 types of, 160 uninterpreted, 159 lattice, 4, 67-70, 68, 71-74, 288-297, 396, 398, 431 and congruences, 71 and Lindenbaum algebras, 4 and ordered algebras, 75 as a relational structure, 68 bounded,69,438 complemented, 85-88, 89, 300 complete, 68, 256 distributive, 93-98 equational definability of, 73, 412 modular, 99 power set, 66 with operators, 314, 318,361,398 lattice of sets, 93 lattice-ordered groupoid, 109,314,406 law of contraposition, 78, 90, 109 law of importation-exportation, 108 law of the left residual, 109, 280, 407 right residual, 109, 280, 407, 420 LC,390 algebras and G6del algebras, 390 least element, 65 lub(S),66

least upper bound, 66, 255 left identity, 282, 420 left lower identity, 421 left residual, 109,421-423 positive paradox for, 113 left-cancellation in semi-groups, 126 left-residuated partially ordered groupoid, 413 Leibniz algebra, 150 Leibniz atlas, 153 Leibniz congruence, 252 atomic, 253 Leibniz operator, 252 Leibniz, G. W, 2, 252 Lemmon, E. J., 184,356 Lewis, C. 1., 106, 356, 381 Lewis, D., 106,381 Lincoln, P., 224 Lindenbaum algebra, 4, 46, 274, 338, 361, 370, 374,389,399,413,423,432 Lindenbaum atlas, 257, 273 Lindenbaum matrix, 212, 257, 260, 273 Lindenbaum, A., 3, 210

464

linear ordering, 57 logic, 184-225 absoluteness and categoricity of, 194 algebra of, 8 algebraic and logistic approaches to, 2 algebraic semantics for, 273 and consequence, 184 and lattices, 5 and natural deduction, 184 and ordered algebras, 77 and tonoids, 77 and valuations, 169-172, 189-191 axiomatic characterizations of, 216 bivalent, 156 classical, 321 consistency of, 263 countable asymmetric consequence, 261 equivalential, 186 formal,208 Hilbert-style presentation of, 218 intuition is tic, 380 Lindenbaum algebra of, 8 modal,356 multi-valued, 148 non-truth-functional, 149 of quantum mechanics, 5 relevance, 274 and algebraic semantics, 275 soundness and completeness in, 191 structural, 208 substructural,274 typical properties of, 184 logical equivalence, 214 logical indiscernibility, 174 logical operations distribution type of, 399 tonic type of, 78 trace of, 400 logical validity, 181 Los, J" 184,226 Ib(S),64

lower bound, 64 lower semi-Boolean algebra, 349 LSBA,349 L m,227 L w ,233 LNJ,233 Lukasiewicz, J., 91, 108, 148,203,226,233,399 M,333 m,248 m-filter, 248 M-Iogic, 333 and Frege logic, 333 m-reduced product of matrices, 248

INDEX

MacLane, S., 397 Maksimova, L., 379, 390 Malinowski, G., 226 Martinez, N. G., 442 material binary gaggle (MEG), 422 material implication, 105,360 mathematical induction, 14,335 matrix, 152,226,321 atomic definability over, 253 Blok-Pigozzi, 254 characteristic for a unary assertionallogic, 232 for asymmetric consequence, 234 completely standard, 272 congruence on, 250, 257 strong and weak, 250 consequence relations in, 244 contraction of, 247 designation extension of, 246 direct product of, 232, 238, 243 equivalence relations on, 256 expansion of, 247 first-order definability over, 252 Glidel,230 homomorphic image, 238, 247 homomorphism, 237 and preservation of designation, 231, 240 positive and negative, 237 strong and weak, 237 infinite, 233 isomorphism, 228, 231, 238 strong and weak, 238 Kleene and Lukasiewicz, 229 Kripke, 266 Lindenbaum, 260 Lukasiewicz, 227, 235, 245 normal,262 and homomorphism into 2, 262 characteristic of unary assertionallogics, 264 pre-standard, 272 reduced product, 248 relations between, 173 relative of, 247 semantics, 274 Shoesmith-Smiley, 260 standard, 272 strong homomorphism theorem, 251 subdirect product, 238 Sugihara, 230 tautologies of, 232 weak homomorphism theorem, 252 maximal Boolean matrix, 153 maximal element, 65 maximal filter, 121,323,368,433 maximal filter principle, 310, 436

INDEX

maximal filter-ideal pair principle, 439 McCall, S., 5, 226 McKinsey, J., 5, 8, 364, 370, 371, 386 McRobbie, M., 197 medley, 152 meet, 68 meet-normal form, 389 meet-semi-Iattice,68 Meskhi, v., 379 method of equivalence classes, 40 metric, 364 metric space, 364 Meyer, R. K., 197,203,234,277,286,390,394, 414,422 minimal conditions for implication, 108 minimal element, 65 minimal logic, 91, 380 and non-classical complementation, 91 minimal negation, 425 minimally faithful homomorphism, 16 minimum condition for separation, 295 mirror principle, 142 Mitchell, J., 224 mixed consequence, 199 mixed structures, II modal algebra, 358-361, 370-375, 416, 424, 442 modal logic, 277, 313, 356-379, 394 and binary gaggles, 421 and classical logic, 378 and intuitionistic logic, 386 and residuated pairs, 398 axioms of, 356 B, 150,357,415,416 backward operators, 397,416 closure operator, 358 decidability, 371-375 deducibility and validity. 363 finite model property, 374 frames, 363, 415 interior operator, 358 K, 150,356,415 algebra, 371, 372 various axiomatizations, 356 1(5,418 Kripke semantics, 313, 361, 394, 415 (weak) completeness, 364 completeness, 363 model for, 362 normal, 358,415 relations among. 357 S4, see S4, 357 S5, see S5, 357 soundness, 361 strict implication and intuitionistic implication, 418

465

T,150,357,415,416 algebra, 358 ternary accessibility relation. 425 modal operators, 149 model,36 model theory, 36 modular lattice, 99 and modular pairs, 102 distributive triples in, 100 ortho-, 102, 103 varieties of, 100 weak,102 M(b,c),102

modus ponens, 3, 107,351,380 monoid, 129 monomorphism, 17 monotonic, 113 Montague, R., 184 MSL,71 multi-valued algebra, 148 mUltiple successor algebra, 129 multi set, 40, 197 natural deduction, 184 necessity, 149. 356,402 negation, 85 gaggle treatment, 427 negation consistency, 429 negative cone, 120 non-classical complementation, 91 non-classical distribution, 98 non-classical logics, 5, 113,394 normal atlas, 262-266 normal extensions of Le, 392 normal matrix, 262-266 and homomorphism into 2, 262 normal modal Boolean algebra, 415 normal operation, 315 normality of a matrix, 262 Ockham lattices, 442 one-point compactification theorem, 370 Ono, H., 290 open cover. 179 open function on a topological space, 435 open set, 179,365,433 operation, 11 additive, 314 normal,315 operational homomorphism, 18 operational structure, II operational syntax, 33 operator, 314 order variety. 76 ordered algebra, 74

466

and lattices, 75 discrete, 74 varieties theorem for, 76 ordered quasi-varieties, 76 ordered quotient algebra, 76 ordered Stone spaces, 438 ordering, 58 and the principle of duality, 60 Ore, 0., 396 orthocomplement, 86, 91, 429 and classical negation, 88 orthocomplementation, 442 orthogonal, 102 ortholattice, 86, 398, 442 equational characterization of, 87 orthologic, 426, 429, 442 orthomodular lattice, 103 orthonegation, 86, 442 overlap, 188, 197,267 paradoxes of implication, 6, 113, 381 of material implication, 419 of strict implication, 419 partial gaggle, ix, 409 partial ordering, 56, 74,78,255 and associated strict ordering, 59 and regular relations, 62 partially ordered set, 56 partially ordered groupoid, 109,279,406 partially ordered residuated groupoid, 279, 406 partition, 19 and quotient algebras, 20 Peano, G., 184 Peirce, C. S., 406 permutation, 79, III Pigozzi, D., 246, 247, 252, 253, 274 polarity, 398, 444 polynomials on an algebra, 329, 336 pop, 282-284,286,420 Popper, K., 184 poset,56 bounded,65 positive cone, 119 generated by a set, 120 positive De Morgan monoid, 423 positive fragment of classical logic, 348 and Boolean rings, 353 axiomatic characterization of, 354 of intuitionistic logic, 380 of relevance logic, 423 positive Routley-Meyer frame, 423 possibility, 149, 402 possible worlds semantics, 313, 362, 397

INDEX

and algebraic semantics, 150 Post complete, 378 Post inconsistent, 378 Post, E., 8 IfJ(X),66 Pratt, v., 109,407,412 pre-ordered groupoid, 196 pre-ordering, 55, 191 and implication, 55 and partial orderings, 56 generated by a relation, 61 pre-semi-Iattice, 199 pretabularity, 234, 375, 390 Priestley Spaces, 438 Priestley, H. A., 431, 438, 439, 441, 442 prime algebra, 29 prime element, 296, 293 separation principle, 305 prime filter, 116,297,298,368,414,426,423 prime filter principle, 440 prime filter separation principle, 298, 302, 368, 439 prime ideal, 116 principal cone, 120, 192,274,395 principal dual cone, 120, 280, 395 principal filter, 118,318,319,322,404 principal ideal, 119,349 principal pair, 123 principle of bivalence, 156 principle of categorial compositionality, 144 principle of compositionality, 143 projection, 28 proof,216 proper filter, 115,302,316,321,330 proper fragment, 348 proper modal extension, 378 proper normal extension of LC, 392 proposition, 2, 278, 362 as a hereditary subset, 290, 421 as a set of information states, 278, 383, 438 pseudo-assertion, 113,281, 283 pseudo-Boolean algebra, 8, 91, 383, 390 pseudo-complement, 91, 383 pseudo-metric, 364 pseudo-trichotomy, 126 push,282-284,286,420 quantum logic, 88, 102, 104, 186,381 implication in, 382 quantum mechanics, 5, 429 quasi-complement, 399 quasi-congruence, 75 quasi-equation, 49 quasi-equational class of algebras, 50 quasi-equations and Horn formulas, 50

INDEX

quasi-inequations, 77 and implication to no ids, 79 quasi-metric, 364, 383 quasi-metric space, 366 quasi-ordering, 55, 366, 383, 385 quasi-partition, 189 quasi-variety, 49, 50 Quine, W. V. 0.,182,184 quotient algebra, 20, 23, 323, 328 and partitions, 20 quotient matrix, 250 weak and strong, 250 quotient structure, 23 of operational structures, 23 of relational structures, 23 R,272 Rasiowa, H., 4, 88 realizing contraposed distribution types, 411 recursiveness ofaxiomatization, 217 reduced product of algebras, 50 reduced product of matrices, 248 reduct of a fusion, 205 reflexivity, 55 refutable sentences and ideals, 6 regular cardinal, 249 regular relation, 62 orderings induced by, 62 relational homomorphism, 15 variants of, 16 relational structure, 10 relative frame for S5, 367 relative matrices, 247 relative topology, 434 relatives (in a set of operations), 408 relevance logic, 287,413,420,427,428 and binary gaggles, 421 and non-classical complementation, 91 implication in, 381, 394 replacement, 22, 214, 328 types of, 22, 253 replacement theorem, 4 representation, 8, 277-320 of Boolean algebras, 303-305, 310, 435 with operators, 313 of closure algebras, 367 of distributive lattices, 293-300, 438 with operators, 314 of distributoids, 40 I, 403 of gaggles, 409-412, 422, 426, 429 of Heyting algebras, 383-386 of implicational posets, 279 of lattices, 288-292,431-444 with operators, 317-320 of residuated partially ordered groupoids, 281

467

of semi-lattices, 287 of tonoids, 279 of LC-algebras, 391 representation theorems, viii, 277-320 and completeness theorems, viii representation theory, viii, 8 residuated monoid, 130 residuated partially ordered groupoid, 109, 279, 281,401 residuation, 82, 280, 382, 395, 396, 401, 406, 410 abstract law of, 281, 408 and fusion and implication, 287 and Galois connections, 396 and intuitionistic implication, 420 and modal operators, 416 in lattice ordered groupoids, 406 in partially ordered groupoids, 396 laws of left and right, 407 right identity, 283, 420 right lower identity, 421 right residual, 109,421-423 positive paradox for, 113 right residuated partially ordered groupoid, 279, 413 right-residuated distributive lattice ordered groupoid with right identity, 421 ring algebraic, 352 of sets, 67, 93, 287, 300, 315, 352 RM,230 Routley, R., 277, 286, 394,422 Routley-Meyer frame, 286, 428 R*, 61, 315 rule of necessitation, 356, 359 rule prefixing, 78, 279 rule pseudo-permutation, 113, 284 rule suffixing, 78, 279 rule-form permutation, 80, 284 Russell, B., 2, 184 Rutherford, D., 67 S-compact, 177 S4,5,8,150,357,415 algebra, 358, 360, 364, 373 and intuitionistic logic, 386 S5,80,150,357,359,375,377,390,415,418 absolute semantics, 149,367 algebra, 358, 361, 368, 369 extensions, 378 and Henle matrices, 376 axiomatization of, 378 pretabularity of, 375 uniform finite model property in, 375 SA,349 Sambin, G., 442

468

Sasaki implication, 381 satisfaction, 36, 169 by a class of algebras, 36 by an algebra, 36 satisfiability, 170 by a Frege interpretation, 330 SEA,349 Scedrov, A., 224 Schmidt, J., 396 Schroder, E., 2 Scott atlas, 213, 260, 273 Scott, D., 185,202,235,356 Scroggs, S. J., 375, 392 Segerberg, K, 356 self-implication thesis, 283 semantic categories, 142 semantics, 141-183, 193 algebraic (compositional), 167 possible worlds, 277 semi-Boolean algebra, 349 and Boolean lattices, 350 equational definability, 349 semi-group, 40, 126 Abelian, 40 free, 40 trichotomy and cancellation in, 126 semi-interpreted language, 158, 190 semi-lattice, 40, 290 as algebra, 72 meet- and join-, 71 varieties, 73 semi-lattice-ordered monoid, 290 semi-modular lattice, 102 sentential modal operators, 149 separation principle, 278, 287, 289, 293-295, 298 Shankar, N., 224 Shoesmith, D., 185,202,257,261 Shoesmith-Smiley matrix, 260 Shukla, A., 373 Sikorski, R., 4, 300 similarity of operational structures, 11 of relational structures, 10 of tonoids, 83 Simmons, G. F., 433 simple entailment, 170 simple semantic categories, 142 Smiley, T. J., 4, 185,202,257,261 soundness, 7 and asymmetric consequence, 269 and freedom, 51 offormal asymmetric consequence logics, 212 offormal symmetric consequence logics, 213 of formal unary assertionallogics, 210 of modal logics, 362, 367, 416

INDEX

of valuations w.r.t. a consequence relation, 191 of S5, 376, 375 w.r.t. a class of algebras, 51 specialized pseudo-assertion, 284 spectral topological spaces, 438 square-decreasingness, 418 square-increasingness, 197,423 stability and asymmetric consequence, 261 stable consequence relations, 259-261 Stalnaker, R., 106, 381 statement. 187 Stone, M. H., viii, 5, 8, 288, 295, 384, 395, 403, 417,422,431,438,441 Stone space, 435 Stone's prime filter separation principle, 297, 391, 436 Stone's representation theorem, 297-300, 303305,322 and duality, 435 for Boolean algebras, 312, 433 for distributive lattices, 297 stop-gap consequence, 220 Strawson, P. F., 4 strict equivalence of logics, 325 strict implication, 360,418 strict linear ordering, 59 strict ordering, 59 and associated partial ordering, 59 string, 125 strong compactness, 326, 370 of a topological space, 370 strong homomorphism theorem for matrices, 251 subalgebra, 12 generated, 14 subbase for a topology, 434 subdirect classes, 46 and K -free algebras, 46 subdirect product, 29 subdirectly irreducible, 30 sublattice, 115 submatrix, 237 weak and strong, 237-239 subminimal complementation, 89 subspace, 369 substitution, 137 semantic, 166,257 syntactic, 166 substitutionally determined language, 165 substructural logics, 80, 278, 400 and implication tonoids, 79 substructure, 12 subtonoid, 83 subtraction algebra, 349 sufficient idempotence in a hemi-distributoid, 205

INDEX

superstructure, 12 supervaluationallogic, 88, 186 Suppes, P., 184 supremum, 66 and set union, 66, 67 Suszko, R., 184 symbolic logic, I symbols, 33, 125 symmetric algebraic cut, 205 symmetric consequence, 184, 189, 196,325 absoluteness in, 202 and valuations, 199 explicit, 203 logic, 273 symmetric cut, 203 symmetric entailment, 170, 325 symmetric infinitary cut, see cut, symmetric infinitary, 189 symmetric lattice, 102 symmetric modal algebra, 416 symmetric modal Boolean algebra and gaggles, 417 symmetric pre-consequence, 188 symmetric stop-gap consequence, 222 symmetrical sequent calculus, 334 symmetry, 56 syntactic categories, 142 syntactic degree, 33 syntax of standard first-order logic, 134 Szasz, G., 454 Tarski, A., viii, 3, 5, 8,188,210,226,233,313, 364,386,394,395,401,422 tautological consistency, 263, 265, 268 tautological implication, 262, 265 tautology, 87, 262 of a matrix, 232 preservation, 232 theorem, 3, 216 theory, 278 thesis in a derivational calculus, 342 of a calculus, 334 Thomas, L, 393 Thomason, R. H., 381 tonic direct product, 83 tonic homomorphic image, 83 tonic type, 78 TIL, 79 tonoid, ix, 78, 279, 398, 400, 409 inequationallogic, 79 similarity of, 83 varieties theorem for, 82 top element, 198, 206, 257 topological space, 179, 193,364,369,385,433

469

and duality, 434 clopen set in, 179 closed set in, 179 compactness of, 179 Hausdorff, 434 induced by a valuation space, 181 open set in, 179 totally disconnected, 434 totally order disconnected, 438 trace, 400 transitive closure, 61, 255 transitive reflexive closure, 61 transitivity, 55 and implication, 108 triangle inequality, 364 truth-functional language, 169 truth functions, 146 truth set, 152, 262, 268 truth values, 146-148, 226, 227 Turing machine, 137 TV, 356 two-valued quasi-metric, 364 type of a relational structure, 10 of an operational structure, 11 typicality, 42, 41-44 and inclusion posets, 58 and non-identity, 42 U-compact, 177,326 UCLA proposition, 150, 277 Ulrich, D. E., 234, 396 ultrafilter, 50 ultraproduct, 50, 248 unary assertionallogics, 185,267,273 unary calculus, 334, 342 unassailable, 326 uninterpreted language, 159 unique decomposition, 131 and universal freedom, 132 uniquely complemented lattice, 85 unital Boolean logic, 321, 333 and supervaluationallogic, 322 identity with Boolean logic, 325 matrix for, 322, 324 strict equivalence of Boolean logic, 325 strong equivalence to Boolean logic, 322-324 strong equivalence to Frege logic, 324 universal algebra, viii, 10 and algebraic logic, 10 universal freedom, 44, 132 and unique decomposition, 132 UF(n),45 UFK(n),45 unsatisfiable, 170, 326

470

ub(S),64 upper bound, 64 upper identity elements, 206 upper semi-Boolean algebra, 349 Urquhart, A., 224, 230, 235, 277, 288, 318, 439, 443 USBA,349

Vacarro, v., 442 validity, 7, 170 in a frame, 363 in a matrix, 210 valuation, 155-158, 162-172, 189 admissibility of, 158 induced by an interpretation, 157 V(f-),200 V(L,M),157 van Benthem, J. F. A. K., 286 van Fraassen, B. c., 158, 178,322 varieties tbeorem, 36, 76, 82 variety, 36, 323, 350 and distributive lattices, 97 and distributoids, 412 and gaggles, 412 and lattices, 73 and modular lattices, 103 and semi-lattices, 72 Vickers, S., 438 Wajsberg algebra, 442 Wansing, H., 417 Ward, M., 74, 406 weak asymmetry, 56 weak modularity, 102 weakly connected, 59 Wechler, W, 51, 74 WI =,40 Whitehead, A. N., 2, 184,421 WII, 47 Wittgenstein, L., 153 WIK,46 Wojcicki, R., 226, 237 Woodruff, P., 262 word algebra, 33 and freedom in groupoids, 40 Zorn's lemma, 30, 297, 316, 395, 404, 426, 429, 436

INDEX