A Formalization of Set Theory without Variables (Colloquium Publications)

Obviously this function maps the set of all systems in :f in a one-one way onto the set of all theories in :f. Thus we could entirely eliminate systems from our discussion, replacing everywhere a system S by its theory TS. However we find it convenient to have the present notion of system available for our discussion. 2 In certain situations it proves desirable to single out from among all possible bases for a given system S a definite base

1.6{v)

FORMALISMS AND SYSTEMS

21

However, in this book we shall also concern ourselves with systems in formalisms of predicate logic whose theories are not recursively enumerable. To obtain one such example consider the algebra

21= (A,+,·) where A is the set of natural numbers, and + and· are respectively the familiar operations of addition and multiplication on A. An appropriate apparatus for studying 21 is the formalism of predicate logic with two nonlogical constants, the binary operation symbols + and '. We are interested in constructing a system A in in which the set of all logically provable sentences consists just of those sentences that are true of 21. The only natural choice for a base of A seems to be the set of all sentences in that are true of 21, i.e., the whole theory 0p21. Thus we take Ae[A] = 0p21. It is known that 0p21 is not recursively enumerable, and consequently that there is no recursive or even recursively enumerable base for A. Instead of 21 we could of course take many other algebraic structures or classes of similar algebraic structures. One could also think of introducing the notion of a system developed in an uninterpreted formalism i.e., a relativization of to some set of sentences. Such relativizations turn out again to be uninterpreted formalisms, contrary to the situation for relativizations of interpreted formalisms. Thus, consideration of such systems does not lead us to anything new. In consequence, we shall only use the term "system" in application to interpreted formalisms.

CHAPTER 2

The Formalism ,c +, a Definitional Extension of ,c

We shall now discuss a formalism £+ which is obtained from the formalism £ by slightly enriching the logical base of the latter. £ + is not a formalism of first-order logic in the strict sense in which we introduced this term in §1.5, but it may still be regarded as a variant of a first-order formalism. 2.1. Symbols and expressions of £ + We obtain the vocabulary of £ + by including in that of £ five new logical contants: the absolute sum, or absolute addition, symbol +, the complement symbol -, the relative product, or relative multiplication, symbol e, the converse symbol ..... , and the second identity symbol =. The first four symbols are (second-order) operation symbols; for simplicity we shall almost always call them operators; +, e are binary operators (operators of rank 2), and -, ..... are unary ones (operators of rank 1). While in £ just two binary predicates, E and i, are available, in £ + , using E, i, and the operators, we construct infinitely many symbolic expressions which are also regarded as binary predicates (i.e., as designations of binary relations). The new identity symbol, =, is not to be confused with the old one, i; when combined with any two predicates, yields a formula, and hence it can be regarded as a predicate of higher order. The intuitive meanings of the new constants will be clarified in §2.2. The variables in £ + are the same as in £. The membership symbol E continues to be the only nonlogical constant. However, £+ contains new compound expressions that do not occur in £, namely (compound) predicates.

=

(i) E and i are called atomic predicates (or relation symbols) of £ +. An expression is called a predicate (or relation term) of £ + if it belongs to every set f of expressions of £+ such that (a) E, i E f, ((3) A+B, A-, AeB, A ..... E f whenever A, BE f. The set of all predicates is denoted by

"n [£ + 1" 23

or simply by

"n" .

THE FORMALISM .c,+ , A DEFINITIONAL EXTENSION OF .c,

24

2.1(ii)

In analogy with the definitions of a formula and of derivability, the above definition of a predicate carries with it a method of showing that every predicate in L + possesses a certain property; this method of proof may be referred to as proof by induction on predicates (cf. remarks after 1.3(ii)). By the definition of predicates just stated, we have A + B , A - , A <:> B , E II whenever A, BEn . Thus +, 0 can be treated as binary operations, and -, as unary operations, on and to predicates. In terms of these we define further operations of the same kind, and also, using i , we single out some particular predicates by setting:

= (A-+B-t

(ii)

A·B

(iii)

o=i-,

and

l=i+i- ,

AeB

and

= (A-0B-)-

for any

A , B E nil

o=(i+i-)-.

e are referred to as absolute multiplication and relative addition. Regarding the use of parentheses in symbolic designations of predicates, we shall adhere to the following rules. Whenever parentheses indicating the order of operations are totally or partially lacking, it will be understood that: (1) each of the unary operations, - and has priority over each of the binary operations; (2) each of the two multiplications, . and <:> , has priority over each of the two additions , + and e ; (3) in case neither (1) nor (2) is applicable, the operations should be performed successively from left to right. Each of the binary operations mentioned above extends by the usual recursion to an operation on finite sequences of predicates. Thus, for every sequence (A o, ... , An-I) of predicates we set The operations . and

Ao+ '" + A n -

l

= 0

if n = 0,

=Ao

if n = 1,

= (Ao+ · · · + A n - 2 )+A n -

l

if n = 2,3, .. . .

By changing, respectively, in these formulas" + " to ". ", " e ", "<:> ", and "0 " to "1", " {) ", "i", we obtain recursive extensions of the operations " e, <:>. Exclusively in the case of 0 we shall use exponential notation; i.e., we set

B n = Ao 0 . . . 0 A n -

l

in case each of the predicates A o, ... ,An - l coincides with B . The expressions xAy and A = B, where x, yET and A , BEn, are the atomic formulas of L +; arbitrary formulas are constructed from atomic ones just as in the case of L. The notions of free and bound variables, canonical sequence, and closure of a formula extend automatically from L to L +. Notice that some sentences of L +, as opposed to sentences of L, are quantifier-free. For instance, the atomic formulas A = B, which are sometimes called predicate I'Thus, in opposition to its use in Henkin- Monk- Tarski [1971], here the symbol " 61 " does not denote the operation of symmetric difference from Boolean algebra.

2.2

DERIVABILITY AND SEMANTICAL NOTIONS OF £+

25

equations or simply equations, are quantifier-free sentences; in general, quantifierfree sentences are just those formulas which can be obtained from equations by a repeated application of the operations - and -,. Inclusions are treated as special kinds of equations: A B is defined by the formula

(iv)

= (A+B=B)

for any A,BEII.

and :E[£+] are, respectively, the set of all formulas and that of all sentences of £ +. Usually, however, occurrences of "£+" in metalogical symbolic expressions will be omitted, and the expressions will be provided instead with a conveniently placed superscript "+"; e.g., we shall write instead of £ +] ". (In the case of "II [£ +]" we have agreed to use simply the abbreviation "II", since no corresponding symbol has been introduced for the language £.) 2.2. Derivability and semantical notions of £ + The set A+ (or A[£ +]) oflogical axioms of £ + is constructed in the following way. First, we include in this set all the sentences described in the same way as the logical axioms of £ (cf. §1.3)- with the provision, however, that every reference to formulas, or atomic formulas, occurring in this description is interpreted as a reference to formulas, or atomic formulas, of £ +. Secondly, we supplement the set thus obtained with all the sentences of the following five kinds, where A, B are arbitrary predicates:

(DI)

VzlI[xA+By ++ (xAy V xBy)],

(DII)

VzlI(xA-y ++ -,xAy),

(DIll)

V zlI [xA0By ++ 3 z (xAz A zBy)],

(DIV)

VzlI(xA..... y ++ yAx),

(DV)

A

=B

++

VzlI(xAy

++

xBy).

Thus, are new axiom schemata which have been adjoined to those for £ to form the full set of logical axiom schemata for £ +. Because of their form and content, can be referred to as possible definitions or, more precisely, possible definition schemata in £ of those logical constants which belong to the vocabulary of £+ but not to the vocabulary of £. They obviously clarify the intuitive meanings of those constants. In fact, we see from that the operators +, 0, -, and ..... are intended to denote, respectively, certain familiar operations on and to binary relations: the formation of unions (absolute sums) R U S and relative products RIS of pairs of relations R, S, as well as the formation of complements and converses R- 1 of single relations R. (The complement is usually taken with respect to some fixed unit relation U xU.) Similarly, by (DV) the new identity symbol is intended to denote the relation of (extensional) identity between binary relations. As a consequence of the above observations the intuitive meanings of the operations on predicates and of the special predicates defined in 2.1(ii),(iii) also become clear. We see, e.g., that A·B denotes the intersection (absolute product)

=

THE FORMALISM .G+, A DEFINITIONAL EXTENSION OF .G

26

2.2(i)

R n S of the relations Rand S denoted respectively by A and B, and that () denotes the diversity relation Di, usually restricted to elements of a fixed set U, i.e., the relation (U x U) n Di. The relation of derivability between a set \II and a sentence X E is expressed symbolically by "lIt I- + X", and is defined for ,c + in terms of logical axioms just as this was done for L in 1.3(ii). The notions and results discussed in §1.3, including the deduction theorem, extend automatically from L to L + . The theory 81]+0, the logic of L+, may be called the extended predicate logic of one binary relation.

We shall state a few interesting logical equivalences which hold between quantifier-free sentences of L+ (where A, B are assumed to be arbitrary predicates in II).

(i)

(A

=B)

(ii)

(A

= 1 - B =1)

=+ (A·B+A- ·B-

=1).

=+ (10A- 01+B = 1).

(iii) (-,A=l) =+ (10A-01=1). (iv) (A = 1 V B = 1) =+ (AeOeB = 1). (v)

(A=lAB=l) =+ (A·B=l).

The proof of (i)-(v) is quite elementary. From (i)- (iii) we easily derive by induction on formulas the following important theorem.

(vi) For every quantifier-free X E

there is aCE II such that X =+ (C = 1).

The results (ii)-{vi) originate with Schroder [1895], pp. 150- 153. We now deal briefly with the problem of introducing semantical notions for the language L + . As opposed to L, L+ contains two kinds of compound expressions which are intended to be meaningful: formulas and predicates. As a consequence, in developing the semantics of L + we need two basic notions: the notion of satisfaction applying to formulas and that of denotation applying to predicates. Actually we first introduce the latter notion, and with its help we define the former. Since the binary predicate E occurs in both Land L + as the only nonlogical constant, the same structures (U, E) with U =f: 0 and E U x U are used as realizations for both languages. Consider any such structure II = (U, E). Keeping 2.1{i) in mind, we first define by a recursion on predicates what is meant by saying that a given relation R U x U is denoted (in ll) by a given A E II. In case A is an atomic predicate, we stipulate that the relation denoted by A is E if A = E, and is U1Id if A = 1. Then, in full agreement with the content of Axioms (DI)-{DIV), we make four recursive stipulations, one for each of the four operators occurring in L +. Thus, for example, we stipulate that, in case A = B + C (B, CEIl), the relation denoted by A is the union R U S of the relations Rand S respectively denoted by Band C.

THE EQUIPOLLENCE OF

2.3(ii)

[,+

AND [,

27

Just as in the case of £." the notion of satisfaction is defined for £., + by a recursion on formulas. In defining this notion for atomic formulas we make essential use of the notion of denotation. In fact, we stipulate that an infinite sequence x = (xo, ... , x n , ... ) of elements of U satisfies (in a given realization II = (U, E)) the formula VkAvl (with k, I = 0, 1,2, ... and A E n) iff the relation R denoted by A holds between Xk and Xl, that is, in set-theoretical notation, iff (Xk' Xl) E R. Similarly, X satisfies A B with A, BEn iff A and B denote the same relation. The remaining parts of the definition of satisfaction, referring to nonatomic formulas, are routine; cf. Tarski [1956], p. 193. Other semantical notions, such as truth and model, are defined in terms of satisfaction just as in the case of £.,. The completeness theorem and its consequences stated in §1.4, as well as the subsequent remarks in that section, easily extend to £., + .

=

2.3. The equipollence of £., + and £., We have thus completed the description of the formalism £., +. As we have seen, £., + has been constructed by extending the original formalism £., in two directions: five new logical constants, say C 1 -C5 , have been included in the vocabulary of £." and an equal number of new axiom schemata, 8 1 -85 (actually (DI)-(DV)), have been adjoined to the set of logical axiom schemata for £.,. Because of its form, each schema 8k, k = 1, ... ,5, can be regarded as a possible definition (schema) of the corresponding constant Ck in £.,; more specifically, as a possible definition of Ck in terms of logical constants of £.,. We can sum up the situation by saying that, in a certain sense, £., + is a definitional extension of £.,. (For the notion of definitional extension in different but related contexts compare 2.4(xiii) below and also Tarski [1968], p. 277.) It is a matter of common belief that when passing from a formalism to its definitional extensions we do not enrich the means of expression and proof of the formalism involved. In the case of semantically complete formalisms this belief appears to be fully justified, and the method by which it can be established for £., and £., + is routine. (In §3.7 we shall come across some semantically incomplete formalisms for which the belief is, in all likelihood, not justified. 2 ) Nevertheless, we shall state here the relevant results in a formal and detailed way, since this discussion will serve as a model and a source of reference for analogous but more involved discussions in later chapters, particularly in Chapter 4. We begin with some obvious statements to the effect that £., is a subformalism of £.,+ (in the sense of §1.6), and hence £., is at most as powerful as £.,+ in means of expression and proof.

(i)

E

E+ and

(ii) For every W

.

E and X E E, if W f- X, then W f-+ X.

2'This has indeed been shown to be the case; cf. footnote 8*, p. 66.

28

THE FORMALISM

£, +,

A DEFINITIONAL EXTENSION OF

£,

2.3(iii)

To show that ,c + is not richer than ,c we define recursively an appropriate translation (or elimination) mapping G from ,c+ into ,c.

(iii) G is the unique function F satisfying the conditions (a) - (lJ) (where x , y E i, A , BEll, and X, Y E 4)+): (a) DoF = 4)+; (f3) F(xi y ) = xiv and F(xEy) = xEy; b) F(xA+By) = F(xAy) V F(xBy); (8) F(xA0By) = 3 z [F(xAz) A F(zBy)J, where z zs the first variable

(e) (I,') (rJ) (0)

which is different from x and y; F(xA-y) = .,F(xAy) and = F(yAx); F(A B) = VzlI[F(xAy) ++ F(xBy)] ; F(X - Y) = FX - FY and F(.,X) = .,FX; F(VxY) = VxFY.

=

In a detailed presentation, the above definition would be preceded, of course, by the proof that there exists a uniquely determined function F satisfying conditions (a )- (0) . The next theorem states some simple properties of the function G. They are easily derived from the definitions of the notions involved by induction on formulas and predicates in ,c+ or ,c.

(iv) (a) G is a recursive function. (f3) G X = X if X E 4).

b) itjJGX = itjJX for every X (8)

(e) (I,')

E

4)+.

G maps 4)+ onto 4), and E+ onto E. GX =+ X for every X E 4)+. G*w =+ W for every w<;;; E+.

In connection with (I,') recall that G*w (the G-image of w) = {GX: X E w} .

The main property of the function G is stated in the following theorem.

(v)

For every W <;;; E+ and X E E+, we have W f-+ X iff G* iII f- GX.

To obtain the implication from left to right in (v) we proceed by induction on sentences derivable in ,c + from a given set W, using the recursive definition of f-+; cf. §§2.2, 1.3. More specifically, we set 11 = {Y: Y E E+ and G*w f- GY}, and we show that 11 satisfies conditions (a) - b) in 1.3(ii) (replacing E and A by E+ and A+); we conclude that X E 11 whenever iII f-+ X. The implication in the opposite direction follows easily from (ii) and (iv)(e),(S"). In informal remarks we shall refer to (v) as the main mapping theorem for ,c + and ,c, and we shall extend this terminology in analogous situations to other pairs of formalisms.

THE EQUIPOLLENCE OF ,c+ AND ,c

2.3(xi)

(vi) C*er,+W =

er,C*w =

er,+W n

29

for every W

In fact, using (iv)(.8) and (v), we easily establish the inclusions:

C*er,+w

e"c*W

(er,+w)

n

C*er,+w.

From the results in (iv) and (v) we shall now derive some simple corollaries which do not explicitly involve the auxiliary function C and are more directly related to the problem of comparing the expressive and deductive powers of the formalisms £, and £, +. This will give us an opportunity to clarify the precise meaning of this problem, thus suggesting a notation which can be applied in similar contexts to other formalisms as well. A direct consequence of (iv)(8),(c) is

(vii)

For every X E

such that X =+ Y.

there is aYE

For reasons which seem intuitively clear we regard (vii) as a formal statement of the fact that the expressive power of £, + at most equals that of £'. Since, on the other hand, £, is a subformalism of £'+, the expressive power of £, at most equals that of £, +. To express this formally we can use the following statement, which is dual to (vii) and is a trivial consequence of (i).

(viii) For every Y

E

there is an X E

such that X =+ Y.

By (vii) and (viii), £, and £,+ have equal expressive powers, or, as we shall say more frequently, are equipollent in means of expression. For this reason the conjunction of (vii) and (viii), or even (vii) alone, may be called the first equipollence theorem. As a consequence of (iv)(.8) and (v) we obtain at once (ix)

For every W

and X E

if W f-+ X, then

wf- X.

We regard (ix) as a formal expression of the fact that the deductive power of £,+ at most equals that of £'. The converse of (ix), i.e., the fact that the deductive power of £, at most equals that of £, +, is stated formally in (ii), and thus is simply one of the two conditions characterizing £, as a subformalism of £, +. Hence (ix) and (ii) express together the fact that £, and £, + are equipollent in means of proof For this reason the conjunction of (ix) and (ii), or even (ix) alone, may be called the second equipollence theorem.

(x)

er,w =

er,+w n

for every W

L

Indeed this is just another way of expressing the fact that £, and £, + are equipollent in means of proof. (xi) ffiP is a theory in £'+, then is a theory in £', and iP Moreover, iP will satisfy one of the following six conditions:

(a) consistency, (.8) completeness,

=

30

THE FORMALISM

b) (8) (c)

.c+, A DEFINITIONAL EXTENSION

OF

.c

2.4

finite axiomatizability, decidability, essential undecidability, hereditary undecidability,

relative to the formalism £, + iff cI> n E satisfies the same condition relative to the formalism £'.

The proof, using (x) and most of the results in (iv)- (vi), is straightforward. Consider now an arbitrary system S (not necessarily a system of set theory) formalized in £'. Following the suggestion in §1.6 we treat S as a relativization of the formalism £', obtained by replacing the notion f- of derivability for £, with a stronger notion f- of derivability relative to some base set cI> E. We correlate with S an extended system S+ which is formalized in £,+ and has again the base cI>. If, in particular, S is an axiomatic system in £, with the axiom set Ae (cf. §§1.3 and 1.6), then S+ is the axiomatic system in £,+ with the same axiom set, i.e., Ae+ = Ae· Obviously the equipollence theorems (vii)- (ix) and (ii) continue to hold if the relations ===+, f-+, and f- involved in them are relativized to S+ or S, i.e., are replaced by ===t, f-t, and f- respectively. In this sense we say that systems Sand S+ are equipollent in means of expression and proof. The second equipollence theorem suggests the possibility of some simplification in our symbolism. It follows indeed from (ii) and (ix) that the formulas \II f- X and \II f-+ X are equivalent whenever the first of them has a well-defined meaning, i.e., whenever \II E and X E E. Hence we could refrain from using the symbol "f-" altogether and replace it everywhere by "f-+ ". We prefer, however, partly for typographical reasons, to change the symbolism in the opposite direction: we extend the use of "f-" so as to cover all the situations in which we have employed "f-+" until now. This includes, in particular, such contexts as "\II f- X". We also make an analogous stipulation concerning "=== " and " ===+ " . We may, of course, return to the original symbolism in some exceptional cases, so as to enhance the clarity of the text. Notice that 8q\ll cannot be identified with 8q+\II , even if \II E.

t

2.4. The equipollence of a system with an extension As we have said before, we are not interested in developing a general theory of formalisms. However, the notion of equipollence between two formalisms and, more generally, between two systems, is of such fundamental importance for our work that it seems proper to state some observations regarding this notion in a general and precise way. This will be the task of the remaining part of the present chapter. The systems with which we shall concern ourselves are systems developed in interpreted formalisms in the sense of §1.6 (although most of our observations will apply to uninterpreted formalisms as well). Furthermore, the formalisms in which these systems are developed, and hence the systems themselves, satisfy

2.4(ii)

THE EQUIPOLLENCE OF A SYSTEM WITH AN EXTENSION

31

postulates (FI)- (FV) in §1.6 and are semantically sound. We assume that the notion of recursiveness and related notions such as recursive enumerability have been appropriately extended to sets of sentences, and to relations between and operations on these sentences. In particular, we assume that in each formalism the set of all sentences is recursive. We do not assume, in general, that the finitary part, or even the singleton part, of the derivability relation of a formalism is recursively enumerable (cf. §1.3, where this terminology is introduced). Whenever this assumption is essential, it will be explicitly stated. However, it is worth pointing out that this assumption does indeed hold for almost all formalisms discussed in this book. In this section we consider an arbitrary system S(1) = (E(l), f-(l), RE(l), MO(1)) and some extension of it,

(cf. §1.6). We define: (i)

(a) S(1) and S(2) are said to be equipollent in means of expression if for every X E E(2) there is aYE E(1) such that X =(2) Y, and also for every Y E E(l) there is an X E E(2) such that X =(2) Y . ((3) S(1) and S(2) are said to be equipollent in means of proof provided, for every IJ! E(1) and X E E(l), we have IJ! f-(2) X iff IJ! f-(l) x. h) s(1) and S(2) are simply called equipollent if they are equipollent in means of both expression and proof.

In view of the fact that S(1) is a subsystem of S(2) we can easily simplify the above definitions by noticing that (ii) (a) S(1) is equipollent with S(2) in means of expression if for every X E E(2) there is aYE E(1) with X =(2) Y. ((3) S(1) is equipollent with S(2) in means of proof provided, for every IJ! E(1) and X E E(1), that IJ! f-(2) X implies IJ! f-(1) X.

The symbol "=(2),, in (i) (a) may be interpreted either syntactically or semantically. In case S(2) is semantically complete (e.g., when S(2) = .c+) these two interpretations obviously coincide. We shall frequently deal, however, with systems which are not semantically complete, and in which, therefore, the two interpretations cannot be identified. Thus we have to distinguish between syntactical and semantical equipollence in means of expression. Since S(2) is assumed to be semantically sound, the semantical notion of equipollence is, in general, weaker than the syntactical one. One might be inclined to regard the semantical notion as more proper. In this work, however, the discussions will be based upon the syntactical interpretation of the equivalence symbol, unless explicitly stated otherwise. In many cases we shall use a semantical argument, but we shall do so only if it can be replaced by a syntactical one.

32

THE FORMALISM

.c +,

A DEFINITIONAL EXTENSION OF

.c

2.4(iii)

It may seem dubious whether Definition (i) (a) (either in its syntactical or semantical form) really captures the intuitive content of equipollence in means of expression. We shall try to explain what we mean, using as an example the formal languages ,c and ,c+. As we know, ,c and ,c+ satisfy (i) (a) (with 8(1) = ,c and 8(2) = ,c+); thus one could say that every statement which can be formulated in one of these two languages can be formulated in the other as well. However, this does not seem to imply that every notion expressible in one of these languages is expressible in the other, e.g. , that every binary relation definable in ,c + is also definable in ,c. If, nevertheless, we feel that ,c and ,c + are equipollent in means of expression, it is due to the fact that, in application to these two languages, (i)(a) continues to hold if X and Yare assumed to be, not necessarily sentences, but arbitrary formulas with the same free variables; this follows immediately from 2.3(iv)(J) ,(c). Extending these observations to arbitrary systems, we may infer that, to capture better the intuitive content of equipollence in means of expression, we would have to introduce a more elaborate concept of formal language, including in the list of fundamental components such notions as formulas, free variables, etc. We do not introduce such a concept, if only because of the fundamental role which is played in this book by formalisms without variables (and hence without formulas that are not sentences). Furthermore, it may also seem dubious whether (i) (,B) by itself really captures the intuitive content of equipollence in means of proof. For example , set 8(2) = ,c, let S be some fixed logically provable sentence of ,c , and take 8(1) to be the subsystem of ,c whose only sentence is S. Thus E(I) = {S}, and III r(1) X iff III E(I) (Le., III = 0 or III = {S}) and X = S; also RE(I) = RE[,c] and MO(1)S = MOS[,c]. 8(2) is clearly an extension of 8(1) that is equipollent with 8(1) in means of proof in the sense of (i)(,8). The deductive apparatus of 8(1), however, has none of the power and complexity which is possessed by the deductive apparatus of 8(2). In a situation such as this, where (i)(a) fails, we are inclined to treat (i)(,8) merely as a necessary condition for the equipollence of a system 8(2) with one of its subsystems 8(1). We do not know any precise statement that would characterize equipollence in means of proof in a more adequate way. If systems 8(1) and 8(2) are (syntactically) equipollent in means of expression, i.e., satisfy (i)(a), then these doubts concerning the interpretation of (i) (,8) seem to dissipate. In establishing the equipollence of a system 8(2) with one of its subsystems 8(1), we shall usually construct a translation mapping e from 8(2) to 8(1) as was done in §2.3. A precise definition of this notion follows.

(iii)

e

is a translation mapping from a system

(a) (,8)

8(2)

to a subsystem

8(1)

if

e e

is a recursive function; maps E(2) into E(1); (J) ex = X for X E E(1); (8) ex =(2) X for X E E(2); (c) for every III E(2) and X E

E(2),

if III

r(2)

X, then

e*1lI r(l) ex.

THE EQUIPOLLENCE OF A SYSTEM WITH AN EXTENSION

2.4(vi)

33

In case G satisfies conditions (11) - (c), but not necessarily (n:), we shall refer to G as a generalized translation mapping.

A useful modification of (iii) is given in the following corollary.

(iv) For G to be a translation mapping (respectively, a generalized translation mapping) from 8(2) to 8(1), it is necessary and sufficient that G satisfy conditions (iii)(n:) ,(J1) ,h),(c) (respectively, (iii)(J1) ,h),(c)) , as well as the following converse of (c): (10') for every W

and X

E

if G*w

f--(1)

GX, then W f--{2) X .

That a translation mapping or a generalized translation mapping G must satisfy (10') follows from condition (iii)(8) and the fact that 8(1) is a subsystem of 8(2). Suppose now that G satisfies (iii)(n:),(I1),h),(c), or just (iii)(I1),h),(c) , we have GGX =(1) GX by as well as condition (10') above. For every X E (iii)(J1),h), so GX =(2) X by (10') . Thus G satisfies (iii)(8) , as was to be shown. The next corollary of (iii) shows that we can replace the notion of a translation mapping by a weaker notion.

(v)

IfG satisfies conditions (iii)(n:) ,(I1) ,(8) ,(c) (respectively, (iii)(I1),(8) ,(c)) , as well as the following condition (trivially weaker than h)):

h')

GX =(1) X for X E

then there is a translation mapping (respectively, a generalized translation mapping) e from 8(2) to 8(1) which agrees with G on

Indeed we construct e by stipulating that ex = GX for X E and Clearly e satisfies conditions (iii)(n:)- (8), (respectively, (iii)(J1)- (8)) . To verify (iii)(c), suppose that W X E and W f--(2) X. Then G*w f--(1) GX by hypothesis. Now using h') it is easy to check that GX =(1) ex and G*w =(1) e*w . Hence we conclude that e*w f--(1) ex, which completes the proof of (v). ex = X for X E

It may be noticed that (v) remains true if we replace in it condition (iii)(8) by (iv)(c') . The proof is a straightforward combination of the proofs of (iv) and (v). The role of the notion of translation mappings in the discussion of equipollence is clarified by the next theorem.

(vi) Let

8(1)

be a subsystem of

8(2).

Then the following two conditions are

equivalent:

(n:) 8(1) and 8(2) are equipollent in means of expression and proof; (J1) there is a generalized translation mapping from 8(2) to 8(1). In case the singleton part of f--(2) is recursively enumerable, each of these conditions is equivalent to:

h) there is a translation mapping from

8(2)

to

8(1).

34

THE FORMALISM

.c+, A DEFINITIONAL EXTENSION

OF

.c

2.4(vi)

The proof that ((3) implies (a) is straighforward and indeed completely analogous to the proof of the corresponding result for £, and £,+ given in 2.3(ii),(vii) (ix). To prove that (a) implies (fJ) we assume that the members of I:;(l) are arranged in a simple infinite sequence Yo, ... , Yn , ... . We define a function e from I:;(2) to I:;(l) as follows: ex = X for X E I:;(1) and ex = Yn for X E I:;(2) I:;(1), where n is the smallest natural number such that X =(2) Yn ; that such an n exists follows from the equipollence of g(l) and g(2) in means of expression. Clearly e satisfies (iii)(fJ)-(8). To verify part (e) of (iii) suppose that III I:;(2) , X E I:;(2) , and III f-(2) X. Since e satisfies (iii)(8) we obtain e*1lI f-(2) ex. From (i)(fJ) and (iii)(fJ) we conclude that e*1lI f-(1) ex, as was to be shown. The proof that (a) implies ('1) under the assumption that the singleton part of f-(2) is recursively enumerable proceeds similarly to the proof that (a) implies (fJ), only the construction of e is more involved. We first arrange the singleton part of f-(2) in a recursive sequence (Uo, Va), ... , (Un, Vn ), ... . We then set ex = X in case X E I:; (1) , and ex = Y in case the following three conditions are satisfied:

(1)

X E

and Y E

(2)

(X, Y) and (Y, X) both occur in our recursive sequence;

(3)

if Z E I:;(1), then (X, Z) and (Z, X) do not both occur before either (X, Y) or (Y, X) in our recursive sequence.

One can show without great difficulty that e is a well-defined recursive function. The verification of properties (iii) (fJ)-( e) is straightforward. Since (J) clearly implies (fJ), and hence (a), this completes the proof of our theorem. In concrete cases it often happens that the construction of translation mappings considerably simplifies the proof of the equipollence of a system g(2) with a subsystem g(l). In practice the set I:;(2) and the relation f-(2) are defined recursively, or are easily reduced to notions that are so defined (for example, sentences in £, + are characterized as special cases of formulas); this leads to an inductive procedure for proving facts about sentences and about derivability. It enables us, for example, to establish the hypothesis of (ii)(fJ)-and hence also equipollence in means of proof-by induction on the recursive definition of derivability; no such procedure seems to be available for a direct proof of this equipollence. Moreover, if g(2) is semantically complete (as will most often be the case in the sequel), the hypothesis of (ii)(a)-and hence also equipollence in means of expression-may be established by a fairly simple semantic argument using induction on the recursive definition upon which the notion of a sentence is based. The efficiency of the method based on translation mappings will be confirmed by some developments in later chapters.

2.4(ix)

THE EQUIPOLLENCE OF A SYSTEM WITH AN EXTENSION

35

Several theorems in §2.3 (in particular 2.3(x),(xi)) can now be immediately generalized from the formalism [, and its equipollent extension [, + to an arbitrary system S{1) and anyone of its equipollent extensions S(2). As an example we formulate here in (vii) and (viii) the generalization of 2.3(xi).

(vii) in

Let S{1) be any system, S(2) any equipollent extension of it, and

f7(2) (

S(2).

(viii) Under the hypothesis of (vii), if

In connection with the last part of (viii), notice that, by (vi), the hypotheses of (vii) ensure that there certainly will be a translation mapping from S(2) to S{1) in case the singleton part of f-(2) is recursively enumerable. Theorems (vii) and (viii) imply there is a one-one correspondence between the theories of S(2) and those of S(l) which preserves various important properties (as well as many others not involved in our of theories, among them (a discussion). This correspondence F is determined by the stipulation: F

n

for every theory

S(2);

its inverse F- 1 is defined by F- 1
= f>f7 (2)

S (1) .

A basic property of the equipollence relation between a system and its extension is its transitivity whose exhaustive formulation is given in the following theorem, due to Givant. (ix) Let

be three systems such that S(l) is a subsystem of S(2), and S(2) of S(3). If anyone of these systems is equipollent with the other two, then those other two are also equipollent. S{1), S(2), S(3)

In proving this we restrict ourselves to the case when S{1) is equipollent to and S(3), since the remaining cases are trivial. By (vi) there are generalized translation mappings G and H from S(3) to S{1) and from S(2) to S(1) respectively. We shall show that G satisfies conditions ((3), (8), (e) in (iii) and condition b') in (v), with S(l) and S(2) replaced everywhere by S(2) and S(3) respectively. Indeed, the first three of these conditions are immediate consequences of the fact that G is a generalized translation mapping from S(3) to S{1). To verify (v)b') consider any X E Then S(2)

(1)

since (iii)(8) holds with G replaced by H;

(2)

by (1), since S (2) is a subsystem of S (3);

36

THE FORMALISM ..c+, A DEFINITIONAL EXTENSION OF..c

2.4(ix)

(3)

eHX =(1) ex

by (2), since e satisfies (iii)(c) (with replaced by and f-(2) by f-(3));

(4)

HX =(1) ex

by (3), since HX E (iii) h);

(5)

HX =(2) ex

by (4), since S(1) is a subsystem of S(2);

(6)

ex =(2) x

by (1), (5).

and e satisfies

We can now apply Theorem (v) to conclude that the mapping K defined by the stipulation KX

= ex

for X E

and

KX

=X

for X E

is a generalized translation mapping from S(3) to S(2). In view of (vi) this completes the proof of the theorem. Suppose now that :7"(2) is a formalism, :7"(1) an equipollent subformalism of :7"(2), and a generalized translation mapping from :r(2) to :r(1). Given any system S(1) in :7"(1) with base

e

THE EQUIPOLLENCE OF A SYSTEM WITH AN EXTENSION

2.4(xii)

37

of the semantical completeness theorem, a necessary and sufficient condition for :r(l) to be a subformalism of :r(2) (in the sense of §1.6) is that the set of nonlogical constants of :r(1) be included in that of :r(2). The system S(1) will be a subsystem of S(2) (in the sense of 1.6(v)) iff the theory of S(1) (i.e., the set of sentences derivable in :r(1) from a base of S(1)) is included in the theory of S(2). Notice that, if S(1) is a subsystem of S(2), then not only is every sentence of S(l) a sentence of S(2), but also every formula of S(1) is a formula of S(2). (x) Let S(1) be a subsystem of S(2). Then S(l) and S(2) are said to be strongly equipollent in means of expression if, for every X E there is aYE such that Y<jJX = Y<jJY and X =(2) Y.

There seems to be no need, and indeed no possibility, of an analogous strengthening of the notion of equipollence in means of proof. When saying that S(1) and S(2) are strongly equipollent we mean that they are strongly equipollent in means of expression, and equipollent, in the original sense, in means of proof. (xi) if

e

is a strong translation mapping from a system

S(2)

to a subsystem

(a) e is a recursive function; (f3) e maps into and Y<jJex = Y<jJX for X E (() ex = X for X E (8) ex =(2) X for X E (c) for every \II and X E if \II H2) X, then e*\II

HI)

S(1)

ex.

In case e satisfies conditions (f3) - (c), but not necessarily (a), we shall refer to as a generalized strong translation mapping.

e

A notion that plays an important role in the study of systems developed in first-order formalisms, especially in the discussion of their equipollence, is that of a definitional extension. Before introducing this notion, we clarify what we mean by a possible definition of a nonlogical constant C in a system S to which C does not belong. (xii) Let C be a (first-order) nonlogical constant that does not occur among the symbols of a (first-order) system S.

(a) If C is a relation symbol of rank n, then a possible definition of C in S is any sentence of the form VXO .. . Xn_l

(f3)

(C(xo,···, xn-d

++

X),

where X is a formula of S, and xo, ... , Xn-l are the distinct free variables of x. If C is an operation symbol of rank n, then a possible definition of C in S is any sentence of the form V XO .. . Xn

(C(xo,···, xn-dixn ++ X),

THE FORMALISM ,e+, A DEFINITIONAL EXTENSION OF,e

38

2.4(xiii)

where X is a formula of S, XQ, ... , Xn are the distinct free variables of X, and the sentence VXO",xn_13xn

(X A V Xn + 1 (X[Xn/Xn+1J - xnixn+d)

is derivable in S whenever x n+1 is a variable not occurring in X . (Recall from §1.2 that X[Xn/Xn+1J is the formula obtained by substituting Xn+1 for Xn in X.)

(xiii) A system

S(2) is said to be a (first-order) definitional extension of system if the set a(2) of nonlogical constants of S(2) includes the set a(1) of nonlogical constants of S(1), and, moreover, for each C in a(2) a(1) there is a possible definition, Ye, of C in S(1) such that the set

S(l)

e U

is a base for

S(2),

where

e

{Ye: C E

a(2)

a(1)}

is the theory of S(1).

It is easy to verify that in (xiii) we may equivalently replace

for

e

by any base

S(1).

The strong equipollence of a first-order system with one of its (first-order) subsystems can be characterized in terms of the notion of a definitional extension, as the next theorem shows.

(xiv) Let

S(1)

be a subsystem of

S(2).

Then the following three conditions are

equivalent:

(0:) S (1) and S (2) are strongly equipollent; ((3) S(2) is a definitional extension of s(1); b) there is a generalized strong translation mapping from

S(2)

to

S(1).

Moreover, if S(2) has a recursive base, then each of the above conditions is equivalent to the following one: (8) there is a strong translation mapping from S(2) to S(1). To prove that ((3) implies (,) one defines by recursion on formulas a generalized strong translation mapping following the lines of our construction of the strong translation mapping G from £,+ to £'; see §2.3. The details of the construction and the proof that the resulting mapping has the desired properties are well known, and we leave them to the reader. Also the proof that b) implies (0:) is straightforward, and in fact analogous to the proof of the corresponding result for £, and £,+; see §2.3. To prove that (0:) implies ((3) suppose, for i = 1,2, that the system SCi) is developed in the formalism :rei) of predicate logic with a(i) as its set of nonlogical constants, and let e(i) be the theory of S(i). Since S(l) is a subsystem of S(2) we have a(1) a(2). For each C in a(2) a(1) we readily construct a possible definition of C in S(1). In fact, in view of the strong equipollence of S(1) and S(2) in means of expression, we may correlate with C a formula Xc of S(l) such that one of the following sentences, (1), (2), is derivable in S(2):

THE EQUIPOLLENCE OF A SYSTEM WITH AN EXTENSION

2.4(xiv)

(1)

39

VXo",xn_l(C(xo, ... ,xn-d -XC)

in case C is a relation symbol of rank n, and

in case C is an operation symbol of rank n; here the distinct free variables of Xc are just Xo, ... , Xn-l in the first case, and Xo, ... , Xn in the second. Obviously, in the first case, (1) is a possible definition of C in S(1). In the second case, using the equipollence of S(1) and S(2) in means of proof, it is easy to verify that (2) is a possible definition of C in S(1). Let 11 be the set of these possible definitions. Thus, we have (3)

11

8(2).

Moreover, from the fact that

S(1)

is a subsystem of

S(2)

we get

On the basis of the possible definitions in 11 we may construct a mapping H from C)[:r(2)] to C)[:r(1)] with the following properties: (5)

HX

=0

X

[:r(2)]

and

Y
for every X E

C)[:r(2)].

Now (3) and (5) yield

Since each HX is in C)[:r(1)], and since proof, we get from (6) that

S(1)

is equipollent with

S(2)

in means of

Therefore, using also (5), we arrive at

Together, (3), (4), and (8) show that 8(1) u 11 is a base for S(2), and hence that S(2) is a definitional extension of S(1). It remains to show that (a) and (8) are equivalent in case S(2) has a recursive base. The proof is analogous to that of the equivalence of (a) and h) in (vi); the details will be omitted. In various portions of this work we shall want to use the terms "possible definition" and "definitional extension" in application to formalisms other than those of the first order. For example, although L is a first-order formalism

40

THE FORMALISM

.c +,

A DEFINITIONAL EXTENSION OF

.c

2.4(xiv)

and £+ is not, we have suggested that £+ should be regarded as a definitional extension of £, and (DI)- (DV) in §2.2 as possible definitions in £ of the (secondorder) constants +, -, 0, .... , and =, which belong to £+ but not to £. In thus extending the usage of the term "possible definition" , it seems that we can no longer consider just single sentences as was done in (xii). Indeed, since £ + contains no variables representing binary relations, we see no way of explicating the intended meaning of a constant such as 0 in terms of the symbols of £ unless we admit schemata as possible definitions. Thus, for example, the letters "A" and" B" in (DIll) are not variables of £ +, but metalogical variables ranging over the predicates of £ +. Each time we take for A and B specific predicates, we get a sentence Y in £+ which is an instance of (DIll) and which should be regarded as a possible definition of the corresponding predicate A 0 B in terms of A and B. (If we wished to obtain a possible definition of the predicate A0B explicitly in terms of a formula of £, it would be necessary to replace the formulas xAz and zBy in Y by the formulas G(xAz) and G(zBy) respectively, where G is the function defined in 2.3(iii).) The intended meaning of 0 is explicated by the (infinite) set of all sentences which together constitute for us the possible definition of the symbol 0 itself. Under this modification of the notion of possible definition, £+ obviously proves to be a definitional extension of £, and hence we conclude that £ and £+ are equipollent, just as was shown in §2.3. In several portions of the subsequent discussion we shall be confronted with a situation analogous to the one with which we have just been concerned. That is, we shall consider a formalism :J and one of its extensions :J', and we shall attempt to show that :J' is a definitional extension of :J, and hence equipollent with :J in means of expression and proof. The vocabularies of both :J and :J' are assumed to contain no variables of order higher than 1. The procedure outlined above can be applied with obvious modifications in case :J is a firstorder formalism and :J' is obtained by enriching the vocabulary of :J with new symbols that are either operation or relation symbols of second order. In other cases some different modifications may be needed. We do not attempt here to give a precise explication of a notion of possible definition which would be adequate for all situations that may arise, and we hope that the observations in the present section will help the reader to understand the nature of such modifications which will be needed. The conclusion that a formalism :J is equipollent with anyone of its definitional extensions :J' is based upon the tacit assumption that :J has a sufficiently rich logical basis. If this is not the case, then the equipollence of :J' and :J may not be a trivial consequence of the fact that one is a definitional extension of the other, and in fact may not even be true. For a concrete example, the formalisms £3 and as introduced in §3.7 do not seem to be equipollent in means of proof, even though is a definitional extension of £3.3

£t

3- Andreka

p.66.

£t

and Nemeti have recently shown that this is indeed the case; cf. footnote 8* ,

2.5(iv)

EQUIPOLLENCE OF SYSTEMS RELATIVE TO AN EXTENSION

41

2.5. The equipollence of two systems relative to a common extension We are often interested in comparing the expressive and deductive powers of two systems 8(1) and 8(2), neither of which is a subsystem of the other. The problem of clarifying the meanings of these notions is rather involved and we shall not discuss it here in detail. It appears that, for the purposes of the present work, the most convenient approach is to consider the equipollence of two systems 8(1) and 8(2) only in those situations in which they are treated as subsystems of a given third system 8(3); in other words, we shall not use here the binary relation of equipollence between 8(1) and 8(2), but the ternary relation of the equipollence between 8(1) and 8(2) relative to a system 8(3) which is assumed to be a common extension.

(i)

We say that 8(1) and 8(2) are equipollent relative to 8(3) in means of expression if 8(3) is a common extension of 8(1) and 8(2) that is equipollent with each of them in means of expression (in the sense of 2.4(i)); analogously for equipollence in means of proof.

(ii) In case 8(1) and 8(2) are equipollent relative to 8(3) in means of both expression and proof, then we say simply that 8(1) and 8(2) are equipollent relative to 8(3), and we refer to 8(3) as a common equipollent extension of 8(1) and 8(2). We formulate here some basic properties of our new notion of equipollence, beginning with its connection with the notion of equipollence discussed in the preceding section. (iii) Let 8(1) be a subsystem of 8(2). Then the following conditions are equivalent:

(0:) ((3)

8(1)

b)

8(1)

8(1)

and and and

8(2) 8(2) 8(2)

are equipollent in the sense of 2.4(i); are equipollent relative to 8(2); are equipollent relative to 8(3) for some formalism

8(3).

(iv) The relation of equipollence between two systems relative to a given system is an equivalence relation on the class of all subsystems of words, for arbitrary subsystems 8(1), 8(2), 8(3) of 8(4) we have:

8(4)

(0:) 8 (1) is equipollent with 8 (1) relative to 8 (4) ; ((3) if 8(1) is equipollent with 8(2) relative to 8(4), then

b)

8(4).

8(2)

lent with 8(1) as well; relative to 8(4), if 8(1) and 8(2) are equipollent and also then 8(1) and 8(3) are equipollent as well.

In other

is so equipol8(2)

and

8(3),

Theorem (iii) is an immediate consequence of 2.4(ix) , while (iv) is trivial. We can of course pass from the tf'rnary relation of relative equipollence to a binary relation of equipollence by stipulating that two arbitrary systems 8(1) and 8(2) are equipollent iff there is a third system relative to which they are equipollent. By (iii) this binary notion of equipollence is of course an extension of the notion of equipollence defined in 2.4(i), and in fact the latter notion is

THE FORMALISM .£:+, A DEFINITIONAL EXTENSION OF .£:

42

2.5(v)

just the restriction of the former notion to pairs S(l) and S(2) such that one of these systems is a subsystem of the other. It should be pointed out that the binary relation of equipollence just introduced exhibits certain properties which are hard to reconcile with the intuitive concept of equipollence. In particular, this relation is not transitive: we can construct first-order systems ']"(1), ']"(2) , ']"(3) such that 'J(1) and 'J(2) are equipollent relative to some first-order system 'J(4), while 'J(2) and 'J(3) are equipollent relative to some first-order system 'J(5) , and yet 'J(1) and 'J(3) are not equipollent relative to any system. We construct 'J(1) and 'J(3) as certain systems formalized in L. As bases for 'J(1) and 'J(3) we take respectively the sentences 81

=

V zll (xEy

++

xiv)

and

83

=

Vzll (xEy

++

.,xiy ).

We construct 'J(2) as a system in a formalism obtained from £., by replacing everywhere the symbol E with a binary predicate symbol of a different shape, say E'. As a base for 'J(2) we take the sentence 8 2 = V zll (xE'y ++ xiv). Both 'J(4) and 'J(5) are constructed in the formalism of predicate logic with just two nonlogical constants, E and E'. As bases for 'J(4) and 'J(5) we take respectively the sets {81 , 8 2 } and {82 , 8 3 }. Obviously 'J(4) is a definitional extension of both 'J(1) and 'J(2), while 'J(5) is such an extension of 'J(2) and 'J(3). But 'J(1) and 'J(3) do not have a common equipollent extension. Indeed, if a system S were an equipollent extension of both 'J(1) and 'J(3), then the sentence 8 3 would be logically provable in S, since 'J(3) is a subsystem of S. But then 8 3 would have to be logically provable in 'J(1), since 8 3 E and S is equipollent with 'J(1) in means of proof. Hence 'J(1) would be inconsistent, which contradicts the fact that every structure (U, U1 Jd) is a model of 'J(1). For the subsequent discussion of the ternary relation of equipollence, we restrict our attention to formalisms of first-order logic and systems developed in these formalisms. Let S(1) , S(2), and S(3) be systems developed in formalisms of first-order logic. We define: (v) (a)

((3)

and S(2) are first-order definitionally equivalent, or simply definitionaliy equivalent, relative to S(3) , if S(3) is a definitional extension of both S(l) and S(2). S(l) and S(2) are (first-order) definitionally equivalent if there is a system developed in a formalism of first -order logic which is a common definitional extension of both of them. S(1)

From 2.4(xiv) we at once get the following characterization of the definitional equivalence of two systems relative to a third system. (vi) The following conditions are equivalent to each other: (a) S(l) and S(2) are definitionally equivalent relative to S(3) ; ((3) S(3) is an extension of S(l) and S(2) which is strongly equipollent with each of them; h) 3(3) is an extension of 3(1) and 3(2), and there are generalized strong translation mappings from S(3) to S(l) and to S(2) .

2.5(vi)

EQUIPOLLENCE OF SYSTEMS RELATIVE TO AN EXTENSION

43

Moreover, if 8(3) has a recursive base, then each of the above conditions is equivalent to the following one: (8) 8(3) is an extension of 8(1) and 8(2), and there are strong translation mappings from 8(3) to 8(1) and to 8(2). It is immediately seen from (vi) that if 8(1) and 8(2) are definitionally equivalent relative to 8(3), then they are also equipollent (in means of expression and proof) relative to 8(3). The binary relation of definitional equivalence introduced in (v)(jJ) may seem to be a proper candidate for a binary relation of equipollence restricted to the domain of first-order systems. It is a strong notion applying to an important class of systems, and may be used to obtain, as the case may be, either positive or negative solutions to the equipollence problem concerning the systems involved. However, it shares an important defect with the binary relation of equipollence discussed in the remarks following (iv): it is not transitive (and hence not an equivalence relation in the domain of first-order systems). In fact, we actually showed above that the first-order systems T(1) and T(2) are definitionally equivalent, as are T(2) and T(3); however, T(1) and T(3) clearly have no common definitional extension. There is another, weaker notion of definitional equivalence between two systems which is transitive, and hence is an equivalence relation on the class of all systems developed in first-order formalisms. To define this notion, let us agree almost identical if there (temporarily) to call two first-order formalisms and is a one-one correspondence between the nonlogical symbols of and which maps predicates, respectively operation symbols, of to predicates, respectively operation symbols, of with the same rank. Such a correspondence extends in a well-determined way to a one-one correspondence F between the expressions of and in particular, for every expression X, the expression F X differs from X only in the shape of its nonlogical constants. 8 being any system in with base , the system 8' in with base F* is said to be almost identical with 8. We can now say that two first-order systems 8(1) and 8(2) are definitionally equivalent in the wider sense if there are systems 8(1)1 and 8(2)1 which are almost identical with 8(1) and 8(2) respectively, and such that 8(1)1 and 8(2)1 have no common nonlogical constants and have a common definitional extension. The reader may notice that the systems T(1) and T(3) constructed above are definitionally equivalent in the wider sense, even though they are not definitionally equivalent in the original sense. There seem to be many interesting problems connected with the notion of definitional equivalence in the wider sense, even if we restrict ourselves to systems developed in our original formalism £. However, this notion will not playa role in our further work.

CHAPTER

3

The Formalism ,ex without Variables and the Problem of Its Equipollence with ,e

We are now approaching the main task of this work. We shall construct a new formalism, £ x, and study its relationship with the formalisms discussed in the preceding sections. Just as £, the formalism £ x proves to be a subformalism of £ +. The formal structure of £ x is much simpler than that of £ and £ +. In fact, the vocabulary of £ x contains no variables, no quantifiers, and no sentential connectives. It will be shown that £ x is poorer in means of expression and proof than either £ or £ +; it will also be shown that £ x is equipollent in means of expression and proof with a formalism £3 which, roughly speaking, has been obtained from £ by eliminating all but three variables from the vocabulary of the latter while otherwise leaving intact the formal structure of £. On the other hand, we shall see in the next chapter that for each set-theoretical system S which is formalized in £ and satisfies certain weak conditions (fulfilled by practically all systems known from the literature), an equipollent system SX can be constructed in £ x . 3.1. Syntactical and semantical notions of £ x The vocabulary of £ x contains just seven different symbols. Six of them are logical constants; these are two identity symbols, i and =, and four operators, +, -, <:), and '"'. The membership symbol E is again the only nonlogical constant of our formalism. From the atomic predicates, i and E, we construct arbitrary predicates in the same way as this is done in £+; thus ll[£X] = ll. The formulas of £x are just those formulas of £ + which contain no variables, quantifiers, or sentential connectives; in other words they are just the equations A B, where A, BEll. Since there are no variables in £x, the formulas coincide with the sentences. Thus the set of all sentences in £x, is the set of all equations A = B. (When introducing symbolic notation referring to £ x we proceed as in the case of £+; cf. the final remarks in §2.1.)

=

45

46

THE FORMALISM

(i) Let X

=

(A

= B)

£,x WITHOUT VARIABLES

be a sentence in

3.1(i)

We set:

(a) Xl (the left side of X) = A; ({3) X' (the right side of X) = B; (,) X t = Xl if X' = 1; xt = Xl ·X' +Xl (6)

·xr- otherwise; X is called a tautology if Xl = X', i.e ., if A coincides with B.

As logical axioms of £, x we select certain equations which can easily be shown to be logically valid in £, +. Their number is infinite, but all of them are particular instances of a few simple axiom schemata. (ii) S is a logical axiom of £,X, in symbols SEA x, iff, for some A, B, CEIl, S coincides with one of the following equations (BI)-(BX):

(BI)

A+B = B+A,

(BII) (BIll)

A+ (B+C) = (A+B) +C, (A-+B)-+(A-+B-)- =A,

(BIV)

Ae(BeC) = (AeB)eC,

(BV)

(A+B)eC=AeC+BeC,

(BVI)

A e i = A,

(BVII)

A---- = A,

(BVIll)

(A + B)"" = A .... +B""",

(BIX)

(A e B)"" = B .... e A"",

(BX)

A.... e(AeB)-+B- =B-.

For obvious reasons, in defining derivability in £, x we cannot follow directly the lines of 1.3(ii) and §2.2. To obtain an appropriate definition, notice that within £, x we are confronted exclusively with deriving equations from sets of equations. Hence, in the process of derivation we can apply methods familiar from algebraic practice and formally described in equational logic; cf. for instance Tarski [1968]. More specifically, we make the following stipulation.

(iii) For any given X E

x and \II x, X is derivable from \II (in £, X), in symbols \II f-x X, iff X belongs to every set 0 satisfying the following conditions:

(a) ({3) (!) o contains all tautologies; (8) if B, c, DElI and B = C, B = D E 0, then C = D E 0; (e) if B,C,D E II and B=C E 0, then B+D=C+D, D+B=D+C, B- = C-, BeD = CeD, DeB = DeC, and B'"' = C'"' also belong to O. It should be pointed out that there is no essential difference between ({3) and (!) in their logical character and in their role in the definition of derivability.

SYNTACTICAL AND SEMANTICAL NOTIONS OF .G x

3.1(iv)

47

Probably it would be more consistent to omit b) altogether and to include the tautologies in A x (i.e., to enrich the set of logical axioms by the schema "A A"). However, our present formulation seems to be somewhat more suitable for the specific purposes of the present work. It may be mentioned that, with the essential help of ({3), we can show that part b) of this definition is superfluous and that two equations in the conclusion of (10), D+B D+C and D0B D0C, can be omitted. As it stands, however, the definition has the virtue that it does not depend on the specific form of the logical axioms. On the other hand, it can easily be shown that, independent of the form of the logical axioms, (b) and (10) together can be replaced by the following.

=

=

=

(iv) If A,B,C,D E n, ifC occurs as a part in A or B, and if A=B, C=D are in 0, then the equation which we obtain from A occurrence of C by D also belongs to O.

=B

when we replace some

The rule of inference embodied in condition (iv) is referred to as the rule of replacement (of equals by equals). Thus, assuming we include all tautologies in the set of logical axioms (as suggested above), we can say that the rule of replacement can suffice in L x as the only direct rule of inference for deriving logically provable equations from the logical axioms. If, on the other hand, the original definition of derivability in L x is preserved, then the rule of replacement is an example of an indirect, or derivative, rule of inference. The notions defined in terms of derivability for L (cf. §1.3) extend to LX by referring their definitions to the new notion of derivability, f-x. In particular, 8t] x 0 is the logic of LX, i.e., the set of all sentences logically provable in LX; it can be called the algebraic or equational logic of one binary relation. LX itself can be referred to as the formalism of such a logic. If we extend L x by admitting any number of additional nonlogical constants (which are all, like E, binary atomic predicates), but without changing the structure of the formalism otherwise, then the logic of any formalism thus obtained can be referred to as an algebraic or equational logic of binary relations. Since predicates and sentences in L x are also predicates and sentences in L + , we can apply to them automatically the semantical notions of denotation and truth defined for L+ (cf. §2.2). Since, on the other hand, LX has no formulas with free variables, the notion of satisfaction becomes trivial and useless when applied to LX. If we wish, we can define the basic semantical notions for L x directly, without referring to L +. The procedure is very simple: we first define denotation just as this was done for L+, and then we stipulate that a sentence X E :Ex is true of a realization II of L x iff the relations denoted respectively in II by Xl. and xr are identical. We shall return to the discussion of semantical notions for L x in Chapter 6.

48

THE FORMALISM LX WITHOUT VARIABLES

3.2

3.2. Schemata of equations derivable in LX The formalism L x is probably less familiar to the reader than L or L +, and formal derivation in LX appears to be more difficult. To give full proofs of various results stated in the subsequent discussion we would have to use a rather large number of simple metalogical lemmas to the effect that every instance of a given equational schema is logically provable in L x or is derivable from appropriate instances of other equational schemata. No method of proof is known which would be applicable to a large number of such lemmas and could be regarded as routine. (As we shall see in §3.4, the completeness theorem does not hold for LX.) In this situation the following remarks are of interest, since they permit us to take advantage of some results which are available in the literature. We can easily notice a close relationship between the set of schemata (BI)(BX) in 3.1(ii) used to describe the logical axioms of LX and a set of postulates characterizing a class of abstract algebraic structures, namely the so-called relation algebras discussed, e.g., in Chin-Tarski [1951]. In fact, forgetting for a while our stipulations concerning the range of variables "A" , "B", ... and the meaning of the symbols occurring as logical constants in LX, we see from op. cit. , pp. 344- 346, that relation algebras can be characterized as those algebraic structures (U, +, -, 8, i) with two binary operations, + and 8, two unary operations, - and and a distinguished element i, in which the equations stated in (BI)- (BX) are identically satisfied. Clearly, we assume here that "A", " B", .. . range over arbitrary elements of U and that " has the meaning of the ordinary identity symbol. (We disregard some symbolic differences between our present notation and the one in op. cit., as well as a divergence in the treatment of i, which in op. cit. is not regarded as a distinguished element of the algebra.)

="

As a consequence of this observation, various mathematical results in the theory of relation algebras can be interpreted as metalogical results concerning the formalism LX, and conversely. In particular, every equation which is shown to be identically satisfied in every relation algebra yields a schema of which all the particular instances (obtained by substituting predicates for variables) are sentences logically provable in LX. To justify this remark, we apply the completeness theorem for equational logic due to Birkhoff [1935], p. 441 , and we compare it with the definition of f-x in 3.1(iii). For the same reason, every result by which a given equation is implied by other equations can be interpreted as a metalogical result to the effect that every sentence in E X of a given form is derivable from some sentences of related forms (cf. Theorem 8.5(iv)). We list below the simple lemmas needed for the subsequent discussion. They can all be obtained by interpretation ofresults stated in Chin-Tarski [1951] or at least can be easily derived from lemmas so obtained (possibly with the help of 2.1(ii)- (iv)). A, B, C, D in these lemmas are assumed to be arbitrary predicates.

49

SCHEMATA OF EQUATIONS DERIVABLE IN LX

3.2(xx)

(i)

rX

(ii)

rX

(iii)

r X {(A 0B)- = A- eB- ,

=

=

=

A---- = A}.

(A eB)- = A- 0B-,

= (iv)

r X {A0(B 0C) = (A 0B)0C,

(v)

r X {A 0 (B +C) = A 0B+A0C,

(vi)

r X {A e (B·C) = (A eB)· (A eC) ,

(vii)

A

B

r X {A 0C

=

A e(BeC)

B 0C, C 0A

= (A eB)eC}.

(B +C) 0A (B·C) eA

= B 0A+C0A}.

= (B eA)· (C eA)).

C 0B , A eC C eA

(viii)

r X A,C 0(B·D)

(ix)

r X {A 0i =A ,

B eC,

C eB,

A 0B,(C0D).

i 0A

=A ,

A

A 01, A A eO

(x)

r X {A 0C·(B0Cr

(xi)

r X {i

(xii)

A 0B

(xiii)

r X {l eA=l,

(xiv)

r X {1 0A=Oe(10A),

0, i

A,

A, 101 = 1, oeo = O}.

A·B- 0C, C 0A,(C0B)-

A-

C- = x

10A, OeA

A-

B- =x

C 0(A·B-)}. ·A-

O}.

A-.

A e1=1, 00A=0, A 00=0}. A 01=(A01)eO, OeA=10(OeA), A eO = (A eO) 01}.

(xv)

r X {A 0B·C

(xvi)

r X {1 0(A0B·C) =

A 0B·C

Oe(AeB+C)

=10A01,

(xvii)

r X {A

(xviii)

r X A 0 (B eC)

(xix)

A eS-- = 1 = x OeA+B = 1 = x OeA+ (OeB) = 1.

(xx)

A 01

=1 =x i

A 0BeC.

=10A01}.

THE FORMALISM LX WITHOUT VARIABLES

50

(xxi)

f-x A eB · iel

(xxii)

A :S

(xxiii) {A

i

= A·S- e1. = A, A e A = A}.

f-x

i,

3.2(xxi)

c

i} f-x {A .C e (B .D) = A e B,(C 0 D),

B.D e {A·C) = B e A·(DeC)}. (xxiv) f-x (i .A e 1) - = i·A - e 1. (xxv)

{i :s A,

A, A e A:S A} f-x A e (A e B)-

i,

(xxvi)

i

(xxvii)

i}

f-x {A e (B ·C)

= (A e B)-.

(A e B) :S

f-x

1.

= A e B· (A e C), =

i

(xxviii)

[B· (A e C)]

f-x

= = C·(B e A)}.

1.

(xxix) f-x (xxx)

i, A e1 = I}

f-x {(A e B)-

(xxxi)

i, S- e B

i, A e 1· (B e l) = O}

:S

=B-

= AeB-,

(A+B) :S

f-x

1.

As is easily seen, the inclusions i :S A, A, A e A :S A, and e A :S i respectively express the fact that the relation denoted by A is reflexive, symmetric, transitive, and functional. This helps to grasp the content of (xxii) and (xxv) - (xxxi). It should be noticed that the same relationship which holds between the set of logical axiom schemata for ,£; x and the set of postulates for relation algebras holds also between the set of the first three of these schemata, (BI) - (BIII) in §3.1, and a familiar set of postulates for Boolean algebras (treated as structures (U, +, -)) which is due to Huntington [1933], [1933a]. Hence, all the identities and implications between equations which are known from the theory of Boolean algebras yield analogous results concerning ,£; x. In this way we get for instance:

= B) :=x (A· B + A - . BX:=x (X t = 1) for every X E

(xxxii) (A

(xxxiii) 1 = 0 f-x X for every X E

= 1) for any A, B

E II,

or equivalently,

i.e., 1 = 0 f-x

While we shall have the opportunity of referring to (i) (xxxi), analogous lemmas obtained by interpretation of various well-known results from the theory of Boolean algebras will not be referred to here in any explicit way. When such a lemma is applied in the subsequent discussion, this fact is sometimes indicated

51

A DEDUCTION THEOREM FOR LX

3.3(ii)

by "BA" , and in simpler cases even this indication is omitted. Also, some of the simpler lemmas among (i)- (xxxi) will be occasionally used without any explicit reference; this applies, in particular, to (i)- (iv), (vii), and (ix). The distributive laws (v) , (vi), and (xxvii) can clearly be extended by induction from two predicates, Band C , to any finite sequence (Do, . .. , Dm) of predicates, and in this extended form they will be applied in some arguments. 3.3. A deduction theorem for £, x We shall now state some results which apply to £, X and are closely related in content to the deduction theorems in §1.3. (i) For any W

E X and X , Y E E X, we have

WU{X} r X Y

(ii) For any A , BEn and W WU {A

=I}

iff W r

X 10X t -01+y t =1.

E X, we have

r X B =1 iff

W r x 10 A - 01 + B

=1.

Apparently, (i) is more general than (ii), since the latter is partly restricted to equations X with xr = 1. However, in view of 3.1(i)(-y) and 3.2(xxxii), each of the two statements can be directly derived from the other. One half of (ii) , the implication from right to left, is rather trivial; we use 3.2(xiii) and BA (in addition to some general properties of derivability mentioned in §1.6) . To obtain the implication in the opposite direction we apply induction on sentences derivable in £, x from the set W U {A I}. In fact , we set

=

[2 =

{X : X E E X and W r X 10A- 01+Xt

=I} .

We then show that [2 satisfies 3.1(iii)(a)-(c:) , with" w" replaced by "wU{A = I}", and hence contains every sentence X such that W U {A I} r X X. The argument is based primarily on BA. In addition, we use various lemmas from §3.2: 3.2(vii) in proving that A 1 E [2 (a part of 3.1(iii)(a)) ; 3.2(i),(ii) ,(xvii) in proving that B'-' E [2 whenever B C E [2 (a part of 3.1(iii)(c:)); 3.2(iv),(vii), (x) in proving that B0D=C0D,D0B=D0C E [2 whenever B=C E [2 (another part of 3.1 (iii ) (c:)).

=

=(}'-'

=

=

A close relationship between (i), (ii) , and 1.3(iv) is obvious. If (ii) is referred to £, + instead of to £, x , it becomes an immediate corollary of Theorem 1.3(iv) (which extends to £,+ according to the remarks preceding 2.2(i)) and Theorem 2.2(ii). In view of this relationship we refer to (i) and (ii) as the deduction theorems for £,X . The last remark suggests the idea of introducing the general notion of a deduction theorem for a given formalism, and thus of singling out formalisms (provided) with a deduction theorem. Let 3" be any formalism for which the notion of derivability r has properties (FI)- (FV) listed in §1.6, and to which the notion of recursiveness has been properly extended (cf. §1.3). We assume, in particular, that the set E[31 is recursive and that every theory in 3" with a recursively

52

THE FORMALISM

£,X

WITHOUT VARIABLES

3.3(iii)

enumerable base is itself recursively enumerable. By a deduction theorem for we understand any theorem of the following form: For any X, Y E E[:J] and any Ilt \II U {X}

r

Y [:7]

E[:J], we have

iff

\II

r

(X - Y) [:7].

Here is some appropriately chosen binary recursive operation on and to elements of E [:J]; for instance, in the case of £ x we can take for the operation defined for any X, Y E EX by the formula

(iii) X _ Y = (10Xt- 01+ y

t

=1),

in agreement with (i). A deduction theorem has important consequences whenever it can be established. In particular, Theorems 1.3(vi) ,(vii) apply to every formalism with a deduction theorem, and the same is true of 1.3(viii) since, by (FV), the inconsistent theory E[:J] has a finite base. Thus, by (i) and 3.2(xxxiii), Theorems 1.3(vi)- (viii) hold for £x. In view of (iii), it is seen from (i) that the binary operation - plays essentially the same role in £ x as the operation _ in £. We can also consider a unary operation .." on sentences of £ x defined by (iv) .."X

=

(10X t -01

=1).

This operation plays essentially the same role in £x as the operation.., in £, as is seen, for instance, from the following theorem. 1

(v) For any set Ilt

EX and any sentence X E EX, the following two condz'tions

are equivalent:

(a) X is incompatible with Ilt, ((3) Ilt f- X .."x. If, in addition, Ilt is a theory in £ x, then the conditions (a) and ((3) are also equivalent to

b) .."x E Ilt. In fact, it is easily shown by means of 3.2(xxxiii) that (a) is equivalent to the condition Ilt U {X} f- x 1 0, and it follows from (i), (iv) that this latter

=

1* Actually, it is not difficult to prove that every instance of Axioms (AI) - (AIII) in §1.3, with " ..... " and ".," replaced by " -. " and "...,." , and with "X", "Y", "Z" ranging over sentences in x, is logically provable in £, x, and that modus ponens (formulated in terms of ..... ) is a valid rule of inference in £, x . This can be shown directly, with the help of the lemmas in §3.2. Alternately, using (i), (v), and 3.2(iii),(xiv), it is easy to verify that Axioms 1- 5, 6* - 10* from Tarski [1956], Article III, p. 32, are satisfied, with "S" , "en", "c", and "n" replaced there by "I: x ", "9" x ", " ..... ", and "..,,"; the desired result then follows from Theorem 3 * and Axiom 7* of ibid. In consequence, all instances of true sentences of the sentential calculus (with " ..... " and ".," replaced by " ..... " and "..,,", and with variables replaced by sentences of X) are logically provable in £, x . This observation justifies our remark that the operations ..... and.." play essentially the same role in £, x as do the operations ..... and., in £' . (However, the observation will not be used in the subsequent discussion.)

THE INEQUIPOLLENCE OF

3.4{iii)

,(,X

WITH

,(,+

AND ,(,

53

condition is equivalent to ((3). In case \II is a theory, the equivalence of ((3) and (1) is obvious. For later use we give here an elementary theorem based upon (v). (vi) Let 8 be a theory in ["x; let 8', or 8", be respectively the set of sentences incompatible, or compatible, with 8. If anyone of the three sets 8,8',8" is recursive, then the remaining two are also recursive. Indeed, by (iv), (v) we have for any X E (1)

X E 8' iff

,.X E 8.

Primarily with the help of 3.1(i)(J), 3.2(ix),(xiii),(xiv),(xxxii), we easily derive X ,.,.X, whence, using also (1),

=x

(2)

X E8

iff

,.X E

8'.

Since the mapping ,. is recursive, we see from (1) that the recursiveness of 8 implies that of 8', and from (2) we derive the implication in the opposite direction. Finally, 8", i.e., 8', is obviously recursive iff 8' is recursive. 3.4. The inequipollence of L x with L + and L

We turn now to the problem of comparing the formalism LX with Land L+ in regard to expressive and deductive powers. It is easily seen that L x is a subformalism of L + in the following sense.

(i) (ii)

For every \II

and X

E

if \II f-X X, then \II f- X.

Theorems (i) and (ii) are analogues of 2.3(i),(ii); because of the peculiar structure of LX, we do not refer in (i) to arbitrary formulas. Notice that in formulating the conclusion of (ii) we apply the convention stated at the end of §2.3. A consequence of (i) and (ii) is:

(iii) If 8 is a theory in L+, then 8 n

is a theory in LX.

Another consequence of (ii) is that the formalism LX, just like L + and L, is semantically sound; cf. §1.6. As a subformalism of L+, LX is at most as powerful as L+ in means of expression and proof. Since, as we know from Chapter 2, L+ is equipollent with its subformalism L , we can also say that LX is at most as powprflll as L (relative to L+); cf. §2.5. The problem naturally arises whether LX is at least as powerful as L+ and L, and whether, therefore, the three formalisms are equipollent. This problem divides into two parts, the first of which concerns the means of expression and the second the means of proof.

THE FORMALISM

54

.ex

WITHOUT VARIABLES

3.4(iv)

In agreement with the stipulations in §2.4, to show that £, + is equipollent in means of expression with its subformalism £, x we would have to prove that every sentence in £, + is logically equivalent with some sentence in £, x. It has been known for a long time, however, that this is not the case. The result is essentially due to A. Korselt, and was reported in Lbwenheim [1915]' p. 448. Using our terminology, it can be formulated as follows. such that X == Y does

(iv) There are sentences X E and actually X E not hold for any Y E For instance, the sentence

is of this kind.

The proof is not difficult. In view of the semantical soundness of £, +, (iv) can be interpreted not only syntactically but also semantically: there is no Y E such that Sand Y have all models in common. On the other hand, it is easily seen that S is almost semantically equivalent with some Y E in the following sense: there is a natural number n such that for every structure II = (U, E) with lUI 2: n, II is a model of S iff it is a model of Y. Indeed, S expresses the fact that the universe has at least 4 elements, and hence every structure II with lUI 2: 4 is a model of S. Consequently, S is almost equivalent with every Y E which is logically provable, e.g., Y = (1 = 1). It is known that every X E in which the nonlogical constant E does not occur is almost equivalent with some Y E . In this connection, Tarski has modified and supplemented Korselt's argument to obtain the following improvement of (iv).

(v) The sentence

is not almost semantically equivalent with any Y E

.

The solution of the equipollence problem in its application to means of proof is also negative, as was first pointed out by Givant. To show that £,X and £,+ are not equipollent (in the sense of §2.4), it suffices to exhibit a set W of sentences in £,X and a sentence X in £,X such that wf- X holds while W f-x X fails. It turns out that this can be achieved by taking the empty set for W. The problem proves to be closely related to the representation problem for relation algebras. 2 It is well known that the solution of the latter problem is negative; cf. Lyndon [1950], [1956]. A very simple solution of the problem has been presented in McKenzie [1970], where a nonrepresentable relation algebra with one generator and 16 different elements has been constructed. An analysis of this construction leads to the following conclusion. hSee §8.3 for a brief discussion of this problem.

3.4(vii)

THE INEQUIPOLLENCE OF LX WITH L+ AND L

There are sentences X E instance, the sentence

such that f- X holds while f-x X fails. For

(vi)

T

55

= [10 (EeE+ [E2 + i+ (E+E'"')-2 .E'"'-]. E- + (E0E'"')-)

01

= 1]

is of this kind. 3 ,4 Sentence T is simpler than the one that Givant constructed directly from McKenzie's argument. (The simplification was obtained jointly by McNulty and Tarski.) Still, T is rather involved. It would be interesting to find the shortest possible sentence which could serve the same purpose. An equivalent formulation of the first statement in (vi) is

(vii) is While e" x 0 is the set of all logically provable sentences in [,X, 8,,+0 n clearly the set of all logically true sentences in [, x. Thus (vii) shows that the formalism [, x is not semantically complete. 3- A similar result, formulated for relation algebras, was recently published in J6nsson [1982]' Theorem 7.9. 4 - We sketch a proof that f-- x T fails , using the terminology and theorems of §8.5 that allow us to interpret results about relation algebras as results about LX. We begin by constructing McKenzie's relation algebra. Let (A , + , -) be a sixteen element Boolean algebra with the four atoms a , b, c, d. We define a binary operation 0 and a unary operation on these atoms via the tables b c d 0 a b a a a a c d 1 b c b b b b+d c b 1 c c c c+d d d b+d c+d a+b+c d d

m

Both 0 and extended in the obvious ways to all of A so that the distributive laws, (Ra V) and (Ra VIII) from 8.2(i), are valid. Let Qt be the algebra of type (2,1,2,1,0,0) defined by Qt

= (A,

+, - , 0

, , i , E),

where i = a and E = a + b = i + b. It is a simple matter to check that all the axioms defining the class of relation algebras are valid in Qt . To show that T fails in Qt it suffices, by 3.2(xiii), to show that (1)

It is simple to see that

Since we get that Therefore, = E·E- = 0,

which verifies (1) . Since the set S of axioms of RA is valid in

Qt ,

we can now conclude that

(2)

But

e"X0 =

e"EAs n

by 8.5(iv), so the nonderivability of T in LX follows immediately from (2).

56

THE FORMALISM £,x WITHOUT VARIABLES

3.5

By (iv) and (vi), the formalism £,x is not equipollent with, and actually (in view of (i), (ii)), is poorer than £,+ in means of both expression and proof Consequently, £, x is not equipollent with £, relative to £, +. These negative results lead in a natural way to several other problems which will be discussed in the remaining part of the present chapter.

3.5. The inequipollence of extensions of £, x with £, + and £, The question arises whether the inequipollence of £, x with £, and £, + is simply due to an overly restrictive selection of both the logical operators of £, + and £, x, and the logical axiom schemata of £, x. Is it possible, in fact, to extend the formalisms of £, x and £, + so as to secure their equipollence with each other and with £', while simultaneously keeping intact their basic structural features? In discussing this problem, we shall assume that the extended formalisms l, x and l,+ with which we shall deal are obtained from £,X and £,+ by means of the following modifications. (1) The vocabularies of £, x and £, + are enriched by including finitely many new constant symbols. Just as the symbols of £,X (with the exception of =), the new symbols are atomic predicates denoting some binary relations, or else operators denoting operations on and to binary relations, each operator having a definite rank, but not necessarily rank 1 or 2. (To simplify the formulations , we treat the atomic predicates as operators of rank 0.) Moreover, we assume that, on the basis of the semantical definitions adopted for l, x and l, +, and specifically on the basis of the definition of denotation (in terms of which the meanings of the new constants in each realization are specified), all these new constants prove to be logical symbols in a sense to be made precise below. (2) The set of logical axiom schemata (BI)- (BX) of £,X, given in §3.1, is supplemented by finitely many new schemata which, loosely speaking, have the same structure as the original ones - i.e., all the particular instances of these schemata are equations between predicates (of l, X) - and which are logically valid on the basis of the semantical definitions adopted for l, x . Conditions (1) and (2) imply some further minor changes of a rather obvious nature in the definitions of predicates and derivability given in 2.1(i) and 3.1(iii). The formal structure of £, x remains otherwise unchanged. As regards l, +, we do not impose on it a condition analogous to (2), for example, that its set of logical axiom schemata be obtained from that of £,+ by adding finitely many new schemata which have the same structure as Schemata (DI)- (DV). Indeed, we shall obtain certain negative results that hold even when l,+ is not a definitional extension of £'. Since E is the only nonlogical constant of l, x and l, +, the realizations of these two formalisms are the same structures il = (U, E) that are realizations of £, x and £, +. Relative to these realizations we define the basic semantical notions applicable to l, x and l, + (denotation and truth), proceeding analogously as in the case of £,X and £,+; cf. §3.1, last paragraph, and §2.2. A detailed formulation

3.5{ii)

THE INEQUIPOLLENCE OF EXTENSIONS OF V

WITH L + AND L

57

of the semantical definitions depends, of course, on the intended meaning of the new constants included in the vocabularies of Zx and Z+ . The assumption that these new constants are logical symbols will play an essential role in our discussion. It is based upon a conception of logical symbols which we believe was suggested for the first time in Lindenbaum- Tarski [1936]. (An English translation of this paper occurs as Article XIII in Tarski [1956]. See also Tarski [1986].) Due to the breadth of this conception, the results of the discussion can be expressed in a strong and general form. Consider any nonempty set U (which can be regarded as the universe of a realization U = (U, E) of .(,X). Thinking in the framework of the theory of types, we can construct from this basic universe U various derivative universes of higher types, e.g., the universe U' of all n-ary relations among elements of U, the universe U" of p-ary operations on and to members of U', the universe U'" of q-ary relations among members of U' (where n, p, q are fixed positive integers), etc. Every permutation P of the basic universe (i.e. , every one-one mapping of U onto U) induces in a natural way a uniquely determined permutation P of any given derivative universe U. A member M of U is said to be invariant under P if P(M) = M. We now stipulate:

(i) Given a basic universe U, a member M of any derivative universe U is said to be logical, or a logical object, if it is invariant under every permutation P of U. (Strictly speaking, since an object M can be a member of many derivative universes, we should use in (i) the phrase "is said to be logical, or a logical object, as a member of U" .)

(ii) A symbol S of the formalism Zx is said to be logical, or a logical constant, if, for every given realization U of this formalism with the universe U, S denotes a logical object in some derivative universe U. On the basis of (i) one can show, for example, that for every (nonempty) U there are only four logical binary relations between elements of U: the universal relation UxU, the empty relation 0, the identity relation U1 Id, and the diversity relation (U x U) n Di. On the other hand, if U is infinite, then there are infinitely many, and in fact nondenumerably many, logical unary, or binary, operations on and to relations between elements of U. Examples of such op·erations are those denoted by the operators of .(,X, i.e. , formation of unions, complementation, relative multiplication, and conversion. The symbol i in .(" '('+, and .(,X is logical in the sense of (ii), and so are the symbol = and the four operators in .(, + and .(, x . As a matter of fact , the remaining symbols in .(, + and .(, x which were referred to from the start as logical constants, i.e., the implication and negation symbols and the universal quantifier, can also be subsumed under logical constants in the sense of (ii); this would require, however, some expansion and elaboration of our terminology.

THE FORMALISM

58

.ex

WITHOUT VARIABLES

3.5(iii)

It may be pointed out that the exact definitions of logical objects and symbols given in (i) and (ii) are essentially involved only in the results of the present section. Elsewhere in this work we use the terms "logical symbol", "logical notion", etc., in an informal way which seems, however, to always be compatible with the definitions given above. The problem of whether any of the extended formalisms L x is equipollent with £., and £., + (or L +) splits again into two parts concerning respectively the means of expression and those of proof. As regards the first part, we shall show that LX can never be equipollent with £., or £.,+ (and hence cannot be equipollent with L+) in means of expression. Notice that, in general, neither of the formalisms £.,+ and LX is a subformalism of the other, if only for the reason that neither of the corresponding sets of sentences, E+ and EX, includes the other. Since, however, £.,+ and LX have the same realizations, at least the problem of their semantical equipollence in means of expression has a well-defined meaning: £.,+ and LX are so equipollent just in case for every X E E+ there is aYE EX, and for every Y E EX there is an X E E+, such that in all realizations of £., +, X is true iff Y is true. An analogous definition applies to £., and LX. To obtain a negative solution of the problem we first introduce notations for several particular predicates in IT. C 1 = E-.i, C 2 = C 0 010C 1 , (iii) co=E.i, C 4 = C 0 010Co '0, C 5 = C 1 010C1 ,0.

(iv) Dy = Cka + ... +Ckn _ 1 whenever Y = {k o, ... , kn-d (so that n 6).

°<

< ... <

ko

kn -

1

< 5

and

Next, using these notations, we establish three lemmas, (v) - (vii).

(v) The following equations are all logically valid:

=

=

=

=

(0:") D0 0, D6 1, D{O,l} i, D{2, ... ,5} 0, D{k} ((3) D y +Dz D yuz , D y ·D z D ynz , Dy

=

=

=

=Ck for k = 0, ... ,5; for any Y, Z

6.

This follows directly from (iii), (iv), and 2.2(DI)-(DV). (Recall that {O, ... , 5} 6.)

=

Now consider any formalism LX of the kind described, with new logical operators 0 0 , ... , Om-I. where each Oi has rank r(i). Let n be the set of all predicates of LX. Thus we have Oi(A o, ... , Ar(i)-d E n whenever A o, ... , A r (i)-l E n. In the remaining part of this section we frequently use realizations U = (U, E) of £.,X in which E U1 Id; for brevity we shall refer to them as Id-models.

(vi) For every Id-model U and every A E holds in U.

n

n there is a Y

6 such that A

= Dy

To prove this, fix an A E and let R, So, ... ,S5 be the relations on U denoted respectively by A, Co, ... , C 5 . We have:

3.5(vii)

(1)

THE INEQUIPOLLENCE OF EXTENSIONS OF

if Sk

n R =I 0, then Sk

<;;; R,

£,X

WITH

£,+

AND

£,

59

for each k E 6.

Indeed, suppose (Xo, Xl) E Sk n R, and let (Yo, YI) E Sk' It is easy to construct an automorphism F of 11 which maps Xi to Yi for i = 0,1. To give an example, if Xo = YI and Xl =I Yo, then we define F by stipulating:

Fxo

= Yo,

Fyo

= Xl,

FXI

= YI,

and Fz

=z

=I XO,XI,YO·

for z

From XOSkXI and YOSkYI and (iii) we easily see that Xo = Xl iff Yo = YI, and XiExi iff YiEYi for i = 0,1. Using this and the fact that 11 is an Id-model, one readily checks that F is a well-defined automorphism of 11. Since R is the relation denoted by A E ft, it is clear that R is invariant under F, and hence from XORXI we conclude YORYI . Now (Yo, YI) in Sk is arbitrary, so Sk <;;; R, as was to be shown. Let Y be the set of k E 6 for which Sk <;;; R. By (1) and (iv), (v), we obtain that R = U{Sk: kEY}, i.e., that A D y holds in 11, as was to be shown.

=

More involved is the proof of the next lemma, which presents an essential improvement of (vi).

(vii) There is a set 6. <;;; ft satisfying the following conditions: (a) 6. is finite; ({3) for every A E ft there is aBE 6. such that A B holds in all Idmodels of ZX.

=

To prove (vii) we first specify a natural number n and we construct n classes Ko , ... , Kn - l satisfying the following conditions:

(1)

U Kj

is precisely the class of all Id-models;

j
(2)

given A E ft, there is, for each J' = 0, ... , n - 1, a Y <;;; 6 such that A Dy is in 8pKj (the theory of Kj ).

=

We proceed as follows. The set Sb6 = {Y: Y <;;; 6} has, of course, cardinality 64, and thus can be represented in the form

(3)

Sb6 = {So,···, S63}'

Put M

= {(i, X) : i < m and X = (Xo, ... , Xr(i)-l)

E r(i)Sb6}

(where m is the number of new logical operators Oi, and r(i) is the rank of Oi). M is of course finite, so that we can specify n by setting

Consequently, the members of MSb6, i.e., the functions from M to Sb6, can be arranged in a finite sequence of lengt h n, so that

THE FORMALISM LX WITHOUT VARIABLES

60

3.5(vii)

MSb6 = {Fo, ... , Fn-d.

(5)

Using the functions F j we now define the classes Kj, j = 0, ... , n - 1, by letting Kj = {ll: II is an Id-model and Oi(Dxa, ... ,Dxr(i)_J=DFJ(i,X) E 0pll forall

(6)

(i,X) EM}.

With every given Id-modelll we can correlate a definite function Fj from (5) such that II E K j . In fact for each (i,X) EM there exists, by (vi) and (3), an Sk for which

The function Fj is completely determined by stipulating that F j (i, X) = Sk, where k is the least number for which (7) holds. Hence, by comparing (6) and (7), we see at once that II E Kj . Thus the classes Ko, ... ,K n - 1 satisfy condition

(1) . To show that they satisfy condition (2) as well, we let 0 be the set of predicates A E IT for which the conclusion of (2) holds, and we show by induction on predicates that all members of IT are in O. Clearly i, E E 0 since, by (iii), (v), and (1), we have i D{o,l}, E D{o} E 0pKj for j = 0, ... , n-l. Now let Oi, with i < m, be an arbitrary operator in ZX and A o , ... , Ar(i)-l arbitrary predicates in O. Given any j = 0, ... ,n - 1, we can find, by the definition of 0, sets Xk 6 with k = 0, ... , r(i) - 1, such that Ak DXk E 0pKj. Therefore

=

°

=

=

Oi(Ao,···, Ar(i)-d

=Oi(Dxa ,···, DXr(i)_J

E 0pKj,

so that, by (6), Oi(Ao, ... , Ar(i)-d

=DFJ(i,X)

E 0pKj,

and, finally, Oi(A o, ... ,Ar(i)-d EO. This completes the induction argument. We have thus shown that IT 0 and that, in consequence, Ko, .. . , Kn - 1 satisfy condition (2) . In view of (2) we can correlate with each A E IT a sequence GA = (GoA, ... , Gn-1A) E nSb6

(where n is defined as in (4)), by stipulating: for each j < n, choose the least k such that A DSk E 0pKj and set GjA = Sk. From this definition we see that, for any A, BE IT with GA = GB, we have A B E 0pKj for every j < n.

=

=

Therefore, by (1), we get: (8)

whenever A, BE Id-model.

IT

and GA

= GB, then

A

=B holds in every

As is well known, the set IT is obviously denumerable and thus can be represented as the range of an infinite sequence of type w. For each Y E G*IT let HY be the

3.5(viii) THE INEQUIPOLLENCE OF EXTENSIONS OF

.ex

WITH

.e+

AND

.e

61

first predicate in this sequence such that G(HY) = Y. Hence, by taking H(GA) for B in (8) we conclude that (9)

A

= H(GA)

holds in all Id-models for every A E ft.

If we now put = H*(G*ft) (so that is the range of H), we obtain a set satisfies (0:) since G*ft which satisfies the conditions of our theorem. In fact, is included in the finite set nSb6, while (9) directly implies that satisfies ((3) as well.

We can now state the main result concerning the equipollence of and L + in means of expression. 5

Zx

with L

(viii) Given any formalism ZX (of the type discussed in the beginning of the and hence also an X E such that X is not section) there is an X E semantically equivalent with any Y E In fact, there is a natural number p such that in the infinite sequence of sentences So, ... , Sn, ... in determined by the condition Sn = VVo"'Vn3Vn+l(-,voivn+l" ... "-,vnivn+d, every sentence Sn with n

Indeed, let

p has this property.

be as in (vii) and assume

=

q.

Then there are at most

q distinct sentences among the Sn's which are equivalent with some sentences in x. For suppose, to the contrary, that the q + 1 sentences Sno' ... ,Sn q are, For each i = 0, ... ,q there is, by respectively, equivalent to Yo, ... , Yq E = q, (vii), a predicate Bi E for which Y/ = Bi holds in all Id-models. Since there must be distinct i, J' with 0 ::; i < j ::; q such that Bi and B J coincide. Thus Y/ = and hence also Si ++ SJ, holds in all" I d-models. But this contradicts the fact that Si ++ SJ is false in the Id-model (i + 2, i + 21 Id). The theorem now

Yi,

easily follows. Theorem (viii) is a far-reaching generalization of Korselt's result 3.4(iv); Theorem 3.4(v) can also be generalized in an analogous way. The first part of (viii) states in a formal way that there is no formalism Zx whose expressive power at least equals that of L or L +. Thus, a fortiori, L, L +, and Z+ are not (semantically) equipollent with Zx in means of expression. Some generalizations of (viii) are known. In fact, it can be shown that (viii) extends from formalisms of type Zx to a considerably wider class of formalisms. To obtain formalisms of this class we include in the vocabulary of LX any finite number of logical constants denoting relations (not necessarily binary) among elements of the universe, operations on and to such relations, as well as relations among such relations, all of them of arbitrary but well-determined finite ranks. 5This result was found by Tarski in 1940 and was announced in a somewhat more general form in Tarski [1941]' p. 89.

62

THE FORMALISM

.c x

WITHOUT VARIABLES

3.5(ix)

We can even provide such formalisms with finitely many logical connectives (like those occurring in £'), as well as with finitely many variables and logical quantifiers (like the universal quantifier )- thus giving them the appearance of formal languages more similar to £, + than to £, x. (Some simple examples of such formalisms will be discussed below in §§3.7- 3.1O.) Theorem (viii) applies to all extensions of £,X thus obtained. Notice that the proof of (viii) is completely independent of the way in which derivability has been defined for the formalism discussed. The problem whether any formalism £,X (of the type described at the beginning of the section) can be equipollent with £,+ in means of proof is still open, even if we restrict ourselves to the case when £, + is a definitional extension of £, (in the same sense as is £,+) . In fact, the problem is even open in the special case when £,+ and £,+ coincide, and £, X is obtained from £,X by leaving the vocabulary unchanged and by adjoining to Ax finitely many new schemata, all instances of which are, as before, logically valid in £, +. Clearly such a formalism £,X is a subformalism of £'+, and the notion of derivability in it is, in general, stronger than that in £, x. This special case of the problem would, of course, be solved negatively if we managed to show that (ix) There is a sentence X E

for which 1-+ X holds while I- X [£, X] fails. 6

We do not know whether (ix) is true. If, however, the vocabularies of the formalisms involved contained not just the nonlogical atomic predicate E, but rather infinitely many nonlogical atomic predicates, then (ix) would be an easy consequence of a result of Monk [1964] to the effect that the equational theory of representable relation algebras is not finitely based. However, allowing infinitely many such predicates would destroy one of those essential structural features of £, x which secure its eminently finitistic character. 3.6. £, x -expressibility

We return to the negative results stated in 3.4(iv),(v). A sentence X in £,+ will be referred to as expressible in £, x or £, x - expressible, if X == Y for some Y E We shall use analogous terminology with respect to other formalisms as well. In this case the relation == is to be understood, in general, as semantical equivalence. (If, however, the formalisms discussed are subformalisms of £, +, then == can of course be equally well interpreted as syntactical equivalence in £'+.) By 3.4(iv),(v) we have: (i) S, S' are not £, x -expressible.

The arguments used in establishing (i) have a restricted range of application and, in general, do not work for other sentences in or with equally simple 6'This special case was recently solved negatively in Maddux [1987]: the set of all true equations of .c x is not finitely schematizable. His proof proceeds by showing that the set of all equations with just one variable that are true in all representable relation algebras is not finitely based.

,c x - EXPRESSIBILITY

3.6(iii)

63

structures. Recently, Kwatinetz [1981] published a fairly general method (discovered around 1972) which enabled him to show that several simple sentences with interesting mathematical contents are not L x -expressible. We give here a few examples of such sentences.

(ii) 8 0 ,81 ,82 are not L x -expressible, where 80

V xlI 3 z [zEx A zEy A Vu(uEx A uEy - uEz)],

81

V xlI 3 z V u (uEz

82

V xlI [3 z (xEz A yEz) - 3 z Vu (uEz

++

uEx V uEy), ++

u1x V uiy)].

If interpreted in a structure (U, E) where E is a partial ordering relation, 8 0 obviously expresses the fact that any two elements of U have a greatest lower bound (in the order established by E). Sentences 8 1 and 8 2 are provable in practically all set-theoretical systems known from the literature, and frequently occur in axiom sets of these systems; 8 1 is the union axiom and 8 2 the restricted pair axiom. Some other related sentences have been shown not to be L x -expressible by Ulf Wostner; see Wostner [1976]. On the other hand, we know a great many sentences in E which are LX_ expressible; by 2.3(iv)(o),(E) all the sentences GY with Y E EX have this property. The problem of whether a particular sentence X in E or E+ is L x -expressible may present essential difficulties. As an example of a simple sentence for which the problem is open we mention

8 3 = VxlI 3 z Vu (uEz

++

u1x V uEy).

This sentence is well known in its set-theoretical interpretation and is provable in various familiar systems of set theory. Around 1971, Kwatinetz (settling a problem stated in Tarski [1941]' p. 89) showed that there is no general method which enables us to solve each particular case of this problem in finitely many steps; see Kwatinetz [1981]. In other words,

(iii) The set of all sentences in L, as well as the set of all sentences in L + , which are L x -expressible is not recursive. In view of this last result, the problem of finding partial criteria for LX_ expressibility and L x -inexpressibility becomes interesting and important. This is the problem of constructing various comprehensive and simply defined recursive sets which can be shown to consist exclusively of LX-expressible or LX_ inexpressible sentences. At present we do not know any interesting criteria for LX-inexpressibility, but we know some for LX-expressibility. We shall become acquainted with such criteria in the subsequent discussion; cf. §§3.1O and 4.7. The negative results stated in 3.4(vi) give rise to some problems analogous to those which were just discussed. While 8170 and 8f7+0 are well known not to be recursive, the question arises whether the sets 8f7 x 0 and 817+0nEx are recursive. We shall see later, in §4.7,

64

THE FORMALISM ,c/ WITHOUT VARIABLES

3.7

that the answer in both cases is negative. Hence, we may be interested in partial criteria for a sentence to belong or not to belong to either of the two sets, and especially to 01] x 0. In particular, it seems likely that useful results could be obtained (perhaps with the help of computers) to the effect that every sentence belonging to 01]+0 n :E x and having a sufficiently simple structure (e.g., b eing a particular instance of a sufficiently simple schema) belongs also to 01] x 0 . No interesting results of this type are known at present. The simplicity of a sentence (or a schema) is not, of course, a precisely defined or intuitively unambiguous notion. We could agree, for instance, to take the length of a sentence (i.e., the number of occurrences of symbols) as the measure of its simplicity. In this case the results of the type proposed could lead us to the solution of a problem mentioned towards the end of §3.4, in fact, to the construction of the shortest possible sentence which belongs to 0'1+0 n:E x but not to 01] x 0 . 3.7. The three-variable formalisms £'3 and

£,t

Since £, x is poorer than £, in means of expression and proof, the problem naturally arises whether a subformalism of £, could be constructed which would be equipollent with £, x in both respects. A possible solution of this problem is suggested by the following considerations. By 2.3(iv)(c), for every X E :Ex there is aYE :E such that X and Y are logically equivalent (in £, +) and hence also semantically equivalent; in fact , Y = GX is such a sentence. Moreover, from the definition of G in 2.3(iii) it is easily seen that, in case X E :Ex, the sentence GX contains at most three different variables, namely x, y, and z. On the other hand, by considering various sentences Y E :E satisfying this last condition, we observe that each of them can easily be transformed into an equivalent sentence X E :Ex. We give here two examples:

Vx 3l1 Vz (zEy

++

zix) == E· (i$E-) 01

= 1,

3 z Vxll (xEy" yEz - xEz) == 10[(E-'-'$E-'-'$E).ij01 = 1. Assume now we have succeeded in showing that our observation applies to all sentences in :E which contain no more than three different variables. It then becomes plausible that a subformalism of £, which is equipollent with £, x can be constructed simply by restricting the number of variables available to three and by otherwise leaving the structure of £, virtually unchanged. Such a subformalism will certainly be equipollent with £, x in means of expression. We recall from §2.5 that we have decided not to discuss in this work the equipollence of two formalisms in means of proof, unless they are presented as subformalisms of a third formalism, and at least one of them proves to be equipollent with the latter. In the present case, a subformalism of £,+ obtained by a similar restriction of the number of variables appears to be a natural candidate for this third formalism. We shall try to realize the idea outlined above. The subformalisms of £, and £, + with which we shall deal will be denoted respectively by "£'3" and "£,t"·

3.7{i)

THE THREE-VARIABLE FORMALISMS L3 AND Lj

65

£,t

is conceived here as the formalism obtained from £'3 in exactly the same way in which the formalism .G+ was obtained from.G. With this in mind we shall restrict ourselves for the most part to £'3 in this section. The extension is straightforward, and therefore in the subsequent of our observations to discussion we shall use them in reference to both £'3 and The vocabulary of £'3 contains just the three variables x, y, and z; we set i 3 = {x, y, z}. The constants and predicates are the same as in £'. The definitions of formulas and sentences are virtually unchanged; c)3 and 1:3 are the sets of formulas and sentences in £'3. The set A3 of logical axioms is determined by the formula A3 = An 1: 3. The definition of derivability, r3, for .G 3 differs from the corresponding definition for £, only in that it refers to 1:3 and A3 , and not to 1: and A. As the realizations of £'3 we take just the realizations of £'. Hence, the notion of semantical consequence in £'3 coincides with the notion of semantical consequence in £, restricted to sentences and sets of sentences in 1: 3 . £'3 turns out to be a formalism with significant deductive power. Many theorem schemata which are known to be logically provable in £, (more precisely, all instances of which are known to be so provable) and which play an important role in the development of this formalism continue to be provable in £'3 when restricted to sentences in 1: 3 . Examples of such schemata are provided, for instance, by Lemmas 1- 17, 20, and 21 in Tarski [1965]. To convince oneself that these lemmas can be applied to £'3, it suffices to study the details of their proofs, and to check that, in the situation when the lemmas refer exclusively to sentences in £'3, all the formal derivations explicitly or implicitly involved in the proofs can be carried out without using any variables different from x, y, z. The same observation extends to Lemmas *113-*170 in Quine [1951]. To see this, notice that the only axiom schema in Quine's system which is not among the axiom schemata of £, is the one formulated in * 104. However, by Lemma 15 in Tarski [1965], * 104 is logically provable in £'. Therefore, Lemmas * 113- *170 hold for £, and hence, as before, when properly restricted they hold for £'3 as well. In the sequel, when speaking of elementary properties of £'3 we shall have in mind the properties obtained as restrictions of the lemmas in Tarski [1965] and Quine [1951] mentioned above, and perhaps a few further properties of a related nature. As an example of a frequently used elementary property of £'3 which we obtain in this fashion, we state the following result.

£,t

£,t.

(i) Consider the schema (R)

[X ++ X' ] - [Y

++

Y' ],

where X, X', Y, Y ' are formulas such that X occurs as a subformula in Y, and Y' differs from Y only in that one such occurrence of X in Y has been replaced by X'. If S is any instance of (R) with X,X', Y, Y ' E c)3, then r3 S; hence Y =3 Y' whenever X =3 X'.

66

THE FORMALISM LX WITHOUT VARIABLES

3.7(il

Indeed, by Quine's Lemma * 120, Schema (R) restricted as in (i) is logically provable in £, and the argument carries over to £3' (R) is called the schema, and (i) the law, of equivalent replacement. We shall often use (i) above without referring to it explicitly. ,(,3 and'('d have been introduced as candidates for formalisms equipollent with £ x. In the next chapter it will be shown that £ x is adequate as a framework for the formalization of set theory, and hence in a sense for the whole of mathematics. From this point of view, £3 (as well as £j) proves to have serious defects as a candidate for this task. 7 Actually it seems unlikely that any two of the three formalisms £3,£j, and £x are equipollent with each other.8 The main basis for this conjecture is the fact that, in spite of our previous observations concerning the significant deductive power of £3, we see no way of establishing for this formalism certain fundamental metalogical laws that hold for £ and are used constantly in the development of formalized mathematical theories within predicate logic. In establishing these laws for £ we make essential use of the fact that for any formula in £, there are variables in the vocabulary of £ which do not occur in this formula; in the case of £3 such variables may not be available. For similar reasons, other such laws are not even expressible in £3 unless they are severely restricted or undergo a substantial reformulation. The laws to which we have referred above deal primarily with renaming bound variables and with substituting new variables for free variables. They usually have the form of statements asserting the validity of all instances (in a given formalism) of some general schemata. To give some examples, without stating any general law on renaming bound variables (such as * 171 in Quine [1951]), consider the following sentence, which has the form of a biconditional: V xy 3 z (xEz A zEy)

++

V zy 3 x (zEx A xEy).

Clearly each side of this biconditional is obtained from the other by renaming bound variables. Obviously the sentence is logically provable in £. It seems, however, quite unlikely that it is also provable in £3. 9 As regards laws on substitution, the situation is more involved. Consider the following general schema of simple substitution (i.e., substitution for single variables): (AVI')

[VxX - X[x/y]],

where X is any formula and x, yare any variables. (Recall that X[x/yJ is the formula obtained from X by substituting y for X; see §1.2.) (AVI') is, of course, 7oHowever, see the footnote, p. 143. s"Recently, Andreka and Nemeti have communicated to us that L3 and Lt , in this form, are not equipollent in means of proof, and that, in fact, the sentence Vxy (3 z [zix A 3 x (xiy A 3 y [yiz A 3 z (xEz A zEylJ)] 3z [ziy A 3 y (yix A 3 x [xiz A 3 z (xEz A zEyl])] )

is derivable in Lt, but not in L3. On p. 68 we point out why Lt is not equipollent with LX. g"Recently, Nemeti showed that this sentence is, in fact, not derivable in L3 .

3.7(i)

THE THREE-VARIABLE FORMALISMS L3 AND Lj

67

a generalization of the axiom schema (AVI). It is well known that every sentence in £ which is an instance of (AVI') is logically provable. If we restrict (AVI') to those cases when the substitution does not require any renaming of bound variables (see §1.2), then all instances of (AVI') so restricted which are in E3 are provable in £3' But in other cases, the formula X[x/y] may contain variables different from x, y, and z, so that the corresponding instance of (AVI') is not even expressed in £3' For instance, if

then by §1.2,

X[x/y] = Vu(yEu) " Vz 3z (xEz). It proves possible, however, to reformulate the definition of substitution in such a way that the formula X[x/y] in the new sense is always logically equivalent in £ to the formula X[x/y] in the old sense, and that, at the same time, X[x/y] is an expression in £3 whenever X, x, and yare expressions in £3. In formulating this new definition, it is convenient for technical reasons to first treat the general case of three variable substitution, i.e. , of substituting variables uo, UI, U2 for distinct variables xo, Xl, X2 in a formula X, and then to treat simple substitution as a special case of three variable substitution. The definition is constructed by recursion on formulas, along the lines of Monk [1971], p. 354, as follows. Let X E (13 and XO,XI,X2,UO,UI,U2 E 1'3, where XO,XI,X2 are distinct. In case X is atomic the substitution is performed simply by replacing everywhere in X the variables Xo, Xl, X2 with uo, UI, U2, respectively. (Of course, in this case at most two of the variables Xo, XI, X2 actually occur in X; if X has the form A = B, and thus no variable occurs in it, then the result of substitution is clearly X itself.) The recursive procedure in case X is of the form .,Y or Y - Z is obvious. Finally, if X has the form, say, VX1 Y, then we set

X[xo/uo, XI/UI, X2/U2] = Vv(Y[xo/uo, xI/v, X2/U2]), where v is the first variable different from Uo and U2. The cases when X has the form VXo Y or VX2 Yare treated analogously. (Since {Xo, Xl, X2} = 1'3, this exhausts all possible cases.) Other substitutions applicable in £3 easily reduce to the one discussed above. Thus, if xo, Xl, Uo, UI E 1'3 and Xo i= XI, we stipulate that X[xo/uo, xI/uI] is X[xo/uo, XI/UI, X2/X2], where X2 is the unique variable in 1'3 different from XO , XI. Similarly, for any Xo,Uo E 1'3 we take X[xo/uo] to be the formula X[Xo/Uo, xI/XI, X2/X2], where XI,X2, with inXI < inx2' are the first two variables different from Xo. The conventions regarding the notations X[u], X[u, v], X[u , v , w] remain the same as in §1.2. From now on, whenever we Bhall Bpeak about substitution in a formalism with only three different variables, we shall use this notion in the sense just defined. In particular, this applies to the schema (AVI'). It seems very likely, however, that, under this new definition, various sentences in £3 which are instances of (AVI') are not logically provable in £3 '

68

THE FORMALISM .c,x WITHOUT VARIABLES

3.7(i)

In addition to (AVI'), we wish to consider another schema of a similar character: (AIX')

[xiy - (X - X[xjy])].

It is closely related to, but much stronger and more general than, (AIX). Strictly speaking, (AIX') is not an extension of (AIX), since there are instances of the latter which are not instances of the former; for example, the sentence Vxy(xi y - (x Ex - yEx)) is such an instance. However, all instances of (AIX) are easily derivable with the help of (AIX'). The statement asserting the validity of all instances of (AIX') (in a given formalism) may be called the general Leibniz law. As in the case of (AVI'), it seems very likely that not all instances of (AIX') are logically provable in L 3 . In addition to, and independent of, the defects discussed so far, L3 and Lt exhibit a further defect of a perhaps more special character. It comes to light when one tries to establish the equipollence of Lt with L X and, specifically, to prove that L x is a subformalism of Lt. Indeed, it turns out that this is not t he case. One of the axiom schemata for LX, namely (BIV) in 3.1(ii), has instances which are not logically provable in Lt .10 This can be established by adapting and modifying an unpublished construction due to J. C. C. McKinsey, who used it to show that the associative law for relative products, which is one of the postulates for the theory of relation algebras, is not derivable from the remaining postulates for this theory. Instead of the axiom schema (BIV) we can consider the following associativity schema:

(AX) [3 z (3y(X[x, y] A Y[y, z]) A Z[z, y])

++

3z (X[x, z] A 3x(Y[z , x] A Z[x, y]))) ,

where X, Y, Z are three arbitrary formulas with just two free variables, x and y. While Schema (BIV) is applicable to LX and Lt, Schema (AX) can be applied to L3 and Lt. Assume for the moment that all the defects of L3 and Lt considered above which do not involve (BIV) have been removed. We can then show that the sets of all instances of (BIV) and (AX) are logically equivalent in Lt. We can conclude that the instances of (AX) in the formalism Lt are not all logically provable in that formalism, and we can extend this conclusion to L3.11 From this discussion it appears clear that the equipollence of L3 and Lt with L x cannot be achieved without substantially modifying the first two formalisms , and indeed without strengthening the notion of derivability in both of them. It turns out that to achieve the desired equipollence, it suffices to strengthen the sets A3 and At of logical axioms by replacing in them the instances of (AIX) with all those of (AIX') and by adjoining all the instances of (AX). The first modification removes all defects connected with the procedures of substitution lO*In Nemeti [1985J it is shown that (BIV) is not even derivable in some rather strong O-systems (see §4.5) formalized in .c,j. lloHenkin [1973] contains a proof of the following closely related result: in certain threevariable subformalisms of a first-order formalism with at least three nonlogical atomic binary predicates, there are instances of (AX) that are not logically provable.

THE THREE-VARIABLE FORMALISMS L3 AND Lt

3.7{i)

69

and renaming bound variables. This enables us to carry out the proof of the along the lines of the equipollence proof for £ and equipollence of £3 and £+ in §2.3. It also enables us to carry out all steps in the proof that £ x is a sub formalism of except for the derivation of (BIV). The second modification overcomes this last difficulty and paves the way for the full proof of the equipollence of and £ x . It may be noticed that, for the modified formalisms £3 and it still remains true that can be obtained from £3 in exactly the same way as £+ from £ . The modifications just described have some negative aspects. The notion of substitution has been essentially involved in the construction of the logical axiom set A3 and thus, indirectly, in the definition of derivability for £3' In addition, the restricted number of variables available in £3 forces us to use a notion of substitution that is much less natural and much more complicated than the original notion used in the development of £. In consequence, our formalization of £3 is deprived of a feature that is a real virtue of the formalization of £ outlined in Chapter 1 (see §1.3), namely the elimination of substitution from that definition. As the reader will see, this leads to complications in the metamathematical study of £3 and , in particular, in some inductive arguments concerning the notion of derivability. Unfortunately, we see no way of formalizing £3 along the lines of the construction in Chapter 1. 12 Somewhat unexpectedly, however, we do know an equivalent formalization of along the lines of Chapters 1 and 2. This equivalent formalization is obtained by changing the set of axiom schemata. The new set, At' , consists of the following schemata: (AI)- (AIX) from 1.3(i), restricted to formulas in ()t ; (DI), (DII) , (DIV) , (DV) from §2.2; the schema

£t

£t,

£t

£t,

£t

£t

(DIll')

Vxy[xA0By

++

3z(xAz" zBy)]'

where x , y, z E T3 and z is the first variable different from x and y ; and finally, (BIV) from 3.1(ii), instead of (AX). (In Schema (DIll') , the variables x , y from T 3 are arbitrary, while in the weaker schema (DIll) from §2.2, x , y , z are, in order, the first three variables of T. For uniformity we could also replace (DI) , (DII) , and (DIV) by corresponding stronger schemata (DI') , (DII') , and (DIV') , analogous to (DIll').) Obviously the notion of substitution is not involved in t he construction of At'. The proof that the two formalizations are equivalent, i.e., that the notions of derivability are the same, is not obvious. In fact , the proof that all instances of (AIX') and (AX) are logically provable on the basis of At' is rather involved. The axiom set At' will not be referred to in our further discussion. However, we may mention that, if the modified formalism were based upon At' , t hen the proof of the equipollence of Lt with L X (outlined in §3.9) would become simpler, while the proof of its equipollence with the modified £3 (outlined in §3.8) would become more complicated.

£t

120See

the last four paragraphs of the footnote , p . 71.

70

THE FORMALISM L x WITHOUT VARIABLES

3.7(i)

Borrowing an idea used for related purposes in Maddux [1978a], p. 188, we can conjecture the possibility of an equivalent axiomatization of '£'3 (and '£'j) which is rather simpler than the one adopted here in so far as it does not involve the complicated notion of substitution defined above. To describe this axiomatization we agree, for any given formula X and variables y, z, to denote by Xyz the formula obtained from X by transposing y and z, i.e., by replacing every occurrence of y (whether free or bound) with z, and every occurrence of z with y. The axiom schemata (AI)-(AVIII) remain unchanged. Instead of (AIX') we adjoin the original schema (AIX) and also a new transposition schema, (AIX")

[xiy - (X - X xy )],

where X is any formula and x, y any variables. (AX) is replaced by a closely related schema formulated in terms of transposition instead of substitution: (AX') where X, Y, Z are any three formulas, each with just two free variables, x and y. It is easily seen that all instances of (AIX") and (AX') in '£'3 are logically derivable using (AI)-(AVIII), (AIX'), and (AX) as logical axiom schemata. (In fact, the derivability of (AIX") follows at once from Theorems 3.8(i),(iv) in the next section.) We have not given thought to the problem of whether (AIX') is derivable from (AI)-(AIX), (AIX"), and (AX'). Even if the answer to this problem is affirmative, we are not sure that the formalism '£'3 with the modified axiom set would provide a more convenient basis for our subsequent discussion. 13

£,t

13 * Recently, Givant has shown that one can equipollently formalize £'3 and using (AIX") (and (AX')) instead of (AIX') (and (AX)). We give here a sketch of the proof. Let r be the set of all instances (in !;3) of Schemata (AI)-(AVIII), (AIX'), and the set of all instances of Schemata (AI)-(AIX), (AIX"). By 3.8(iv), all instances in of (AIX") are derivable from r, and the instances of (AIX) are easily seen to be so derivable. (Schema (AX) are derivable from r. To is not involved in the proof of 3.8(iv).) Therefore, all sentences in prove the converse it suffices to derive all instances of (AIX') from First of all, on the basis of we can prove the theorem on alphabetic variants, 3.8(vii). Indeed, 3.8(vii) rests on 3.8(v), which, in turn, is proved by applying 3.8(iv), i.e., by using various instances of (AIX"). Next, one proves the following lemma.

(1)

For any distinct X,y,z E 1'3 with iny

< inx , and for any Y E .3 , the

formula (VxY)[y/z] is an alphabetic variant of Vx(Y[y /z]). The proof of (1) is not difficult. It uses the fact that any such Y must be a Boolean combination of atomic and universally quantified formulas. Combining (1) and the theorem on alphabetic variants we obtain: (2)

For any distinct x,y,z E 1'3 with iny

< in x, and for any Y E . 3, the

formula (VxY)[y/z]-Vx(Y[y/z]) is derivable The final step is to show, for every X E (3)

.3

and every x, yET 3, that

[xiy ..... (X - X[x/y])] is derivable from

The proof of (3) is by induction on formulas . Most of the steps are rather straightforward and use only some lemmas from Tarski [1965]. Of course the key cases are when X is of the form VuY and x"l y. If u = x, then X[x/y] = X and (3) is trivial. If u = y, then

X[x/y]

= (VyY)[x/y] = (VyY)[x/y,y/x] = X xy ,

3. 7{i)

T HE THREE-VARIABLE FORMALISMS £3 AND £j

71

To conclude, we wish to point out that the text of this section contains several statements, the wording of which makes it clear that they are to be treated merely as conjectures. We are inclined to believe that the conjectures are correct, but we have made no attempt to demonstrate them. These conjectures have some interest in their own right, but their validity does not substantially influence our further discussion. by 3.8{i) ,{ii) , so that (3) is an easy consequence of (AIX"). If u i- x , y and inu < in x , then X[x/y] = Vu{Y[x/yJ), so that (3) follows from the induction hypothesis and Lemma 11 in Tarski [1965]. Finally, if u i- x, y and in x < in u, then (3) follows from the induction hypothesis and (2) above. The fact that we can use as a logical base for £3 simplifies the statement and proof of 3.9{iv), as well as the proof of 3.9{v) . Indeed, it appears that parts (as), (a6), and ((3) can be dropped entirely from 3.9{iv), and that the proof of 3.9{v) in case X E is an instance of (AIX") or (AX') requires the consideration of many fewer cases than for the corresponding schemata (AIX' ) and (AX). Quite recently, Andreka and Nemeti have informed us that they have found a set B of axioms for £3 in which the notion of substitution is not directly involved. The idea behind their axiomatization is the observation of Tarski [1965], p. 62, that the formula 3 x {xi y II X) can serve as an adequate substitute for the formu la X[x / y] in the development of first-order logic. (See also Tarski [1951] and Henkin- Tarski [1961]' p . 86.) To define B it is helpful to introduce some abbreviations. For any formulas X , Y E 4-'3 in which the variable z does not occur free, we set

X S Y = 3z[311 {yiz II X) 113x {x iz II Y)], Xo = 3z {z iy II 311 [yix 113 x {xiz II X)]). The formulas X S Y and XO are respectively equivalent to the formu las 3z {X[x , z] II Y[z , yJ) and Xlv, x] in £, and thus can play the role of t he "relative product" of two formulas X and Y , and of the "converse" of a formula X (cf. Henkin- Tarski [1961]' pp. 86, 103) . The definition of X O is slightly complicated owing to the difficulties involved in using Tarski's idea to define certain simultaneous substitutions. The set B consists of all instances in £3 of the following thirteen schemata: (AI)- {AIX) , the schema

[xiy ..... (3 x {x i y II X) ..... X)] (for X, Y E . 3 and x, y E 1'3) , which , in view of the observations made above, is closely related to Schema (AIX') , and the schemata

[(X S Y) S Z - X S {Y S Z)l. [(X S Y)O _ yO S X°l. [XOO - X],

.3

where X and Y are arbitrary formulas in in which the variable z does not occur free . Although this axiomatization shows that, in principle, we can avoid the notion of substitution in defining the set of logical axioms of £3, it does not seem to be a particularly simple axiomatization. It is true that if we used it as t he basis for our discussion in this and the next section , then Lemma 3.9{iv) would become superfluous. However, it is not clear whether the proof of 3.9{v) would become substantially simpler than, say, its proof using the axiom set mentioned above. It would be quite interesting to find a simple and natural axiomatization of £3 {and of the formalism obtained from £ 3 by deleting all instances of Schema (AX)) that does not involve t he notion of substitution.

THE FORMALISM

72

.ex

WITHOUT VARIABLES

3.8

3.8. The equipollence of L3 and Lt Henceforth, by L3 and Lt we shall always understand the formalisms obtained from those described in the first part of §3.7 by replacing (AIX) with (AIX') and adding (AX) in the definitiontj of A3 and At. Our main task now is to establish the equipollence (in means of expression and proof) of the three formalisms L 3, Lt, and LX, and, more specifically, to show that L3 and LX have Lt as their common equipollent extension. To carry out this task we shall need to know that certain basic metalogical theorems which are known to hold for L , especially those of the types discussed in §3.7, can be extended to L3 (and Lt). A simple theorem of this character, the law of equivalent replacement, was stated, with a hint of its proof, in 3.7(i). Two further theorems of this kind, concerning Schemata (AIX') and (AX), have become available simply because all instances of these schemata have now been included in A3 . Another such theorem, in fact the general law of simple substitution (to the effect that all sentences in which are instances of (AVI') are logically provable in L 3 ) admits now a straightforward proof along the lines of the proof of Lemma 14 in Tarski [1965]. Among the remaining theorems of the same type, the most important for our purposes is the law of renaming bound variables, which we are going to take up now. We begin with a series of lemmas. The first two of them do not involve the notion of derivability in the formalism L 3 , but are merely consequences of the definition of substitution adopted in this formalism.

(i) Let P be any permutation of 1'3 = {x, y , z}. Then for every X E C) 3, the formula X[Px, P y , pz] coincides with the formula obtained from X by simultaneously replacing everywhere x, y , and z with Px, P y , and pz respectively. This can be proved by a straightforward induction on formulas.

(ii) If 1'3 = {u, v, w} and X is a formula in

C) 3

beginning with Vu , then

(a) X[u/v, v/u] = X[v/u], ((3) X[v/w, w/v] = X[v/u][w/v][u/w]. This theorem obviously expresses the fact that a simultaneous substitution for two variables reduces to a series of simple substitutions, i.e., substitutions for single variables. To prove it, let X = VuZ. Applying our definition of substitution and (i) above, we get

X[v/u] = (VuZ)[u/u, v/u, w/w] = Vv(Z[u/v, v/u, w/w]) = X[u/v, v/u], which gives (a). Applying (a) three times, and using (i), we obtain ((3).

(iii) For every X

E C) 3

and v, wE 1'3 we have

1-3

[vi w - (X

++

X[v/w]) ).

This is easily derived by applying (AIX'), first to the given formula X, and then to .,X.

THE EQUIPOLLENCE OF £..3 AND

3.8(iv)

(iv) For every X

E 4)3

and v, wE

T3

we have

f-3

£..t

73

[viw - (X ++ X[v/w, w/v])].

When v = w, (iv) is trivial, in view of (i) . Assume, therefore, that v =f. w, and proceed by induction on formulas. Let {1 be the set of all formulas X in 4)3 that satisfy the conclusion of (iv). In case X is atomic we set

(1)

X' = X[v/w, w/v]

and observe that

(2)

X[v/w] = X/[v/w].

By (iii) we get (3)

h {[viw - (X ++ X[v/w])], [viw - (X'

++

X' [v/w])]).

Clearly, (1) - (3) imply (4)

X E 11 whenever X E Cb3 and X is atomic.

We obviously have (5)

.,Y, Y - Z E {1 whenever Y, Z E {1.

Consider still the case when X begins with Vu , where u is any variable in T 3 . If u = v or u = w , then the desired conclusion follows easily from (ii)(a) and (iii). Suppose now that u =f. v, w, so that u, v, and ware all distinct. By three consecutive applications of (iii) we arrive at f-3

[uiv" viw - (X ++ X[v/u][w/v][u/w])],

and hence, by (ii) (.8), at (6)

h [uiv " viw - (X ++ X[v/w, w/v])].

Since X begins with Vu , we have u Thus, from (6) we conclude that f-3

T
[3 u (uiv) " viw - (X ++ X[v/w, w/v])],

and consequently, by (AVIII) , f-3

[viw - (X ++ X[v/w, w/v]))'

This last formula shows that (7)

X

E {1

whenever X

E 4)3

and X begins with V.

The conclusion follows directly from (4), (5), (7).

T
THE FORMALISM .c,x WITHOUT VARIABLES

74

3.S(v)

(v) For any X E ()3, if P is a permutation of T 3 which is the identity when restricted to T4>X , then X =3 X[Px,Py , Pzl·

We first consider the case when P leaves at least one variable fixed, say x , so that Px = x . In view of (i) , the conclusion obviously holds if P is the identity on all three variables. We can therefore restrict ourselves to the case when Px = x, Py = z, and pz = y, and thus to the case when P is a transposition. It follows then by the hypothesis of (v) that y , z occur free neither in X nor, by (i), in X[y/z,z/yl. Since by (iv) we have

r3

[viz - (X

++

X[y/z, z/y])] ,

we can conclude

r3 [3 yz (yiz) -

(X

++

X[y/z, z/y])] ,

whence

X

=3

X[y/z,z/yl·

A moment's reflection suffices to see that this is just the conclusion of our theorem in the case under consideration. We still have to consider the case when P leaves no variable fixed . Hence, by hypothesis, T4>X = 0, i.e., ·X E }:;3 . Also, as is well known, in this case P is a composition of two transpositions, say Q and R, so that P = Q 0 R. Now, as has been shown above, the conclusion of our theorem holds for transpositions, so we have

X

=3

X[Rx , Ry, Rz],

and

X[Rx, Ry , Rzl

=3

(X[Rx , Ry, Rz])[Qx, Qy, Qzl,

whence

X

=3

(X[Rx, Ry , Rz])[Qx, Qy, Qzl.

Since P = Q 0 R, it is clearly seen from (i) that the last formula yields directly our conclusion, and the proof is complete. To formulate the law of renaming bound variables we need the notion of an alphabetic variant of a given formula X. (The term "alphabetic variant" originates with Quine [1951]' p. 111.) This notion is defined as follows . (vi) Let X, Y E

()3 .

(a) Y is called a direct alphabetic variant of X if there is a permutation P of T 3 such that P is the identity when restricted to T4>X and Y = X[Px,Py,Pzl. ({3) In general, Y is called an alphabetic variant of X if there is a finite sequence (Zo, ... , Zm) of formulas in ()3 such that X = Zo, Y = Zm, and, for every i with a :::; i < m, the formula Zi+l is obtained from Zi by replacing a single occurrence of some subformula of Zi with some direct alphabetic variant of this subformula.

3.8(ix)

THE EQUIPOLLENCE OF £'3 AND £'j

75

This definition of the notion of an alphabetic variant of X differs from the ones implicitly or explicitly used in the literature in that its formulation does not necessitate the use of new variables not occurring in X. The law on renaming bound variables can now be formulated in the following way.

(vii) If X, Y E 4.i3 and Y is an alphabetic variant of X, then X =3 Y. In case Y is a direct alphabetic variant of X, Theorem (vii) is an immediate consequence of (v), (vi)(a). This extends to arbitrary alphabetic variants by a straightforward induction based upon (vi)(.B) and using 3.7(i) .

£,t.

We take up the proof of the equipollence of £'3 and We shall frequently use here the fact that various results stated in Chapter 2, especially in §2.3, continue to hold if the notions involved in these results that refer to the formalism £, or £, + are now referred to £'3 or more specifically, this means that symbols like " 4.i ", " 4.i+" , "f-", "f- + " , etc., are now provided with a subscript "3". For brevity, we shall refer to statements obtained in this fashion as appropriately modified statements. We see at once that the appropriately modified statements 2.3(i) ,(ii) hold.

£,t;

(viii) (a) E3 Et and 4.i3 4.it . ((J) For every \[I E3 and X E E 3 , if \[I f-3 X, then

\[I

£,t. £,t

f-t X.

Thus, £'3 is a subformalism of As a translation mapping from to £'3 we shall use the translation mapping G defined as in 2.3(iii), but restricted in its domain to 4.it ; we shall use the same symbol "G" (without any subscript) to denote this restricted mapping. It is obvious that the appropriately modified results 2.3(iv)(a)- ({) are valid for the restricted mapping G. This applies also to 2.3(iv) (8) if we recall Definition 2.3(iii) and notice, in particular, the exact wording of 2.3(iii)(8),(S-). As regards 2.3(iv)(e), suppose that a detailed proof of this theorem for £, and £, + is available and that this proof has a purely syntactical (proof-theoretical) character, avoiding any application of the semantical completeness theorem. There is no difficulty in supplying such a proof on the basis of the definitions of A and A+ given in §§1.3 and 2.2, and then to modify it so that it applies to £'3 and We proceed by induction on predicates and formulas, and we use some of the lemmas in Tarski [1965] and Quine [1951] which, according to the remarks in §3.7, extend to the formalisms presently discussed; we also use the law on renaming bound variables, (vii). The fact that the appropriately modified 2.3(iv)(e) holds in obviously implies that the same is true of 2.3(iv)(<;). Consequently, the appropriately modified 2.3(iv) is valid.

£,t.

£,t

(ix) The restricted mapping G has the following properties.

(a) G is a recursive function .

THE FORMALISM LX WITHOUT VARIABLES

76

3.8(x)

((3) GX=XifXEiJ)3'

h) i¢(GX) =i¢X for everyXEiJ)t· (b)

(e)

G maps iJ)t onto iJ)3 , and I:t onto I: 3 . GX =t X for every X E iJ)t. G*IJI =t IJI for every IJI I:t·

To proceed further we need the following lemma. (x) G(X[UO ,Ul,U2]) =3 (GX)[UO,Ul,U2] for all X E iJ)t and UO,Ul,U2 E i 3 .

This is proved by induction on formulas; the case of atomic formulas requires a preliminary induction on predicates and makes use of (vii). We can now establish the main mapping theorem for £3 and that

(xi) For every IJI

I:t and X E I:t , we have IJI f-t X

£t by showing

iff G*IJI f-3 GX .

The proof of this is entirely analogous to that of 2.3( v). Some attention should be given to that portion of the proof in which we deal with Axiom Schemata (AIX') and (AX), and show that, for every Y E I:t which is an instance of one of these two schemata, we have f-3 GY . We use here in an essential way Lemma (x). From (ix)({3) and (xi) we easily derive the appropriately modified statement 2.3( vi), just as in §2.3. As was pointed out in 2.4(vi), the main mapping theorem (together with such statements as 2.3(i),(iv)({3),(b),(e)) implies directly the proper equipollence theorems 2.3(ii),(vii)-(ix), and this applies not only to £ and £+ , but also to any formalisms :r and :r+. Thus, in particular, the proper equipollence theorems hold for £3 and

£t·

(xii) (a) For every X E I:3 there is aYE I:t such that X =t Y, and for every X E I:t there is aYE I:3 such that X =t Y . ({3) For every IJI I:3 and X E I: 3 , we have IJI f-t X iff IJI f-3 X.

3.9. The equipollence of £ x and

£t £t,

We now take up the problem of the equipollence of £ X and We shall still refer here to the argument in §2.3 as a paradigm for proofs of equipollence between a formalism and one of its subformalisms. Since, however, £x contains no variables, the notion of a formula (with free variables) is not applicable to it, and in consequence, certain portions of §2.3 in which this notion occurs will be disregarded. First of all, £x is a subformalism of (in the sense of §1.6).

£t

(i)

I: x atomic.

I:t ; more specifically, I: x is the set of all sentences of I:t that are

THE EQUIPOLLENCE OF £/ AND

3.9(ii)

(ii) For every W

EX and X E EX,

..ct

77

if wI-x X, then w I-t X.

The proof of (i) is obvious. That of (ii) is also rather straightforward; we make essential use of Axioms (DI)- (DV) in §2.2 and apply to them several times the law on renaming bound variables, 3.8(vii). We now construct a function H on CJ.>t which, when restricted to Et, proves to be an appropriate translation mapping from .et to .e x. Its definition is complicated and must be formulated with care. We give here enough hints for constructing such a definition, without formulating it precisely in all details. H is defined by recursion on formulas, and maps CJ.>t into the set of all quantifier-free formulas in CJ.>t. At each recursive passage we distinguish several cases dependent on whether the number of variables occurring free in the formulas involved is 0,1,2, or 3. The definition of H is designed so that we always have T(jJHX = T(jJX and HX =t X; in case IT(jJXI < 3, HX is an atomic formula; in case IT(jJXI = 3, i.e., T(jJX = T3 = {x, y, z}, HX is a conjunction of a finite sequence of formulas Uo, ... , Un, where each Uk is of a special form,

Uk = xAky V XBkZ V yCkz , with appropriately chosen predicates A k , B k , C k . Formulas of this special form will be referred to in this section as canonical formulas. The definition of HX for an atomic X runs as follows: if X = uAv (where inv, and HX = vA'-'u u , v E T 3 and A E II) , then HX = X in case inu otherwise; if X = (A B) (with A, BE II) , then HX = X. Assume X = .,Y;HX is defined in terms of HY . If IT(jJYI = 3 and HY is a conjunction of n canonical formulas, then H X proves to be a conjunction of 3n canonical formulas; H X is constructed from HY by performing on .,HY certain simple transformations based primarily on some well-known laws of sentential logic (De Morgan law, distributive, commutative, and associative laws); the order in which these transformations are performed is unambiguously determined. For example, let n = 2, so that

=

HY = (xAoY V xBoz V yCoz) A (xA1y V XB1Z V yC1z) .

Then we set

UoA ... AUs ,

HX

where

Uo U1

Us

xAo +A1y V xOz V yOz , xAoY V xBlz VyOz ,

= =

xOy V xOz VyC o +C1z.

If IT(jJYI is 1 or 2 and HY uAv, set HX HY E E X, set HX = [10(Hy)t-01 1].

=

=

uA-v. If IT(jJYI

= 0,

so that

78

THE FORMALISM

£,X

WITHOUT VARIABLES

3.9(ii)

Now assume X = Y - Z. This part of the definition is the most involved in that we have many cases. Each case is determined by the ordered pair of two sets, "f(j>Y and "f(j>Z. Hence, in principle, we have to consider 64 different cases (however, some of them differ very little from one another). Out of these 64 cases, only six will be examined here. For typographical convenience we set Y' = .,Y and notice that "f(j>Y' = "f(j>Y . HX can be expressed in terms of HY' and HZ. In our first four cases we assume that "f(j>Z = {x,y,z} and that HZ is a conjunction of q canonical formulas; in actual examination of these cases we take, for simplicity, q = 1, and in fact we let (1)

HZ = xDyVxEzVyFz .

We consider the cases when "f(j> Y' coincides with one of the four sets {x, y, z}, {x,z}, {y}, 0. HX is constructed by transforming HY' V HZ in a way similar to the way in which .,HY was previously transformed. Case 1. "f(j>Y' = {x, y , z} and HY' is a conjunction of p canonical formulas; then H X is a conjunction of p. q such formulas. (The fact that, by the preceding part of our definition, p must be of the form 3n plays no role in our discussion.) For simplicity take p = 3, and in fact let

(2)

HY'

=

(xAoY V xBoz V yCoz ) A (XAl Y V XBIZ V yClz) A (xA 2y V XB2Z V y C 2z).

Since we want H X to be logically equivalent to .,HY V HZ , and hence to HY' V HZ , we set HX

(xAo+Dy V xBo+Ez Vy Co +Fz) A (xA I +Dy V xB I +Ez VyC I +Fz) A (xA 2 +Dy V XB2 +Ez V y C 2 +Fz ).

Case 2. "f(j>Y' = {x,z}; say HY' = xAz . Then we set HX = xDy V xA+Ez VyFz. Case 3. "f(j>Y'

= {y};

say HY'

= yAy.

Then

HX = x(10(A.i)+D)y V xEz Vy(A.i01+F)z. Case 4. "f(j>Y' = 0; say HY' = (A

= B). Then

HX = x[Oe(A·B+A- .B-)eO+D]y V

x[Oe(A·B+A- .B-)eO+E]z V y[Oe(A·B+A- .B-)eO+F]z.

We also wish to consider two cases when both Y' and Z have less than three free variables.

3.9(iii}

THE EQUIPOLLENCE OF

Case 5. "ftPY' We t hen set

= { y}

and "ftPZ

.e x AND

= { x , z} ; say

HY'

.et = yAy

79

and HZ

= xEz.

and H Z

= xE z .

H X = x1e (A . i )yVxEzVyA.ie1z. Case 6.

"ftPY'

= {z } and "ftPZ = {x , z };

say H Y'

= zAz

Then HX

=

x [1e (A.i ) + El z .

We still have to define H X in case X = VtX. Ifu "ftPY , we set H X = HY . Assume now that u E "ftPY . Then I"ftPXI < 3 and H X must be an atomic formula. Consider the case l"ftPYI = 3, so that H Y has the form H Y = (xAO Y V xBoz V yCoz) A ... A (x AnY V xBnz VyCnz).

We have u E "f 3 ; let, e.g., u = y . We then set

In the case l"ftPYI = 2, assume, e.g., H Y = xA y and u = y ; set H X In case l"ftPYI = 1 and H Y = uAu, set H X = (A +O = 1).

= xA eOx .

We hope the above outline gives an adequate idea of the definition of H . We now state formally several properties of H .

(iii) (a) H is a recursive function . (f3) H X = X f or every X E EX.

b) Assume X

bd

E

tlt ·

If I"ftPX I = 3, then there exists a uniquely determined number nEw and three uniquely determined sequences A, B, C E n+1II such that HX

=

(xAoY V xBoz V yCoz) A ... A (xAnY V xBnz V yCnz).

(2) If 1 :::; I"ftPXI :::; 2, then there exist a uniquely determined

(0)

A E II and uniquely determined u, v E "f 3, with in u :::; in v and {u, v} = "ftPX, such that H X = uAv. (3) If I"ftPX I = 0, then there exist uniquely determined A, B E II such that HX = (A = B). (od H maps the set into the set of quantifier-free formulas in tlt, and the set Et onto EX.

tlt

(02) "ftPHX = "ftPX for every X E tlt.

(c) HX =t X for every X E tlt. Parts (a) and (f3) follow immediately from the definition of H. b) is established by an easy induction on formulas in tlt. (0) is an obvious consequence of b), at one point with the help of (f3). The proof of (c), again by induction on formulas in tlt and with the help of some elementary properties of Lt, is straightforward.

THE FORMALISM ,ex WITHOUT VARIABLES

80

3.9(iv)

..et

From (8) and (c) we see that H maps every formula in to a logically equivalent formula containing no quantifiers. Thus, using frequently employed terminology, we can say that H permits us to eliminate quantifiers from any given formula in Before proceeding further, we state three preliminary results, (iv)- (vi), that will be involved in our discussion. The proofs of these results do not present any essential difficulties. Certain portions of the arguments require a familiarity with the fundamental equational laws of ..ex that are listed in §3.2, and some skill in deriving further laws of a similar nature. The proofs of (iv) and (v) are long and tedious. These are typical "proofs by cases" , and the number of cases involved is so large that a complete presentation of the arguments would certainly wear out the reader. 14 Therefore, in each proof we shall restrict ourselves to illustrating the argument by means of two examples of average difficulty. Only (vi) will be proved in full. The content of (iv) can be approximately described in the following way: given any formula X and a substitution instance Y = X[r, s, t], then Lemma (iv), in combination with (iii) ("t), shows how HY is determined by H X and the variables r, s, and t. The formulation of (iv) is quite involved. To make it somewhat less cumbersome we introduce auxiliary notation. Let X be any given formula in If 11'ct>XI = 3, then we shall use "n(X)", "A (X)", "B(X)", "C(X)" to denote respectively the uniquely determined number n and sequences A, B, C that satisfy the conclusion of ('d in (iii). Analogously, if 1:::; 11'ct>XI:::; 2, then we shall use "ACX)", "u CX )", "vCX)" to denote respectively the uniquely determined predicate A and variables u, v that satisfy the conclusion of (,2) in (iii). (An analogous notation in case 11'ct>XI = 0 will not be needed.)

..et.

c)t.

(iv) Let X (a)

E

c)t and r, s, t E 1'3'

We set Y = X[r, s, t].

If I1'ct>X I = 3 and I{r, s, t}1 = 3, then n CY ) = n(X) , and there is a permutation P of {O, ... , n(X)} such that for each i = 0, ... , n(X) the following conclusions hold:

(ad f-X (a2) f-x (a3) f-x (a4) f-x

(a5) I- x

ct)

= = =

=

Bt)

=

ct)

=

in case (r, s, t) = (y, x , z),

= ... , Bt)

in case (r , s, t) = (x, y , z),

ct)

=

=

=

... }

in case (r, s, t) = (x, z, y) , ct)

=

in case (r, s, t) = (z, y , x), .... , CiCY)

=

in case (r, s, t) = (y, z, x), 14'See the fourth paragraph of the footnote, pp. 70-71.

THE EQUIPOLLENCE OF

3.9(iv)

=

(a6) I-x

BJY)

.ex

AND

=

.et

81

=

ct) (BJ;;)) ...... } in case (r, s, t) = (z, x, y).

((3) IfI T 4>XI = 3 and 1:::; l{r,s,t}l:::; 2, we set 1° .... 1+ B(X) + C(X)) . .. . (A(X) .i r.- 1+ B(X) + d X )) E -- (A(X) a ' ':' a o' n(X) '"' n(X) n(X),

F

=

.i 0 1+

(A&X)

...... ) .... . (A(X) + B(X) n(X) n(X)

.i 0 1+ (dnX() X) ......)),

G= .... .

.i)),

X) ) X 1+ B(X) + d n(X), - (A(X)+B(X) a a + d a )) ..... (A(X) n(X) n(X)

and we conclude ({3d I-x A(Y) E ((32) I-x A(Y) E'"' ((33) I- x A(Y) F ((34) I-x A(Y) F'"' ((35) I-x A(Y) G ((36) I-x A(Y) G'"' ((37) I- x A(Yl.i = I · i

= = = = = =

h) If IT4>XI = 2 and u(Y)

=1=

in case tn case in case tn case tn case tn case in case

in r = ins < int, inr = ins> int, inr = int < ins, inr = int > in s, inr < ins = int, inr>ins=int, inr = ins = into

v(Y), then we have

bd

I-x A(Y) = A(X) in those cases when (u(X),v(X)) and (u(Y), v(Y)) coincide respectively with (x , y) and (r, s), or with (x, z) and (r, t), or with (y, z) and (s, t); (2) I- x A(Y) = (A(X)) ..... in the remaining cases.

(8)

If 1 :::; IT 4>X

(c)

If

IT4>XI

I :::; 2

and u(Y) = v(Y), then 1 I-x A (yl. i = A (xl. 1.

= 0, then H Y =:x H X.

A detailed proof proceeds by induction on formulas X. Thus, the proof divides into four parts according as (a) X is atomic, or X has one of the forms (b) X = (c) X = W - Z, or (d) X = VzZ. In each of these four parts we must consider all the particular cases of (iv) specified by which variables occur free in X and which variables are taken for r, s, and t- thus cases (ad-(a6), ((3d-({37), bd, (2)' (8), and (c) described in the statement of (iv). As mentioned above, we shall illustrate the method of argumentation with only two examples. Let 0 be the set of formulas X in such that, for any choice of r, s, t E T 3 , all the conclusions of (iv) hold. Our first example belongs to part (b) of the proof. Thus, the proof of this part is based on the following inductive premise:

C)t

(1)

X =

and Z E 0;

our task is to show that (1) implies

(2)

X E O.

82

THE FORMALISM

.ex

WITHOUT VARIABLES

3.9(iv)

To specify the particular case which we wish to treat, we let

(3)

11'4>XI = 3,

(4)

(r, s,t) = (z,x,y);

hence we are dealing with the case (CX6) in (iv). Clearly, by (1) and (3) we have (5)

11'4>ZI =

3;

from (4) and the definition of Y we get

(6)

Y = X[z, x, y] .

Setting

(7)

V = Z[z, x, y],

we obtain

(8)

Y =.,V

by (1), (6), (7), and the definition of substitution in §3.7. For notational simplicity we let

(9)

n(Z)

= O.

By (4), (5), (7), (9), we see that our premise Z E 11 in (1) reduces to

= n(Z) = 0,

(10)

n(V)

(11)

there is a permutation Q of {O} such that (CX6), with P,X, Y replaced respectively by Q, Z, V, holds.

Since Q is obviously the identity permutat ion, we can rewrite (11) as

In view of the definition of H outlined above, we see from (1), (8), and (9) that

(13) (14)

n(X)

= =

A1X ) = (15)

n(Y)

= 2,

d

X ) = (Cb Z ))-, and Bi X ) = (Bb Z ))- , X X ) = 0, = Bb ) = = Ca X ) =

ci

the formulas in (14) continue to hold if X and Z are replaced everywhere by Y and V.

83

THE EQUIPOLLENCE OF LX AND Lt

3.9(iv)

Let P be the permutation of {O, 1, 2} given by PO = 2, PI = 0, and P2 Using (12), (14), (15) we then easily check with the help of 3.2(i),(ii) that (16)

f-X {A(Y)=d X )

•

P'"

P'"

P.

= l.

for i=O 12 ' , .

From (16) we see that (a6) holds; hence, in view of (13), all the conclusions of (iv) that are relevant for our particular case as specified by (3) and (4) do hold for X. (The remaining parts of (iv), namely (ad- (a5), (f3d - (f37), (--rd, (--r2), (8), (e), hold vacuously since the premises involved in these parts are incompatible with (3), (4).) Thus, we arrive at (2), and this was just our task. Our second example belongs to part (c) of the proof. Thus, we assume as the inductive premise (17)

X=W-Z and W,ZEO;

our task is again to derive (2) from (17). It turns out that part (c) involves a complication which does not occur in other parts of the proof. To avoid inessential distractions we choose for our example the same particular case of (iv) as was chosen for our first example, i.e., (a6)' Thus, we assume (3) and (4). However, because our inductive premise in part (c) involves two formulas, Wand Z, and not just one as in parts (b) and (d), in the discussion of any particular case in this part we need to know not only which variables occur free in X, but also which of them occur free in Wand which in Z. In other words, within a particular case we may have to consider a large number of sub cases (indeed, 27 sub cases under assumption (3)). For our example we select one of these subcases, namely the one specified by the condition (18)

Ytj>W={x,y} and Ytj>Z={z}.

Clearly, (6) continues to hold. Setting (19)

u = W[z, x, y]

and V

= Z[z, x, y],

we get

u-

(20)

y =

V,

(21)

Ytj>U = {x, z} and Ytj>V = {y},

by (17), (6), (19) and the definition of substitution in §3.7. From (4), (18), (19), (21) we see that the second part of (17) reduces to

In view of the definition of H, we get from (17), (18), (20), (21) that

THE FORMALISM

84

(23)

n(X)

.ex

WITHOUT VARIABLES

3.9{v)

= n(Y) = 0,

(24)

= (A(W))- ,

B6 X ) = 10 (A(Z) .1),

(25)

= 10(A(V) .1),

B6Y ) = (A(U))- ,

cg X ) = 10(A(Z) .1), cgY ) = A(V) ·101.

Using (22) , (24) , (25) and 3.2(i) ,(ii),(iii) ,(xxii) we easily conclude that

From (26) we see that (a6) holds, provided we take P to be the identity permutation on {O} . Hence, in view of (23), all relevant conclusions of (iv) hold for X , and we arrive at (2). This completes our discussion of (iv). (v) f- x HX for every X

EAt. that At

In fact , we first recall consists of all sentences of Et which are instances of the schemata (AI)- (AVIII), (AIX') , (AX), and (DI) - (DV) (cf. §§1.3 , 3.7, and 2.2). In principle, a separate proof is needed for each of these schemata. Given a particular instance X E Et of one of these schemata, we first compute H X on the basis of the definition of H (this usually requires distinguishing many cases). Then we show that f-x H X. To this end we apply some of the lemmas listed in §3.2; we also use implicitly a number of analogous laws of a Boolean algebraic character which are not listed in §3.2. Moreover, in the cases of (AIX') and (AX) we must use Lemma (iv). As in the proof of Lemma (iv), we shall illustrate the methods of argumentation by considering two examples. For our first example we take a particular instance X of Schema (AIV) . Thus X has the form

where u, v E T 3 and Z E Cbt . To specify the particular example which we shall consider, we make the additional assumptions: ITiPZI = 3 (if ITiP ZI < 3, the argument is quite simple) and u = y , v = x . Thus, by bd in (iii),

(2)

H Z = (xAo yVxBo zVyCo Z) A . . . A (xAn yVx Bn zVyCnz)

for some nEw and some A, B , C E n+ln , and by (1),

Setting

(4)

D = (AO' e Bo+ Co) · · · · . (A,;;, e Bn + Cn ), E = (Ao e Co +Bo)··· · . (A n e Cn + Bn),

THE EQUIPOLLENCE OF L X AND Lj

3.9(v)

85

we obtain by the definition of H and (2),

H (VyVxZ ) = zOeDz , H(VxVyZ)

= zOeEz ,

and hence by (3).

By 3.2(vi) we get

= [o e (A'Q' eBo+ Co)] ' ... . [oe (A,;; eBn + Cn )], f- x oeE = [Oe(Ao e Co+Bo)]' ... . [Oe (An e C n + Bn)].

(6)

f-x OeD

(7)

Now 3.2(xvi) yields f-x Oe (A; eBk+ C k ) =Oe (Ak e Ck +Bk ),

for k = 0, ... , n, whence by (6) and (7),

(8)

f-x OeD

=oeE.

From (5) and (8) we conclude at once with the help of BA that

(9)

f- x HX,

and this is just what we wanted to prove. For our second example we take a particular instance X of Schema (AIX').15 Thus, let

(10)

X = [ui v - (Z - Z[u/v])],

where u , v E 1'3 and Z E Thus, by (10),

(11)

c1>t,

Again we assume 11'¢ZI = 3 and u = y , v = x.

X = VXVyVz[yix - (Z - Z[y/x])].

From (/d , (/2) in (iii), and from the definition of H , we see that

(12)

H(yix) = xi.... y ,

(13)

HZ = (xAoY V xBoz VyCoz) 1\ ... 1\ (xAnY V xBnz Vy Cnz)

for some nEw and some A, B, C E n+1II, and

(14)

H(Z[y/x]) = xFz

for some FEll. For notational simplicity we assume n IS'See the fourth paragraph of the footnote, pp. 70- 71.

= 0, so that (13) becomes

T HE FORMALISM

86

(15)

.ex

WITHOUT VARIABLES

3.9(v)

HZ = xAoY V xBoz V yCo z .

Using t he definition of H , we obtain step-by-step:

(16)

H (Z - Z[y jx]) = (xAo Y V xO+ F z VyO z) A (xOy V xBo + Fz VyOz ) A (xOy V xO + F z V YCo z) by (14), (15);

(17)

H (yix - (Z - Z [y j x])) =

+ AoY V xO+ Fz VyOz) A +Oy V xBo + Fz VyO z) A +Oy V xO+ Fz VY Coz) by (12), (16);

(18)

+ Ao +

HX =

.

+0+

+O+ (B o

eo+o = 1) by (11), (17).

We now use Lemma (iv), with X, Y, (r, s, t) replaced respectively by Z, Z[y jx], (x, x,z). Since I'l'¢Z I = 3, 1'l'¢Z[yj xll = 2, and inx = in x < in z, we see that case ({3t} of (iv) applies. In view of (14) and (15) we therefore have (19)

f- x F = Ao . i 01 + Bo + Co.

Setting

we obtain successively: (21)

f-x G = (i.A o)- + (F eO)

by (20), 3.2(ii);

(22)

f-x G

(i .Ao)- + (Ao' i01eO)

by (19), (21), 3.2(vii);

(23)

f-x G

(i . Ao) - + Ao . i 01

by (22) , 3.2(xiv);

(24)

f-x G

(i. A o)- + Ao· i

by (23) , 3.2(ix);

(25)

f-x G =1

by (24), BA.

In an analogous fashion we get (26)

f-x

= 1,

(27)

f- x

= 1.

THE EQUIPOLLENCE OF £x AND

3.9(ix)

£t

87

(To obtain (27) we use, in particular, 3.2(xi).) From (18), (20), (25)- (27), and 3.2(xiii) we easily arrive at (9). With this , our discussion of (v) has been completed.

(vi) If X, Y E 'J:,t , then {HX, H(X _ Y)} I- x HY. Indeed, by (iii)(8) we have HX , HY E 'J:, x , and hence by 3.2(xxxii) ,

(1)

HX

=x[(HX)t = 1],

HY

=x[(Hy)t = 1].

On the other hand, we obtain from the definition of H,

By 3.2(xiii) and BA, (1) and (2) imply at once the conclusion of (vi). We come to the main mapping theorem for ,c x and ,ct .

(vii) For every W

'J:,t and X E 'J:,t , we have W I-t X

iff H*w I- x HX.

The proof is entirely analogous to that of 2.3(v). In establishing the implication from left to right we make essential use of (v) , (vi) ; in the opposite direction we use (ii), (iii)(e) . We can now use the observations in §§2.3 and 2.4 to derive from (vii) (with the help of (i) , (iii)(,B),(8)) the proper equipollence theorems for ,c x and ,ct .

(viii) For every X E 'J:,t there is aYE 'J:,x , and for every Y E 'J:, x there is an X E 'J:,t , such that X =t Y.

(ix)

For every W

'J:,x and X E 'J:, x , we have W I-t X iff wI- x X.

We can summarize less formally the main results obtained in §3.8 and the present section by saying that the formalisms ,c3 and ,c x have the formalism ,ct as their common equipollent extension, and hence ,c3 and,c x (treated as subformalisms of ,ct) are equipollent in means of expression and proof. The construction used here to establish these equipollence results has clearly some serious defects, if only from the point of view of mathematical elegance. Actually, this applies to the proof of the equipollence of ,ct and ,c x . The splintered character of the definition of the translation mapping H, with its many cases, is a principal cause of the fragmented nature of certain portions of the argument; the involved notion of substitution (which we have to use because of the restricted number of variables in our formalisms) is another detrimental factor . As a final result , the construction is so cumbersome in some of its partsculminating in the proofs of (iv) and (v)- that we did not even attempt to present them in full. A different construction that would remove most of the present defects would be very desirable indeed. As easy corollaries of the equipollence results we get the following.

THE FORMALISM

88

(x)

0173 III = 0f7tlll n

(xi)

017 X III = 0f7tlll n

.ex

WITHOUT VARIABLES

3.9(x)

for every III for every III

(xii) If if! is a theory in £t, then if! n is a theory in £3 and if! n I; X a theory in £x , and if! = 0f7t(if! n = 0f7t(if! If, moreover, any of the theories if!, if! n or if! n x satisfies one of the conditions (a) - (c;) in 2.3(xi) , then the remaining two theories also satisfy this condition.

To prove (x) - (xii) we argue in exactly the same way as we did in deriving 2.3(x),(xi); compare also 2.4(vii),(viii) , and the subsequent remarks. An immediate consequence of the equipollence results is that the formalisms £3 and £t, just as £x, are not equipollent with £ and £+ , and indeed are actually poorer than them in means of both expression and proof. In fact, the sentences S, S' of £ given in 3.4(iv),(v) are not logically equivalent with any sentence of £3. The sentence T of £t given in 3.4(vi) is logically provable in £+ , but not in £t, and hence the correlated sentence GT is provable in £, but not in £3. From this last observation it follows also that £3 and £t are not semantically complete. Nevertheless, it will be seen from our results in the next chapter that £3 can serve as an adequate framework for formalizing the whole of set theory, and hence, in essence, the whole of mathematics. It seems appropriate to conclude this section with some remarks of a historical nature. The proof of the equipollence of £3, £t, and £x was first outlined by Tarski in essentially the following form. For £3 and £t he took the formalisms described in the first part of §3.7, but enriched with appropriate associativity schemata. As translation mappings he used the same mappings G and H that were used above. In presenting his argument (which followed the lines of the development given above) he used the assumption that various basic metalogical laws holding for £ and £+ can be extended to £3 and £t by analyzing carefully their original proofs; however, he treated this assumption as a "working hypothesis" which required careful checking.16 In working out a detailed presentation of Tarski's argument, Givant realized that the "working hypothesis" was unwarranted, and might well prove to be false if the formalisms £3 and £t were left unchanged. He suggested, therefore, a rather radical extension of these formalisms consisting in the replacement of (AIX) by (AIX') on the respective lists of logical axiom schemata. As a result of this modification the notion of substitution becomes essentially involved in the description of £3 and £t . For obvious reasons the original definition of substitution given in §1.2 is not suitable in this context, and he suggested using the variant given in Monk [1971] instead; to secure the usefulness of this notion of substitution, Givant proved a special theorem, (iv), which establishes a connection between substitution and the translation mapping H . In consequence, i6·Cf. Henkin- Tarski [19611 , p. 73.

3.10

SUB FORMALISMS OF L, L + WITH FINITELY MANY VARIABLES

89

he succeeded in obtaining a full and exact proof of the equipollence results. This is just the proof that has been sketched here in the last few sections. (Givant also observed that, while the modification of L3 discussed above seems to be essential, in the case of Lt a far less radical one is adequate for our purposes; we have in mind the modification briefly mentioned at the end of §3.7.) The reader will see in the next section that Givant's idea can be used to obtain an interesting and reasonable formalization of logics Ln with n variables, for n > 3. 3.10. Subformalisms of Land L + with finitely many variables In this final section of the chapter we shall concern ourselves with some problems that do not belong to the main stream of our discussion, but are closely related to it. The inclusion of the associativity schema (AX) in the lists of logical axiom schemata for L3 and Lt was motivated exclusively by our desire to construct subformalisms of Land L+ that are equipollent with LX (cf. §3.7). We may therefore consider the (standardized) formalisms LS3 and Lst obtained from L3 and Lt by deleting (AX) in both A3 and At . These standardized formalisms are undoubtedly more natural and more interesting in their own right than L3 and Lt. Clearly, they differ from L3 and Lt as originally described in §3.7 only in that (AIX) has been replaced with (AIX'). By analyzing the arguments in §3.8 we easily see that LS3 is an equipollent subformalism of Lst. Two problems naturally arise concerning LS3 and Lst. Assume that LoX is the formalism obtained from L X by removing (BIY) from the list of logical axiom schemata in §3.1. The first problem is whether LoX is equipollent with the formalisms LS3 and Lst. From a result of Maddux [1978a], Chapter 11, it follows that the solution of this problem is negative. Maddux has shown that the formalism Lw X obtained from L X by weakening the associative law for 0 , and in fact by replacing it with a special case, (BIY')

(A 0 B) 0 1 = A 0 (B 0 1),

is equipollent with Lst , and that (BIY') cannot be derived from the remaining axiom schemata of 'cw x . Thus, 'co x is weaker than 'cw x , and hence also weaker than LS3 and Lst , in means of proof. Maddux has also shown that (BIY') can be replaced on the list of axiom schemata for Lw x with a still more special case of the associative law, namely with (A 0 1) 0 1 = A 0 (1 0 1)

or, equivalently, with (A 0 1) 0 1 = A 0 1.

Finally, he has proved that (BIY) cannot be derived in Lw x, so that Lw x IS actually weaker than L x in means of proof. The second problem, somewhat related to the first, but still open, is more essential for our main purposes. We should like to know whether the formalism

90

THE FORMALISM LX WITHOUT VARIABLES

3 .1O(i)

Lw X provides an adequate basis upon which our fundamental results (to be presented in the next chapter) could be reconstructed and our main goal, a formalization of set theory without the use of variables, could be achieved. 17 Even if the answer to this question is affirmative, the proof of it may be a hard task. The development of the logic and metalogic of LX, to the extent that this is needed for our purposes, is not easy even in the presence of the full associative law, and would probably become much more involved if this law were not available. By 3.9(viii), for every X E Ej there is aYE E X which is logically equivalent with X in Lj. Since Lj is obviously a subformalism of L +, the sentences X and Yare a fortiori logically equivalent in L + . Thus, using the terminology introduced in §3.6, we obtain

(i) Every sentence in Ej, and hence every sentence in E 3, is LX -expressible. The sets E3 and Ej are obviously recursive, so (i) provides partial criteria for LX -expressibility of sentences in Land L +. We shall show how these criteria can be widened. To this end we single out some new sets of formulas and sentences in Land L+. Let () (3) (to be distinguished from ()3!) be the set of all X E () such that 11'4>YI 3 for every subformula Y of X. The sets E(3)' and are defined analogously. Notice that these definitions impose no restrictions either on the total number of variables occurring in formulas involved or on the indices of variables occurring free. We can prove the following.

(ii) For every X

and hence also for every X E () (3), there is a quantifierfree Y E ()+ such that X == Y and 1'4>X = 1'4>Y; if, moreover, I1'4> X I 2, then Y is atomic. E

To show this, we define a function H' which is closely related to the function H introduced in §3.9 (and actually coincides with H when its domain is restricted to () j). H' is defined by recursion on recursion is warranted by the obvious fact that all the subformulas of a formula in belong themselves to A detailed definition of H' differs from the definition of H only in that the variables involved are not assumed to coincide with x, y , or z. For example, by a canonical formula we now understand any formula of the form uAv V uBw V vOw,

where u, v, w E l' and in u < in v < in w . The proof that H ' has the properties strictly analogous to 3.9(iii)({),(8),(c), with ==j replaced by ==+, presents no difficulty, and (ii) is an immediate consequence of these properties. Thus, we see that the mapping H ' permits us to eliminate quantifiers from any formula as H does with formulas in () j (see §3.9). in 17 * Recently, in Nemeti [1985J, it has been shown that the answer to this question is affirmative: Lw x provides an adequate basis for the formalization of set theory (see the footnote, p. 143) . On the other hand, in Nemeti [1987J it is proved that LoX does not provide an adequate basis for such a formalization (see footnote 6* , p. 138) .

3.1O(v)

SUBFORMALISMS OF £ , £+ WITH FINITELY MANY VARIABLES

91

As a particular case of (ii) we obtain

{iii} Every sentence in Et), and hence every sentence in E(3)' is £ x -expressible. Since the sets E(3) and Et) are recursive, (iii) provides new partial criteria for £ x-expressibility which are wider than those in (i). In the next theorem, (iv) , by supplementing (i) and (iii) we obtain new full characterizations of £ x -expressibility (Le., necessary and sufficient conditions, but not necessarily of a recursive character) . In formulating (iv) we find it convenient to speak not of the sets E(3) and Et) , but of the corresponding formalisms £ (3) and £ t). These two formalisms are, respectively, assumed to be subformalisms of £ and £+ whose sets of sentences coincide with E( 3) and Et)" Further information concerning £(3) and £t) (such as the definitions of derivability) is not needed for understanding (iv) and will not be given here.

{iv} If any sentence in £ + (or in £) is expressible in one of the five formalism s £ X, £3 , £t , £(3)' £t), then it is expressible in the remaining four. In fact , the passages from £ x to £t , from £3 to £ (3) ' and from £ (3) to £t) are obvious; to pass from £t to £3 we use 3.8(ix)(8) ,(c) , and to pass from £t) to £ x we use (iii) . An analogous result of a more general character, concerning formulas and not just sentences, can be established; however, £ x-expressibility must be replaced by expressibility in terms of a quantifier-free formula of £t. It may be interesting to observe that , by (iv) , £ x-expressibility and £3expressibility are equivalent properties (of sentences in £) , even though the formalisms £ x and £3 differ so radically from each other in their formal structures. In this connection we may note the following reformulation of the result of Kwatinetz [1981] stated in 3.6(iii) . {v} The set of all sentences in £ (or £+) which are £3-expressible is not recursive. To conclude, we should like to point out that , following the pattern of §§3.7 and 3.8, formalisms £n and £;t can be constructed for every integer n > 3. The study of these formalisms may have some intrinsic value for finitistic trends that are present in contemporary foundational research. We shall not concern ourselves with formalisms £n and £;t for n = 1 or 2. To construct £n and £;t for n > 3, we first define () n and ();t to be respectively the sets of formulas in () and () + in which no variables different from Vo , ... , Vn - l occur, and we set En = ()n n E, E;t = ();t n E+ . We consider the sets of logical axiom schemata for £ and £+ , with (AIX) replaced by Schema (AIX') from §3.7; the sets of all instances of these schemata that belong to En and E;t, respectively, are taken as An and A;t, the sets of logical axioms for £n and £;t .

92

THE FORMALISM

£,x WITHOUT VARIABLES

3.10(vi)

One can convince oneself without great difficulty that various important metalogical laws concerning Land L + can be extended to the formalisms Ln and L;i just described. We have in mind such laws as the general law of substitution and the general Leibniz law- both of them for multivariable, and not just simple, substitution- as well as the law of renaming bound variables. (For the case n = 3, cf. §§3.7 and 3.8.) In particular, notice that, in opposition to the case n = 3, the associativity schemata (AX) from §3.7 has not been used in the construction of An and A;i for n > 3. The reason is that all instances of this schema turn out to be easily derivable from An and A;i (by means of modus ponens alone). From this remark and the description of Ln and L;i, it is obvious that, for any n, mEw with 3 :::; n :::; m, Ln and L;i are respectively sub formalisms of Lm and The formalisms just described lead, in a natural way, to the extension of problems previously discussed for L3 and LX to formalisms Ln for n > 3. We shall give here a short survey of some results obtained so far in this direction, and of open problems that seem to us interesting. We consider first the problems of expressibility. Notice that, in view of (iv) , the results of 3.4(iv),(v) can now be reformulated as

(vi) There are sentences in L 4 , e.g., sentences S, S' in 3.4(iv),(v), which are not L3 -expressible.

The question naturally arises whether (vi) remains true if L3 and L4 are respectively replaced by Ln and Ln+l' where n is an arbitrary positive integer 2: 4. It has been shown that this is indeed the case; see Kwatinetz [1981]. The fact that for every n 2: 4, there are sentences in L (not necessarily in Ln+d which are not Ln-expressible is also a simple consequence of Theorem 3.5(viii) in its general form. (Compare here the remarks after 3.5(viii), where formalisms Ln and their extensions L;i are implicitly referred to.) By the results of Kwatinetz [1981]' Theorem (v) remains true when applied to arbitrary formalisms Ln with n 2: 4. On the other hand, it seems to be an open problem whether, for any given n 2: 3, the set of sentences in Ln+l that are Ln-expressible is recursive. We ·turn now to the problems of provability. It will be seen that here there are still many unanswered questions. As already noted at the end of the last that are valid, but not provable in L 3 , i.e., section, there are sentences in

Does this formula continue to hold when we replace 91]30 by 91] n0 n 2: 4, i.e., do we have the inequality

91] n0 n

=I- 0'10 n

for every n 2: 4? Maddux [1918a] has shown that 91]30 = 91]40 n

n

for

3.10(vi)

SUBFORMALISMS OF

.c, .c+ WITH F INITELY MANY VARIABLES

93

so that the inequality holds when n = 4. It is an open problem whether it continues to hold for every n > 4. Related to this problem is the one of determining for which values of n the inequality

8f7n 0 n E3 f. 8f7n+1 0 n E3 holds. For n = 3 it fails by the above result of Maddux. On the other hand, it holds for n = 4; indeed the sentence GT mentioned at the end of the last section is in 8f7 5 0 n E 3 , but not in 8f7 3 0. Therefore, in view of Maddux's result , GT is not in 8f740 n E 3 . Again, it is not known whether this inequality holds for n > 4. 18 Finally, it will follow from our results in §4.7 (cf. remarks after 4.7(vi)) that n E X is not recursive for any n 2: 3. Therefore we can infer from 3.9(xii) and the equipollence of .en with .e;t that 8f7n0 n E3 is not recursive for any n > 3. We do not know whether the set (8f7n+1 0 8f7n0 ) n E3 is recursive for any n > 3. We may consider problems of a still more general character, involving two or even three varying indices; for instance, the problem for which triples (k , m , n) with k :::; n < m the inequality

8f7n0

n Ek f. 0'7m0 n Ek

holds. However, it is easily seen that, for any fixed n, if this inequality holds in the particular case when k = 3 and m = n + 1, then it holds for all k and m with k:::; n < m. To conclude, we mention that Maddux actually established the results attributed to him in this section for a somewhat different axiomatization of .en; cf. Maddux [1978a]. However, he has communicated to us that his methods and results can also be applied to our particular axiomatization of .en.

IS "Maddux [1987] h as recently shown t h at

e"n0 n for all n > 4, and t hat for infinitely many n .

=I e,,0 n

CHAPTER 4

The Relative Equipollence of /:; and /:; x , and the Formalization of Set Theory in /:; x

From the results in §3.4 we know that £., x is poorer than £., + and £., in means of expression and proof. In this chapter we shall be able, however, to establish a kind of relative equipollence between £., + and £., x (and hence also between £., and £., X) . More specifically, we shall single out a certain set I; of sentences in £., X and we shall show that , for every sentence X in I; , the formalisms £.,+ and £., X when relativized to X (or, more properly, to {X} , cf. §1.6) turn out to be equipollent in means of expression and proof. This means that every sentence in £., + is equivalent with some sentence in £., x relative to X, and whenever a sentence Y in £., x is derivable in £., + from a set IJt of sentences in £., x, then it is also derivable in £., x from IJt relative to X . Sentences forming the set I; have a rather specialized character, and therefore the results obtained may not seem very interesting from a general metalogical point of view. We shall see, however, that some sentences of I; occur as provable sentences in practically all systems of set theory known from the literature, when formalized in £., + ; this applies also to systems formalized in £." except that we have to consider, instead of sentences of I; (which are not sentences in the language £.,) , some logically equivalent sentences formulated in £., . As a consequence, our general results , which establish a relative equipollence between £.,- or £., + - and £., x , carry with them the full equipollence of every such settheoretical system S formalized in £." or system S+ formalized in £"+ , with an appropriately constructed system S x in £., x .1 4.1. Conjugated quasiprojections and sentences

Q AB

The set I; referred to above consists of all sentences QAB correlated with arbitrary ordered pairs of predicates A, B E II by the following formula: lThis result was explicitly stated in Tarski [1953]. It was implicitly m entioned in Chin [1948]' p. 2, as well as in Chin- Tarski [1951]' pp. 341- 343, with references t o Tarski's seminar at the University of California at Berkeley in 1945, where the result was presented and discussed . 95

RELATIVE EQUIPOLLENCE OF

96

.c

AND

.c x

4.1(i)

To grasp the meaning of QAB , notice some obvious consequences of (i), where A, B are, as before, arbitrary predicates.

S i,

S i,

= I}.

(ii)

QAB

==X

(iii)

QAB

== Vzyz[(zAy A zAz) V (zBy A zBz) - yiz] A Vzy 3z(zAz A zBy).

In derivations within LX we shall refer more frequently to (ii) than (i) . From (iii) it is clearly seen that QAB expresses the following facts: the binary relations F and G (between elements of a set U) which are respectively denoted by A and B in a given realization (U, E) of our formal language are functions; moreover, for any x, y E U there is a z E U such that Fz = x and Gz = y. Two relations F and G with these properties will be referred to as coniugated quasiproiections or simply quasiproiections (on a set U). Thus, (U, E) is a model of QAB iff the two relations denoted in (U, E) by A and Bare quasiprojections on U. A familiar example of conjugated quasiprojections is provided by the ordinary projections defined over ordered couples. Consider a (nonempty) set U satisfying the condition: U x U U, i.e., (x, y) E U whenever x, y E U. By proiections over ordered couples we understand, as usual, the two functions F and G whose common domain is U x U and which correlate with every ordered pair z = (x, y) of U x U its first term Fz = x and its second term Gz = y . F and G can be jointly referred to as coniugated proiections over ordered pairs of elements of U. The conjugated projections over ordered pairs are obviously conjugated quasiprojections in our sense. Notice the following two simple sentences in LX which express certain properties of conjugated projections that do not apply in general to conjugated quasiprojections. (iv)

S i , A 01=B01.

As the reader may notice from the later discussion, the fact that quasiprojections are not required to satisfy the above sentences will considerably facilitate our presentation. The problem naturally arises of characterizing those sets U on which conjugated quasiprojections exist.

(v) In order that there exist two coniugated quasiproJ·ections on a set U, it is necessary and sufficient that U have at most one element or be infinite.

In fact, from the definition of quasiprojections we can conclude that the exislUI. tence of quasiprojections on a set U is equivalent to the condition IU x UI On the other hand, it is well known that the latter condition holds iff U is infinite or has at most one element. These two observations lead at once to (v). (The proof of sufficiency requires the axiom of choice.) We establish some consequences of QAB in proofs in this chapter.

LX

which will be used in various

CONJUGATED QUASIPROJECTIONS AND SENTENCES

4.1(viii)

(vi)

QAB

== x

QBA

97

QAB

for every A, BEll .

This is an immediate consequence of (ii) and 3.2(i)- (iii). (vii)

QAB

f- x {1 0 A

=1,

10B

=1,

A'-'01

B'-'01

=1,

=1,

A'-'0A

=i ,

B'-'0B

=i}

for every A, BEll.

To prove this on the basis of (ii) we notice that , by 3.2(vii) and BA, we have QAB

f- x {A'-'01

=1,

10B

=1} ,

and therefore, by (vi) above, QAB

f- x {B'-'01 = 1, 10A = 1}.

Hence, finally, by 3.2(xx) ,(i) we obtain QAB

f-x {A'-'0A =

i,

B'-'0B = I} .

The proof of the next theorem is the first long and rather involved derivation in this chapter within the formalism [, x . In connection with such derivations, the reader may recall the closing remarks of §3 .2. The particular proof of this theorem presented below is due to Maddux.

(viii)

QAB

f- x (Co 0 D o),(C1 0 Dd

=

[(Co 0A'-')· (C1 0 B'-')] 0 [(A 0 D o)' (B0D1)] for every A , B,Co,C1 , Do , D 1 Ell.

Let Co, Cll Do, D1 be arbitrary predicates and set

We obtain successively:

(2)

f-X Co 0 A'-'· (C1 0 B'-') 0 [A 0 Do' (B 0 Dd]

Co 0 A'-' 0 A 0 Do ' (C1 0 B '-'0 B 0 Dd by 3.2(viii); (3)

QAB

f-x Co 0 A'-'· (C1 0 B'-') 0 [A 0 D o ' (B 0 Dd]

L

by (1) , (2) , (ii) , 3.2(vii),(ix) . To derive the reverse inclusion it proves convenient to use the following abbreviations: (4)

R = Co 0 B '-' ·A'-', S = B 0 Do ' [A 0 C 1 0 D 1], T = B 0 A'-' · (A 0 C1 0 B'-').

RELATIVE EQUIPOLLENCE OF ,(, AND

98

,(,X

4.1(viii)

Proceeding step-by-step, we obtain:

f-x {R .....

(5)

A, T .....

A 0 B ..... }

by (4), 3.2(i),(iii);

(6)

(JAB I- x Co=A ..... 0 B·Co

(7)

QAB

f-x Co

(8)

QAB

f-x L

R 0 (B 0 Do)' (Cl 0 Dd

by (1), (4) , (7) , 3.2(vii);

(9)

QAB

f- x L

R 0 (B 0 Do' [R ..... 0 (Cl 0 Dd])

by (8) , 3.2(xv) ;

(10)

QAB

f-x

by (ii);

Co 0 B ..... ·A ..... 0 B

by (6), 3.2(xv) ;

by (4), (5), (9), 3.2(vii);

(11)

f-x S

A 0 Cl 0 Dl·(B 0 1)

by (4), 3.2(vii) ;

(12)

f-x S

B 0 10Dl·(A 0 Cd 0 Dl

by (11) , 3.2(xv) ;

(13)

QAB

f-x S

B 0 A ..... 0 B.(A 0 Cd0D l

by (12), (ii), 3.2(vii);

(14)

QAB

f-x S

A 0 C l 0B ..... · (B0A ..... ) 0B 0 Dl

by (13), 3.2(xv) (with A replaced by B0A ..... and C by A 0 Cd,3.2(vii);

(15)

QAB

f-x S

T 0 (B 0 Dl) · (B 0 Do)

by (4) , (14) , BA;

(16)

QAB

f- x S

T 0 (B 0

by (15) , 3.2(xv) ;

(17)

QAB

f-x S

T 0 [B 0 Dl·(A 0 Do)]

by (5) , (16), (ii), 3.2(vii);

(18)

QAB

f-x L

R 0 T 0 [A 0 Do·(B0Dd]

by (10) , (17), 3.2(vii);

Dl .

[T ..... 0 (B 0 Do)])

f-x Co 0 B ..... · A ..... 0 [B o A"'" . (A 0 C l 0 B ..... )]

(19)

(Co 0 B ..... 0 B o A""') . (A ..... 0 A 0 C l 0 B ..... ) by 3.2(viii) ;

(20)

Q AB

f-x R 0 T

(21)

QAB

f-x L

Co 0 A ..... . (C1 0 B ..... )

by (4) , (19), (ii) , 3.2(vii);

Co 0 A ..... ·(Cl 0 B ..... ) 0 [A 0 Do·(B 0 Dl)] by (18), (20), 3.2(vii).

From (1), (3), and (21) we conclude that (viii) holds.

CONJUGATED QUASIPROJECTIONS AND SENTENCES

4.1{x)

QAB

99

(ix) For any A, B, C, DEn we have

(a) ((3)

QAB

b)

QAB

QAB

f-x A'"'e (AeCeB'"' . i.D) eB = A'"'e (i.D) eB·C, f-x A'"'e(AeceB'"' .i)eB=C, f-X 1e(AeCeB'"' .i)e1 = 1eCel.

In fact, applying successively both parts of 3.2(xxviii) we get QAB

f-x A'"'e (AeCeB'"' . i.D) eB = A'"'e (i.D). (CeB'"')eB,

QAB

f-x A'"'e (i.D) . (CeB'"') eB = A'"'e(i'D) eB·C,

which yields (a). Taking 1 for D in (a) we obtain ((3), with the help of (ii) and 3.2(ix). By using repeatedly 3.2(vii), we derive from ((3): QAB

f-x C

1e(AeCeB'"'.i)e1 ,

and we obtain directly f- x 10 (A 0 C 0 B'"' . i) 01

10 C e1.

These two formulas easily lead to (I)' The following consequences of QAB are given mainly because of their intrinsic interest. 2

(x)

For any A, B, C, DE II we have

(a) ((3)

QAB

b)

QAB

QAB

=

f-x CeD Ie (D'"'eA'"'· B'"')· (CeA'"') eB , f-x CeD = Ie (DeB'"' ·A'"') · (CeA'"')eB, f-X C'"' = Ie (CeA'"' ·B'"') ·A'"'eB.

We first derive (a): (1)

QAB

f-x CeD

(2)

QAB

f-x C eD

CeA'"'e(AeD ·B)·l

(3)

QAB

f-x CeD

Ie (D'"'eA'"' ·B'"')· (C eA'"') e (AeD ·B)

= Ce(A'"'eB·D)

by (ii); by (1), 3.2(xv), BA; by (2), 3.2(xv) (with B replaced by AeD·B, A by CeA'"', and C by 1);

(4)

QAB

f-x CeD

Ie (D'"'eA'"' ·B'"') · (CeA'"') eB by (3), 3.2(vii);

(5)

QAB

f-x Ie (D'"' eA'"'· B'"')· (C eA'"') S CeA'"'e (A eD ·B) ·le (D'"' e A'"' .B'"' ) by 3.2(xv);

2·See the remark at the end of the second paragraph of §8.6 to the effect that ORA's can be construed as algebras of type (2,1,1,0,0).

100

(6)

RELATIVE EQUIPOLLENCE OF LAND LX

QAB

4.1(xi)

f-x 1e(D ..... e A ..... ·B..... )·(CeA ..... )eB

S CeA ..... eAeDeB ..... eB by (5), 3.2(vii);

(7)

QAB

f-x 1e (D ..... eA ..... ·B..... )· (CeA ..... ) eB S CeD by (6), (ii), 3.2(vii),(ix).

Steps (4) and (7) immediately yield (a) . From (a) we get ((3) at once by applying 3.2(xvi). To derive b) we replace C and D in (a) with i and C ...... In formulating and establishing the main results of this chapter we shall replace given predicates A and B by predicates AD and B O such that from QAB we can derive not only QA 0 BO, but also A D e1 1 and B O e1 1. To this end we stipulate:

=

(xi)

CO =

c+i.(c- .o)

=

for every CEll.

Notice that if C denotes a function F in a given realization (U, E), then CO denotes the function on U which coincides with F on DoF and with the identity function on U", DoF. (xii)

f-x A Oe1=1

and

QAB

f-x

QA oBo

for every A, BEll.

Indeed we obtain successively:

(2)

f-x i· (A- .0) e1 = [i. (Ae1)e1r f-x i.(A-.O)e1= (Ae1t

by (1), 3.2(xxi);

(3)

f-x A Oe1 = 1

by (xi), (2), 3.2(v);

(4)

f-x Ae1 .[i.(A-.O)e1] =0

by (2);

(5)

f-X [i . (A- .o)] ..... e [i. (A- .0)] S i

by 3.2(ii),(vii),(ix);

(1)

Si

by 3.2(xxiv);

(6)

QAB

f-x AO..... eA o

(7)

QAB

f-x BO ..... eB o S i

by (vi), (6);

(8)

QAB

f-x 1 = AO..... eB o

by (ii), (xi), 3.2(vii).

by (4), (5), (ii), (xi), 3.2(xxxi);

Theorem (xii) follows immediately from (3), (6), (7), and (8), by (ii). 4.2. Systems of conjugated quasiprojections and systems of predicates PAB In this and the next section we consider two arbitrary predicates A, BEll, assumed to be fixed throughout the discussion. In terms of A and B we define an infinite sequence of predicates

PAB = by stipulating

pi";}, ···)

SYST EMS OF QUASIPROJECTIONS AND PREDICATES P AB

4.2(vi)

p l"l} = A m 0B

(i)

101

fo r each mEw.

The significance of these predicates under the assumption of QAB is brought out in the following statements.

(ii)

Q AB

f-

(iii)

Q AB

f-

A VVo ... Vm 3 Vm+l

and each sequence (k o , . . . , k m

(iv) If C

=

D

=

)

- v1iv2)

f or each mEw.

A . .. A Vm+l p l'i:) vm ) for each mEw of distinct natural numbers.

and m , nEw with m =j:. n, then Q AB I- X Q eD.

Theorem (ii) is a straightforward consequence of (i) and 4.1(iii). The proof of (iii) can easily be reduced to the special case when ki = i for i = 0, .. . , m . To prove the special case one proceeds by induction on m , using 4.1(iii) ; as usual, the completeness theorem for £., + may be used to simplify the argument. Theorem (iv) may be derived from (i) and 4. 1(ii) ,(vii), with the help of some elementary lemmas from §3.2, by a double induction on m and n . In connection with (ii) and (iii) we may generalize the notion of (a pair of) conjugated quasiprojections: for each mEw a sequence (Ho , .. . , Hm) of binary relations between elements of a set U will be referred to as a sequence of m + 1 conjugated quasiprojections (on U) just in case all the relations Ho , . .. , Hm are functions, and for any elements Yo, . .. , Ym in U there is an x in U such that Hox = Yo , ... , Hmx = Ym. A familiar example of such a sequence is provided by the projections defined over ordered (m + 1)-tuples of elements of U (presuming, of course, that (Yo , ... , Ym) E U whenever Yo , ... , Ym E U) . From (ii) and (iii) we conclude that, in case A and B are assumed to denote two conjugated quasiprojections, the corresponding relations denoted by , ... , pl'i: ) form a sequence of m + 1 conjugated quasiprojections. It follows trivially from (iv) that 4.1(vii)- (x) continue to hold if we replace (m ) and p AB (n) all occurrences of A and B on the right-hand side of f- x by p AB respectively, whenever m , nEw and m =j:. n. We thus get, in particular, (v)

Q AB

f- x

= 1,

=1,

=i} for every mE w.

Theorem 4.1(viii) can now be generalized as follows. (vi) Let (k o , ... , k m ) be any finite sequence of distinct natural numbers. Th en Q AB

f-x (Co 0Do)·· · · , (Cm 0 Dm)

=

(ko) ..... ) • . . . . (Cm 0P(k (k o) 0· D 0) ' · [(CO 0P · AB · ABm)..... )]0• [( p AB

.. .

(p AB (km ) 0• D m )]

for arbitrary sequences (CO, . .. , Cm) an d (D o, ... , D m) of predicates.

We first establish (vi) under t he hypothesis that k i = i for i = 0, . . . , m . The argument proceeds by induction on m. In case m = 0 the proof reduces to showing that Q AB I- x Co 0 D o Co 0B ..... 0 (B0 Do) ·

=

102

RELATIVE EQUIP OLLENCE OF

.c

AND

.c x

4.2(vi)

This however follows at once from 4. 1(vii). Assume now that the conclusion holds for a given m in w, and let (Co, ... , C m+1) and (Do, .. . , D m +1) be any two sequences of predicates. We introduce two abbreviations: S

(C 1 0P1°J..... )· ....

T

... .

By applying 3.2(xxvii) (extended to finite sequences) we get

Using (i) and 3.2(iii) we also have

(2)

LX

,-

p AB (k) ..... 0· A "'" -- p(k+1) ..... lor c k AB

0 ... , m. =,

From (1) and (2) we therefore obtain

In a completely analogous fashion we can derive

Now by the induction hypothesis applied (C1, ... , Cm+1) and (D1' ... ' D m+1) we see that

to

the

two

sequences

and therefore

From 4.1(vi),(viii) we obtain

Putting (3), (4), (6), and (7) together, we arrive at QAB

f-x (C00Do)····· (Cm+10Dm+d

[(C0 0P1°J ..... )· ....

=

..... )]0

The conclusion continues to hold for m under the special hypothesis.

+ 1.

....

0Dm+d].

This completes the proof of (vi)

SYSTEMS OF QUASIPROJECTIONS AND PREDICATES

4.2(vii}

P AB

103

In the general case let n be t he largest of the natural numbers k o, . .. , k m . Given two sequences (Co, .. . , Cm) and (Do, . .. , Dm) of predicates, we define and ... by stipulating that new sequences (Cb, . . .

= Ci

(8)

= Di

and

(9) Cj = D j = 1 in case 0

for i j

= 0, ... ,m;

nand J' =f. ko, ... , k m ·

By the special case of our theorem which was established above, we have Q AB

I- x

=

... .

. . . . . (Pi]

From this , and by (9) and 3.2(vii),(ix), we obtain

(10)

Q AB

I- x

' ':' [(C ko

• . . . • (C'km':'

AB

"" (ko) D'ko·· ) · · · (p AB (km) "" .:. [( p AB' .:. D'km)] .

AB

On the other hand, using (iv) , 3.2(viii), and 4.1(vii), we derive by induction onm:

(11)

Q AB

I- x

. .. ... .

... .

From (8), (10), and (11) we obtain at once the desired conclusion. The results which will be stated in the subsequent part of this section are of some interest in their own right, but will not be applied until Chapter 7. From (vi) we obtain the following corollary.

(vii) Let m , nEw with n

m and let (ko, ... ,km ) be an arbitrary sequence of distinct natural numbers. Then for any predicates Do , .. . , Dm we have

(a)

QAB

((3)

QAB I- x

I-x [(1 0 Do)····· (10D n -d]·Dn · [(1 0 Dn+d' ... . (10Dm)]

=

0 Do)' . . ..

0 Dm)] ; [(D o 0 1) . . .. . (Dn-101)].Dn·[(Dn+101) . . . . ·(Dm 01)] -- [(D 0':' "" AB "" AB "" (kn ) . • . . . . (D m':' .:. p AB

Indeed,

(1)

QAB

I- x [(10Do)·· ·· · (10Dn- d l· (i0D n )· [(10Dn+1)· .. . . (10Dm)] ....

=

. ... .

... . by (vi) ;

RELATIVE EQUIP OLLENCE OF [" AND

104

(2)

4.2(viii)

["X

Q AB f-x [( 10Do ) ·· ··· (10Dn -d]· D n · [( 10 Dn+d ' .... (10 Dm)] (ko) 0· D 0 ) " . . • (p AB (k m) 0• D m )] < _ AB • [( p AB

by (1) and 3.2 (vii), (ix);

(3)

0D o ) ' . .. . ) • AB 0• D 0"

f-x

< _ (4)

AB

Q AB f-x

0 D m)] AB

. .•

• AB 0• D m ) by BA and 3.2(viii);

. .. .

S [( 10Do )' ... . (10 D n -d]·Dn · [( 10 Dn+d' .... (1 0

D m)]

by (3), (iv) and 4.1 (ii). Part (a) follows at once from (2) and (4). We easily obtain ({3) from (a) by replacing Do, ... , Dm in (a) with Do, . .. , D;;;, and applying 3.2(i)- (iii) .

(viii) Let m, nEw with n ::; m, and let (ko, ... ,km ) be any sequence of distinct natural numbers. Given arbitrary predicates Co, ... ,Cm , set

...

D = and

Then

({3)

= en ·E;

I-QAB

(a) { 10 Ci 01

= 1: i::; m, i =I n} f-QAB

' ..J-} {.,C i = O: 2· < _ m, 2 Tn

(1)

= Cn;

L

'QAB

AB

•

•

AB -- C n'

To prove (a) we first apply (vii)(a), with Di replaced by i = 0, ... ,m, and obtain

for

QAB f-x --

[( 10 • G0

·

. . . . . (10 • Gn-l

AB .

·

AB

. (Gn ·

AB

... . (10 C m

From this we arrive directly at (a) by applying (vii)({3), with Di replaced by =I n, and by Ci for i = n. ({3) is an immediate consequence of (a), and b) follows at once from ({3) with t he help of 2.2(iii). 10 Ci for i = 0, ... ,m and i

With "f- x " replaced by "f-" in (a) and ({3), Theorem (viii) can be somewhat generalized. To this end, with any sequence (Ao, ... ,Am) of predicates (m E w), we correlate the sentence Q (Ao, ... ,Am) defined as follows:

4.2{xi)

SYSTEMS OF QUASIPROJECTIONS AND PREDICATES

PAB

105

(ix) Q(Ao , . . . , Am) = So A ... A Sm A T, where

Sk= VVOV1V2(VoAkVl A voAkV2 - Vliv2) for k=O, . . . , m, and T = VVo "'Vm 3Vm+l (vm+lAovo A ... A vm+lAmvm). Thus Q(Ao, . .. , Am) expresses the fact that relations denoted by Ao ,· . . , Am (in any realization of .c+) are m+I conjugated quasiprojections; cf. (ii), (iii) , and the subsequent remarks. It then turns out that Theorem (viii) (with " f- x" replaced by "f- " ) continues to hold if we change everywhere QAB , , . . . , pi'iJ ) to Q(Ao , ... , Am) , Ao , . . . , Am , respectively. The changes needed in the proof are obvious. Theorem (viii) can clearly be formulated in set-theoretical terms. We state this formulation explicitly for the generalized form of (viii) just indicated. (x) Given any mEw , let {Ho, . . . , Hm} be a sequence of m + 1 conjugated quasiproJ'ections on a given set U; let 0 be the (m + 1) -ary operation from and to (binary) relations on U defined by the condition

O(Ro , ... , Rm) = n{HkIRkIH;l : k::; m}. In case Ro , ... , Rm are nonempty, we then have

Rn=H,;;-lIO(Ro , . .. , Rm)I Hn for n=O, . .. ,m. In other words, when restricted to nonempty relations on U, 0 has the fundamental property of the operation of forming ordered (m + I)-tuples, i.e., O(Ro , .. . , Rm) uniquely determines each of its arguments Flo , . .. , Rm . (x) can be proved directly by an elementary set-theoretical argument. As was observed by Givant, some theorems can be established which (with one trivial exception) improve (viii) and (x) . We state these in (xi) and (xii) below. In the case of (xii) it is easy to explain in what this improvement consists: it turns out that if the operation 0 from (x) is replaced by a more elaborately constructed operation 0' , then the restriction to nonempty relations Flo, .. . , Rm may be deleted provided only that lUI 2: 2. (However, in the exceptional case IU I = 1, (xii) can easily be seen to fail.) (xi) Let m , nEw with n ::; m and let {k o , . .. , km } be any sequence of distinct natural numbers. Given arbitrary predicates Co , ... , Cm , set D' =

..... [pi'iJ) 0 (0+ A 0 Cm and let E be as in (viii). Then (a)

Q AB

f- x

(,8)1 000 1 = 1 f- x

h)

.,0=0

.i) 0 B = Cn · (1000 1+ E) ; .i) 0 B

= Cn ;

f-

The proof of (a) is based on (viii) and proceeds as follows .

106

(1)

RELATIVE EQUIPOLLENCE OF L AND L X

Q AB f- x

. ([10 = 10001+ E

01] ... . . [10 0 1]. '" . [10 by BA, 3.2(v) , and 4.1 (vii);

(2)

Q AB f- x

= (O+ A 0 Cn

(3)

Q AB f- x

.i )0 B

(1 0001+ E ) by (1) and (viii)(a );

= Cn · (4)

4.2{xii)

. i ]0 B) by (2) , BA, 4.1(ix)(a) ;

f- x 10 (10001+ E) 01 = 10001+ E by the definition of E and 3.2(vii)- (ix);

(5)

f- x A 0 [A 0 (10001 + E) 0

. i] 0 B

5 by (4) , 3.2(vii) ; (6)

Q AB f- x 10001+ E

5 by (5) , 4.1(ii) ,(ix)(,8);

(7)

Q AB f- x 10001+ E

(8)

Q AB f- x

=

[( 1000 1+ E) . i ]0 B by (4) , (6) , and 3.2(vii);

.i) 0 B = Cn · (10001+ E) by (3) and (7) .

Part (,8) is an immediate consequence of (a), and b) follows from (,8) with the help of 2.2(iii) . With some changes, the remarks following (viii) apply to (xi) as well. (xii) Given any mEw , let (Ho , . . . , Hm) be a sequence of m + 1 conJ'ugated quasiproy'ections on a given set U , and F, G any pair of cony'ugated quasiproy'ections on U ; let 0' be the (m+ 1) -ary operation from and to relations on U defin ed by the condition

O'(Ro, . .. , Rm) = n {Hkl(Di U FIRkIG- 1)IH;l: k

m} .

If U has at least two elements, then for any relations Ro , . .. , Rm we have Rn=F-11 [(H,:;-lIO'(Ro , .. . , Rm)IHn)nId]IG for n=O , ... , m .

Theorem (xii) is derived from (x) in much the same way as (xi) from (viii) .

pt;;1o,

where AD and For later purposes we state a property of the predicates BD are the predicates obtained respectively from A and B according to 4.1(xi). It is established by induction on m, with the help of 4.1(vi) ,(xii).

4.3(v)

(xiii)

REMARKS ON THE TRANSLATION MAPPING FROM.e+ TO.e x

f-x

107

foreverymEw.

4.3. Historical remarks regarding the translation mapping from L + to LX In establishing the relative equipollence of L + and LX, i.e., the equipollence of the systems obtained by relativizing the formalisms L + and L x to any given sentence QAB , we shall apply the same general method which was used to establish the equipollence of L+ and L in Chapter 2 and of Lj and LX in Chapter 3. 3 As was stated in §3.4, L x is a subformalism of L +, and hence its expressive and deductive powers at most equal those of L +. To show that, conversely, the expressive and deductive powers of L + at most equal those of L x relative to any given sentence QAB, we shall use again an appropriately constructed translation mapping. Actually, we shall construct not a single mapping, but a whole system of mappings KAB indexed (just as the system of sentences QAB) by arbitrary pairs of predicates A, B. Each KAB is defined on the set E+ and maps this set onto EX. Recall that to define the translation mapping from E+ to E in Chapter 2 we first constructed a certain translation mapping G from C)+ to C) with the property that, for each X E C)+ , X and GX have the same free variables; then we took the restriction of G to E+. This approach cannot be utilized in constructing the system of mappings KAB because LX does not contain formulas with free variables. In his original construction Tarski first defined an auxiliary system of mappings LAB with the following properties.

(i)

LAB is a recursive function from C)+ into C)+ .

(ii)

For every X E C)+ with canonical sequence (xo , ... , Xm-l) there exist a variable u and predicates Go, . . . , G m - 1 , H (all of them uniquely determined) such that

(iii) In particular, for every X E E+ there is a uniquely determined HEn such that

(iv) i¢(LABX) = i¢X for every X (v)

LAB X

=QAB

E C)+.

X for every X E C)+ .

As a framework for later remarks, and perhaps for some historical interest, we state here the definition of LAB explicitly (although, as the reader will notice, the notion so defined is not involved in our further development). 3. Thus, the proof we shall give may be regarded as a syntactical proof. In the footnote on p. 244 we discuss briefly a semantical proof essentially due to Maddux.

RELATIVE EQUIPOLLENCE OF

108

£.,

AND

4.3(vi)

£., X

(vi) For any A, B E il , L AB is the unique function F satisfying conditions (0:) - ("') given below. In these conditions X , Y, Z E 4)+; m, n, p, i, J', k are natural numbers; (xo , . .. , Xm-l) , (Yo , ··· , Yn - l) , (zo , " " Zp- l) are respectively the canonical sequences of X , Y, Z ; u is the fi rst variable such that u "f<jJX ; r, s E "f; G , D , H , Go , .. . , G m - 1 , 10 , ... , In-I, J o , .. . , J p - 1 E il .

(0:) DoF= 4)+. ((3) If X = rGs, then FX Vu(rGu V sOu V uOu) FX = Vu(sG'"'u VrOu V uOu) FX = Vu(rG +OuV uOu)

=D), then FX

(!) If X = (G (8) If X

zn case zn case in case

inr < in s , inr > in s, in r=ins.

= Vz (xOEB Xtx) .

.,Y, Vu(xoGou V ... V xm-1Gm-1u V uGu) ,

FY

and

10 (G- . i) 0 [( GO'- 0 (A o 0 B)'"')· . . .. (0;;:-=-1 0 (A m -

H

1

0 B)'"')],

then

=

FX

(e)

Vu[xo(A o 0B )'"'-u V . . . V Xm-l (A m - 1 0B )'"'-u V uHu] .

If Y - Z, = Vr(Yo Ior V ... V Yn-lIn-lr V rGr), Vs(zoJos V . . . V zp-1Jp-1s V sDs) ,

FZ

and

H = (OEB (G +O) EB A'"'- ) + (OEB (D +O) EB B'"'-) , then FX where for each k

=

Vu(xoGou V .. . V xm - 1Gm-1u V uHu) ,

<m

= Ii 0 A '"'+Jj 0 B '"' zn case Xk = Yi = Zj for some i < n , J' < p, G k = Ii o A '"' zn case Xk = Yi =I- Zo, . . . , Zp-l for some i < n , Gk

G k = Jj 0 B'"' ("')

in case Xk = Zj

=I- yo, . . . , Yn-l for some

J'

< p.

If X

=

VrY

and

FY

Vs(YoIos V . . . V Yn-lIn-lS V sGs),

FX

Vu [yoIou V .. . V Yk-lh- lU V Yk+lh+lU V ...

then V Yn-lIn-lU V u(OEB h + G)u] in case r = Yk, k FX

FY

zncase r=l-Yo , .. . , Yn-l .

< n,

and

4.3(viii)

REMARKS ON THE TRANSLATION MAPPING FROM L+ TO L X

109

A consequence of (ii) and (v) is that, on the basis of the sentence QAB, we can partly eliminate quantifiers from any given formula X in more specifically, we can construct another formula in namely L ABX, which is equivalent with X under Q AB and which has only one occurrence of a quantifier. Hence, it easily follows that in case ITc/>X I :::; 2, the quantifiers can be eliminated entirely; e.g., if m = 2 in (ii), then X

=QAB

xo(Go e (o+ H) e G?,)xl'

Obviously, throughout the whole discussion we could use , instead of universally quantified disjunctive formulas, the dual existentially quantified conjunctive formulas. The definition of a system of mappings K AB in terms of the mapping LAB is simple. (vii) For any A, BEll, KAB is the unique function F such that

(a) DoF = (/3) FX = X in case X E

h) FX = (H = 1) in case X E

and H is the uniquely determined predicate for which LABX = Vx(xHx) .

(We could simplify this definition by removing condition (/3) and extending h) to all X E Such a change would exert only a minimal influence on the present discussion, but it would necessitate the use of a different notion of translation mapping than the one introduced in 2.4(iii); cf. 2.4(v).) On the other hand, the proof (by induction on sentences derivable in L +) that KAB has the desired property, namely

(viii) For every W

and every X E KABW

f-QAB

if W f-+ X , then KABX,

turns out to be more complicated than one would expect. Curiously enough, one of the most involved parts of the proof is where we show that KABX is derivable from QAB whenever X is a logical axiom of L+ of the form (AI) , i.e., X has the form of a sentential syllogism (cf. §1.3). There is another construction of translation mappings which leads to some simplification of both the basic definitions and the proofs of the fundamental results. This construction was discovered by Monk around 1960 (but was never published and was not known to the authors) and was rediscovered in 1974 by Maddux in a slightly modified form. With their permission we shall use the new construction as a base for the subsequent discussion, and in fact we shall present it in the form given by Maddux. In particular, the specific proof of (viii) that we shall give in the next section is essentially due to Maddux. To describe the main difference between the original and the new constructions (disregarding inessential details), we should point out that in both cases we can define by recursion a function L AB satisfying conditions (i)- (v) above. In the

RELATIVE EQUIPOLLENCE OF

110

.e

AND

.ex

4.3(ix)

case of the original construction, the predicates GO,"" Gm - l , H occurring in LABX according to (ii) are rather involved expressions which depend on the whole structure of X. In contrast to this, the predicates Go, . .. , Gm - l in the new construction depend only on the set Tr/JX, and in fact they are the predicates th e k' h AB , ... , AB , were i s are th' e m d'Ices 0 f th e correspond'mg variables Xi in the canonical sequence of X. As regards the new definition of H, it is somewhat more involved than the old one in case X is either an atomic or a universally quantified formula, but rather simpler in the remaining cases. Altogether, the new definition of LAB appears to be considerably less involved than the old one. A further simplification in the new construction results from the observation that, for every formula X with specified free variables, LABX is now uniquely determined by the predicate H. We also know from (vii) that the translation mapping KAB can actually be defined in terms of H. Since the mapping LAB plays only an auxiliary role in the construction, it can now be eliminated entirely, provided we define by recursion a mapping MAB which correlates with any given formula X the predicate H = MABX. This is essentially the course we shall follow. If for any reason it should prove convenient, we can obviously reintroduce LAB by setting explicitly

(ix) LABX = Vu (x o p17/""-u V ... V x m - 1 P1'i'-1) ...... -U V uMABXu) for every X E where (xo, ... , Xm-l) i8 the canonical8equence of X and ki = in Xi for i = O, .. . ,m-1. 4.4. Proof of the main mapping theorem for £., x and £., + We begin by introducing some notation. We consider two arbitrary predicates

A and B , which are regarded as fixed throughout this section. For abbreviation we set

(i) Pn = P1r;}BO

for every nEw.

(Notice that in (i) we have used AD,BD, and not merely A, B.) For further reference we restate 4.2(iii)- (vi),(xiii) , using 4.1(xii) and the abbreviation in (i).

(ii) QAB f- Vtlo ...

(Vm+IPkoVO" .. . "Vm+lPkm vm ) for every 8equence (k o , . .. , km ) of di8tinct natural number8.

=i, P;'0Pm =1, Pn 01 =1, 10Pn =1, P;'01 =1, 10P;' =1} for all n, mEw with n =f. m .

(iii) QAB f- x {P;'0Pn

(iv) QAB f- x (Co0Do)···· . (Cm 0 D m)

=[(Co 0Pk;;)' .. .. (Cm 0Pk:J] 0 [(Pko 0 Do)' ... . (Pkm 0 D m)] for every 8equence (ko , . . . , km ) of di8tinct natural number8 and for any two 8equence8 (Co, ... , Cm), (Do,.··, Dm) of predicate8.

PROOF OF THE MAIN MAPPING T HEOREM FOR ,ex AND ,e+

4.4(vi)

111

Every finite set I w determines, of course, a unique strictly increasing sequence of integers (io, ... , i m - l ) with range I . We correlate with I a predicate V(I) by setting

(v)

V:(J)

=

(P

·(P'l.m-l

'1.0

'l.m - l

Some useful consequences of results in §4.2 are listed in the following theorem.

(vi) Let I, J be any finite subsets of w. T hen f-X V(J) V(I) whenever I J, (a) ((3) QAB f-x i V(J), f-X V(I) = V(I)' b) f-x (8) QAB V(I) 0 V(I) V(I), (e) QAB f- x V(I) 0 V(J) V(InJ) , (I,') Q AB f-x V(I) • V(J) V(InJ)·

= =

The proof of (a)- b) is based on (v) and elementary parts of §3.2; in the case of ((3) we also use (iii) and 3.2(xx). The proof of (e) is more involved. Let m be the least natural number such that i < m for i E I U J, and recall that m = {O, ... , m - I}. We define two sequences (Co, . . . , Cm-l) and (Do, ... , D m - 1 ) of predicates by stipulating: for i E I and Ci = 1 for i E m for j E J and D j = 1 for J. E m

Ci = Pi Dj=Pj

I; J.

From this, by (iii) and 3.2(ix), we clearly obtain

=1 for i E P 0D =1 for J. E C 0D =1 for I E

(1 )

QAB

f- x Ci0Pt"'

(2)

QAB

f- x

(3)

QAB

f- x

j

J

l

l

With the help of (v) , we get from (1)- (3) respectively:

=(Co o PO') .. .. ·(C 1 0P;;:;'-1)' V(J ) =(Po 0 D o)· .. . . (Pm - 0D m -t) , V(InJ) =(C 0 D o)···· ·(Cm - 0D m -t).

(4)

QAB

f- x V(I)

(5)

QAB

f- x

(6)

QAB

f- x

m-

1

0

1

Now (4)- (6) , together with (iv), immediately yield (e). Finally, both (8) and (I,') follow directly from (e) , and the proof of (a )- (1,') is complete. Together, statements (vi)((3)- (8) express the fact that in any realization of .c + which satisfies Q AB , every relation denoted by a predicate of the form V(I) (I a finite subset of w) is an equivalence relation. The definition of MAB , constructed by recursion on formulas, runs as follows.

RELATIVE EQUIPOLLENCE OF

112

£,

AND

£, x

(vii) M AB is the unique function F satisf ying conditions (a) bitrary C, D E ll , X, Y E C) +, and

i,J

4.4(vii)

below for ar-

E w:

(a) DoF = C) +; ((3) F(ViCVj) = i .(Pi 0 C 0P7 ) 01 ; h) F(C = D) = oei. [Pa 0 (C .D+C-.D-)0P1 j01 ; (8)

(e)

F( -,X) = (FX) - ; F(X - Y) = (FX)- + FY; F(VViX) = Vc'I) e FX, where 1 = in * Y4>(VViX).

(Recall that in * Y4>VViX = {in x: x E Y4>VViX}.) We list various consequences of this definition.

(viii) M AB is a recursive function mapping C) + into ll . (ix) For every X, Y E C)+ and every i E w we have:

(a) ((3)

h) (8) (e)

I--x M AB(X V Y) = M ABX +MABY; I--x M AB(X A Y) = M ABX · M ABY; I--X M AB(X ++ Y) (M ABX M ABy)t; I--x M AB (3 vi X) = V(I) 0MABX, where 1= in* Y 4>( 3 v,X); M AB(X - Y) = 1 M ABX M ABY; M AB(X ++ Y) 1 M ABX M ABY.

=

=

=x = =x

=

The proofs of (viii) and (ix) are obvious.

The proof is straightforward and proceeds by induction on formulas; we use 3.2(xiv) and notice that, as a simple consequence of it, (A 0 1 = A)

(xi) Let U

=x

(A eO= A) for every A E ll.

= (U, E)

be a model of QAB and X any formula with canonical sequence (Vko' ... ' Vkm_J. Let Rand H a ,.·., H m - 1 be respectively the relations denoted by MABX and Pko , ... , Pk m- 1 in U. Then for every u E U, we have uRu iff the sequence (Hau, . .. , H m - 1u) satisfies X in U.

To understand (xi) notice that, by the hypotheses and (iii) , all the relations Hn are functions with domain U (whence the Hnu's are function values). The

straightforward proof of (xi) proceeds by induction on formulas , using the definition of M AB, the definition of satisfaction, and (ii), (iii), (x). In view of the semantical completeness of ,c+, Lemma (xi), together with (ii) and (iii), leads directly to the following purely syntactical result. (xii) Let X E C) +, let (xa, ... , Xm-l) be any sequence of variables such that Y4>X {xa, .. . , xm-t}, and suppose u is a variable different from all Xi'S. Setting ki = in Xi for i = 0, ... ,m - 1, we obtain:

(a)

X

= Q AB

Vu(xa Pk'a- u V ... V xm-1 P;:::_1 U V UM ABXU);

4.4(xv)

PROOF OF THE MAIN MAPPING THEOREM FOR L x AND L +

X

=QAB

3u (x OPk'""U o A ... A xm-1 Pk'""m - l U A UMABXU);

((3) UM ABXU ((3') UMABXU

= Q AB

VXO "'Xm _1(xo P;;;- U V . .. V xm-1 P;;::_1U V X) ; 3 Xo "'X m_, A ... A xm- 1Pk';,,_1 U A X).

(0/)

= Q AB

113

Theorem (xii) throws light on the role played by the mapping in our discussion. We know from (viii) that M AB is a mapping which correlates with any formula X in ..c+ a predicate M ABX, (xii)((3) shows us how M ABX can be explicitly constructed (more precisely, explicitly defined relative to QAB) in terms of X. (We use here the fact that, in view of (x), the formula UM ABXU in (xii)((3) may be replaced by UM ABXV , where v is any variable different from u .) Conversely, (xii)(a) shows us how X can be explicitly constructed in terms of MABX, In this sense the predicates M ABX may be said to represent, or to be (binary) representatives of, formulas X with any number of free variables. It may be mentioned here that the idea of constructing, for any formula, a predicate which may serve as its binary representative is not new. In particular, it was known to Tarski ; the proofs ofresults announced in Tarski [1954], [1954a], which will be discussed in Chapter 7 below, are based on it.

(xiii) For every X E c)+ with i
or

Po 0 (i. M AB X) 0P1 ·

This is a direct corollary of (xii)(a) ,(a'). (xiv) X

=QAB

(M ABX

=1)

for every X E

Indeed, by (xii)(a) we have X (i M ABX) =QAB (M ABX 1).

=

=QAB

(i

M ABX) , while (x) easily implies

Theorem (xii)(a) ,(a') , as well as Theorems (xiii) and (xiv) , are clearly results concerning the role of M AB in eliminating quantifiers on the basis of Q AB (cf. the remarks following 4.3(vi)). (xv) For every X , Y E c)+ , we have X

= Q AB

Y iff QAB f- MABX

=MABY.

Indeed, (xv) is a simple consequence of (xii) and (x). By (xv) , two formulas X and Yare equivalent relative to QAB iff their binary representatives, M ABX and M ABY' are so equivalent (in the sense that QAB f- M ABX M ABY) ; as a consequence, the many-one mapping M AB from formulas to predicates induces a one-one mapping from equivalence classes of formulas to predicates.

=

In this connection the following observation, due to Givant , may be of some

interest. It is possible to restrict the domain of the mapping M AB , i.e. , the set c)+ , in an inessential way so that this mapping itself (and not just the mapping induced on equivalence classes) is one-one. In fact, let us agree to say that X is a formula without useless quantifiers, for brevity a w.u.q. formula , if X has no

RELATIVE EQUIPOLLENCE OF L AND LX

114

4.4(xvi)

subformula of the form VxY with x ¢. T¢Y. By deleting from any given formula X all useless quantifier expressions Vx we obtain a uniquely determined w.u.q. formula X' which is, of course, logically equivalent to X. Using this terminology we get: (xvi) The mapping MAB with the domain restricted to the set of w.u.q. formulas is one-one. In other words, for any w. u. q. formulas X and Y, we have X = Y if (and only if) MABX = MAB Y. This can be proved by induction on formulas , using the recursive definition of MAB . We shall now study some deeper properties of the mapping from .e + to .e x which correlates with every X in E+ the sentence MABX = 1 in EX. It should be clear from various remarks in §4.3 that this mapping is closely related to the translation mapping KAB. Almost all the results established in (xvii)- (xxxi) below concern the derivability in .e x of sentences of the form MABX = 1 under various assumptions on X. As will be seen, these results will lead directly to the main mapping theorem for .e + and .e x. We begin with a theorem closely related to (xiv). (xvii) X

=QAB

(MABX

= 1)

for all X E E X.

To prove this, we use the notation of 3.1(i)(J) and set, as an abbreviation,

We obtain successively:

o

o

(2)

QAB f-x

(P -ei.C01).(P 0P1 ) by (1), (vii)(J), (iii);

(3)

QAB f-x MABX

p

(4)

1)

f-x i.C01.(i0Pd = i·C0P1

(5)

QAB f-x MABX

(6)

QAB f-x

(7)

MA BX=1 f-xQAB X

(8)

o0(i.C01.P

p

o0(i.C)0P

X f-x MABX= 1

In view of (7) and (8), the proof is complete. The next theorem can be compared to (x).

1

by (2), 3.2(iii),(x); by 3.2(xxiii); by (3), (4); by (1), (5), (iii), 4.1(ix)(,B); by (6), 3.2(xxxii); by (vii)(J), (iii), and 3. 2(xxxii) , (xiii).

4.4(xx)

PROOF OF THE MAIN MAPPING THEOREM FOR .G x AND .G +

In fact, we first show that in case X is a sentence of the form VXO" we have

QAB f-x MABX

'Xm

115

Y, then

=OfBMABY.

The proof uses and the fact that V(0) = 1 by (v). With the help of this result we establish the theorem by induction on formulas; we apply here the definition of MAB in (vii) and the equivalence (O fB G G) (leG G) for every G E n, which is a simple consequence of 3.2(xiv).

= =x

=

From (xviii), using 3.2(ix),(xiii), we immediately obtain: (xix) (MAB[X]

=1) =QAB (MABX =1) for every X E

Our next task is to show that

for every logical axiom S of .c +. By 1.3(i) and the remarks at the beginning of §2 .2, the set A+ of these axioms is divided into fourteen mutually exclusive groups, each of which consists of all instances of one of the schemata (AI)- (AIX) and (DI) - (DV). In principle each group requires a separate argument and may be conveniently treated in a separate lemma. However, in view of the similarity of some of the arguments, we find it possible to occasionally treat two or more groups in the same lemma. In this series of lemmas two auxiliary results of a different character, (xxii) and (xxvi), will be proved separately so as to make them available for the subsequent discussion. (xx) If S is a sentence of one of the forms (AI)- (AIII), then

QAB f- x MABS

=1.

This is actually an immediate consequence of (xix) and (vii)(8),(c). For example, suppose S is an instance of (All), i.e.,

S = [(,X - X) - X] for some X E

By (vii)(8),(c) and BA we obviously have f-x MAB((,X _ X) _ X)

=1.

Hence, by (xix) we obtain directly the conclusion. The argument just completed would be more involved if we did not use Theorem (xix). This theorem will frequently be used in an analogous way in the subsequent lemmas.

RELATIVE EQUIPOLLENCE OF .G AND .G x

116

4.4(xxi)

(xxi) If S is a sentence of the form (AIV), then QAB f-x MABS

= 1.

Assume S = [VxVyX - VyVxX), where X E cf1 + and x, y E 1. Let i = inx, J' = in y, and J = in· l<j)X. We get:

(1)

QAB f-x

$ MABX)-

=

$ MABX)-

by (2)

QAB f-x

$ MABX

=

$ MABX

by

(3)

by (1), (2), and

In view of (xix), the desired conclusion follows directly from (3). (xxii) Let X E cf1 + and I

= in· l <j)X. We then have

QAB f-x {V(I) 0 MABX

= M ABX, V(I) $ MABX = MABX},

To prove this, set

(1)

r = {Y: Y E cf1+ and QAB f-x V(J) 0 MA B Y = MABY, where J = in· l <j)Y}.

By induction on formulas we show that X E r for every X E cf1+ . Consider first the case when X = xCy, with C E n. Let i = in x, and set

i

= in y,

Since by hypothesis 1= {i,i}, we have

(3)

f-x {M AB X = i.T 01 , V(I) 0MABX = Si,Sj 0 (i.T) 0 1} by (v), (vii) (,B).

We get step-by-step: (4)

QAB f-x {Si 0 T = T, T 0 S j= T}

by (2), (iii);

(5)

Q AB f-x Si,Sj 0 (T.i )

by 3.2(viii) and (4);

<

(6)

f-x T·( i0 S) J

(7)

0 (T . i ) 01 Q AB f-x S"S t J

(8)

QA B f-x T·i01

-

<S -

t

J

J

by 3.2(xv);

< T ·i01

by (4)- (6), 3.2(vii);

, S J 0 (T.i )01

by (2), (v), (vi)(,B).

-

In view of (3), (7), and (8) we conclude by (1) that X Erin the case considered.

4.4(xxiii)

PROOF OF THE MAIN MAPPING THEOREM FOR ,ex AND ,e+

117

Secondly, assume that X = (C = D) with C, DE lI . Then I = 0 and therefore V(I) = 1, so that we obtain again X E f, using (vii)({) and 3.2(xiv). Next, we take up the case when X = .,Y and Y E f . Since in this case 1= in* T¢X = in* T¢Y, we have, by (1),

Continuing, we get:

(10)

QAB f-x V(I)0(MABY)-=V(I)0(V(I)0MABYt by (9);

(11)

QAB f-x V(I) 0(MA B Y)-

=(V(I) 0MABYt

by (10) , (vi)(,B)- (8) , and 3.2(xxv);

In view of (vii)(8), we see from (12) and (1) that X E r. The argument in the case when X = Y - Z with Y, Z E f is analogous to that in the preceding case, but uses also (vii)(c) and (vi)(c). We also proceed analogously in the case when X = VxY, applying (vi)(,B)- (8) , 3.2(xxv) (without even using the assumption that Y E r). By combining all the cases above we infer by (1) that Q AB f-x V(I) 0MABX

=M ABX

for every X E CJ.) +;

by (vii)(8) and 3.2(iii), this easily implies QAB f-x V(I) eMABx

=M ABX

for every X E CJ.) +,

and the proof of (xxii) is completed. (xxiii) If S is a sentence of the form (AV), then Q AB f-x MABS

=1.

Assume that S = [Yx(X - Y) - (VxX - VxY) ].

Set 1= in* T¢VxX and J = in* T¢VxY. Obviously (1)

I u J = in* T¢Vx(X - Y) = in* T¢ (VxX - VxY)'

We obtain:

(2)

f-x V(I) eMA Bx

V(IUJ)

e (M ABX +MABY) by (vi) (Il:'), 3.2(vii);

(3)

f-x (v(IUJ)

e [(MABX)- +MABYj) . (V(I) eMA BX) V(IUJ) e ([(MABXt +MABYj. (M ABX +MABY)) by (2), 3.2(vi);

RELATIVE EQUIPOLLENCE OF

118

(4)

.e

AND

.ex

4.4 (xxiv)

f-x (V(IUJ) eMABY)-

(v(IUJ)

e [(MABX)- +MABY]t + (V(I) eMABX )by (3), SA.

Setting now K

(5)

= in· T4JY , so that

(I U J) n K

QAB f-x V(IUJ) eV(I<) eMABY

= J,

we derive:

= V(J) eMABY by

(6)

QAB f-x V(IUJ) eMABY

(7)

QAB f-x MAB(Vx(X - Y) - (VxX - VxY))

by (5) and (xxii);

= V(J) eMABY

=1 by and (1), (4), (6), SA.

From (7) and (xix) we arrive directly at the conclusion. (xxiv) If S is a sentence of the form (AVI) or (AVII), then

QAB f-x MABS

= l.

In both cases the argument, based upon (xix) and is straightforward. When dealing with (AVI) we use also (vi)(.B), while in the case of (AVII) we apply Lemma (xxii) above. (xxv) If S is a sentence of the form (AVIII), then QAB f-x MABS

= l.

Suppose S = [-,Vx(-,xi y)), where x,y are distinct variables. Let i = inx and j

= iny. We have

< p':"' 0 J

(1)

QAB f- x i

(2)

QAB f-x Pj 01

(3)

QAB f-x 1 = Pj 0PT0[i.(Pi 0PT)]01

by (2), (iii), 3.2(xxi);

(4)

QAB f-x 1 = MAB(-,Vx(-,xi y ))

by (3), (vii) (,8), (8),

(P' .J P)

by (iii) and 3.2(xv) (with PT , Pi' i taken for A,B,C respectively) ;

Pj 0PT0 (Pi .PJ )01

by (1), 3.2(vii);

By (xix), step (4) yields at once the conclusion. (xxvi) QAB f-x [i. (Pi 0C0 Pk') 01J- = i.(Pi0C-0Pk')01 CEll andi,kEw.

Indeed from 3.2(xxiv) we get f-x [i.(Pi0C0Pk')01r

= i.(Pi0C0Pk')- 0l.

for every

4.4 (xxix)

PROOF OF THE MAIN MAPPING THEOREM FOR L X AND L+

119

Hence, using (iii) and 3.2(xxx) , we easily arrive at (xxvi) . (xxvii) If S is a sentence of the form (AIX), then QAB f- x M ABS = 1. Assume that (1)

S

= [x i y - (xCz - yCz) ,

where C E TI and X,y,z E T . Let i = in x , j = in y , and k = in z . We get consecutively:

(2)

f-X i · (P,·0P":"') < J -

(3)

f- xi · (Pi 0 Pt )

(4)

f- x i ,(Pi 0Pt)

(5)

f- x i. (Pi 0Pt ) 01

,

J

Pi . p ) 0 [( C -

J

+ C) 0 Pk ]

by 3.2(xv); by (2) , (iii) (with C - + C replacing 1) , 3.2(vii); by (3) , 3.2(v) ,(vii);

i .(Pi 0 C -0Pk ) 01+ i.( P) 0 C 0 Pk ) 0 1

by (4) , BA, 3.2(v) ;

(6)

by (5), (xxvi) and (vii) ({3) , (c:) .

From (6) , (xix) and (ix)(c:), we obtain the conclusion of our lemma under the assumption (1). By hypothesis, S is an instance of (AIX) , and hence it is either of the form (1) , or of the form obtained from (1) by replacing xCz with zCx and yCz with zCy, or, finally, of the form obtained from (1) by replacing both xCz and yCz with C = D (C, D E TI ). The first case was settled above; the second can be dealt with in an entirely analogous manner. T he proof in the third case is quite simple; one uses (xix), (vii) (c:) , (ix)(c:) , and BA. We turn now to the axiom schemata (DI)- (DV). (xxviii) If S is a sentence of one of the forms (DI), (DII) , (DIV) , then QAB f-x M ABS = 1.

The proof of (xxviii) is based upon (xix) and (ix) (<;). In addition, one needs (vii) ({3), (ix) (0:') , and 3.2(v) in the case of Schema (DI) , (xxvi) and (vii)({3),(8) in the case of (DII) , and (vii)({3) as well as 3.2(i) ,(iii),(xxii) in the case of (DIV) . The details are straightforward. (xxix) If S is a sentence of the form (DIll), then QAB f- x M ABS = 1. By hypothesis S = [xC 0Dy ++ 3z (xC z A zD y )),

120

RELATIVE EQUIPOLLENCE OF

.c AND .c x

4.4 (xxix)

for some C, DElI. Set (1)

E = MAB(XC0Dy). In su ccessive steps we obtain:

(2)

by (1) , (vii) (,B), 3.2(xxi) (taking Po 0C and D 0P1 for A and B);

(3)

QAB f- x E=Po 01, (P I 01 ) ,[Po 0C,(PI 0D"'-)01] by (2) , (iii) ;

(4)

QAB f- x E=Po 0Po,( PI 0Pi""'),[Po 0 C,(PI0D"'-) 0P2']01 by (3) , (iv) , (iii);

(5)

QAB f- x E

=Po0Po ' (Po0 C0P2') . (PI o Pi""') . (PI o D"'- 0 P2') 01 by (4) , (iii) , 3.2(xxvii) ;

(6)

QAB f- x Po 0Po ,(Po 0C0P2')

=Po0[Po 0i,(C0P2')] by (iii), 3.2 (xxvii );

(7)

QAB f- x p o 0i ,(C 0P2' ) =po 0[i ,(Po 0 C 0P2')] by (iii) , 3.2(xxviii) ;

(8)

QAB f- x Po0Po,(Po0C0P2')=Po0Po0[i,(Po0C0P2')] by (6), (7).

By a fully symmetric argument we get statements (6')- (8') differing from (6)(8) only in that the subscript 0 is replaced everywhere with the subscript 1 and C with D"'-. From (8'), by virtue of 3.2(xxii) , we obtain

We continue: (10)

QAB f- x E =Po 0Po 0 [i, (Po 0 C 0P2' )] . (PI 0Pi""'0 [i. (P20D0Pi""')]) 01 by (5), (8) , (9) ;

(11)

QAB f-X E

=Po0Po ' (PI o Pi""') o [i. (Po 0 C 0P2' )' i· (P2 0D0Pi""')] 01 by (10) , 3.2(xxiii) ;

PROOF OF THE MAIN MAPPING THEOREM FOR L x AND L +

4.4 (xxxi)

(12)

121

Q AB f- x M AB(XC 0Dy ) = M AB (3 z (xCz A zDy))

by (1), (11) , 3.2(xxiii), and (v) , (vii) (f3) , (ix)(f3),(b) . The desired conclusion follows directly from (12) by means of (xix) and (ix)(<;) . (xxx) If S is a sentence of the form (DV) , then QAB f-x MABS

= 1.

Suppose where C, DEn. Set

=

(2)

Fo

(3)

Co = P o 0D0P1 ,

Po0C0Pl'

Fl

c1

=

Po 0C-0P1 ,

= P o 0D-0P1 ·

We have:

(4) (5)

by (1) , (vii)(f) ; QAB f-x M AB(VxlI (XCY ++ xDy))

= 0$ [(I ·Fo 01). (i 'C o 01 ) + (i. Fl 01)· (i 'C 1 01)] by (2) , (3) , (xviii) , (xxvi), (vii)(f3), (ix) (f) ;

(6)

by (iii), 3.2(v ),(xxvii);

(7)

QAB f-x E=(i.Fo 01).(i·C o 01)+(i.F1 01).(i,c 1 01) by (6) , 3.2(v) ,(xxiii);

(8)

QAB f-x MAB (C

= D) = MAB(VxlI(XC Y ++ xDy)) by (4) , (5), (7) .

The conclusion follows from (8) by virtue of (ix)(<;). We have thus completed the proof that QAB f-x MABS = 1 for every S E A+. The next result is a preliminary form of the main mapping theorem. (xxxi) For every \lI and sufficien