STUDIES IN LOGIC AND
THE FOUNDATIONS OF MATHEMATICS
Editors L. E. J. BROUWER, Laren (N.H.)
A. HEYTlNG, Amsterdam A. ROBINSON, Los Angeles P. SUPPES, Stanford
Advisory Editorial Board Y. BAR-HILLEL, Jerusalem K. L. DE BOUVERE, Amsterdam
H. HER M E S, Munster if W. J. HINTIKKA, Helsinki A. MOSTOWSKI, Warszawa J. C. SHEPHERDSON, Bristol E. P. SPECKER, Zurich
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
FORMAL SYSTEMS AND RECURSIVE FUNCTIONS PROCEEDINGS OF THE EIGHTH LOGIC COLLOQUIUM OXFORD, JULY 1963
Edited by
J. N. CROSSLEY Fellow of St. Catherine's College, Oxford and
M.A.E.DUMMETT Fellow of All Souls' College, Oxford
~It
m ~ 1965
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the publisher
PRINTED IN THE NETHERLANDS
PREFACE In July 1956 the first Logic Colloquium was held in Oxford thanks to the efforts of Professor A. Prior. It was a fairly small gathering. Since that time, however, the Colloquium has grown considerably. In 1963 the second colloquium to have a larger international membership took place and there were nearly 100 logicians present. The Colloquium was recognized as a meeting of the Association for Symbolic Logic. Financial support was provided by NATO and the Colloquium was a NATO Advanced Study Institute with a Symposium on Recursive Functions sponsored by the Division of Logic, Methodology and Philosophy of Science of the International Union of the History and Philosophy of Science. We are grateful to NATO for their generous contribution towards the publication of this book. MICHAEL DUMMETT JOHN
Oxford, June 1964
N.
CROSSLEY
I. FORMAL SYSTEMS
SOME MODAL CALCULI BASED ON IC R. A. BULL Wadham College, Oxford, UK
The motivation for this work was an attempt to find modal logics acceptable to intuitionists. While I do not know whether any of the systems described here would in fact satisfy an intuitionist philosopher, they seem to me to have some formal interest. A fundamental difficulty is that the customary modal equalities
yield
Met
= NLNet and La. = NMNa. CNNLpLp,
which is an intuitionistically implausible thesis. Of the two approaches described here, the first accepts this thesis, and the second avoids it by having only L and not M. The first system formalises the position that, while contingent propositions obey intuitionist logic, necessary propositions obey classical logic. I must confess that I know of no philosopher who actually maintains such a view. The first system is suggested by the Wajsberg completeness proof of S5 with respect to the Henle model (see [4]). It is obtained by adding to IC f- a. => f- Let
CLpp CLpLLp CLCpqCLpLq ALNLpLp Met = dfNLNa.
4
R. A. BULL
(cf. the Godel axioms for S5 given on p. 312 of [3]). The model I use for this system has elementsxc., 02' 03' •.. ), where each Gi is an element of i) x ,1) with ¢, ¢, ¢, ... ) designated. The values of A, K, C, and N are found by applying i) x to the corresponding terms of the sequences. The word L\I. has value <¢, ¢, ¢, ... ) when \I. has value <¢, ¢, ¢, ... ), and value
<
\I. -
CLp\I., CNLP\I. - CLELPCf3f3\1., CLELf3NCf3P\I.,2)
and that parts of words can be replaced by their necessary equivalents. Applying these rules, to each word there corresponds an equivalent set of words, each of the form
Cia, CL\l.2· .. CL\l.mCNLf3lCNLf32' .. CNLf3nY, where the \I.;'S, the f3/s, and y have no modal operators. I shall call words of this form normal words. If any of the words Cal Ca2' .. CamP j
1 ::; j ::; n
Cal Ca2' .. C\l.mY
is a thesis of IC, then the normal word Cl.a, CLa 2 . . . CLCl.mCNLf3lCNLf32' .. CNLf3nY
can be derived from it in our axiom system. Let us suppose, on the other hand, that none of these words is a thesis of Ie. Then there are allocations of values of i) x which give IX l, Cl. 2, ... , Cl.m value ¢ and in turn give Ph 132, ... , 13m Ynon-designated values (see the Lemma below). Taking these allocations in rotation gives sequences which reject the normal word A word is derivable if all its normal words are derivable, and is rejected if one of its normal words is rejected, since the model verifies the axiom system. This gives the required completeness proof.
fl x is a topological model for IC, the designated element being ,p, and being H. In fact any normal model for IC with the property that if ca{3 is not a thesis of IC then there is an allocation verifying a and rejecting (3, will suffice. I append a proof that fl x has this property to the end of this paper. Not all models for IC have it, for example the Lindenbaum model does not. 2) I use ~ as an equivalence relation between sets of words which can be derived from each other. 1) See [1].
',p
SOME MODAL CALCULI BASED ON IC
5
The second approach is suggested by Lemma 1. 11 of [2]. Let us consider as models for modal calculi the octuples (H, K, {I}, + ,., -'-,0, I), where
+,., -'-,0) is a Heyting algebra with unit 1. (2) K is a sub-set of H such that there is a greatest element of K below each element of H. (3) For each x in H, Ix is the greatest element of K below x. (1) (H,
All such models verify the axiom system, which I call 1M, given by adding to IC R I- O:x{3 ~ I- CLr:xL{3 1 CLpp
2 CLpLLp.
I shall say that a model (H, {I}, +,., -'-, 0, I) for 1M is normal when
it strongly verifies the rules, and H is partially ordered under
x
~
y when y -'- x = 1.
A normal model for 1M has the properties (1) (H, +,., -'-,0) is a Heyting algebra with unit 1. (2) K = {Ix I each x in H} is a sub-set of H such that for each element in H there is a greatest element of K below it. For given x in H, and
Ix Iy
~
~
x by 1,
x implies Iy
~
Ix by Rand 2.
(3) For each x in H, Ix is the greatest element of K below x. Thus all the normal models for 1M are of the form described above. It can be shown that 1M has a characteristic normal model, using the equivalence classes of words under I- Er:x{3. We can now show that 1M has the finite model property, and is therefore decidable. Suppose that a word r:x is rejected by a normal model (H, K, {I}, +,., -'-, 0, I) its parts taking the values aI' a z, ... , am' Take (HI, +,.) as the sub-algebra of (H, +,.) generated by 1, aI' az, ... , am, 0, and take K
I
=
trr, K.
6
R. A. BULL
Now (1)
x -'- 'y = x -'- y.
°
°
(Cf. Theorem 1.10 of [2].) (2) is in H' and K, therefore is in K', so there is an element of K ' below each element of H'. If x and yare in H' then (x+y) is in H'. Since x is in K if and only if
Ix = x, and
ELALpLqALpLq is a thesis of 1M, if x and yare in Kthen (x + y) is in K. So if x and yare in K' then (x+y) is in K '. It follows that there is a greatest element of K'
below each element of H', namely the sum of those elements of K' below it. (3) We can now define I' on H' by taking I'x as the greatest element of K' below x, for each x in H'. Thus
IX =>
I- LIX
corresponds to
CKLpLqLKpq corresponds to ANpp corresponds to
1
««.
K closed under .,
+,., -'-, 0)
Boolean.
The systems obtained by adding combinations of these to 1M can be shown to be complete with respect to the finite models with the corresponding restrictions, using appropriate modifications of the proof given above. In my opinion the most plausible of these systems for the role of intuitionist logic of necessity is that obtained by adding to IC the Godel rule and axioms for S4:
SOME MODAL CALCULI BASED ON IC
7
I- ex => I- Lex CLpp
CLpLLp CLCpqCLpLq (cf [3], p. 312), this being the extention of 1M with I- ex => I- Lex
CKLpLqLKpq. If Cex/3 is not a thesis of IC then there is an allocation of values which verifies ex and rejects /3.
LEMMA:
of i> x
(I use the notation of [1]). Suppose that the variables of ex and /3 are Pl' P2' ... , Pk. Since Cex/3 is not a thesis of IC and i> x is characteristic for IC, there is an allocation of the values of i> x which gives Cex/3 a nonempty set for its value. Suppose that ex, /3, and PI' P2' ... , Pk have values A, B, and P l , P 2 , ••• , Pk under this rejection, and that (m, n) is a point in B ...:... A. It follows that (m, n) and the points below it are not in A, but that (m, n) is in B. Given any value X of i> x , with Y as its sub-tree under (m, n), 1 form a value X' of i> x as follows: PROOF.
(1) For each i the sub-tree of X' under (i, n) is a copy ofY. (2) X' is the closure of these points. Allocate value P/ to each Pj: it will be found by induction on the length of words that ex gets value A' and /3 gets value B'. Clearly A' is empty but B' is not, so this allocation verifies ex but rejects /3. References [1] M. A. E. Dummett and E. J. Lemmon, Modal Logics between 84 and 85. Zeitschr. f. math. Logik und Grundlagen d. Math. 5 (1959) 250-264. [2] J. C. C. McKinsey and Alfred Tarski, On Closed Elements in Closure Algebras. Annals of Math. 47 (1946) 122-162. [3] A. N. Prior, Formal Logic, 2nd ed. (Oxford 1962). [4] M. Wajsberg, Ein erweiterter Klassenkalkul. Monat. f. Math. Phys. 40 (1933) 113-126.
THE LOGIC OF INTERROGATIVES M. J. CRESSWELL Victoria University of Wellington, New Zealand
This paper attempts to give a formal analysis of a concept which bears many resemblances to "p is the answer to d" where p is a statement and d a question. Let Spd represent this concept. The sense of "answer" which it characterizes is such that the following laws hold; Al
(Spd. Sqd)
~
(p = q)
Every question has at most one answer. This will exclude answers which give too much information. E.g., if! ask "Is it raining?" and receive the reply, "No it isn't; in fact the sun is shining." Then this would usually count as an answer. But in the sense of "answer" characterized by S the only correct answer to "Is it raining?" would be, "It is not raining" though the statement, "It is not raining and the sun is shining" obviously entails the answer to "Is it raining?" A2 (tlp)Spd, Every question has at least one answer. A3 Spd ~ p, The answer to a question is true. S then designates the true or correct answer to a question. A4
[0 (p == q). Spd]
~ Sqd,
If a statement is the answer to a question then any statement logically equivalent to it answers that question. This does make S diverge from many ordinary senses of "answer". It is a sense which, e.g. is of no use to the mathematician, since all mathematical truths are logically equivalent. But it does represent "answer" in that given a p which answers d wecan deduce an answer which satisfies any stricter criterion for "answer".
9
THE LOGIC OF INTERROGATIVES
There are cases where the same statement answers different questions. If I ask "Are both Mary and John here?" and someone else asks "Which of Mary and John are here?" then if both Mary and John are here the answer to both questions is "Both Mary and John are here" (though in the first case we would express it by saying "yes" and in the second by saying "both".) But they are different questions; for suppose Mary is here but John isn't. Then the answer to the first is, "Mary and John are not both here" and the answer to the second is "Mary is here but John isn't." (This last is an "answer" to the first question which gives too much information.) Identity of questions is defined,
Def=
(d
=
e)
=
df
(P) 0 (Spd
=Spe)
The following question-forming operator on questions is such that; Q
(d v e) is a question whose answer is the conjunction of the answer to d and the answer to e. If I ask, "Which of Mary and John is here?" then the possible answers are,
"John is here and Mary is here" "John is here but Mary isn't" "Mary is here but John isn't" "Neither Mary nor John is here" Q
and these are the only possible answers. The axioms for v are; A5
Q
(Spd.Sqe) => S(p.q) (d v e) Q
Q
A6 (d v e) = (e v d) Q
Q
Q
Q
A7
((d v e) V f) = (d v (e v f)
A8
(d
= e)
=>
Q
Q
[(fv d) = (fv e)J
The simplest kind of question is the yes/no question, i.e, the question which has the form "Is it the case that p'I", If I ask "Is it raining?" there are only two possible answers, "It is raining" "It is not raining"
Letting Qp = "Is it the case that p?" we have A9
SpQp v
S~ pQp
10
M. J. CRESSWELL
from which we may prove (using A3) SpQp == p, S"'pQp == "'p, and Qp = Q'" p. But yes/no questions form only a subclass of questions. A much more extensive class may be formed by introducing a questionforming quantifier. We use (Qa)A(a) to mean, "for which of the a's does A hold?" where a represents any variable and A a wff in which a occurs. I might ask, (Qp)fp, for which p does the propositional function/hold?
or
(Qx)¢x, which x's ¢?
or
(Q¢ )¢x, what properties does x possess? etc.
The answer to (Qa)A(a) will be the true conjunction of A(an)'s or '" A(an)'s for every an" To express this formally we express what it is for a proposition p to entail and be entailed by such a conjunction. p entails such a conjunction iff, (a) [p ~ A(a) . v -P -'5 '" A(a)] p is entailed by such a conjunction iff it is entailed by every q which
entails such a conjunction (since among these q will be the conjunction itself) i.e. iff (q){(a)[p ~ A(a) => q -'5 A(a):. :p -'5 '" A(a) => . q -'5 '" A(a)] => (q -'5 p)} 0
0
0
p is the answer to (Qa)A(a) if both these are satisfied and p is true.
AlO p. (a)[p
~
A(a). v.p ~ '" A(a)]. (q){(a)[p ~ A(a). => . q -'5 A(a):. :
p ~ '" A(a). => . q -'5 '" A(a)] => (q ~ p)).:
== :. Sp(Qa)A(a)
If we consider the class of questions definable in terms of (Qa) and restrict our logic to these we can define the expression (Sa)pA(a) (i.e. "p is the answer to the question 'which a's A?''') as; p.(a)[p -'5 A(a).v.p ~ ",A(a)].(q){(a)[p ~ A(a). =>.q ~ A(a):.: p -'5 ",A(a). =>.q -'5 ",A(a)] => (q
~
p)}
A wider class of questions can be considered by defining the schema Sp(Qna) [A 1(a), ... , An(a)] where (Qna) [A 1(a), ... , A,,(a)] would be equivQ
Q
alent to (Qa)A 1(a) v ... v (Qa)An(a). It is then possible, substituting (Qa)A 1(a)/d, (Qa)Aia)/e, etc., to prove as theorems the equivalents of Al-AlO by means of this definition.
THE LOGIC OF INTERROGATIVES
11
This method is sufficient to define, in a modal system with quantification, the concept "p is the answer to d" (in the sense of "answer" outlined) whenever the question d can be interpreted as asking, "Which a's A 7" I.e. it is sufficient for questions where a set of possible answers is so definable that the true member of the set is the answer to the question. If we can interpret questions like "When - - - - 7" as asking, "At what time- - - - 7", "Why- - - - 7" as asking, "For what reason - - - - 7", "How- - - - 7" as asking "By what means- - - - 7", etc. then the account given will suffice as an account of question logic in respect of "p is the answer to d", References [1] C. L. Hamblin, Questions. Australasian Journal of Philosophy 36 (1958) 159. Discussion-Questions aren't Statements. Philosophy of Science 30 (1963) 62. [2] David Harrah, Communication: A Logical Model (M.LT. Press 1963). [3] Henry S. Leonard, Interrogatives, Imperatives, Truth, Falsity and Lies. Philosophy of Science 26 (1959) 172. A Reply to Professor Wheatly. ibid. 28 (1961) 55. [4] J. M. O. Wheatly, Note on Professor Leonard's Analysis of Interrogatives. ibid. 28 (1961) 52.
SOME GENERALIZATIONS AND APPLICATIONS OF A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULP) RONALD HARROP University of Newcastle upon Tyne, UK
1. Introduction In an earlier paper [2], a study was made of the possibility of reducing propositional calculi of a general structure to ones with equivalent decision problem of a fairly simple structure. As an application a proof was given of a result, a particular case of which was the theorem of Post which states the undecidability of the problem of testing for completeness of a finite set of tautologies under substitution and detachment. To avoid many technical and notational complexities in the proofs of the relativization results, the theory was in the main developed only far enough to enable the application mentioned above to be made. This had, however, the effect that several restrictions were imposed which were unnatural and which, from the point of view of the truth of the main theorems, were completely unnecessary. It was stated that many of these restrictions would be removed in the present paper. It was also said that further applications of the theory would be given, including the construction of sequences of decidable and of undecidable calculi satisfying certain conditions. The present paper will be found to make frequent reference to [2] not only when use is made of lemmas and theorems proved there but also in order to avoid the necessity of excessive repetition of definitions and of notational conventions introduced there. 1) This paper, together with [2], covers the results outlined in the paper presented to the Logical Colloquium at Oxford in July 1963 under the title: A relativization procedure for propositional calculi and some applications.
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
13
2. Generalization of the relativization result
For technical simplicity in the proof of Theorem 2 of [2], a propositional calculus L was considered such that (i) its set of formulae was defined in terms of an infinite set of propositional variables and a finite set of binary connectives and (ii) its provable formulae were obtained using a finite number of axiom schemes and two-premise rules, no axiom scheme or premise or conclusion of a rule being a vaf (variable for arbitrary formula). We generalize the theorem in two stages corresponding roughly to the use instead of L of the calculi K, J described below. For each of K, J it is shown that there is a calculus L with equivalent decision problem, the calculus satisfying the conditions imposed on L in [2] except that if K, or J, contains infinitely many axiom schemes or rules or J has infinitely many constants or connectives, then L also has infinitely many axiom schemes or rules. The corresponding calculus L + has in these special cases infinitely many axiom schemes. Let K be such that (i) its formulae are built up from an infinite set of (propositional) variables a 1 , a 2 , ••. together with a finite, or denumerably infinite set a 1 , •.. , rx" or a 1 , ... , of (propositional) constants, by means of a finite or denumerably infinite set 't'1' ••. , 't'so or 't'1' .•• , of connectives. For each i for which 't'i is a connective, 't'i shall be ti-ary for some t i > O. It is important to notice that when we later use the phrases "of r, form", "for some a/' we will automatically assume that we are only concerned with values of i, j for which 't'i is a connective and rx j a constant. Other similar phrases involving suffixes which belong to a set which may be finite should be similarly construed. Only at a few isolated places in the paper, where special emphasis seemed desirable, is direct reference made to this type of restriction on the values of suffixes. We use A 1 , ••• to denote vafs of K. By a formula scheme of K we will mean an expression constructed from vafs and constants by means of connectives . .5/'X for a formula X and Y tp; JfqJ for a formula scheme qJ will be defined as in [2], it being noticed that substitution can take place for vafs or variables but not for constants. As in [2], we will think of substitution or taking instances as being done simultaneously for all vafs or variables occurring in the formula under consideration. Certain vafs or variables may of
14
RONALD HARROP
course be subject to an identity substitution, that is, be unchanged Suppose
for some formula schemes
({Ji'
In the case when there are infinitely many constants, connectives, axioms or rules, we presuppose the satisfaction of certain recursivity conditions with respect to some "standard" type of Godel numbering. We assume that, following normal techniques using primitive recursive functions, Godel numbers are attached to formulae and formula schemes in such a way that, again using primitive recursive procedures, we can test whether or not a number is the Godel number of a formula or formula scheme and, if it is, can obtain the formula or formula scheme concerned, and, if desired, can determine its main connective, subformulae or subformula schemes etc. We suppose that the axiom schemes are recursively enumerable in the sense that their Godel numbers can be given as the set of values of a recursive function or of a partial function everywhere undefined. A more restrictive condition is imposed on the presentation of the rules if infinite in number, namely, that they shall be given so that, for each i, the vafs occurring in R i shall be exactly Au . . . , As; for some s i and further that it shall be possible to test recursively, in terms of Godel numbering,
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
15
whether or not a given arbitrary finite set of formula schemes t/t l' . . . , t/t n are, in order, the n-I premises of some (n-I) premise rule (in the precise form in which they are given in the statement of the rule) and the conclusion of that rule. If they are of such a form then the number of the rule concerned must be uniquely determined and recursively obtainable. It will be noticed that these requirements have the effect that we can test recursively whether or not a set of formula schemes form a variant form of some rule and more generally whether or not a set of formula schemes form the premises and conclusion of some application of a rule. In cases when there are finitely many constants, connectives, axiom schemes or rules, the number of such shall in each case be explicitly known and it will be supposed that the Godel numbers of members of such finite sets will, if required, also be known. The calculus L to be associated with K shall have propositional variables ai' ... , and connectives a, b, C, 0, all binary. We shall use Ai' ... , for vafs of L. For d anyone of a, b, C, and D the corresponding one of A, B, C, we define DiJ for any formula or formula scheme J of L and any i 2:: 1 by
DJ = D iJ = JdJ D i + 1 J = Jd(DiJ.dJ) all i 2:: 1.
(2)
We will adopt the standard convention for omission of brackets in a formula that reinsertion of omitted brackets shall take place so as to keep the scope of unary connectives as small as possible. For this purpose A, B, and C will be considered as unary connectives. We will often be concerned with expressions obtained by iterated application of the connective O. If E 1 , ••• , En (n 2:: 1) are formulae (or, similarly, formula schemes) of L, possibly with 0 as main connective, then (3)
will denote an arbitrary one, possibly fixed through a limited context, of the formulae which are obtainable from E 1 0 ... 0 En by the insertion of (n-I) pairs of brackets. The inserted brackets will show the method of construction of the formula from E 1 , ••• , En. The statement that a formula F is ofform E 1 0 ... 0 En will mean that Fis one of the formulae representable as E 1 0 ... 0 En. The expression (4)
16
RONALD HARROP
will be used to indicate the particular formula represented by £1 O ... OEn which is obtained by inserting the required brackets by association from the left. For notational convenience, we generally in this section of the paper use
H,;(X l '
... ,
(formula or formula scheme case respectively)
Xt,) = BtHX 10 ... 0 HXtJ*, all connectives 'i and formulae Xl' ... , Xt"
H';(Xl, ... , Xt) = BtHX10 ... 0 HXtJ*,
(5)
aU connectives 'i and formula schemes Xl> ... , Xt,'
Since H is an operator from expressions of K to expressions of L and not a connective of L, it should cause no difficulties in respect of the insertion of omitted brackets. For easy visual appreciation of the structure of formulae it can, however, be considered as a unary connective. The calculus L shall have an axiom scheme corresponding to each axiom scheme ofK, the axiom scheme corresponding to Wi being (6)
Among the axioms of L are those of the form Cal 0 (Cal 0 HJw i) which may be considered as being the simplest axioms arising directly from the transforms of the axioms of K. Not all the axioms of L are however of this form though all can be written in the form 9'(Ca 1 0 (Cal 0 HJw;)). We have not used an expression based on this form in the actual statement of the axioms since it would not have been immediate that the axioms would then have been expressible in terms of axiom schemes, one for each value of i. The desire to express the axioms of L
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
17
conveniently in terms of axiom schemes, which were formula schemes, is partly responsible for the use in this paper of vafs, such as A l , A z , as well as of expressions for arbitrary formulae such as X, Y. As rules of L we take the following, noting that technically the premise of R2 i needs to be repeated so that the rule can become a two-premise rule, a requirement to be satisfied by the rules of L:
Rl R2· ,
CAl 0 A z CA 10A3 CA10 (A z 0 A 3 )
CAl 0 [( CAl 0 H CPil) 0 ... 0 (CAl 0 HCPik,)]* CAl 0 (CAl 0 Hcp;)
n) (8)
L will be seen to have infinitely many axiom schemes or rules if and only if K has respectively infinitely many axiom schemes or rules. LEMMA 1 (i): With the notational conventions given above, any formula scheme which can be written as H!7cp can also be written as !7Hcp, and if !7cP is obtained from cP by replacing A j for relevant j by ljJ j then Y Hip is obtainedfrom Hcp by replacing A j + 1 for the samej by HljJj' (ii) The formula scheme CAl 0 (CAl 0 H!7w i ) is a substituted case of the axiom scheme (6) ofL. PROOF. Result (i) is immediate from (5) and (ii) follows from (i) and the fact that in the construction of Y Hip from Ho no substitution is made for A l • A formula scheme 0 of L will be said to be potentially provable (of PP form) if and only if it is of the form
for some w (~ 1) and some formula schemes 0 1 , 000' Ow of L, A formula X of L is said to be of PP form if and only if it can be written in the form
for some w, 0 1 ,
••
0' Ow as above or, equivalently, in the form
where Xl> 0.0' X w are formulae ofL.
18
RONALD HARROP
LEMMA 2: If 0 is a PP formula scheme, then, with the above notation, w, the bracketing to be inserted in (9) to obtain 8, and, under the same substitution 1) as is used in (9), Y AI> YOlo' .. , YOw are all uniquely determined by 8. A similar result holds for PP formulae.
We use induction on the length of O. It follows from the hypothesis of the lemma that 8 must be of the form (CeO 0 «(to (2) where ( is the formula scheme substituted for A 1 under the substitution used in (9). If (1 is (e(, then w = I, Y8 t = (2 and no inserted bracketing is required. If (1 is not (e(, then «(cO 0 (1' «(eO 0 (2 are PP formula schemes which are shorter than O. Further, any expression in form (9) of can be considered as trivially arising from similar expressions for these schemes. The required result now follows by suitable use of the induction hypothesis. We say that a formula scheme ofL is of-t i form (with the usual restriction, if necessary, on the range of i) if it is of the form YB i[A I 0 ... 0 AtJ* and of tu form ifit is of the form YA iA I . The formula scheme AilJ will be called the IJ-rt i formula scheme. PROOF.
°
LEMMA 3: Noformula scheme can be of t , and «.form or of i .form for two distinct values of i or of «, form for two distinct values of i or the IJ-rti formula scheme for more than one choice of the ordered pair (IJ, i). PROOF.
Immediate from (2), (3) and the definitions of the terms in-
volved. The next two lemmas give a result which corresponds to Lemma 6 of [2]. Their proofs are similar to the corresponding parts of the proof of that lemma but there are some additional complications. These originate from the fact that the operator H used in this paper is less simple than the corresponding operator T of [2]. LEMMA 4: Suppose t/J l' . . . , t/J m; XI' .•. , Xm are formula schemes of K, 8,8 1 , .•. , Om are formula schemes ofL and 5', 5" substitutions for vafs of L in each of which AI' if it occurs, is replaced by 8. Suppose further that (i) 0i = YHt/Ji' I :::;; t s; m, under 5', (ii) OJ = YHXb I :::;; i s; m, under 5", (iii) no vaf in any Ht/Ji' HXi' I :::;; t s. m, except possibly AI' gets 1) Strictly, the restrictions of the substitution to the vafs in the formula schemes concerned.
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
19
replaced under S', SIt respectively by a Tj formula scheme or by the B-~j formula scheme for any j in the appropriate ranges, (iv) in S' and also in SIt, distinct formula schemes are substituted for distinct vafs A j' A k (j ~ 2, k ~ 2). Then l/Ji and Xi' 1 :<:::: i s; m, are variants of each other under some common substitution. PROOF. The result is proved by induction on the sum of the lengths of the l/Ji' If, for any i, l/Ji is a vaf then Xi must also be a vaf for that i for otherwise either Xi has a connective or is a constant and Y"HXi is thus of T j form for some j or is the B-~j formula scheme for some j. In either case this would involve Bi being obtained from Hl/Ji by a substitution which would violate (iii). Similarly, if, for some i, Xi is a vaf, are vafs then, for that i, l/Ji must also be a vaf. Further, by (iv), if l/Ji" Xit' Xi2 are occurrences of the same vaf if and only if l/Ji" l/Ji2 are then occurrences of the same vaf. By similar use of (iii) in conjunction with Lemma 3, we can show that for any i for which one of l/Ji' Xi is C1.q the other of l/Ji' Xi must also be IXq • The truth of the lemma for all cases in which none of the l/Ji has a connective follows at once. Suppose now that some 1/Ii' say l/Jio is neither a vaf nor a constant. It must have a main connective, say Ta. Y"Hl/Jio must be of Ta form. By our previous results, Xio must have a main connective which must, by (ii) and Lemma 3, also be T a • We thus know that for certain formula schemes l/J~, ... , l/J;a' X~, ... , X;a of K
v.,
BiO = Y"HTa(l/J~,
and Bio =
Y"HTa(X~,
,
l/J;J with substitution S'
, x;J with substitution SIt.
} (12)
Hence, by (5) and (2), Y"Hl/J~ under S' is the same as Y"HX~ under SIt for all p such that 1 :<:::: P :<:::: tao Combining this fact with (i) and (ii) restricted to the cases i #- i o and applying the induction hypothesis noting that the combined sum of the lengths of the l/Ji (i #- io) and of the l/J~ (1 :<:::: P :<:::: ta) is less than the sum of the lengths of all the l/Ji' we deduce that, under a common substitution, l/Ji is a variant of Xi (i #- i o) and l/J~ of X~ (1 :<:::: P :<:::: ta)· From the form of l/Jio' Xio (see (12)), we deduce at once that under the same substitution l/Ji is a variant of Xi (1 :<:::: i :<:::: m). This completes the proof of the lemma.
20
RONALD HARROP
For notational simplicity, we will in the statement and proof of the next lemma use the term ()-substitution to refer to a substitution in which the vaf A 1 , if it occurs in the formula scheme in which substitution is beingmade, is replaced by (). LEMMA 5: Suppose l/J1' ... , l/Jn are formula schemes ofK, (), ()1' ... , ()n are formula schemes of Land () i = /7 Hl/J i' l s i s n, under a common 8-substitution S. Then there exist l/J't, l s i s n, such that vt> /7l/Ji under a common substitution and in addition (a) ()i = /7 Hl/Jf under a common 8-substitution S* and (b) if 8 i = /7Hl/J~ under a common ()-substitution S' for some formula schemes l/J~, •.. , l/J~ of K, then l/Jf = /7l/J~ (l s i s n) under a common substitution.
Although the fact that l/Jf = /7l/Ji under a common substitution will not be proved this way, it can be noticed that it would follow immediately from (b) by taking l/J; as l/Ji (1 s i s n). Suppose that the distinct vafs which occur in the l/J;, 1 s i s n, are Art' ... , A rq and that the formula schemes substituted for the corresponding vafs of the Hl/J;, that is, for Art + l' . . . , A rq + 1> under S are (1' ... , (q. For each j, 1 s j s q, replace A r j throughout the l/Ji by Are where e is the least integer such that (e, (j are the same formula scheme. Further, replace in the formula schemes just obtained from the l/J;, A rr by llh for eachf, if any, for which there is an h such that (I is the O-llh formula scheme. Denote the resulting formula schemes by l/J11), ... , l/J~1). We know that the l/J~1) are obtained from the l/Ji by a common substitution and that forl s i s 11,8 i = /7Hl/J~1) by a common O-substitution, say S(1). Let I.51) denote the list of distinct formula schemes substituted under S(l) for vafs other than A 1 in the Hl/J~1). None of the members of I.51) will be a ()-rxh formula scheme for any h. Distinct vafs A p , A q (p, q ;::; 2) of the Hl/JP) will have different members of I.5 1 ) substituted for them under S(1). Suppose that with corresponding notation we have defined up to L(ml, s», l/Jlm), ... , l/J~m) knowing that l/J~m) = /7l/Ji' 1 s i s n, under a common substitution, ()i = /7Hl/J~m), 1 s i s n, under the common 8-substitution s», no member of I.5m) is a 8-a h formula scheme for any h, and that distinct vafs A p , A q (p, q ~ 2) of the Hl/J~m) have distinct members of I.5m) substituted for them. If no member of I.5m) is of 't"j form PROOF.
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
21
for any i. we take t/Jim), ... , t/J~m) for t/Ji, ... , t/J: respectively and S(m) for S*. Otherwise, suppose , is the longest, or one of the equal longest, of the members of 15m) which is of 'Cj form for some j. Suppose it is of 'C a form and is YB a[A 1 0 ... 0 AtJ*, and that it is substituted for A e + 1 under S(m). We obtain t/J~m + 0, ... , t/J~m + 1) by replacing A e throughout the t/J~m) by 'CiXl' ... , Xt) where Xl' ... , Xt a are formula schemes of K chosen as follows: If in the formation of' from Ba[A 1 0 ... 0 At.]", AI is replaced by the member of L(m) which is substituted for A c + 1 under s». then XI is taken as A c • If A I is replaced by the 8-rx g formula scheme, XI is taken as rx g • Any of Xl' ... , Xt a which are not determined by the above two methods are taken as "new" vafs, that is, as ones not occurring in any of the t/J~m), the same vaf being used for XI and Xg if and only if the same formula is substituted for A I' A g in the formation of , from B a[A 1 0 ... 0 AtJ*. It is immediate that t/J~m + 1) = Yt/Ji' 1 :<::: i :<::: It, under a common substitution and that 8i = YHt/J~m + 1), 1 :<::: i :<::: It, under a common 8-substitution which we may denote by s(m + 1). Further, if 15 m+ 1) is the list of distinct formula schemes substituted under S(m + 1) for vafs other than A 1 in the t/J~m + 1), then no member of 15 m+ 0 is a 8-rxh formula scheme for any h, and distinct vafs A p , A q ofthe Ht/J~m + 1) (p, q ;;::: 2) have distinct members of 15 m + 1) substituted for them. Since the construction given for I5m + 1) will have the effect that I5m + 1) will have arisen from 15 m ) by the replacement of' by a set (possibly null) of formula schemes") the total length of which is less than the length oH, the process can be effectively iterated until further iteration is impossible. We take the final t/J~m), ... , t/J~m) as t/Ji, ... , t/J~ and the final s(m) as S ", We have thus shown the existence of formula schemes t/J1, ... , t/J': such that (i) t/J't = Yt/Ji under a common substitution, (ii) 8i = YHt/Ji under a common 8-substitution S*, (iii) distinct vafs A p , A q (p, q ;;::: 2) of the HljJ; get replaced under S* by distinct formula schemes and no formula scheme substituted under S* for a vaf other than A 1 is of 'Cj form for any j or the 8-rxj formula scheme for any j. 1) Namely the formula schemes which under S(m+l) replace vafs of the Htp/m+l) which correspond to any "new" vafs used in the specification of the XI' 1 :s; f:S; tao
22
RONALD HARROP
Finally suppose that Oi = Y'Hl/J; (1 S; i S; n) under a common I)-substitution S'. By the above result there exist l/J;* (1 S; i S; n) such that (i), (ii), (iii) hold with l/Ji, l/Ji, S* replaced by l/J;, l/J;*, S'* respectively. It follows immediately by Lemma 4 that l/Ji, l/J;* (1 S; i S; n) are variants under a common substitution and hence, using (i) for l/J;, l/J;\ that l/J't = Y'l/J; under a common substitution. This completes the proof of Lemma 5. THEOREM 1 (i): A formula scheme can be written in the form
e is provable
in L
if and
only if it
for some provable formula schemes l/Jl' ... , l/Jw(w 2 1) ofK. (ii) A formula P is provable in L if and only if it can be written in the form for some provable formula schemes l/J l' ... , l/J w( w 2 1) of K. (iii) A formula or formula scheme of L which is provable is potentially provable.
(ii) is immediate from (i) and the fact that a formula P is provable in L if and only if the corresponding formula scheme, which is obtained by replacing a, by Ai for each Qi which occurs in P, is provable in L. (iii) is immediate from (i), (ii) and the definition of potentially provable. We now show, by induction on the length of proof of l/J in K, that if a formula scheme l/J is provable in K, then CAl 0 (CAl 0 Hl/J) is provable in L. If l/J is Y'OJ i for some axiom scheme OJi of K, the required result is immediate by Lemma 1 (ii). Suppose l/J is obtained from Xl' ... , Xk, by an application of the rule R, of K. By induction hypothesis, CAl o(CA l o HXr) is provable in L for each r, 1 S; r S; k, and thus, by iterated application ofR1, so is PROOF.
CAl 0 [(CAl OHX1)O ••• O(CA 1 0 HXk,)]*'
(14)
Let Q denote the set of vafs occurring in R, as specified in (1). Let S denote the common substitution, known to exist, such that Xr = Y'qJiro 1 S; r S; k i , l/J = YqJi and suppose that, under S, A g is replaced by X~, all A g E Q. Considering the substituted form of rule R2 i of L in which A 1
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
23
is unchanged and, for all g such that A g E Q, A g + 1 is replaced by H X~, and using Lemma l(i) and the provability of (14) in L, we see that CAl 0(CA 1 OHljJ) is provable in L. Thus, for any ljJ provable in K, CAl 0 (CAl 0 HljJ) is provable in L. Hence, using appropriately iterated application of rule R1 of L and the fact that the set of provable formula schemes ofL is closed under substitution, we can deduce that any formula scheme of type (13), for W ;;::: 1 and ljJ 1, •.. , ljJw provable formula schemes of K, is provable in L. To complete the proof of the theorem we now show by induction on the length of proof of () that if () is a provable formula scheme of L then it can be expressed in form (13) with the stated conditions on w, ljJ l' . . . , l/Jw all satisfied. If () is a substituted case of an axiom scheme of L, the required result is trivial (see (6». Suppose () is the conclusion of an application C(10(Z
C(1 0 ( 3
C(1 0«(ZO(3)
(15)
of rule R1 of L. By induction hypothesis, the premises of the rule can be written as and where the l/Jij' i = 1,2, 1 ::;; j ::;; Wi, are provable in K and where we can assume without loss of generality that the vafs in (16) other than A 1 do not occur in (17). Since it follows from (15) that the formula scheme substituted in (16), (17) for A 1 must in each case be (1' we can deduce that () can be written as 9'(CA 1o ([(CAl
o HljJll) 0
'" 0(CA 1 o Hl/J1w)] 0
[(CAl
o HljJZ1) 0
... 0(CA 1o HljJzw,)]»
and this can easily be expressed in the required form. Finally, suppose () is the conclusion of an application 9'(CA 1 0 [(CAl 0 Hrpil)O ... 0 (CAl 0 HrpikJJ*) 9'(CA 1 0 (CAl 0 HrpJ)
(18)
of rule R2 j of L where the substitutions indicated are common and will
24
RONALD HARROP
be denoted by S. By induction hypothesis the premise of (18) can be written as .9"(CA10 [(CAl OHt/JDo •.. O(CA 1
OHt/J~,)J)
(19)
for some t/J~, ... , t/J~, provable in K and some substitution S'. It follows at once from Lemma 2 that k, = w', that [J can be replaced in (19) by []* and that there exist formula schemes 0*, 0 1 , ••• , Ok, such that (0 S, S' are O"'-substitutions, (ii) OJ = Hq>ij, 1 ::; j ::; k i, under S and (iii) ()j = Ht/Jj, 1 ::; j ::; k i, under S'. Let Ok, + 1 = .9"Hq>i under S and let t/J~, + 1 be a variable other than A 1 which does not occur in any t/Jj, 1 ::; j ::; k;, so that we can extend S' so that Ok, + 1 = .9"Ht/J~, + 1 under S'. By Lemma 5, there exist t/Ji, ... , t/J:, + 1 such that (iv) t/Jj = .9"q>ij, 1 ::; j ::; k i, t/Jk' + 1 = .9"q>i under a common substitution, (v) t/Jj = .9"t/Jj, 1 ::; j ::; k, + 1 under a common substitution, and (v i) OJ = .9"Ht/Jj, 1 ::; j ::; k, + 1 under a common 0'" -substitution. It follows from (v) that the t/Jj, 1 ::; j ::; k i , are provable formula schemes of K and hence, using (iv) and (1), that + 1 is provable in..K. We can deduce from the definitions of 0*, Ok, + l' that 0, which is the conclusion of (18), can be written as CO* 0 (CO* 0 Ok, + 1)' Thus, using the case j = k i + 1 of (vi), 0 can be written as .9"(CA10 (CA 10 Ht/Jt. + 1)) which is of the required form. This completes the proof of Theorem 1.
»:
THEOREM 2: The decision problems of K, L are with standard Giidel numbering primitively recursively equivalent to each other. PROOF. We first note that both in the calculus K and in the calculus L, a formula is provable if and only if the corresponding formula scheme, obtained by replacing a, by Ai for all occurring i, is provable. If X is provable in K, then, by Theorem 1 (ii) and (5), Cal 0 (Cal 0 HX) is provable in L. On the other hand, if Cal 0 (Cal 0 HX) is provable in L, then so is CAl 0 (CAl 0 Hip for the formula scheme q> which corresponds to X. Using Theorem 1 (i), it follows that this formula scheme can be written in form (13) for some substitution S and some formula schemes t/Jl' ... , provable in K. By Lemma 2, w must be 1, S must be an Acsubstitution, and, with substitution S, Hip must be .9"Ht/J r- Hence, by Lemma 5, there exists t/J* such that t/J* = .9"tp, t/J* = .9"t/Jl and Hip = .9"Ht/J* under an A1-substitution. Since, by
v;
A RELATlVIZATlON PROCEDURE FOR PROPOSITIONAL CALCULI
25
Lemma l(i), Ht/J* (= HYcp) can be written in the form YHcp under an AI-substitution, it follows that H cp and Ht/J* are variants of each other, the substitution employed in obtaining Htp from Ht/J* and that employed in obtaining Ht/J* from Hip being an Acsubstitution. A trivial application of Lemma 4, taking m = 1 and e, t/Jl' Xl' el as AI' cp, t/J*, Hip respectively, now shows that ip, t/J* are variants of each other. Hence cp is provable in K. Thus a formula X is provable in K if and only if Cal 0 (Cal 0 HX) is provable in L. To complete the proof of the theorem we show that we can determine whether or not a formula P is provable in L either directly or in terms of the provability or otherwise in K of certain effectively obtainable formulae of K the Godel numbers of which would, under standard Godel numbering, be primitively recursively bounded in terms of the Godel number of P. P is provable in L if and only if the corresponding formula scheme e is provable in L. Using Theorem 1(iii), the fact that we can determine effectively whether or not a formula scheme is potentially provable, Lemma 2 and Theorem 1(i), we note that we can test whether or not e is provable directly, or reduce the problem to determining whether or not certain primitively recursively obtainable formula schemes (1' ... , (w (yet> ... , yew of Lemma 2) can be expressed as YHt/Jl' ... , YHt/Jw under a common substitution with a given substitution for Al (Y Al of Lemma 2), for some t/Jl' ... , t/Jw provable in K. We can assume for simplicity and without loss of generality that there are no variables in common between any t/J hand t/J iz for which i l =f. i 2 • Our problem thus reduces to determining, given a formula scheme ( of L, whether or not ( can be expressed in the form Y Ht/J for some provable formula scheme t/J of K. Since, for any formula scheme t/J of K with at least one connective, Ht/J is longer than t/J, and, except for fairly trivial substitutions, YHt/J is longer than Ht/J, the consideration of ( reduces to the determination as to whether or not at least one of a finite number of primitively recursively obtainable formula schemes of K restricted so that no two are variants of each other, is provable in K. Since such a determination can be made by considering the provability or otherwise in K of the corresponding formulae, this completes the proof of Theorem 2. Suppose given a calculus K satisfying the conditions imposed at the beginning of this section of the paper. We can construct from it the
26
RONALD HARROP
calculus L described above and from L can construct, in the manner described in [2], calculi L', L * (ambiguously denoted by L +) which can, if desired, by taking ::J as the only connective, be made subsystems of positive implicational logic. L + will have detachment as its only rule. If K has only finitely many axiom schemes and rules, the number of connectives and constants possible being infinite, then L will satisfy the conditions required of that calculus in [2] and so, by Theorem 2 of the present paper and Theorem 2 of [2], the decision problems of K and L + are equivalent, and in fact are primitively recursively equivalent. If K has infinitely many axiom schemes or rules, then L + , as constructed following the method described in [2], will have respectively infinitely many axioms of type 1+,2+ (there will never be infinitely many axioms of type 3 + since L has only finitely many connectives). -The presence of infinitely many axiom schemes in L does not affect the proofs of Theorems 1, 2 of [2] and we can still assert that the decision problems of K, L + are primitively recursively equivalent. In cases when there are infinitely many rules in K, and therefore in L, although the proof of Theorem I of [2] is unaffected, the proof of Theorem 2 of [2] requires some change. In the consideration of the cases ({3+), (y+), we have to note that by our assumptions concerning the form of the rules ofK, and their effect on the form of the rules ofL, we can test recursively whether or not the members of a set of formula schemes form variants, under a common substitution, of the premises and conclusion of some rule R i of L and if they are can determine which rule it is. We can thus still show that the decision problems of K and L + are recursively equivalent but our proof will not show that they are primitively recursively equivalent. We have thus proved THEOREM 3: (Extension to K of Theorem 2 of [2]). With the above notation the decision problems of K and L + are recursively equivalent. L + has finitely many axiom schemes and rules except in cases when K has infinitely many axiom schemes or rules. L + has, in such cases, infinitely many axiom schemes.
The calculi so far considered have all had their formulae built up by general iterated application of connectives of definite order to the members of an infinite set of propositional variables together possibly with the members of a set of propositional constants. They have had axiom
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI 27
schemes which were formula schemes, and rules, the premises and conclusions of which were formula schemes. This has meant that there has been a simple duality between the provability offormulae and offormula schemes in that a formula was provable if and only if its corresponding formula scheme was provable and vice versa, that is, a formula scheme was provable if and only if its corresponding formula was provable. Considerable use has been made of this fact. To conclude our discussion of systems more general than the system L of [2] to which the relativization process applies, we develop in outline the theory for a system J for which the duality properties are not available. There are other examples of similar type which could have been used instead of J. Consider a system J which is defined exactly as K except that there shall only be finitely many propositional variables, say ai' ... , a q and that there shall be a recursivity condition imposed on the axiom schemes of J similar to that imposed on the rules of K, and therefore of J, namely that the axiom schemes shall form a recursive set and that the ith axiom scheme shall involve exactly the vafs Ai' ... , Ar; for some rio We can still use A 1, ••• , for vafs, there being no need for them to be finite in number. Let (i' R; denote respectively the ith axiom scheme and the ith rule of J (for appropriate values of i). An example of a system of type J which illustrates the differences which can exist between such a system and one of type K is the following: propositional variables - just a i ; propositional constants - none; connectives - just ~ (binary); axiom scheme A 1 ~ Ai; rules
Ai (A 3
~
~
(A 3 A z)
~
~
A z)
Ai
It is immediate that the set of provable formulae of the system is the set of formulae of the form U ~ V where U, V are formulae, that is, it is the set of instances of Ai ~ A z . The formula scheme Ai ~ A z is not itself provable. If in the definition of this system, the conclusion of the second rule is replaced by Ai ~ A z, no change would occur in the set of provable formulae but Ai ~ A z would become provable. It will be
noticed that the axiom schemes and rules of the system both in original and revised form are independent. We associate with J a calculus K which satisfies the conditions we
28
RONALD HARROP
imposed on a calculus denoted by K. Its propositional variables will be ai' .... It will have the same constants as J, and, for connectives, the connectives of J together with F, G (unary) and * (binary). We define a mapping E from formulae of J to formula schemes of K by Ea,
= F~l
EIX i
=
1 ~ i ~ q,
each constant
IXi
IXi
of J,
I
E(-r:iUl' ... , Uk)) = TiEU1' ... , EU k) each connective Tj } (20) (krary) and all formulaej U 1 , ••• , U k j of J. In this definition, F iA 1 is defined by') F 1A1 = FA 1 Fe + 1 A 1 = F(F eA1 )
all c z-L
K has one axiom scheme corresponding to each variable of J and one corresponding to each constant of J. The axiom schemes corresponding to a, (1 ~ i ~ q) and IXi (relevant i) are respectively (21)
K has one rule corresponding to each connective of J, one corresponding to each axiom scheme of J and one to each rule of J. The rules corresponding, for relevant values oft, to the connective T; (k i ary), the axiom scheme ~ i and the rule R; are respectively FA 1 *F A z .,. FA 1 *FA k i + 1 FA 1 FA 1
* FA z
.. , FA 1
FA 1
* G~;
where the vafs occurring in ~i are A 1 , by replacing A r by A r + 1 for each r, 1 FA 1
* FA z
... FA 1
* FA
ti
(22)
* FT;(Az, ... , A k i + 1)
+ 1
FA 1
••• ,
~
r
FA 1
* Gp;
* FA
si
+ 1
(23)
A r i and ~; is obtained from ~i r.; and
~
* Gp;l
... FA 1
* Gp;h, (24)
1) Note that F is an explicit unary connective of J, not part of an abbreviated form of an expression involving a binary connective like D in (2).
A RELA TIVIZA TION PROCEDURE FOR PROPOSITION AL CALCULI
29
where R; is
, Pil ... Pih,
R~
(25)
are all the vafs which occur in R;, and P;j (1 ::;; j ::;; hJ and P; are obtained from Pij (1 ::;; j ::;; h;) and Pi by replacing A r by A r + I for each r, 1 ::;; r ::;; Si. The special restriction imposed on the form of the rules of K is automatically satisfied in view of the satisfaction of the AI' ... , As,
corresponding conditions imposed on the form of the axiom schemes and rules of J.
We can now prove the following results: (a) A formula is provable in K if and only if the corresponding formula scheme is provable in K and vice versa. (b) Given formula schemes sp, l/J of K, we can effectively determine whether or not cp can be expressed in the form :/EU for some formula U of J under a substitution in which Al if it occurs at all is replaced by l/J. If it can be so expressed then U is uniquely determined by cp, l/J and can be effectively obtained. The substitution will also be fully determined by cp, l/J since no vaf other than Al can arise in EV for any formula V of J. (c) We can effectively determine whether or not a formula scheme ofK can be expressed in one of the forms :/(FA I * FEU), :/(FA I * GEU) for some formula U of J. Ifit can be so expressed then the form concerned and the formula U involved are uniquely determined and can be effectively obtained. (d) If U is a formula of J then FA I * FEU is a provable formula scheme ofK. (e) If U is a provable formula of J then FA I * GEU is a provable formula scheme of K. (f) If cp is a provable formula scheme of K then it must be either of the form :/(FA I * FEU) for a formula U of J or of the form :/(FA 1 * GEU) for a provable formula U of J. (This result can be proved by induction on the length of proof of cp in K, making considerable use of (c)). It follows from (a), (c), (d), (e) and (f) that the decision problems for provability of formulae in J and K are primitively recursively equivalent. Hence, using Theorem 3, we obtain the following theorem:
30
RON ALD HARROP
THEOREM 4: (Extension to J of Theorem 2 of [2]). With the above notation the decision problems of J and L + are recursively equivalent. L + has finitely many axiom schemes and rules except in cases when J has infinitely many constants, connectives, axiom schemes or rules. In such cases L + has infinitely many axiom schemes.
3. Iteration of the relativization process - preliminary results Later in this paper we construct by iteration of the relativization process, which led in [2] from L to L +, certain sequences of distinct calculi. The proofs that the calculi are distinct rest on some results now to be obtained which are concerned with the relation of the structure of provable formulae of L + to that of provable formulae (if any) of L which have ~ as main connective. We will use, as far as possible and often without comment, definitions and notations used in [2] but, since we are working towards an iterative application of relativization, it will not be possible to keep quite as rigidly as in [2] to the use of particular letters to denote formulae or formula schemes of particular calculi. For example, in [2], formulae such as TX, Y'TX would be "recognized" as formulae of L', L* and L + even before they were stated to be so. Further, the equation Q = Y'TX would look "reasonable" whereas the equation Y = Y'TX would look "unreasonable". We now consider explicitly some effects of the possibility of ~ being a connective of L and will on occasions wish to consider expressions such as TX, Y'TX not only as formulae of L', L* and L + but also as formulae of L. Under such circumstances any attempt to have a rigid distinction between the use of Q and of Y would probably lead to confusion. 1) Since considerable reference will be made to theorems, lemmas and displayed formulae of [2], we will through the remainder of this paper denote the use of such by means of an asterisk, for example, Theorem 1*, Lemma 1*, (1)* will refer to Theorem 1, Lemma 1 and formula (1) of [2]. Suppose L, L +, T satisfy the conditions imposed on the calculi L, L + and the transformation T in [2] and that, in addition, Land L + 1) The possibility of ::J being a connective of L and of TX being a formula of L as well as of L + was naturally present throughout [2]. The use of the notational conventions referred to relied on the fact that we never wished in that paper to consider TX other than as a formula of L! (or L', L*).
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
31
both have :::> as their only connective, this having the effect that s = I and (J t, is :::> in the definition of T (see (1)*). Let S, l: denote respectively the set of formulae, formula schemes, of L, and therefore also of L + • Suppose XES. We define the rank of X through the following conditions, used inductively; (i) If X cannot be expressed in the form !/'TY. :::> ZZ for some Y, Z E S, Y having at least one connective and ZZ denoting Z :::> Z as in (2)*, then the rank of X is I, (ii) If X can be expressed in the form !/'TY:::> ZZ for some Y, Z E S, where Y has at least one connective and has rank r, but cannot be expressed in the form !/'TY' :::> Z'Z for Y', Z' E S where Y' has at least one connective and has rank greater than r, then the rank of X is r + I. Using the fact that, for any U, V, and any substitution, !/'TU:::> V Z is longer than U, we can prove inductively that the rank of a member of S is a uniquely determined effectively obtainable integer and that the rank of a member of S is equal to the rank of any variant of that member. By changing X, Y, Z, S to cp, t/J, x, l: respectively, throughout the definition of rank of a formula, we obtain a corresponding definition for the rank ofa formula scheme. Suppose M is a calculus which has S as its set of formulae. M is said to be offinite rank if it does not possess provable formulae of arbitrarily large rank. The rank of M will be defined to be zero if M has no provable formulae and to be r (> 0) if M has provable formulae of rank r but none of greater rank. If M is not of finite rank it is said to be of infinite rank.
6: If X, YES, Y = !/'X and X is of rank r, then Y is of rank greater than or equal to r. LEMMA
If r = I, there is nothing to prove. If r > 1, then there exist W, Z E S, with W having at least one connective and being of rank r-l, such that X = !/'TW. :::> ZZ. Hence Y can be written as !/'(!/'TW. :::> ZZ). By Lemma 5(i)*, this formula is of the form !/'TW. :::> Z,Z for some substituted form of TW and some Z' E S. The required result can be obtained immediately. We note trivially that for some formulae, for example at, a substituted form can have a greater rank than the original formula, while for others, for example at :::> at, all substituted forms have the same rank as the original formula (in this case, I; see Lemma 2 (i) (aJ (,yz)*). PROOF.
RONALD HARROP
32
THEOREM 5: Suppose L, L +, T are as above. If L has finite rank, so has L +. More precisely, if the rank of L is r, then that of L + is r+ 1. PROOF. By Theorem 1*, a formula P is provable in L + if and only if it satisfies condition D+. By Lemmas 1(ii)* and 3* and the definition of rank of a formula, if P is provable in L + then it is of rank 1 unless it satisfies D + by (a+). Hence if r = 0, that is, ifL has no provable formulae, then L +, which will have some formulae which satisfy D+, but none of them by (a+), will be of rank 1. Suppose that r > and that P, of rank b greater than or equal to r+ 1 satisfies D+. Then, (i) since P satisfies (a+), it is of the form !JYTX.::> y2 for some provable formula X of L and some YES and (ii) since P has rank b it can be written in the form !JYTX'.::> Y'2 for some X', Y' E S where X' has at least one connective and has rank b - 1. By Lemma 7*, P can be written in the form YTZ. ::> y 2 for some provable formula Z of L which is of the form !JYX'. Hence, by Lemma 6, L has a provable formula of rank at least b - 1. Thus b - I :s; r and the rank of L + is not greater than r + 1. Since the rank of Lis r, there is a provable formula W of L of rank r. The provable formula TW. ::> ai of L + which satisfies D+ by (a+), has rank at least r+ 1. Hence the rank of L + is at least r+ 1 and thus is exactly r+ 1. This completes the proof of Theorem 5. Our final results in this section are concerned with a calculus which is essentially the "union" of the relativizations of two "disjoint" calculi. Suppose that L 1 , L 2 are calculi which satisfy the conditions imposed on Lin [2] and that they have disjoint sets of binary connectives which we will denote by CT1' ••• , CTs ; CTs + 1, ..• , CTk respectively. Let S1' I 1 ; S2' I 2 denote the sets of formulae and formula schemes of these calculi and let S, I denote the set of formulae, formula schemes constructed from a l' . . . ; A 1, .•. , by means of the single binary connective ::>. Let T be the mapping defined as in (1)* with respect to the complete set CT 1, , CTk of connectives. Tthus maps the set of formulae constructed from aI' , by CT 1, . . . , CTk' and thus also S1 and S2' into S. We construct as usual, using the restriction of T to 8 1 , II' and consider the case in which Lt has no connectives other than c . There is a slight complication in the construction of the corresponding relativized form of L 2 since the definition of L + in [2] used (1)* and this depended,
°
t.;
A RELA TlVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
33
through its third line, on the connectives of the calculus concerned being numbered consecutively as the 1st, ... , sth connectives, for some s, whereas the restriction of T as defined above to Sz, X z treats the connectives of L z as if they were numbered as the (s+ l)st, ... , kth connectives.' ) Let Li(s) denote the relativized calculus, with ~ as its only connective, which has axiom schemes and rules formally the same as those given for Li in [2], that is, for L + with L replaced by L z , except that in the statement of 3*, 1 ::;; r ::;; s, 1 ::;; t ::;; s should be replaced by s + 1 ::;; r ::;; k, s + 1 ::;; t ::;; k respectively. T should be considered as the restriction of the mapping Tdefined above to the sets Sz, 2,'z. It will not be surprising to find that Li(S) behaves like Li (obtained by first renaming the connectives as 1st, ... , (k-s)th), since the main aspects of (1)*, (4)* are retained. These are the possibility of carrying over the structure of the formulae of L fully into L + in such a way that formulae of some recognizable structure not previously used (namely (4)*), would be available in L + for use in the construction of the axiom schemes of L +. We denote by Ltz the calculus with formulae S, formula schemes X, axiom schemes the union of the axiom schemes of L 7, Li (s) and detachment as its only rule. Similar notation is used in corresponding cases when calculi other than Lj , L z are involved. The calculi Ltz and Li.1 will, in general, naturally be distinct. It is shown later that their decision problems are primitively recursively equivalent to each other. Let »t, denote the condition obtained from D+ by making the following changes: (i) replace everywhere, D+ by Dtz and S+ by S. (ii) replace (ct+), (r), (y+) by (atZ)l' (cttz)z, Wi.Z)l, (f3tz)z, (ytZ)l' (ytz)z, the conditions with suffix i having L i for L, i = 1,2. The changes in (y+) are consequential on those in (r) in the sense that (ytZ)i shall include a reference to (f3tZ)i' i = 1,2, where (y+) includes a reference to (r). (iii) (0+), (e+), (C+) shall be renamed (otz), (etz), ("i.z) respectively. The special condition imposed on (otz), which has consequential effects ') The third line of the definition of T restricted to the connectives of L 2 reads T(EaiF) = TE :J ((TE)3+i :J (TF)2), s-} 1 :::; i ::;; k. For normal theory it would read T(Ea s + i F) = TE :J ((TE)3+i :J (TF)2), 1 :::; i :::; k-s.
RONALD HARROP
34
on (Btz), (n.z) shall be modified so as to read "it is required that Ql' Qz should be of the form fT(A l urA 3 ) , fT(A z utA 4 ) respectively, for some r, t such that either both 1 ~ r ~ sand 1 ~ t ~ s or both s + 1 ~ I' ~ k and s + 1 ~ t ~ k:" THEOREM 6: With the above notation, a formula N is provable in Li.z
if and only if it satisfies
»t.:
PROOF. The result follows by consideration of the details of the proof of Theorem 1* that is, of the proof of Lemmas 4*, 9*. In obtaining, following the proof of Lemma 4*, a proof that if N (E S) satisfies »i, then it is provable in Li.z, the main cases are (ai.Z)l and (ai.zh- These, however, lead the consideration straight back to proofs of corresponding results for Lt, Li(s) and these can be obtained by direct use of the method employed in the consideration of (a +) in the proof of Lemma 4 *. The restrictions imposed in the definition of axiom scheme 3* are satisfied at places where the axiom scheme is required for use both in the (ai.Z)l and in the (at2)Z case. The cases (Y:'z); are referred back to the use of cases of axiom scheme 2 + of L:'z, these being cases due to Lt or to Li(s) according as i = 1 or 2. The remaining cases are trivial, noting where necessary the special restriction imposed on (btz), (atz) and (n.z)·
We now show that if N is provable in Lt2 then it satisfies »i; This trivially reduces (compare the proof of Lemma 9*) to showing that if P, P :;) R are provable in Li.z and satisfy Di.z then R, which is provable in Ltz, also satisfies D7.2' Due to the splitting of each of (a+), (r), (1'+) of D+ into two parts when Dtz was formed, there are now 81 instead of 36 cases to be considered. Cases corresponding to those in parts (i), (ii) and (iii) of the proof of Lemma 9* can still be shown to be trivially impossible by use of the structural considerations previously employed, or to lead easily to the conclusion that R satisfies Di.2' This accounts for 36 + 27 + 12 (= 75) cases. Consider now cases of type (iv), that is, cases in which P satisfies D7.z by (Ci.2) and P :::> R satisfies D7.z by a method other than (arZ)l' 2 (C7.2)· P is of the form Ql A Qz. :;) M where, in the case of D'1'.z, Ql, Q2 are respectively of the form JT(A lurA 3 ) , fT(A 2utA 4 ) and either 1 ~ r ~ s, 1 S t ~ s, or s + 1 S r S k, s + 1 s t s k. From Lemma 2(i)(a;)(p)*, P :::> R cannot satisfy D1.z by (btz), (S'1'.2)' If P :::> R satisfies
«:»;
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI 35 D~.z by (b~.z), (e~.z) then R satisfies D' by (e~.z), «(~.z) respectively. We know, by Lemma 2(i) (IX;) (f3)* that P :::J R cannot satisfy Dtz by (f3i.Z)i' i = 1, 2. Suppose P :::J R satisfies Di.z by (yi.Z)i' Then R is of the form R t :::J M Z. Since P satisfies ot; it follows, using Lemma 3*, which with trivial notational modification applies to »i; that both Qt :::J M Zand Qz =:> M Zsatisfy Dtz by (IXtz)t, (IXtz)z or Since Qt, Ql' u, are of the form JT({Jt, JT({Jz, JTt/J (under a common substitution) for some rule ({Jt ({Jz/l!J of L i, it follows, again using lemma 3*, that Qt =:> M 1 , Qz =:> M 1 must satisfy Dtz by (IXtz)t or (IXtz)z. Further, by using Lemma 1(iii)* modified so as to apply to Ltz, L t, L z, we see that Qt, Qz, R are respectively of 1';" 1';2' T, form where all of it, i z, j are in the range 1, ... , s if i = 1, and in the range s + 1, ... , kif i = 2. Further application of lemma 1(iii)* shows that Qt =:> M 1 and Qz =:> M Z must both satisfy »t, by (IXtZ)i' Using Lemma 6* as in the proof of Lemma 9*, we can now show that R t :::J M Z satisfies Dtz by (IXi.Z)i by following directly the last part of the consideration of case (iv) in the proof of Lemma 9* substituting r, for Land (IXtZ)i for (IX+). This completes the proof of Theorem 6.
«z,»
THEOREM 7: With the above notation and any standard type of Giidel numbering, the predicate corresponding to provability in Ltz is primitive recursive in the corresponding predicates for L t , L z, and, for i = I, 2, the predicate corresponding to provability in L; is primitive recursive in the corresponding predicate for Li.z. PROOF. This is similar to the proof of Theorem 2* preceded by the proof of a result which corresponds to Lemma 10*. Suppose X E S, and that X is provable in L i, then TX . =:> ai satisfies »t, by (IXi.Z)i and is provable in Ltz by Theorem 6. Suppose now that X E S, and that TX . =:> ai is provable in Ltz. By Theorem 6, the definitions of (lXi.z)t, (lXi.z)z, and Lemmas 3*,1 (ii) (iii)* (having put them in a form applicable to Ltz, Lj , L z), we can deduce that X is not a variable and that there exists X' which is a provable formula of Si' and is thus also not a variable, such that TX = !/TX'. Hence, by lemma 8*, which generalizes to the new context, X = !/X' and therefore X is provable in L; Hence, for i = 1, 2, if X E Si then X is provable in L, if and only if TX . =:> ai is provable in Li.z. Thus the decision problems of L, , L z are primitive
36
RONALD HARROP
recursive in that of Li.z' The proof that the decision problem of Li.z is primitive recursive in those of Lj , L z can be obtained, using Theorem 6, by trivial modification of the corresponding part of the proof of Theorem 2*. Using the notation of Theorem 7 we can obtain the following two corollaries the first of which is trivial and the second of which can be obtained by noting that the decision problem of Li.z is primitive recursive in the decision problems of L t, L z each of which is primitive recursive in the decision problem ofL;.t. r ) Corollary 1. Ltz is decidable if and only ifL 1 andL z are both decidable. Corollary 2. The decision problem of Li.z is primitively recursively equivalent to that of L;.l'
Suppose L 3 is a calculus of type L and that t is an integer greater than zero. Consider the calculi Lj and Lj(t). It seems likely that a proof of the primitive recursive equivalence of the decision problems for these calculi could be obtained by constructing explicitly a condition (I) for Lj
o;
o;
Corollary 3. The decision problems of L;
A RELA TIVIZA TION PROCEDURE FOR PROPOSITIONAL CALCULI
37
4. Iteration of the relativization process - applications
In the following applications of iterated relativization, we construct sequences of calculi with specified decision properties. The motivation for the examples, and to some extent for the relativization theory, has been the fact that until comparatively recently most of the decidability and undecidability results for propositional calculi have been related to individual calculi, possibly of philosophical interest, and there seemed to be a shortage of general mathematical results which covered in one theorem or construction infinitely many propositional calculi. The applications of iteration to be given, like the application of relativization to prove Post's theorem in [2], are probably not the most general or the most interesting consequences of the processes described in [2] and in the present paper. It is thought though that they, and the methods used in obtaining them, might prove useful as a basis for later more advanced work. Each of the calculi in the sequences constructed in applications (i)(iii) below will be a subsystem of positive implicational calculus, the set of formulae of which we will again denote by S. Further, each of the calculi will have its axioms specified through a finite number of axiom schemes and will have detachment as its only rule. Since we wish to consider the calculi as calculi of type L, and since such a calculus cannot have a vaf as a premise or conclusion of a rule, it is necessary for us to consider detachment as written in the form At ;:)
Az
(At:::J
A3
A z)
:::J
A4
:::J
(A 3
:::J
A4 )
(26)
This form can be used since all provable formulae of the calculi with which we will be concerned will have :::J as main connective. All calculi constructed during (i)-(iii) as relativized forms of other calculi will have ;:) as their only connective and will be formed by using the simpler of the two methods of relativization, that is, the method denoted by the superscript ' in contrast to that denoted by the superscript *.
3R
RONALD HARROP
(i) Construction of an infinite strictly increasing sequence of (distinct)
decidable calculi:
Let L o denote the calculus with formulae S and with no axioms and with detachment (in form (26)) as its only rule. The sequence to be constructed will be{L{nl}, n z 1, where L{l} = L o and L{r +1l = (L{r})', all r z 1. The transformation T which arises in the determination of L{r + 1l from L lrl will automatically be defined, see (1)*, by Ta, = a.; TA i = Ai' all i; T(E ~ F) = TE ~ «TE)4 ~ (TF?) for all E, F which are either both formulae or both formula schemes of Lid. By induction on n we can show that for any n z 1, the axiom schemes of Lin) include the axiom schemes of L{m} for all m (if any) such that 1 :::; m < n. It will be noticed that the axiom schemes ofL l 2} are the schemes denoted by 2', 3' in the definition of L' in [2]. These are trivially in all the L(n), n z 2. Since the L(n} all have detachment as their only rule and VI} has no axioms, it is easy to show that the calculi form an increasing sequence and also, using induction and Theorem 5, that the rank of L{n} is n - 1, all n 2: 1. Hence, the calculi are distinct and thus strictly increasing. The calculi can be proved to be decidable by induction, using Theorem 2* and the fact that L(l} is decidable. (ii) Construction of an infinite sequence of distinct undecidable calculi:
In [1], an example is given of an undecidable calculus which is of type K and has finitely many axiom schemes and rules. By Theorem 3, there is thus an undecidable finitely axiomatizable calculus with formulae S and with detachment as its only rule. Replace ~ throughout the definition of this calculus by * and denote the resulting calculus, which will be of type L, by L 1 . Let L o be the calculus defined and denoted by L o in construction (i) above. Consider L O. 1 • By Corollary 1 of Theorem 7, this calculus is undecidable. Further, since no formula is provable in L o and all provable formulae of L 1 have * as main connective, it follows by Lemmas 3* and 1(iii)* (modified to meet the new context), that no formula of the type .~Tcp. ~ M 2 , where cp has ~ as main connective, satisfies D~.l' and therefore, by Theorem 6, that no such formula is provable in L~.l' In JTcp . ~ M 2 , as just used above, T denotes the transformation used during the construction of L~.l from L o and Lj, that is, T is defined as in (1)* using connectives 0"1' 0"2 which are ~, * respectively. Thus, although L~.l has some provable formulae, it has
A RELA TIVIZA TION PROCEDURE FOR PROPOSITION AL CALCULI
39
none of the form .'7TX :::J y2 for members X, Yof S where There denotes the transformation, used in the definition of rank of a formula, which is defined as in (1)* using just the connective (J 1 which is :::J. Hence, L~.1> which is a calculus with S as its set of formulae, has rank 1. Further, it has a finite number of axiom schemes, has detachment as its only rule, and, as stated above, is undecidable. Hence the calculi L
If the sequence L
1,
40
RONALD HARROP
(c) (T(A I =:>A 2 ) . =:> B 2 ) =:> .(T(A 1 =:>A 2 ) A T«A I =:>A 2 ) =:>(A 3 =:>A 4 ) ) . =:>B 2 ) 2 =:> (T(A 3 =:> A 4 ) . =:> B ) (d)(T(A l*A 2).=:> B 2 ) =:>.(T(A l*A 2 ) 2 =:> (T(A 3 * A 4 ) . =:> B )
(e) (AI
=:>
B2)
=:>. (A 2 =:>
B2 )
=:>
(AI
A
A
T«A l * A 2)*(A 3*A 4)).=:> B
2
)
A 2 • =:> B 2 ) .
Here, T is defined as in (1)* with s = 2 and with the connectives being =:>, * respectively. For each n ~ 5, L, will have detachment as its only rule. By Theorem 6 and Lemmas 3* and l(iii)*, the only formulae of Lm n ~ 5, which have rank greater than 1, that is the only ones of the form YT(o/l =:> 0/2)' =:> M 2 (or equivalently .?T(X 1 =:> X 2 ) . =:> M 2 ) ) , where T is temporarily restricted to act only on S so that the rank of a formula will be defined, are those which satisfy condition (et~.3)n of D~.3.1) These are exactly the formulae of the form .?TX. =:> M 2 where X is a provable formula of L, _ i - Hence, by Theorem 1*, the provable formulae of Ln , n ~ 5, of rank greater than 1 are exactly the provable formulae of L~ - 1 of rank greater than 1. If we can show that L 4, which is L~(l) has rank 1, we can deduce at once, by induction and Theorem 5, that L, will have rank n - 3 all n ~ 5, and therefore that the calculi of our sequence are distinct. Let L o now be used to denote the calculus with S as its set of formulae and with no axiom schemes and no rules. Then, compare the proof of Corollary 3 of Theorem 7, L~.2 will be the same calculus as L~(l). Since, by Theorem 6 and Lemmas 3* and 1(iii)*, there are no provable formulae of L~.2 of rank greater than 1, while, on the other hand, L~.2' which has =:> as its only connective, does have some provable formulae, it follows that L~.2 has rank 1. Hence the calculi L m n ~ 5 are distinct. The calculi Ln , n ~ 5 can be seen to be undecidable by induction using Corollary 1 of Theorem 7, the definition of L, and the fact that L 3 which is equivalent to L 2 is undecidable (see the definition of L~.l (=L 2 ) in construction (ii)). Hence, to complete the proof that our construction is satisfactory, it is now sufficient to show that the calculi L m n ~ 5, form an increasing sequence. We first show that the axiom schemes of L 6 include those of Lj. Since schemes (b), (c), (d) and (e) are automatically 0"1,0"2
1) (ex' n.3)n denotes the condition related to L n , L. in the way in which (C/1.2)1 is related to L 1 , L. in the definition of L1.2'
A RELATIVIZATION PROCEDURE FOR PROPOSITIONAL CALCULI
41
in L 6 all we need to prove is that the axiom schemes of type (a) in L, are axiom schemes of L 6 • Since L, = L~.3' the schemes involved are the axiom schemes of type I' (see the definition of L' in [2]) of L~, that is of (L~(1))'. Now the axiom schemes (b), (d), (e) of L, are the axiom schemes of L~(l) and these will provide the required axiom schemes of (L~(1»)' as axiom schemes of L S . 3 , that is, of L 6 , of type (a). Hence the axiom schemes of L 6 include those of L s. Suppose now that, for some r ~ 5, the axiom schemes of L, are included among those of L, + r- The axiom schemes of L, + 2 consist of (b), (c), (d), (e) and the schemes of type (a) which arise from the axiom schemes of L, + r- Thus by our hypothesis, the axiom schemes of L, + 2 include (b), (c), (d), (e) and the schemes of type (a) which arise from the axiom schemes of L" that is, they include the axiom schemes of L, + i - Hence, by induction, the axiom schemes of L, + 1 include those of L, for all n ~ 5. Since the calculi L m n ~ 5, all have detachment (in form (26)) as their only rule, they form an increasing sequence. This completes the proof that the constructed calculi satisfy the required conditions.
References [I] R. Harrop, On the Existence of Finite Models and Decision Procedures for
Propositional Calculi. Proc. Camb. Phil. Soc. 54 (1958) 1-13.
r2]
R. Harrop, A Re1ativization Procedure for Propositional Calculi with an Application to a Generalized Form of Post's Theorem, Proc. London Math. Soc. 14 (1964) 595-617.
A METHOD FOR PRODUCING REDUCTION TYPES
IN THE RESTRICTED LOWER PREDICATE CALCULUS H. HERMES AND D. RODDING') University of Munster (Westf.), Germany
1. A class T of formulae with unsolvable decision problem for validity
The method is based on an appropriate description of a Semi-Thuesystem in the language of the restricted lower predicate calculus. No effort has been made to optimize the resulting reduction type. We hope that our method will yield better results in the future. We start with the well-known notion of a Semi-Thue-system S. The words W (including the empty word 0) are built up from the letters aI' " ., aN' The relations W =;Os W' (W immediately produces W') and W ~s W' (W produces W') are introduced in the usual way by the use of a finite system (L k , R k ) (k = 1, ... , M) of rules. The task of the wordproblem for S is to find a method to decide for arbitrary words W, W', whether or not W ~s W'. This problem is known to be unsolvable for many S. Now let us assume that we have a method to associate effectively to each triplet S, W, W' (W, W' are words in the alphabet of the SemiThue-system S) a formula (f.s; w, w' of the restricted lower predicate calculus such that (*)
W
~s
W' iff (f.s; w, w' is (universally) valid.
Let us further assume first that the word-problem for S is unsolvable and second that for all W, W' in the alphabet of S the resulting formula (f.s;w,w' belongs to a class T of formulae. Then it is obvious that the decision problem for validity is unsolvable for T. 1) Communicated by H. Hermes.
A METHOD FOR PRODUCING REDUCTION TYPES
43
2. A reduction type T for validity
The completeness of the restricted lower predicate calculus can be expressed in the following way: There exists a Turing machine M and an initial complete configuration Co of M, and a function C which associates effectively with every formula o: a complete configuration C(ct) of M, such that ('1'*)
Co gives rise to C(ct) iff o: is valid.
Every Turing-machine M can be described in terms of a suitably chosen Semi-Thue-system SM' To every complete configuration C there corresponds a word W(C). If C1 and Cz are arbitrary complete configurations, we have (***)
C 1 gives rise to
c,
iff W(C 1) -+ SMW(CZ)'
(*), (**) and (***) imply that
ct is valid iff cts M ; W(Co), W(C(~)) is valid. Here the formula ctSM; W(Co), W(C(~)) can be effectively calcu ited from ct. The last result shows that T is a reduction type for validity. 3. Construction of the formula ct s ;w, w'
Let S be a Semi-Thue-system over the alphabet at> ... , aN' Let the rules of S be (L k , R k ) (k = 1, ... , M), where L k , R k are words in the alphabet of S. One can assume that none of the L k , R k are empty and that every word LkR k consists of at least three letters (because all SM can be chosen in this manner). The formula ct s; w, w' contains at most the following N + 2 predicate variables: Singulary predicate variables E j (j = 1, ... , N), Binary predicate variables A, T. We shall now give the standard interpretation of a closed formula built up from these predicate variables: The individual domain is the class of all words in the alphabet of S (including the empty word D). The interpretation of E j is the property of ending with a j' of A is the binary relation which holds between two words W, W' iff W' == Waj for a suitable aj
44
H. HERMES AND D. RODDING
of T is the binary relation which holds between two words W, W' iff W~SW'.
We define Cjxy by (1)
Cjxy
~
Axy
A
Ejy
(j
= 1, ... , N),
Furthermore we give inductive definitions for Pwx (for every W) and for Cwxy (for non-empty W) by (2)
(3)
PoX ~ , Elx
A ... A'
ENx,
PWaox ~ V(PwZ ACjZX) } z
(j = 1, ... , N),
Cajxy
(j
~
= 1, (j = 1,
Cjxy
Cwaoxy ~ V(CwXZ } z
A
Cjzy)
, N), , N).
The standard interpretation respectively associates with P w the property of being identical with W,
Cw the binary relation which holds between two words
W', WI! iff W" == W' W.
Now let
IXo
be the conjunction of the following formulae (i, j = 1, ... , N), (k = 1, ... , M), where
IX4 ij ' IXs , IX6' IX7' IX8k IXl
~ VPox x
IXs ~
AAA(Txy A Tyz
Gt6 ~
AA(Pox
x y z
Gt 7 j ~
x y
A
Poy
AAAA(Txy x y z w
A
~
~
Txz)
Txy)
CjXZ
A
Cjyw
~
Tzw)
IXl, IX1j' IX3j,
A METHOD FOR PRODUCING REDUCTION TYPES
45
It is easily seen that IlC O is valid under the standard interpretation. Now we are in a position to give the following definition:
(0)
IlC S-
w
.,
W'
~ AA(IlC OA x y
Pwx
A
Pw'y ~ Txy),
4. Proof of (*)
We have to show (a) If IlCs;w,w' is valid, then W ~s W', (b) If W ---"s W', then IlCs;w,w' is valid. For (b) see section 5. (a) can easily be shown as follows: Let cxs; w. W' be valid. Then it holds for the standard interpretation. Hence the formula IlCO A Pwx A Pw'y ~ Txy holds when arbitrary words are associated with x and y. If we associate W with x and W' with y, the formulae Pwx and Pw,y are valid. Since llC o is also valid, we find that Txy holds. This means that W ~s W'. 5. Proof of (b) (cf. section 4)
We start with the following lemmata, where the premise llC o has been suppressed. LEMMA
1:
vr;«
LEMMA
2:
AVCwxy
LEMMA
3:
AAA(PwXA PwY
LEMMA
4:
AA(Pyx ---" (Cwxy
LEMMA
5:
AA(Pwx
LEMMA
6:
AAAA(Txy A CwXW
x
xy
x y z
xy
xy
xywz
A
A
(W =1=
D)
(W =1=
D).
CjXZ ---" Cjyz)
~
PywY))
PwY ~ Txy) A
Cwyz ---" Twz)
(W =1=
D).
We omit the proofs which are in all cases straight-forward by induction on W. To prove (b) we show by induction on n, that, if we suppose that
46
H. HERMES AND D. RODDING
=>S W 2 =>S ... =>s Wn , then Txy is deducible from lXo, P W1x, Pwny· For n = I see Lemma 5. Let us assume that the proposition is proved for n (induction hypothesis). Now we have to show that if we suppose that W1 =>s W2 =>s ... Wn =>s Wn + 1 ... , then Txz is deducible from lXo, P W,x, P Wn + 12. We shall show in section 6 that Tyz is deducible from lXo, PwnY, P Wn + lZ, provided that Wn =>s Wn + r- Using the induction hypothesis together with IXs we get TX2 from IXO, P W,x, PwnY, P Wn + 1Z, provided that W1 =>s ... =>s Wn + l' Now it is obvious from Lemma 1 that we can eliminate pwny.
W1
6. Deducibility of Tyz Wn + 1 means that there exist words U and Vand a k such that Wn == U L k V and Wn + 1 == U R; v: According to Lemma 1 there exists a U such that Puu. According to Lemma 2 there exist an I and an r such Wn
=>s
that CLkul and CRkur (we should keep in mind that L k i= 0 and R k i= D). With IXSk we get Til'. Lemma 4 yields PUL) and PURkr. Case 1: V == D. Then Wn == UL k , Wn + 1 == UR k , P w ) , P w " + 11'· Lemma 5 yields Tyl and Trz. This, together with Til' and (xs, Implies Tyz. Case 2: V i= D. Lemma 4 gives Cyly from PUL) and pw"y, and Ccrz from PURkr and p w " + lZ. From Til', Csl y, Ccrz we get Tyz using Lemma 6. 7. Application to the decision problem 1Xl> .•• , (X7j are given in prenex normal form (Po and C, are quantifierfree). Their conjunction has a prenex form of prefix YAyNA 3 . CLk and CRk are contained in IXSk' Let Ilk + 2 be the length of the word LkR k• Ilk ~ 1, accordmg to our assumptions in section 3. Then IXSk can be brought into prenex normal form with prefix Ailk + 3. Let Jl be the greatest of all the Ilk' Then the conjunction of all (XSk can be brought into prenex normal form with prefix All + 3. Hence lXo has a prenex normal form with prefix YAyNAIl+2. Pwx A Pw.y has a prenex normal form with a purely existential prefix. No finite upper bound exists
A METHOD FOR PRODUCING REDUCTION TYPES
47
for the length of this prefix. We denote this type of prefix by the symbol YOCJ . C(O /I Pwx /I Pw,y has a normal form with prefix YOCJAyNI\I<+2. The resulting prefix of a normal form of C(s;w,w' is AOCJYANyl' + 2. The resulting reduction type for satisfiability is hence
where (N, 2) denotes a kernel with N singulary and 2 binary predicates. Using a well-known procedure of Suranyi we can simplify this type to
Chosing appropriate descriptions SM of the Turing machine M (cf. section 2) it is possible to get N = 2 (with a moderate 11) or 11 = I (with a moderate N).
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC JAAKKO HINTIKKA University of Helsinki, Finland
1. The purpose of this paper
The distributive normal forms of first-order logic (functional calculus of first order, quantification theory, predicate calculus) were first described and proved to exist in the author's dissertation of 1953.1 ) Subsequently, some of the basic ideas of these normal forms have been independently rediscovered and used by others (Hanf, Ehrenfeucht, Dana Scott. 2) In this paper, an attempt will be made to explain, in a form more compact and more manageable than that of my original paper, what these normal forms are. We shall also attempt to carry their theory further by studying some of their most important properties. In particular, we shall describe and prove semantically complete a disproof procedure which is connected especially closely with their structure. The properties of the distributive normal forms would seem to repay closer study in several directions. In this paper, however, we are trying to survey the basic features of these normal forms rather than to push their applications to the limit in any particular direction. This paper is restricted to first-order logic, although similar normal forms are easily seen to exist elsewhere, e.g. in higher-order logics and in modal logics. The reason for the restriction is that the properties of these 1) Jaakko Hintikka, Distributive Normal Forms in the Calculus of Predicates. Acta Philosophica Fennica, 6 (1953) (Helsinki, 1953). The notation and terminology used in this work is not identical with that employed in the present paper. 2) See e.g, A. Ehrenfeucht, An Application of Games to the Completeness Problem for Formalized Theories. Fundamenta Mathematicae, 49 (1961) 129-141. Distributive normal forms have also been employed by Francis C. Oglesby in his monograph, An Examination of a Decision Procedure. Memoirs of the American Math. Soc. 44 (1963).
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
49
parallel normal forms are so different as to make a separate treatment advisable. 2. Special cases The distributive normal forms of first-order logic are generalizations of the well-known "complete" normal forms of propositional logic and of monadic first-order logic. The notation and terminology that will be employed in this paper can be conveniently explained in terms of these more restricted normal forms. Consider first propositional logic. Each consistent formula F of propositional logic has a complete disjunctive normal form which is a disjunction of certain conjunctions that will be called constituents. If F does not contain any atomic formulae different from Pl' P2' ... , Pk> then a 'Constituent occurring in its normal form contains for each i = 1, 2, ... , k either Pi or ,...., Pi (but not both) as a member. An arbitrary conjunction of this kind will be referred to in the sequel as k
II Pi'
i = 1
Different conjunctions of this kind may be distinguished from each other by attaching subscripts to II. It will be assumed that these subscripts run from one on so that the same kind of notation can be used repeatedly. By means of this notation the normal forms of monadic first-order logic (without identity) are easily characterized. They are again disjunctions of certain conjunctions that will be called constituents. A constituent of course depends on the predicates which occur in it. If these are P 1X, P 2X, ••• , Pkx, an arbitrary constituent of monadic first-order logic is of the form j = 2k i =k II (Ex) IIj PiX. (1) j=l
i=l
Whenever the limits of our pi-operations are inessential or can be gathered from the context, they may be omitted. Thus instead of (1) we may sometimes write II (Ex) IIj PiX
j=l
i=l
50
JAAKKO HINTIKKA
or even
n
j = 1
(Ex)
n, Px.
The intuitive meaning of (I) is worth noting as it is generalizable to the whole of first-order logic. Using the language of an interpreted logical system, one may say that in propositional logic the constituents list all the different possible states of affairs or "possible worlds" that can be specified by means of the atomic propositions PI (plus propositional connectives). In monadic first-order logic, the constituents (I) likewise describe all the different kinds of worlds (states of affairs) that can be specified by means of the monadic predicates PiX (plus quantifiers and propositional connectives). From (1) we see how these descriptions come about. First all the possible kinds of individuals that can be specified by means of the predicates PiX (plus propositional connectives) are listed. This is what the conjunctions
(where j = 1,2, ... , 2k ) accomplish. Then it is indicated, for each such possible kind of individual, whether individuals of that kind exist or not. This is what the rest of (I) does. On the basis of this intuitive meaning of (l) certain simple observations can be made. For instance, we see that (I) can be written in a different form. Instead of listing all the different kinds of individuals that there exist and also all the different of individuals that do not exist, it clearly suffices to list all the existing ones and then to add that they are all the existing ones. In other words, (I) can be rewritten as follows: (2)
(Ex)nilPx &
(Ex)nhPx & ... &
& (Ux)(nj,px v nhPx v '"
(Ex)njmPx
v njmPx)
where {nil Px, ... , n j m Px} is the set of those conjunctions whose existential closures occur unnegated in (I). Here it is useful to have a shorthand notation which will save us the trouble of writing out the main conjunction and disjunction of (2) explicitly. Using such a notation, (2) may be written (3)
ti; j=l
(Ex)
n,
i=1
PiX & (Ux) a,
n,
j=1i=1
PiX.
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
51
The conventions on which this more compact version is based may be expressed as follows: Given the conjunction IT, Pi' tt,
i = 1
Pi
i
=
1
is the conjunction of all its un negated members, and (J,
i = 1
Pi
is the disjunction of the same formulae. More generally, given two arbitrary functions f and g whose arguments and values are formulae,
n, f(p)
i = 1
is the conjunction of all the formulae f(pJ where Pi ranges over all the un negated members of IT, Pi; and
i = 1
is the disjunction of all the formulae g(p) with the same choice of the arguments Pi. Thus tt, and (J, essentially express the formation of arbitrary conjunctions and disjunctions. The identity of subscripts merely serves to indicate that the same selection of arguments is involved in the two cases. The equivalence of (I) to (3), which was found intuitively obvious, is also readily demonstrable. In order to convert (I) to (3), you may proceed as follows: First replace every combination of symbols ~ (Ex) by (Ux) ~ ; then let all the universal quantifiers thus introduced merge into one (in virtue of the distributivity of universal quantification with respect to conjunction). From propositional logic it follows that the formula which then constitutes the scope of the universal quantifier is equivalent to (J, I1 j PiX j
=1 i =1
where the subscript of the first IT in (1) is assumed to be r. Another observation: In propositional logic, a formula has a (nonempty) normal form if and only if it is consistent. In monadic first-order logic, this remains true, but only with a qualification. This qualification
52
JAAKKO HINTIKKA
pertains to the "only if" part of the equivalence. This part remains valid only if all the constituents (1) are consistent (satisfiable). Now there is one constituent (and one only) which is satisfiable only in an empty domain of individuals. It is that constituent (1) all of whose members are negated. On the basis of the intuitive meaning of (1) it is seen to deny that there are individuals of any kind in existence, i.e. that the universe is empty. Admitting empty domains of individuals on a par with nonempty ones thus simplifies our discussion in that it enables us to say that all constituents of monadic first-order logic are consistent. This simplifying assumption will be made throughout this paper. 3. The definition of distributive normal forms The distributive normal forms of full first-order logic (without identity) are also disjunctions of certain conjunctions which we shall call constituents. In order to define the normal form we therefore have to define a constituent. A constituent depends essentially on the following features which will be called its parameters: (P.1)
The set of all the predicates occurring in it;
(P.2)
The set of all the free individual symbols occurring in it;
(P. 3) The maximal length of sequences of nested quantifiers occurring in it. The parameter (P. 3) will be called the depth of the formula in question. More loosely, the depth of a formula is the number oflayers of quantifiers it contains. If it is stipulated that the scopes of two quantifiers which contain the same bound variable must not overlap, we can characterize (P . 3) in analogy to (P. 1)-(P . 2): The depth of a formula is the number of all the different bound individual variables it contains when this number is made as small as possible by renaming bound variables. The definition of a constituent is facilitated by a few shorthand notations. Given a set (P .1) of predicates, let Ala!> a2' ... , ak) (where i = 1, 2, ... ) be all the atomic formulae that can be formed from the members of (P.l) and from the free individual symbols ai' a2' ... , ai. Furthermore, let Bla!> a 2, ... , ak) (i = 1,2, ... ) be all those atomic formulae so defined which contain at least one occurrence of ak' Then we have, trivially,
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
(4)
II, A;(a1' ... , a k -
i = 1
=
IIs i = 1
A;(a 1, ... , ak - 1) &
l'
53
ak) =
IIt B;(a 1, ... , ak - 1, ak)
i = 1
for each given r and for suitably chosen (with a view to r) sand t, Before defining a constituent with given parameters (P .l)-(P. 3), it is advisable to define a closely related kind of formula which will be called an attributive constituent (in short, an a-constituent) with the same parameters. An attributive constituent with a given fixed parameter (P .1) whose depth is d and whose parameter (P.2) is {a 1, a z, ... , ak} will be referred to as cri«; ... , ak ) . It may be defined recursively in terms of a-constituents of depth d-l as follows: (5)
Ct~(a1' ... ,ak) =
=
IIs B;(a1' ... ,ak) &
i = 1
IIt i = 1
(Ex)Ctt- 1(a 1, ... ,ak-1,ako x).
We have assumed here that indices are used to distinguish different a-constituents with the same parameters from one another. In (5) the first index r is of course a function of sand t. It does not matter what the dependence is as long as the values r can assume are 1, 2, . .. up to some finite number. 1 ) When d = 0, the second member of the right-hand side of (5) vanishes, giving us a basis for recursion. After an attributive constituent is defined, it is easy to define a constituent: (6) Cd(a 1, ... , ak) = II A;(a 1, ... , ak _ 1) & i = 1
Ct d(a 1,···, ak - l' ak)·
For simplicity, the indices of c-, of II, and of Ct d have not been indicated here. The first depends on the other two; it is again assumed that the values it can take run consecutively from Ion. What is thus defined will be called the first distributive normal form. The constituents and attributive constituents just defined will be said to be of the first kind. 1) One especially simple method of defining this dependence would be to put
r = (s-I)+2 n . (1-1)+ 1 where n is the number of different formulae of the form
Bi(ah ... , ak) (with the appropriate parameters). Analogous definitions may be used
elsewhere on similar occasions. We shall not use the details of these definitions, however.
54
JAAKKO HINTIKKA
In monadic first-order logic, every constituent (1) may be rewritten as (3). In the same way, constituents and attributive constituents of the first kind may be transformed so as to become constituents and a-constituents of the second kind. Attributive constituents of the second kind may be defined as follows: (7) i
Ct~(al' ... ,ak) n, (Ex)Ct1=
I(a
1
l, ... ,ak'x)
= II. 8;(a l , i = 1
&
...
&
,ak)
(Ux) a, Ct1-
I(a
i = 1
l,···,a k,x).
Here r depends on sand t as before. The first parameter (P .1) is assumed to be constant throughout (7). Constituents of the second kind may be defined in terms of a-constituents of the second kind in the same way as before, viz. by (6). Alternatively, this definition may be rewritten as follows:
cs«; ... ,ak) =
(8)
i
n, (Ex)Ct1=
1
I(a
l,···,ak,x)
IIs i = 1
Aj(a l, ... ,ak) &
&
(Ux) a, Ct1j
I(a
= 1
l,···,ak>x).
4. The intuitive meaning of constituents and of attributive constituents The normal forms of monadic first-order logic are special cases of the more general normal forms just defined. A comparison of (1) with (5) and of (3) with (7) or (8) shows that these special cases are not entirely unrepresentative. The similarity between (I) and (5) or between (3) and (8) can be further heightened by writing (1) as
(1)*
II (Ex) Ct?(x) j
and (3) as (3)*
= 1
tt, (Ex) Ct?(x)
; = 1
&
(Ux)
(J,
i = 1
Ct?(x).
This similarity helps us to appreciate the nature of the general normal forms in many respects. For one thing, the intuitive meaning of constituents and of attributive constituents can be explained pretty much in the same way as in the monadic case. If we speak as if we were dealing with an interpreted system, we may say that constituents with certain
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
55
fixed parameters (P. l)-(P. 3) list all the different kinds of world that can be described by the sole means of these parameters (plus quantifiers and propositional connectives). Attributive constituents do not describe possible worlds (states of affairs) but rather possible kinds of individuals. (Hence their name.) If the attributive constituents (5) or (7) are considered as complex attributes of the individual referred to by ai, they may be said to list all the different kinds of individuals that can be specified by the sole means of (i) a given fixed set of predicates (P. 1); (ii) the "reference-point" individuals specified by a I' ... , ak _ I; (iii) at most d layers of quantifiers; (iv) propositional connectives. The recursion equation (5) shows how the list comes about: First we list all the kinds of individuals that can be specified by means of (i) the same set of predicates; (ii)' the reference-point individuals specified by a I ' . . . , ak _ I as well as ak; (iii)' at most d - I layers of quantifiers; (iv) propositional connectives. This is what the a-constituents Ctd-I(al, ... , a k_ l, ak' x) do. Then we specify, for each such kind of individuals, whether individuals of that particular kind exist or not. This is what the second pi-operator and the accompanying quantifier accomplish in (5). This adds one more layer of quantifiers, but it also makes the result something we can attribute to the referent of ak' Finally, we specify how the referent of ak is related to those of a 1, ... , ak_l; this is what the first member of the right-hand side of (5) does. Intuitively, (7) is obviously related to (5) in the same way as (3) is to (1). The equivalence of (7) to (5) (the same parameters in both cases) can also be proved by an argument closely reminiscent of the proof which was sketched in section 2 for the equivalence of (3) to (I). The former equivalence is most easily proved by induction on d. Both the case d = I (which gives us a basis for induction) and the inductive step can be dealt with in the same way as the relation of (3) to (1) was dealt with earlier. From (8) we see what a constituent with given parameters (P. 1), (P.2) (= {ai' ... , ak}), and (P. 3) (= d) says. First, it says which possible kinds of individuals, specifiable with reference to ai' ... , ak by means of at most d-l layers of quantifiers, there exist; second, it says how the referents of a l' . . . , ak are related to each other.
56
JAAKKO HINTIKKA
5. The existence of distributive normal forms
The possibility of converting every formula F (of first-order logic without identity) with given parameters (P. l)-(P. 3) into the first distributive normal form with the same parameters (or with certain fixed larger ones) can be proved by induction on the depth d of F. A basis for induction is given us by the propositional normal forms (d = 0). In the general case F is a truth-function of formulae of the following two forms: (a) A;(al' ... , ak), where {al> ... , ak} is the parameter (P .2) of F; (b) (Ex)G, where all the predicates of G are among those of F, where the depth of G is at most d-l, and where all the free individual symbols of G are among aI' ... , ak, x. By the inductive hypothesis G is therefore equivalent to a disjunction of formulae of the form i
IT A;(al' ... , ak) & ee
I
Ct d - l(a l, ... , ak, x).
By well-known laws of first-order logic (the distributivity of existential quantification with respect to disjunction, the irrelevance of the scope of an existential quantifier as far as members of a conjunction which do not contain the bound variable in question are concerned), (Ex)G is then equivalent to a disjunction of formulae of the form i
IT A;(a l, ... , ak) & ~
I
(Ex) Ct d - l(al' ... , ak, x).
But this means that F is equivalent to a truth-function of formulae of the following two kinds: (a) (b)'
A;(a l, ... , ak), as before;
(Ex) Ctd-l(al, ... , ak> x).
The desired normal form is then obtained simply by a transformation to the propositional normal form. This completes the proof of the existence of the first distributive normal form. From the argument just given we can read a set of directions for actually converting each given formula to the first distributive normal form. The term distributive normal form is chosen because of the importance of the distribution of existential quantifiers in this conversion. It is seen that the process by which the distributive normal forms are reached is in a sense opposite to the one by which a formula is converted to the prenex
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
57
form: Instead of pulling the quantifiers out to the beginning of the formula in question they are pushed as deep into it as they will go. The possibility of converting each given formula to the second distributive normal form now follows from the equivalence of the first and the second normal form. In the sequel, we shall consider only the second normal form, usually omitting the word "second". Likewise, only constituents and a-constituents of the second kind will normally be considered unless the opposite is stated in so many words. There also exist normal forms and constituents dual to the ones we have defined. They will not be considered in this paper either. As a special case of the convertibility of every formula into the distributive normal form, we see that each constituent with depth d and with certain given parameters (P. I )-(P . 2) can be converted into a disjunction of constituents with the same parameters (P. I )-(P . 2) but with a greater depth d-i-e, for every e = I, 2, .... These constituents will be said to be subordinate to the constituent in whose normal form they occur. We see that every a-constituent Ct d of depth d may be converted into a disjunction of a number of a-constituents of depth d-i-e with the same parameters (P .l)-(P .2) as These a-constituents are said to be subordinate to Ct". Again, this holds for every e = 1,2, .... The procedure by means of which this is accomplished is, mutatis mutandis, the same as in the case of constituents.
cr.
6. The structure of distributive normal forms
The structure of the distributive normal forms of both kinds is very clear-cut. In the constituents of the first kind there are no disjunctions and no universal quantifiers. All negation-signs are prefixed to atomic formulae or to existential quantifiers. In the constituents and a-constituents of the second kind all negationsigns are prefixed to atomic formulae. Since disjunction, conjunction, and both kinds of quantification are monotonic operations as far as the logical strength of formulae is concerned, it follows that the same is true of constituents and attributive constituents (of the second kind): Whenever a subformula Sl of an arbitrary a-constituent Ct., implies another formula Sz, Ct., implies the result of replacing Sl by Sz in Ct o. The same obviously holds for the simultaneous replacement of non-
58
JAAKKO HINTIKKA
overlapping subformulae by weaker formulae. From this the following omission lemma immediately follows: If we. omit from an a-constituent any number of subformulae of the following kinds: (i) negated or unnegated atomic formulae; (ii) quantified formulae, then the result is implied by the original a-constituent. The same result obviously holds also for constituents (of the second kind). Certain qualifications are needed in both cases, however. It has to be understood that not all the members of any conjunction which occurs as a memberofa disjunction are omitted, i.e. that no member of a disjunction is allowed to disappear altogether. It has to be understood also that connectives which become idle as a result of the omissions are likewise omitted. It is readily seen that whenever an a-constituent occurs in another a-constituent or in a constituent (of the second kind), it also occurs there without being a subformula of any disjunction, universally quantified formula, or (trivially) negation. Now the operations of conjunction and of existential quantification cannot remove an inconsistency from a formula. It therefore follows that whenever an a-constituent is inconsistent, every constituent or a-constituent that contains it is likewise inconsistent. This observation will be called the inconsistency lemma. A formula which is like one of the constituents or a-constituents we have defined except for (i) the order of the conjunctions and disjunctions it contains, (ii) the repetitions of some of the members of conjunctions and disjunctions, or (iii) the naming of the bound individual variables it contains will be called its notational variant. We shall call two constituents or a-constituents different only if they are not notational variants of each other. If this convention is presupposed, we can say that in propositional logic two different constituents with the same atomic formulae are logically incompatible. As a special case, we see that two different constituents or a-constituents of the first kind with the same parameters are logically incompatible. From the equivalence of our two normal forms it follows that the same is true of constituents and of a-constituents of the second kind. (Incompatibility lemma.) Each a-constituent and constituent of the second kind has the structure of a tree in the mathematical sense of the word. The elements of the tree are the a-constituent in question -let us call it Ct~ - and all the a-con-
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
59
stituents of lesser depth occurring in Ctg. If one of them is Ct 4 - e, the elements covering it are all the a-constituents of depth d - e - 1 occurring in it. Turning the tree upside down, the structure of the a-constituent Ct~ can therefore be represented schematically by the following diagram:
The two dividing lines (solid and dotted) will be given a meaning later.
7. The effects of identity So far, we have been dealing with first-order logic without identity. In first-order logic with identity, normal forms similar to the ones we have discussed can be defined. In fact, an attributive constituent may still have exactly the same appearance as it had in first-order logic without identity. The only difference is that quantifiers must now be given an exclusive interpretation. I) That is to say, a formula of the form (Ex)F will now be understood as shorthand for the formula (Ex) (x :F a l & x :F a2 & ... x :F a k & F) where ai' a2' ... , a k are all the free individual symbols of F and where the quantifier is understood in the usual way. The universal quantifier has of course to be reinterpreted in the same way: (Ux)F will now mean the same as the formula (Ux) ((x :F al & x :F a 2 & ... & x :F ak) ::::l F) used to mean on the old interpretation. In (5) and 1) See my paper on Identity, Variables, and Impredicative Definitions, Journal of Symbolic Logic 21 (1956) 225-245. The difference between the two variants of exclusive interpretation which I distinguish in this earlier paper is immaterial in the case of distributive normal forms. For this reason, I shall in the sequel occasionally resort to the solecism of speaking of the exclusive interpretation.
60
JAAKKO HINTIKKA
(7) this reinterpretation has to be applied to all quantifiers, but in all
other respects these definitions will remain intact. A constituent will still be like (6) except that it will always contain as additional members of the main conjunction all the formulae a, =F aj where i =F j, i, j ~ k. It may also contain as additional members of the main conjunction a number of formulae of the form b = a., one for each b, where b is different from all the free individual symbols a, (i = 1,2, ... , k). It is obvious that formulae of the latter form cannot affect the consistency or inconsistency of the constituent in question. Hence they may be disregarded for many purposes (cr. section 8). Any formula (with identity) whose depth is no greater than d and whose free individual symbols are all among the a;'s and b's can be converted into a disjunction of the constituents just described. The argument that was given in section 5 for the corresponding result for formulae without identities can be extended to the case at hand. In fact, the operations by means of which the normal form was reached remain applicable on the exclusive interpretation of quantifiers. Hence essentially the only new thing to do is to transform the given formula first to a form in which the interpretation of quantifiers can be changed. There are no difficulties about this, however. 1) The three lemmata formulated above (the omission lemma, the inconsistency lemma, and the incompatibility lemma) remain valid in firstorder logic with identity. Some qualifications are needed here, however. In the omission lemma the prohibition against omitting members of disjunctions has to be applied also to the disjunction (Ux) (x = al V x = = a 2 v ... v x = ak v F) which is implicit in the universally quantified formula (Ux)F, exclusivelyinterpreted. The incompatibility lemma applies to a-constituents without change, but not to constituents. It does apply to what we shall call the main parts of constituents, i.e. to the constituents just defined minus the identities b = a.. Thus it applies e.g. to closed constituents. 1) Suppose, for instance, that a is the only free individual symbol in (Ex)F. Then this formula is trivially equivalent to a formula of the form (Ex) (x = a & F ,) v (Ex) (x a & F 2) which in turn is equivalent to F,(a/x) v (Ex) (x a & F 2) where F,(a/x) is the result of replacing x by a in F" subject to the usual precautions concerning variables. By using the exclusive interpretation, this may be written simply F,(a/x) v (Ex)FaThe cases in which we have more than one free individual symbol may be treated similarly, and universally quantified formulae dually.
*
*
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
61
8. Distributive normal forms and decision problems In propositional logic and in monadic first-order logic distributive normal forms yield a decision method: If a formula has a non-empty normal form, it is satisfiable, and vice versa; it is logically true if and only if its normal form contains all the constituents with the same parameters as it. In view of Church's undecidability result they cannot do this in the full first-order logic (with or without identity). It is easily seen that this failure is possible only if some of our constituents are in this case inconsistent. In fact, the decision problem of first-order logic is seen to be equivalent to the problem of deciding which constituents are inconsistent. More explicitly, the decision problem for formulae with certain fixed parameters is equivalent to the problem of deciding which constituents with these parameters are inconsistent. For instance, the decision problem for formulae of depth ~ 2 is equivalent to the problem of locating the inconsistent constituents of depth 2. In this case, the problem is known to be solvable.') In fact, it can be shown (although we shall not do it here) that the conditions (A)-(C) which we shall formulate in section 9 give us a decision method in this case. In the case of formulae of depth 3 no decision method can be available, for it is known that these formulae constitute a reduction class. In fact, this remains true even though we impose limitations on the other parameters of these formulae; for instance, we may require that they contain only one fixed dyadic predicate plus an indefinite number of monadic predicates, and that they contain no free individual symbols.") (It is assumed here that no identities are present.) In addition to obtaining reduction classes for the decision problem of the whole first-order logic (without or with identity) we obtain formulations for a number of more specific decision problems. For instance, 1) See G. H. von Wright, On Double Quantification, Societas Scientiarum Fennica, Commentationes physico-mathematicae 16, no. 3 (Helsinki 1952). As pointed out by Wilhelm Ackermann in his review of von Wright's paper (Journal of Symbolic Logic 17 (1952) 201-203), this case can be reduced to cases which have been proved solvable Godel, Kalmar, and Schutte. A generalization of Ackermann's point was put forward by Dana Scott in a paper read at the January 1963 meeting of the Association for Symbolic Logic in Berkeley, California. 2) See Janos Suranyi, Reduktionstheorie des Entscheidungsproblems im Pradikatenkalkul der ersten Stufe (Verlag der ungarischen Akademie der Wissenschaften, Budapest 1959).
62
JAAKKO HINTIKKA
consider a finitely axiomatizable first-order theory the conjunction of whose non-logical axioms is F. Then it is easily seen that the theory in question is decidable if and only if the set of all inconsistent constituents which have the same parameters as F and which are subordinate to one of the constituents C I , C 2 , •.• , Cf occurring in the normal form of F is recursive. For a decision method it even suffices to have a recursive function rF(d) which for each d indicates how many inconsistent constituents (with the appropriate other parameters) there are subordinate to one of the Cj(i = 1, 2, ... ,f). For if we know this number, we can for each d enumerate recursively all the inconsistent formulae until this number of constituents of depth d (with the appropriate other parameters) has appeared in the enumeration; then we know that the rest are all consistent. Such a function rid) of course always exists, although it is not always recursive. It will be called the range function of F. Even when it is not recursive, it can be seen to be of the same degree of unsolvability as the theory axiomatized by F. A theory of range functions might be of considerable interest. For instance, from a recent result of Hanf's it follows that every recursively enumerable degree of unsolvability contains the range function of some formula. I ) Here we shall only list some of the most obvious properties of range functions: (i) The range function of each F is the sum of the range functions of the constituents occurring in its normal form. The range functions of constituents thus constitute a base for all range functions. (ii) In particular, the range function of a constituent Cd of depth dis for every e = I, 2, ... equal to the sum of the range functions of all the constituents of depth d-i-e which have the same parameters (P .l)-(P. 2) as Cd and which are subordinate to c-. When a constituent is split into a disjunction of deeper constituents, its range function is thus split into the sum of their range functions. (iii) The range function of any constituent subordinate to Cd is Turing reducible to that of cr. The higher up we climb along a branch of the tree which is constituted by all the constituents subordinate to the
c;
1) Hanf has proved that every recursively enumerable degree of unsolvability con-
tains the class of the Godel numbers of the theorems of some finitely axiomatizable theory. I have seen his result only in the form of an abstract. - By the degree of unsolvability of a theory we of course .mean the degree of the set of the Godel numbers of its theorems.
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
63
easier our decision problems thus become - in so far as they change at all. (iv) Given two constituents C 1 and C2 with the same parameters (P .l)-(P. 2) neither of which is subordinate to the other, the sum rc,(d)+rc,(d) of their range functions belongs to the join of the degrees to which I'c,(d) and r c2(d) belong in the semi-lattice of degrees of unsolvability. (v) A formula axiomatizes a complete theory if and only if its range function is everywhere = 1. 9. Conditions of consistency The theory of range functions will not be developed any further here. It has been mentioned mainly as an illustration of the interest of the
problem of locating the inconsistent constituents - or at least some of them. As far as the general form of this problem is concerned, we know that the set of all inconsistent constituents is not recursive (Church's undecidability theorem). It is however, recursively enumerable. One thing we may thus hope to accomplish is to find as natural methods of recursive enumeration as possible (i.e. as natural disproof procedures as possible). What we shall do in the next few sections is just to provide such methods of recursive enumeration - methods which are especially closely connected with the structure of our constituents and attributive constituents. This will be done in two stages. First, certain sufficient (but not necessary) conditions of inconsistency for constituents and a-constituents will be formulated. Then it will be proved that these conditions, when used in a certain systematic fashion, really provide us with a method of recursively enumerating all the inconsistent constituents and a-constituents. In other words, it will be proved that every inconsistent constituent and a-constituent has a disproof of a certain simple structure. This structure can be compared with the structure of the disproofs which are in the simple normal forms of first-order proofs that have been established by Herbrand and by Gentzen. Prima facie, the relation of our disproofs to the Herbrand and Gentzen normal forms is not very close. As far as the propositional structure of our disproofs is concerned, they embody the ideals of Herbrand and Gentzen well enough; no applications of modus ponens or of the cut rule are needed. As far as quantifiers are concerned, however, our procedure may seem to be diametrically opposed to that of
64
JAAKKO HINTIKKA
Herbrand: Instead of eliminating quantifiers in our disproofs, as Herbrand does, we introduce new ones all the time. There is a closely related normal form of first-order disproofs, however, which is much more obviously comparable with the Herbrand-type proofs and disproofs. It will be explained briefly in section 18. We shall not examine its exact relation to the Herbrand proofs and disproofs, however. The sufficient conditions of inconsistency (necessary conditions of consistency) which we shall formulate will be given in two different forms. The first set of conditions makes fuller use of the structure of the constituents and attributive constituents than the second. On the other hand, the second set of conditions seems even simpler than the first. Either of them serves the purpose of recursive enumeration; hence there does not seem to be much to choose between them on the level of general considerations. The first will be described and shown really to establish inconsistency in section 10, the second in section 12. 10. Conditions based on the compatibility of different partitions
In order to find suitable necessary criteria of consistency for attributive constituents, let us assume that we are given an a-constituent (7). Each a-constituent of depth d -1 occurring in (7)-say CI~ - l(a l, ... , ai, x)is again of the form (9)
II v B;(a l, ... , ak, x) & TC w (Ey) CIt - 2(al' ... , ab x, y)
i = 1
i = 1
& (Uy) lT w Ct~-2(al' ... ,ak,x,y). i = 1
Consider the simplest case k = 0 as an example. Switching over to the language of an interpreted system once again, we may say that (7) then gives us a list of all the different kinds of individuals that can be specified by means of the parameters of (7) and that are exemplified in the world in case (7) is true. In (9) all the different kinds of individuals that there are in the world are listed again; this time kinds of individuals specifiable by means of the same parameters plus the reference-point individual referred to by x. Since (9) occurs in (7), at least one such reference-point individual will exist if (7) is true. Hence the two lists have to be compatible in order for (7) to be consistent: Every individual mentioned in the "relative" list (9) has to find a place in the "absolute" list (7), and vice
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
65
versa. Moreover, the individual referred to by x will have to find a place in its own "relative" list (9). It is obvious that similar considerations apply to the general case. In every case, the partitions effected by (7) and by (9) have to be compatible in a similar way in order for (7) to be consistent. The three intuitive conditions of consistency which were just explained can be converted into explicit formal conditions, formulated in terms of the structure of the a-constituent in question. Let us consider the intuitive conditions one by one. (A) Assume that we are given (7), (9), and furthermore some a-constituent Ct~ - 2(aI' ... , a k, x, y) which occurs in (9). We shall call the following formula the bough 01'(9) determined by zeal' ... , ak> x, y):
c.; -
(10)
II v Bi(al' ... , ak, x)
i = 1
&
Ct~ - 2(a 1 ,
••• ,
ak> x, y).
The reason for the choice of the term "bough" should be obvious in view of the tree structure of a-constituents. Notice that the first conjunction of (10) is the same as that of (9). We shall apply to boughs criteria of identity similar to the ones we have been applying to constituents and attributive constituents: notational variation does not constitute a reason for calling two boughs different. Hence we may say, in the same way as in the case of a-constituents and for the same reason, that any two different boughs with the same parameters are logically incompatible. Whenever a bough (or one of its notational variants) is determined by one of the a-constituents of depth d - 2 occurring in (9) or in (7) we shall say that this bough is contained in (9) or in (7), respectively. If the roles of x and yare interchanged in (10), we obtain a formula which is again a bough with the same parameters (up to notational variation, which we are here disregarding). This bough will be called the inverse of (10). The operation of forming the inverse will be expressed by " inv" . After these preparations, we may argue as follows: By the omission lemma, (7) implies
(Ex) (Ey) (II v Bi(a 1 , i = 1
...
,ak,x)
&
Ct~-2(al' ... ,ak'x,y»,
for this formula can be obtained from (7) by omitting quantified for-
66
JAAKKO HINTIKKA
mulae and by extending the scope of the quantifier (Ey). In virtue of the permutability of the two existential quantifiers, (7) also implies (11) (Ex)(Ey)(inv(I1 v Bi(a 1, ... ,ak'x) & i = 1
Ct~-2(al' ... ,ak'x,y))).
On the other hand, (7) implies by the omission lemma the formula (12)
(Ux) (Uy) (Bg 1(x, y) v Bgz(x, y) v ... )
where the members of the disjunction are all the boughs of depth d - 2 that are contained in (7). Because of the incompatibility lemma (as applied to boughs), (11) and (12) are compatible only if the inverse of (10) is among the members of the disjunction in (12). But this means that an a-constituent of depth d is inconsistent unless it contains the inversion of each bough of depth d-2 contained in it. This condition is the formal counterpart to the intuitive requirement that each individual mentioned in the relative list (9) has to find a place in the absolute list (7). (B) Assume again that we are given (7) and (9) in (7) and that we are also given another a-constituent Ct:- 1(a b ... , ak, x) which likewise occurs in (7). Then by the omission lemma (7) implies (Ex) (Uy) (BguJx, y) v Bgu/x, y) v ... )
where the members of the disjunction are now all the boughs of depth d-2 that are contained in (9). In virtue of the well-known exchange rule for quantifiers of different kinds (7) also implies (13)
(Ux) (Ey) (inv(BguJx, y)) v inv(Bgu,{x, y)) v ... ).
On the other hand, by the omission lemma (7) also implies (14)
(Ex) (Uy) (Bgq,(x, y) v Bgq2(x, y) v ... ),
where the members of the disjunction are all the boughs of depth d - 2 that are contained in Ct:-1(al' ... , ak, x). In virtue of the incompatibility lemma, (13) and (14) are incompatible unless the two disjunctions share at least one member. But this means that of two attributive constituents of depth d-l occurring in the same consistent a-constituent of depth d one has to contain the inversion of at least one of the boughs of depth d- 2 which the other contains.
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
67
This condition is the formal counterpart to the intuitive requirement that every individual mentioned in the absolute list (7) must find a place in the relative list (9). (C) By universal instantiation the a-constituent (9) implies II v Bla 1 , •.• ,ak'x) & O"w Ctf- 2(a l ' ... ,ak>x,x), i = 1 i = 1
This formula is inconsistent unless
(15)
II v Bi(a 1 ,
i = 1
•••
,ak,x) & Ct~-2(al'" .,ak>x,x)
is consistent for at least one a-constituent ci; - 2(a 1, ... , ak' x, y) occurring in (9). In order for (15) to be consistent, it must not contain any conjunction one of whose members is the negation of another. (This is so for the same reason for which the inconsistency lemma is valid.) Whenever this is the case, we say that the corresponding bough (10) is strongly symmetric with respect to x and y. By the same token, in order for (7) to be consistent it must contain at least one bough of depth d-l which is strongly symmetric with respect to ak and x. Furthermore, by the very same token (7) must for each ai (where i = 1, 2, ... , k) contain at least one bough of depth d-l which is strongly symmetric with respect to a, and x. In general, it may be said that every consistent a-constituent ofdepth d whose outermost bound variable is x must, for every free individual symbol b occurring in it, contain at least one bough of depth d-l which is strongly symmetric with respect to x and b. This is the general, exact form of the intuitive requirement that the individual referred to by x in (9) must find a place in its own list. It is not difficult to see that in a given a-constituent (9) there can be contained at most one bough of depth d-2 (say (10» which is strongly symmetric with respect to x and y, provided that the conditions (A) and (B) of consistency are satisfied. (There cannot be more than one place, we may thus say, which the referent of x may assume in its own "relative" list.) This suffices to explain what the three conditions (A)-(C) of consistency are. When they are said to be applied to a constituent or an a-constituent, it is understood that they are applied to this constituent or a-constituent as well as to all the a-constituents of lesser depth occurring in it. If one of these a-constituents is inconsistent, then so is the given constituent or
68
JAAKKO HINTIKKA
a-constituent by the inconsistency lemma. Even in this extended sense, the question whether a given constituent or a-constituent fulfills the conditions (A)-(C) (or any single one of them) can always be decided in a finite number of steps. The three sufficient conditions of inconsistency (A)-(C) were the main content of chapter 5 of the author's dissertation. 1 ) Here they have been reformulated in a different terminology and notation, and derived from one and the same intuitive principle. 11. The effects of an exclusive interpretation of quantifiers What happens to the conditions (A)-(C) in first-order logic with identity? As was pointed out earlier, a change in the interpretation of quantifiers is the only modification which we have to make here. How, then, do the conditions (A)-(C) fare on the exclusive interpretation of quantifiers? It is easily seen that (A) carries over without any changes. The condition (B) is also seen to apply without major changes. Its applicability has to be restricted to the cases in which the a-constituents 1 (9) and (a 1 , ••• , a k , x) which were assumed to occur in (7) are really different a-constituents. Since two different a-constituents with the same parameters are logically incompatible (by the incompatibility lemma), the individuals satisfying them must be different from each other. And this may be seen to suffice to restore the condition (B). In fact, we shall assume that this restriction is built into the condition (B) itself. This does not make any difference for our purposes. It is true that in the original formulation of (B) we did not exclude the case u = q. However, the force of (B) in this special case is also obtained from (C), as you may easily verify. Hence we may exclude this case from (B), and say that it remains unchanged on the exclusive interpretation. In contrast to (A) and (B), the condition (C) is based entirely on modes of reasoning that are incompatible with the exclusive interpretation of quantifiers. It therefore becomes inapplicable in first-order logic with identity.
Ct:-
12. Omitting layers of quantifiers In this paper the conditions (A)-(C) will not be discussed as much as certain consequences of theirs. These consequences may also be derived 1) See the first footnote of this paper.
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
69
independently in a rather simple manner. They are found by asking: What happens to a constituent or a-constituent when a layer of quantifiers is omitted from it? There are two cases to be considered here: (a) the omission of the innermost layer of quantifiers and (b) the omission of the outermost layer of quantifiers. The other cases in effect reduce to these. In order to eliminate an intermediate layer of quantifiers from an a-constituent (7) - say to eliminate the e-th layer of them - it suffices to omit the outermost layer of quantifiers from every a-constituent cr ::: 1 of depth d - e + I occurring in (7). (a) What happens to (7) when all the subformulae of the form (Ex d) Cto(a 1o
or
(UX d) (J
••• ,
a., Xl' ... , Xd -
crc«; ..., ak> Xl'
l'
... , Xd -
Xd) 10
Xd)
are omitted? An answer is obtained by examining what happens to the subformulae of the form Ct 1(a 1. . . . , a., Xl' ... , Xd _ 1) of (7), i.e. by considering the special case d = 1. In this case, the result is seen to be of the form Cto(aI' ... , ai, Xl' ... , x d _ 1) with the possible exception of notational variation. Because of the way deeper a-constituents depend on shallower ones it follows that in the general case (7) becomes a formula of the form Ct" - 1(a1, ... , ak), i.e. becomes an a-constituent with the same parameters (P .1) -(P .2) but with depth d -1, with the possible and inessential exception of notational variation. From the omission lemma it follows that the resulting a-constituent is implied by (7). (b) Assume that we are given (7) and (9) in (7). A part of 0) is then 1 (Ex) (a 1 , •.• , ai, x). What happens to this part of (7) if all the atomic formulae containing x (and the quantifier (Ex» are omitted from it? The result is obviously of the form
Ct:-
(16)
1r
i
p (Ey)Ctt-
=1
2(a1,
... ,a k, y) & (UY)(JpCtti
=
1
2(a
1, ... ,ak,Y)
except, perhaps, for notational variation. If we add to (16) as an additional member of the conjunction the unquantified part IIs i = 1
B i (a 1 ,
••• ,
ak )
of (7), we obtain a formula which is of the form Ct d- 1(a1, ... , ak), again up to notational variation. In virtue of the omission lemma, this formula
70
JAAKKO HINTIKKA
is implied by (7). It will be said to be obtained from (7) by reduction with respect to (9). When (7) is reduced with respect to the different a-constituents of depth d-l occurring in it, we obtain a number of formulae of the form Ct d- 1(al ' ... , ak). If there are two formulae among them which are different (apart from notational variation), then (7) is inconsistent, for it implies both of these formulae which are mutually incompatible by the incompatibility lemma. In order for (7) to be consistent, the results of reducing it with respect to the different a-constituents of depth d-l occurring in it must all coincide. This gives us a necessary condition of consistency for an a-constituent (7). It will be called condition (D). When in the sequel it will be said to be applied to a given constituent or a-constituent, this will be understood to mean that it is also applied to the a-constituents of lesser depth occurring therein. In the diagram of section 6 the reduction of the constituent illustrated there (with respect to one of the a-constituents of depth d-l occurring in it) is represented schematically by the solid line. As we just saw, the result should be independent of the choice of Ct~-l. Further conditions of consistency are obtained by comparing the results of eliminating the different layers of quantifiers which occur in an a-constituent, say in (7). If (7) is to be consistent, the result is in the case of each layer an a-constituent of depth d-l, as we just saw. It may now be added that all these resulting a-constituents of depth d - I must be identical (up to notational variation); otherwise they would be incompatible, though they are all implied by (7). Hence the omission of a layer of quantifiers must yield the same result, no matter which layer is omitted, if a constituent or a-constituent is to be consistent. This requirement will be called condition (E). In our diagram (section 6) the omission of the last layer of quantifiers is indicated by the dotted line. Notice that the two omissions that are represented schematically in the diagram must yield the same result if the attributive constituent represented by the diagram is to be consistent. Since the result of omitting one layer of quantifiers from a given a-constituent Ct~(aI' ... , ak) is unique if this a-constituent is consistent, we can refer to all these results in one and the same way: Each of them will be called Ct~[-l](al"'" a k). Incase Ct~(al' ... , ak)does not yield a unique result when a layer of quantifiers is omitted, or if the result is not
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
71
an a-constituent, we shall say that Ct~[-l](al' .,., ak) disappears, If it does not disappear, it is easily seen to satisfy the conditions (D)-(E) of consistency and hence to yield a unique result when another layer of quantifiers is omitted. The result of applying the same operation to ca«; ... , ak ) e times will be called Ct~[-e](al' .. " a k). Notice that there is an operation which in a certain sense is the inverse of the operation of omitting a layer of quantifiers. It is the operation of expanding a constituent or a-constituent Cd of depth d to a disjunction of a number of subordinate constituents or a-constituents of depth d+ 1. The procedure which was mentioned earlier in section 5 for converting formulae into the normal form may be assumed to be such that each of the subordinate constituents (or a-constituents) of depth d-s- 1 again yields Cd when the innermost layer of quantifiers is omitted. We may also require that this is the case no matter which layer of quantifiers is omitted from the subordinate constituent or a-constituent in question, for if the result is not C", the subordinate constituent is inconsistent and hence may be omitted from the normal form of c- with depth d-s- 1. The relation of the two sets of conditions (A)-(C) and (D)-(E) (conceived of as conditions of consistency) is straightforward. It may be proved that whenever a constituent or a-constituent satisfies (A)-(C) it also satisfies (D)-(E). In fact, it may be proved that it satisfies (D)-(E) whenever it satisfies (A) plus (B) in its original, strong form. Conversely, it may be shown that a constituent or a-constituent Cd satisfies the conditions (A)-(B) if at least one of its subordinate constituents or a-constituents of depth d-s- 1 satisfies the conditions (C)-(F). In a sense, the two sets of conditions (A)-(C) and (C)-(E) are thus equally powerful for the purpose of discovering inconsistent constituents and a-constituents. In order to apply the latter ones successfully to a constituent, however, we must first explicate its content by expanding it into a disjunction of constituents of depth d+ 1, and apply the conditions (C)-(E) to each of these. These results may be proved rather simply by means of the diagrams of constituents and a-constituents which have been explained in section 6. For reasons of space the proofs are not given here. Suffice it to say that the implication from the satisfaction of (A)-(C) to that of (D)-(E) is proved conveniently by induction on d. That this should be the case is not
72
JAAKKO HINTIKKA
surprising in view of the intuitive meaning of the conditions (A)-(C), for from the intuitive meaning they have it is rather easy to gather that they require (among other things) that the omission of two adjacent layers of quantifiers has to give us the same result. 13. The effects of identity again
So far, we have considered the process of omitting a layer of quantifiers only in first-order logic without identity. What changes are occasioned by the exclusive interpretation in this respect? The omission of the last (innermost) layer of quantifiers can be accomplished as before. The omission of an intermediate layer of quantifiers also reduces to the omission of the outermost quantifier in the same way as before. What cannot be done in the same way as above is the process of relative reduction. Applied to (7) with respect to (9), it no longer gives us a formula which is implied (in all cases) by (7). The reason why the omission lemma fails in this case is the following: In order to get rid of the outermost quantifiers (Ex) and (Ux) we have to omit from (9) not only all the atomic formulae containing x but also all the identities involving x which on the exclusive interpretation are implicit in the quantifiers (Ex) and (Ux) themselves as well as in all the inner quantifiers (Ez) and (Uz) of (9). Simply omitting all these identities does not always fall within the scope of the omission lemma. In some cases it does; thus the part of (7) beginning with (Ux) is omitted altogether in the reduction, eliminating all problems concerning identities occurring in it. Furthermore, all identities implicit in existential quantifiers are easily seen to fall within the scope of the omission lemma. Thus there only remains the problem of dealing with a universal quantifier (Uz) occurring in the inner layers of (7). On the exclusive interpretation, this quantifier really means (Uz) (z =ft x:::> ... ). Intuitively, it is easy to see whence the trouble comes here. We are trying to omit x, that is, we are trying to convert a statement about all individuals different from the referent of x into a statement about all individuals without restrictions. Clearly this is possible only if we add a new clause which takes care of the case in which one of these "all" individuals is the referent of x. The way to do this is as follows: Assume that we are reducing (7) with respect to (9). The at the same time as we omit all the atomic formulae which contain x from (9), we add
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
73
as a new member of the main conjunction of (9) the following formula: (17)
and as a new member of the outermost disjunction x, y) the following formula:
lT w
ct' - 2(a l,
... , ai,
0: - I[-l](al' ... , ai, y).
(18)
The same operation has to be applied to every a-constituent ci; - e(a I, . . . , ai, x, y, ... ) which occurs in (9), and whose outermost quantifiers are (let us say) (Ez) and (Uz). At the same time as we omit all the atomic formulae which contain x from it, we add as a new member of the main conjunction the formula (17)*
(E z)
cr:t p
e[ - 1](
aI' ... , ak' z, y, ... )
and as a new member of the outermost disjunction the related formula (18)*
0: -
Ct~ -
e[-I](a I, ... , a k,
z, y, ... ).
Here l[-I\aj, ... , ab x) and O~ - e[-l](al' ... , ai, x, y, ... ) may be defined as the respective results of omitting the last layer of quantifiers from (9) and from Ct~ - e(a I' . . . , a k , x, y, ... ). The result of carrying out this operation in all the a-constituents which occur in (9) as well as in (9) itself, at the same time as we omit all the atomic formulae containing x from (9) and add as a new member of the main conjunction of (9) the unquantified part
ITs Blal' ... , ak )
i = 1
of (7), will be called the result of reducing (7) with respect to (9) in firstorder logic with identity. This result is implied by (7) (on the exclusive interpretation of quantifiers, of course). This implication will not be proved formally here. No proof is probably needed to convince the reader, for on the basis of the intuitive considerations which led us to modify the process of relative reduction it should be obvious (at least on a moment's reflection) that the modification is just what is needed to reinstate the implication. After the reduction of an a-constituent with respect to another has thus been redefined so as to restore the crucial implication, everything else may be done in the same way as in first-order logic without identity.
74
J AAKKO HINTIKKA
We can define what it means to omit a layer of quantifiers, and we can reformulate the necessary conditions of consistency (D)-(E) for firstorder logic with identity. The relation of these conditions to (A)-(B) (the latter in the weak form) is even more clear-cut than before. A constituent or a-constituent csatisfies (D)-(E) if it satisfies (A)-(B); and it satisfies (A)-(B) if at least one of its subordinate constituents or a-constituents of depth d+ I satisfies (D)-(E). 14. A disproof procedure defined
The conditions (A)-(E) are connected in an intimate and intuitive way with the structure of constituents and a-constituents. It will now be shown that they provide us with a disproof procedure for inconsistent constituents and a-constituents, a procedure which is semantically complete in that every inconsistent constituent is subject to this disproof procedure. The procedure can be described very simply. Given a constituent Cd of depth d, how can we try to find out whether it is consistent or inconsistent? It may be the case that our conditions (A)-(E) suffice to establish its inconsistency. If not, we do not yet know whether Cd is consistent or not. What we can do, however, is to expand Cd into a disjunction of a number of subordinate constituents of depth d+ I (with the same parameters (P .l)-(P. 2)), to which we may apply our conditions. If all of them are inconsistent by our sufficient conditions, their disjunction and therefore Cd itself is likewise inconsistent, and we have an answer to our question. If not, we have to keep on expanding Cd into a disjunction of subordinate constituents of greater and greater depth d-i-e. If during this procedure some constituents are inconsistent by our conditions, they may be omitted in the sequel. If for some e all the subordinate constituents of depth d-i-e turn out to be inconsistent by (A)-(E), then so is C". What we want to show is that for each inconsistent constituent cthere is an e such that this happens at depth d + e. In other words, whatever inconsistencies there may be in a constituent can be brought to light by adding to its depth. And since every formula can be brought to a distributive normal form, this likewise gives us a method of disproving every inconsistent formula. This method consists of the rules for converting
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
75
a formula to the (second) distributive normal form (which also give us rules for expanding a constituent to a disjunction of a number of deeper constituents) plus our sufficient conditions (A)-(E) ofinconsistency. The statement that this method really can be used to disprove every inconsistent formula will be called the completeness theorem of our theory of distributive normal forms. Because of the connection between the two sets of conditions (A)-(C) and (C)-(E) either of these combinations of conditions may be used in the disproof procedure just described (in first-order logic without identity). The only difference between the two sets of conditions is that if we use the former we shall be able to see inconsistencies one step earlier than if we used the latter; i.e. if using the former we have to go down to depth d-i-e (in the case of some particular formula), then using the latter we have to go down to depth d+e+ 1. The same relationship holds in firstorder logic with identity between the two sets of conditions (A)-(B) and (D)-(E). In proving the completeness theorem either set of conditions may be used. Because of the greater simplicity of (D)-(E) they will be used in what follows instead of (A)-(E). 15. Completeness proof (first part)
The disproofs described in the preceding section have the structure of a tree. Since each constituent of depth d has only a finite number of subordinate constituents of depth d+ 1, at each point of this tree only a finite number of branches can diverge. Hence the tree theorem (Konig's lemma) applies, showing that the completeness theorem can be proved by proving the following result: A constituent is consistent (satisfiable) if it can occur in a sequence of constituents which satisfies the following conditions: (i) each member of the sequence is subordinate to its immediate predecessor; (ii) each member satisfies the conditions (C)-(E) of consistency (in first-order logic with identity, the conditions (D)-(E». We may also assume, for simplicity, that the depth of each member of the sequence is d + 1 when the depth of its immediate predecessor is d.
76
JAAKKO HINTIKKA
It clearly suffices to prove that the first member of each sequence which
satisfies (i)-(ii) is consistent. Let us assume that we are given a sequence So of deeper and deeper constituents which satisfies all the conditions just mentioned. We shall show that the first member of So is satisfiable. For this purpose, we shall first construct a sequence of attributive constituents S r- This will be done in such a way that instead of adding to the depth of constituents (as in So) we introduce new free individual symbols. In fact, the depth of all the members of Sl will be the same as that of the first member of So (say d); however, each of them will have one free individual symbol more than its immediate predecessor. Each member of Sl will be chosen in such a way that it occurs in the corresponding member of So, i.e. occurs there with appropriate bound variables substituted for some of its free individual symbols, of course. For the first member of Sl we may take the attributive constituent of depth d which occurs in the first member of So' The main question which remains to be answered is therefore: How is a member of Sl obtained from its immediate predecessor? In order to answer this question, let as assume that (7) is an arbitrary member of S r- Let is also assume that of the free individual symbols of (7) a l' a2 . . . , a j (1 :s; j :s; k) occur already in the first member of So while a j +1' aj +2, ... , a k do not occur there. Then an a-constituent of the form Ct~(a 1, . . . , aj, Xj + 1, . . . , Xk) occurs in the corresponding member of So' Since the next member of So is obtained (as was pointed out in section 12) by adding one more layer of quantifiers, there must be in the next member of So at least one (usually there are several) attributive constituent which is subordinate to the one just mentioned and which is of the form Ct: + teal' ... , aj' X j +1' ... , Xk) or, more explicitly, (19)
II B;(a 1, ... ,aj'x j+ 1, ... ,xk) &
s i = 1
nt' i = 1
(EXk+l)Ct~(al' ... ,a j,xj+l' ... 'X k'Xk+ 1) &
(UXk + 1)
(Jt'
i = 1
Ct~(al> ... , aj' Xj + 1, ... , Xb Xk + 1)'
In order to obtain the next member of Sl' we choose one of the a-constituents of depth d occurring in (19). The principle of selection will be explained later. If the attributive constituent chosen is
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
77
(20) the next member of S 1 is simply (20)* Ct~(a1' ... , a j , a j + l' ... , ai, ak + 1)' where a k + 1 is a new free individual symbol. This a-constituent satisfies by construction the requirement that the corresponding bound-variable formula (20) occurs in the corresponding member of SQ. How is (20) to be selected? We shall first explain one particular method of making the choice. Subsequently, it will be pointed out that this method can be considerably generalized. Let Tk be the set of all the a-constituents of depth d -1 which occur in (7), and let be the set of all the similar a-constituents which contain x j + l' Xj + 2' . . . , X k instead of a j + l' a j + 2, . . . , ak' Each of the a-constituents from among which (20) is chosen arises from a member of through the addition of a new layer of quantifiers. For the members of T: we shall shortly establish a certain seniority ranking. After this has been accomplished, (20) may be chosen to be any of the a-constituents which arise from members of T: of the highest rank. Because of the similarity of Tk and T:, a similar ranking will be automatically induced for the members of Tk , too. The only thing that remains in order to define Sl is therefore to explain how the seniority ranking is established. This ranking will be a linear quasi-ordering, i.e. it will be a linear ordering of the different ranks into which the members of T~ (and ofTk ) will be partitioned. The ranking of the a-constituents of depth d-l occurring in the first member of Sl does not matter. Hence the only thing we have to do is to explain how this ranking is carried over from one member of Sl to the next one. (In each case, we have a ranking of the a-constituents of depth d-l occurring in a member of Sl') In order to explain this, notice that one and the same result is obtained from (19) in two different ways:
T:
T:
(a) By omitting the last layer of quantifiers; (b) By reducing it with respect to (20). This identity follows from the fact that (19) satisfies the conditions (D)-(E). The result of'{a) is simply ca«; ... , aj' Xj + l' . . . , x k ) , which
78
JAAKKO HINTIKKA
is therefore also obtainable through the operation (b), i.e. whose quantified part is obtainable by omitting all reference to X k + 1 from (20). This fact establishes a one-to-many correlation between the members of T{ and the members of the set T~ + 1 of all a-constituents of depth d-1 which occur in (20): each of the former arises from one of the latter by omitting all mention of X k + 1. This correlation will be called the weak correlation between the members of the two sets and conceived of as a symmetric relation. A similar relation, which will be referred to in the same way, obtains between the members of Tk and the members of the set Tk + 1 of all the a-constituents of depth d -1 which occur in (20)*. It is to be observed that this relation is determined as soon as the sequence S1 is given; its definition does not in any way turn on the definition of our seniority ranking. The weak correlation can be strengthened (artificially, so to speak) into a one-to-one correlation between all the members of Tk (or of TD and some of the members of Tk + 1 (or of T~ + l' respectively). All we have to do for the purpose is to choose one of the weak correlates of each member of Tk and assign it to this member as its strong correlate. This can of course be done in a variety of ways in most cases: in the sequel we shall consider some particular way of doing so. One important exception will be made here, however. That member of Tk which gave rise to (20)* (i.e. which is identical with Ct~[-1](a1' ... , a k , Xk + 1) will not be assigned any strong correlate in Tk + 1; instead, we shall say that it is associated with (20)*. The same will of course apply to T~, T{ + l ' and (20). When the strong correlation has been established, the seniority ranking may be carried over from Tk to Tk + 1 by stipulating that strong correlation preserves relative rank, and that members of Tk + 1 which do not have any strong correlate in Tk rank lower than one which has such a correlate. The net effect of the transition from (7) to (20)* (i.e. from a member of S1 to the next one) on the seniority ranking is thus that the ranking is preserved except that (i) one member of the highest rank gets lost and that (ii) a new rank will be created which is lower than all the old ones. From this we can gather what happens in the long run in the sequence S1. Every rank will become empty while lower and lower new ranks will (usually) be created. Every chain of strong correlations will come to an end with an a-constituent which does not have a strong correlate any
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
79
more but which is associated with the next member of St. Conversely, every member of St is associated with an a-constituent of depth d-1 which occurs in its predecessor. Notice also that if we move backwards in St no chain of weak correlations comes to an end until it reaches the first member of St. What has been said so far applies in the first place to first-order logic without identity. In first-order logic with identity, the situation is slightly complicated by complications that were needed in the definition of relative reduction in section 13. They enter the present discussion through the operation (b) which was applied to (19) earlier in this section. The net effect of the complications here is that one of the members of Tk may become weakly correlated, not with any member of Tk + t, but rather with Ct~[ -t](a t, ... , ak , X k + t). This member of T k will, however, be the one which is not going to have any strong correlate anyway. The fact that it may now also lack a weak correlate does not interfere with the way the seniority ordering is carried forward in St. Nor does it interfere with the statements made in the preceding paragraph. This completes our explanation of how the sequence St may be obtained from the given sequence So' We can see that such a sequence can always be obtained. It only remains to show that all the members of St are simultaneously satisfiable. If we can do this, then we have shown that the first member of St is satisfiable. In first-order logic with identity, the satisfiability of the first member of So is thereby made obvious. In first-order logic without identity, the satisfiability Of the first member of So can be established by an argument similar to the one we shall give at the end of section 17. 16. Attributive constituents and model sets
The simultaneous satisfiability of all the members of St may be proved by imbedding them into one and the same model set. t ) If no negation1) For model sets and for a proof of the fact that imbeddability in a model set equals satisfiability (consistency), see my papers, Form and Content in Quantification Theory, Acta Philosophica Fennica 8 (1955) 11-55, and Notes on Quantification Theory, Societas Scientiarum Fennica, Commentationes physico-mathernaticae 17, no. 12 (Helsinki 1955). Cf. also Modality and Quantification, Theoria (Lund, Sweden) 27 (1961) 119-128. - It is convenient to assume here that (c. &) and C. v) have been generalized so as to apply also to conjunctions and disjunctions with more than two members.
80
JAAKKO HINTIKKA
signs are allowed except those which immediately precede an atomic formula, if no identities are admitted, and if the only propositional connectives are r-«, &, and v, then a model set may be defined as a set of formulae - say fl - which satisfies the following conditions: (C.
~)
(c.&)
(C.v) (C.E)
If F G u, then not '" F G fl. If (F & G) G fl, then F G fl and G G fl. If (F v G) G II, then F G fl or G G fl (or both). If (Ex)F G u, then F(a/x) symbol a.
G
fl for at least one free individual
Here F(a/x) is the result of replacing x everywhere by a in F, subject to the usual precautions concerning the binding of variables. The same notation will be used in what follows. (C.V)
If (Ux)F G fl and if b is a free individual symbol occurring in at least one formula of fl, then F(b/x) s fl.
Satisfiability has to be interpreted here as satisfiability in an empty or non-empty domain of individuals. For the exclusive interpretation of quantifiers, the definition of a model set can obviously be modified by changing (C.E) and (C.V) as follows"): (C.Eex)
If (Ex)F 8 fl, then F(a/x) 8 fl for at least one free individual symbol a which does not occur in F.
(C,V ex)
If (Ux)F G fl and if the free individual symbol b occurs in the formulae of fl but not in F, then F(b/x) G fl.
These conditions can be geared more closely to the structure of constituents and attributive constituents. It is easily seen that an a-constituent is imbeddable in a model set and therefore satisfiable ifit can be imbedded in a set A of attributive constituents which satisfies the following conditions (in these conditions (7) is thought of as an arbitrary a-constituent):
(Cict-c )
If an atomic formula all of whose individual symbols are free occurs unnegated in a member of A, then it never occurs negated in any member of A.
1) In the terminology of the paper referred to in section 7 (footnote), p. 59, these conditions formulate the weak fy exclusive interpretation of quantifiers.
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
81
(C.ctE)
If (7) occurs in A, then for every a-constituent (9) of depth d-l which occurs in (7) there is a free individual symbol b such that Ct:-l(a l, ... , ai, b) eA.
(C.ctU)
If (7) occurs in A, then for every free individual symbol b which occurs in at least one member of A there is an a-constituent (9) of depth d-l which occurs in (7) and which is such that \al' ... , ai, b) eA.
Ct:-
A set A of a-constituents which satisfies these conditions will be called a constitutive model set"), For the exclusive interpretation of quantifiers, the condition (C.ctE) and (C.ctU) have of course to be modified in the same way as in (C.E ex) and in (C,U ex ) , respectively, by requiring that b does not occur in (9) - or in (7), which is the same thing. The resulting conditions will be called (C.ctE ex) and (C.ctU ex) ' 17. Completeness proof (concluded)
The simultaneous satisfiability of all the members of the sequence Sl which was defined earlier will be proved first for first-order logic with identity. Subsequently, the modifications needed for the ordinary interpretation of quantifiers will be explained briefly. On the exclusive interpretation ofquantifiers, all we have to do in order to imbed Sl in a constitutive model set is to form its closure with respect to the following operations: (a) The operation of omitting a layer of quantifiers. (b) The operation of omitting a free individual symbol (i.e. of omitting all the atomic formulae which contain this symbol, together with all the connectives which hence become idle). The operation (b) requires a few comments. First, we shall restrict it by requiring that the free individual symbol omitted is not the distinguished 1) Given a constitutive model set A, an ordinary model set fl is easily obtained as the closure of J. with respect to the following operations: (i) Whenever a conjunction occurs in A, adjoin to A all its members. (ii) Whenever (7) occurs in J. and b occurs in the formulae of A, adjoin the disjunction l at Ct/- (a b ••• , ak' b) i= 1
to A. This fl contains Aas a part and has exactly the same free individual symbols as A.
82
JAAKKO HINTIKKA
(last) free individual symbol of an a-constituent. For instance, the symbol
ak must not be omitted from (7). Otherwise (b) might not give an a-con-
stituent as a result when applied to an a-constituent. Furthermore, in first-order logic with identity the operation (b) has to be modified in the same way and for the same reason as the operation (a). Suppose, for instance, that we are omitting the free individual symbol a, (1 ::;: i ::;: k) from (7); and suppose that Ct;-e(a 1 , •.. , ai, x, y, ... ) is an arbitrary a-constituent which occurs in (7) and whose outermost quantifiers are (Ez) and (Uz) (exclusively interpreted, of course). At the same time as we omit all the atomic formulae which contain a, we must add as a new member of the main conjunction the formula (E)C z t dp -
e[ - l ] (
aI' ... , a j
_
l'
z, a i + 1,
... ,
ak , x, y, ... )
and as a new member of the outermost disjunction the formula
cr: p
e[-l]( aI'
... , a j _
1,
z, aj +
1, . . . ,
) a k , x, y, ....
Let the closure of Sl with respect to the operations (a) and (b), so qualified, be A. Then it is easy to see that A satisfies (Cict-«). For the purpose, consider an arbitrary atomic formula which occurs in the formulae of Awith all its individual symbols free. If it occurs (in this way) in a member of A, negated or unnegated, it must likewise occur, in view of the way A was obtained from Sl' in a member of Sl' Hence it suffices to verify (Cict o- ) for Sl alone. Now all the free individual symbols of Sl are aI' a 2, ... ; we assume for simplicity that they are different from all the bound individual variables we are dealing with. If the last member of this sequence which occurs in F is ai; then F occurs (negated or unnegated but not both) in exactly one member of Sl' viz. in (7) provided of course that k > j. But if k ::;: j, F cannot occur in any member of S 1 at all. Hence F cannot occur both negated and unnegated in the members of 8 1 nor therefore in the member of A. It is also easy to see that all the members of A satisfy the other two defining conditions of a constitutive model set provided that all the members of Sl do so in A. Indeed, the relation between (7) and (9) which is mentioned in (C.ctE) and (C.ctU) and hence also in (C.ctEex ) and (C.ctUex) continues to obtain if one of the operations (a) and (b) is applied to both of them. From this it follows in first-order logic without identity that the other (new) members of A satisfy (C.ctE) and (C.ctU)
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
83
if the members of Sl do so. In first-order logic with identity, there are two additional cases we have to worry about. They are caused by the fact that in omitting a free individual symbol on the exclusive interpretation a new attributive constituent of depth d-l is added to each attributive constituent of depth d from which a free individual symbol is omitted. Let the latter be (7) and the former therefore (21) Now), might prima facie fail to satisfy (C.ctEex) because of the presence of this new a-constituent of depth d-l in the modified from of (7), and it might prima facie fail to satisfy (C.ctUex ) because a free individual symbol (viz. ai ) which formerly did occur in (7) does not do so any more and hence seems to open a new possibility of applying (C.ctUex) ' In both cases the violation of one of the two conditions is avoided because A is closed under (a) and hence contains the formula (cf(21)):
Hence the only thing that remains for us to do in order to show that A is a constitutive model set is to prove that the members of Sl satisfy the conditions (C.ctE) and (C.ctU) or (C.ctE ex) and (C.ctUex) ' as the case may be, when they are considered as members of A. This may be proved by means of certain lemmata concerning Sl' These lemmata follow directly from the way in which Sl was constructed. LEMMA 1: Whenever Ct~ - \a 1 , ... , ak, x) e T k is weakly correlated with Ct: - \a 1 , ••• , ak, ak + l ' x) e T k + i - the former results from the latter (up to notational variation, as usual) by omitting the free individual symbol ak + 1 in the sense of operation (b) defined earlier in this section.
This is, on reflection, just what being weakly correlated means. Lemma I can be generalized:
ci;-
1*: Whenever \al' ... , ak' x) e T k is connected by the ancestral of the weak correlation with Ct: - l(a l, •.. , ak' ak + 10 •.. , ak + I' x) e T k + l ' the former results from the latter by omitting the free individual symbols ak + l' ak + 2, . . . , ak + I' LEMMA
Another useful lemma is the following:
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JAAKKO HINTIKKA
LEMMA 2: Whenever Ct~ - 1(a1, ... , ak' x) s T k is associated with the next member (20)* of Sl' it results from the latter by omitting one layer of quantifiers.
This is just what being associated means. By means of Lemmata 1* and 2 we can show that the members of Sl satisfy (C.ctE e x) and (C.ctUex ) in A. For this purpose, assume that (7) is a member of Sl and that (22)
occurs in (7). As pointed out in section 15, the chain of strong correlations which passes through (22) will always come to an end at some a-constituent which is not strongly correlated with any further a-constituent but which is instead associated with the next member of Sl .- say with (23) Then by Lemmata 1* and 2 it follows that (22)* results from (23) by first omitting one layer of quantifiers and by then omitting the free individual symbols ak + 1, ak + 2, . . . , ak + I' But since Ais closed under the operations (a) and (b), this implies that (22)* belongs to A. This means that (C.ctE) is satisfied by (7). Moreover, since ak + 1 does not occur in (7) or (22), it also means that (C.ctE ex) is satisfied. In order to verify (C.ctU ex) , assume that (7) occurs in Sl' Then every free individual symbol of A which does not occur in (7) (nor in the less deep a-constituents which occur in (7)) is of the form a, + I' Consider now the first member of Sl which contains ak + I; let it be (23). Then (23) is associated with some member of T k + 1 _ 1 which is in turn connected by a chain of weak correlations with some member of Tk , i.e. with some a-constituent of depth d-l occurring in (7). Let this a-constituent be (22). Then from Lemmata 1* and 2 it follows in the same way as in the case of (C.ctE e x) that (22)* results from (23) by means of the operations (a) and (b) and hence belongs to A. This suffices to verify (C.ctU ex) for (7). This completes our argument to the effect that all the members of Sl are simultaneously satisfiable for a system with identity. For a system without identity, an additional argument is needed to take care of those cases of (C.ctU) in which b occurs in (7), i.e. in which b is an a, where
DISTRIBUTIVE NORMAL FORMS IN FIRST-ORDER LOGIC
85
I .:::;; i .:::;; k. In the special case of Sl we know that each member (7) of Sl satisfies the condition (C) of consistency. Hence there must be in (7) an a-constituent (9) of depth d-l such that the bough of (7) determined by (9) is strongly symmetric with respect to x and a.. Consider, then, the formula Ct~- teal' ... , ak- l, ak, aJ obtained from (9) by replacing x by a.. By a straightforward argument whose details are here omitted, it can be shown that because of strong symmetry this formula is identical with the result of reducing (7) with respect to (9) and therefore also identical with the result of omitting one layer of quantifiers from (7). (Identity here means, as usual, identity except for the naming of bound variables and the order of conjunctions and disjunctions, and in this case also for the vacuous repetition of some members of conjunctions and disjunctions.) Since A is closed with respect to the operation (a), this implies that Ct~-l(al> ... , ak-l, ai, a i ) e A and shows that the condition (C.ctU) is satisfied also in the cases which do not fall within the scope of (C.ctU ex ) ' 18. General considerations
This brings to an end our completeness proof for the disproof procedure described in section 14, both in first-order logic without identity and in one with it. As a dual of the disproof procedure we obtain a complete proof procedure for first-order logic. Proofs of this form have an easily surveyable structure, and most of their steps are quite innocentlooking. They may be taken to be linear, i.e. each line of the proof is a single formula which is obtained from its immediate predecessor. Most of these formulae are conjunctions; hence we may, if we want, split the proof into branches (Beweisfiiden) which are tied together into the form of a tree by the simple inference rule F, G f- (F & G). It may be worth while to list, by way of summary, the different kinds of step which will occur in these proofs (with the optional exception of the inference rule just mentioned). In listing these steps, we are looking at the proofs not in the direction from the axioms to the formula to be proved but in the opposite direction. The following operations are needed to convert a formula into the dual of its distributive normal form:
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JAAKKO HINTIKKA
(1) Transformation into propositional normal form without affecting the parameters (P .l)-(P. 3) of the formula in question.
(2) The distributivity of the universal quantifier with respect to conjunction. (3) The possibility of reducing the scope of a universal quantifier by omitting from it members of disjunctions which do not contain the variable which is bound to the quantifier in question. We must be able to carry out these operations also within a formula (i.e. to apply them to a subformula of a larger formula). Furthermore, we must of course assume the usual interrelation between the two quantifiers and the possibility of renaming bound variables. These operations also enable us to reach the dual of the second distributive normal form. In adding to the depth of the dual of a constituent we must assume something further: (4) One may add to the depth of a formula by introducing propositionally redundant parts which do not change the other parameters (P .l)-(P .2). Finally, in each branch of the proof or in each conjunct of the first line of the rest of the proof (looking at it now in the direction from axioms to the formula to be proved) we need exactly one initial operation which is the dual of the argument by means of which attributive constituents not satisfying (A)-(C) (or, in first-order logic with identity, (A)(B» were shown to be inconsistent. From section 10 it is seen that in the case of (A) and (B) these operations are, apart from certain inessential preparatory steps, essentially applications of the well-known exchange laws for adjacent quantifiers: f- (Ux) (Uy)F
==
(Uy) (Ux)F.
f- (Ex) (Uy)F ~ (Uy) (Ex)F.
In the case of (C) this initial step is essentially an application of the law
(5h
f- (Ux)F ~ F(a/x)
where a occurs in F. The application of these laws may perhaps be considered as the only
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non-trivial step of the argument. In each initial conjunct, only one application of one of them is needed. In first-order logic with identity, a few unimportant complications arise because of the presence of identity. We shall not discuss them in detail here. We also obtain two dual methods of proofs from assumptions. These proofs are interesting because they are linear; each line of the proof consists of a single formula.'). We may consider that method which turns on our original (second) distributive normal form. In order to prove G from F by means of this method, we may proceed as follows; First, we pool together the parameters (P. I)-(P .2) of F and G, take the maximum (say d) of their depths, and convert F and G into their respective normal forms F d and Gd in terms of the parameters so obtained. Then we expand the normal forms F d and Gd by splitting their constituents into disjunctions of deeper and deeper constituents while their other parameters are unchanged. At each depth we test all the constituents for consistency by means (A)-(C) or (C)-(E) (or, if identities are present, by means of (A)-(B) or (D)-(E)) and omit all the ones which fail the test. Let us call the disjunction of the remaining ones of depth d-s- e, and cr-, respectively. Then if G really follows from F there will be an e such that all the members of F d + e are among the members of c-:». A proof of G from F will then proceed from F to F d to r-: 1 to ... to F d + e to c-:» to G d+ e- 1 to ... to c-: 1 to G d to G. All the steps of this proof except the one from F d + e to Gd+e are equivalences. The same procedure gives us a complete method of equivalence proofs. If F and G are equivalent, then for some e the disjunctions F d + e and Gd+e will have the same members and therefore be equivalent. In this case, all the steps of the proof from F to G will be equivalences. In disproofs, proofs, proofs from premises, and equivalence proofs, we thus have to add to the depth of the formulae we are dealing with. Consider disproofs as an example. In order to disprove F, we bring it to its distributive normal form F d (where d is the depth of F) and keep adding to the depth of the constituents of F d until at some depth d-s- e all the subordinate constituents are inconsistent by our conditions. Here the
r-:»
1) For an interesting discussion of a different standard form of linear proofs from assumptions, see William Craig, Linear Reasoning, A New Form of the HerbrandGentzen Theorem, Journal of Symbolic Logic 22 (1957) 25(}-268.
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difference e between the depth of F and the depth at which the inconsistency of F becomes explicit may be considered as a kind of (rough) measure as to how deeply hidden the inconsistency of F is and therefore also as a measure of the amount of "development" or "synthesis" required to bring this inconsistency into the open. This idea seems to have some philosophical interest. It also applies to the other kinds of proofs we have mentioned. For instance, in the method of equivalence proofs just described the difference between d-i-e and the depth of F perhaps serves as a numerical measure of the amount of "synthesis" which has to be performed on F in order to bring out its equivalence with G 1 ) . It may be objected here that these suggested measures are all relative to the particular conditions of consistency which are being used in the proof in question, and hence not likely to have much general significance. That they depend on the conditions employed is true; but on the other hand it seems to me that in some rather elusive sense our conditions (A)-(E) of inconsistency are as strong as we can possibly hope natural conditions to be. Further work is needed here to clear up the situation. From the way we proved the completeness theorem we can also read directions for a different standard form for first-order proofs which is much closer than the ones we have just mentioned to the earlier standard forms of Herbrand and Gentzen. Consider disproofs first. In this new standard form, each stage of the proof is again a disjunction of constituents, preceded by a transformation to the distributive normal form. Again, each disjunction is obtained from its immediate predecessor. The way in which this happens is now different, however. Instead of adding to the depth of our constituents we add new free individual symbols to them. In order to describe the procedure in more detail, assume that (8) occurs in one of the disjunctions. We choose one of the a-constituents of depth d-l occurring in (8), say (9); the principles of selection will be commented on later. We drop the existential quantifier (Ex) which precedes an occurrence of (9) in (8), and replace x everywhere in this occurrence of (9) by a new free individual symbol ak + l ' This leaves the parameter (P .1) of (8) and its depth unchanged, but adds a new member to the set (P. 2). Hence we may transform the formula we 1) If the depths of F and G are identical, exactly the same amount of "synthesis" is required to convert F into G by the method just described as is required to disprove ~ (F == G), if these amounts are measured in the suggested way.
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have obtained into a disjunction of constituents with the same (P .1) and the same depth as (8) but with one more free individual symbol ai: This procedure corresponds to the expansion of a constituent of depth d into a disjunction of subordinate constituents of depth d-s- 1. To each of the new constituents we may apply the conditions (A)-(E) and eliminate those which turn out to be inconsistent. Each of the remaining ones is related to (8) in the same way as each member of St was related to its predecessor. We can set up strong and weak correlations as well as associations in the same way as before and also define a seniority ranking among the a-constituents which occur in (8) and which have depth d-l in the same way as in the completeness proof. For the particular a-constituent of depth d-l in (8) into which the new free individual symbols was first introduced we may simply choose one of the highest ranking ones. This disproof procedure is complete in the same way as the earlier one: Either at some stage of the procedure all the constituents are inconsistent by our conditions (A)-(E) or else we have a sequence of constituents which is just like the sequence St of a-constituents and which can be shown to be satisfiable in the same way as St. The only new feature is that there is not just one way of going from one stage to the next one, for there usually is some choice in the selection of that a-constituent of depth d - 1 in (8) into which the new free individual symbol is first introduced. In fact, both here and in the completeness proof there may even be more choice than we have allowed so far. The situation is the same in the two cases; hence we may consider St by way of example. Suppose that we construct St without any regard to the selection of (20) in (19) (section 15). Then we can define weak correlation and association as before. We can also define strong correlation in a variety of ways in most cases. From section 17 (from the argument which follows the lemmata) we can see that there is only one further thing which is needed in order for us to be able to carry out the completeness proof. This is that every chain of strong correlations eventually comes to an end when we proceed further and further in St- The use of the seniority ranking was simply an artifice for securing this end. If it can be obtained by other means, so much the better. In any case, the conventions concerning the seniority ranking need only be adhered to from some (arbitrarily late) stage on.
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This suffices to describe the new disproof procedure. Similar procedures for proofs from premises and for equivalence proofs are obtained as a consequence, and a similar proof procedure as its dual. The number of new free individual symbols which are introduced in one of these proofs may be said to indicate the amount of synthesis performed in it. The situation is now more complicated than before in that a given formula F may now have different disproofs with different numbers of free individual symbols introduced in them. The smallest of these numbers may be taken, however, to indicate the amount of synthesis needed to bring out the hidden inconsistency of F. It may be shown that this measure of the amount of synthesis required to disprove F coincides with the measure suggested earlier. Similar considerations apply to the other types of proofs. We have described a proof procedure in which the free individual symbols remain unchanged but in which the depth of our formulae grows steadily, and one in which the depth of our formulae remains intact while new free individuals are introduced. These are really the two ends of a long spectrum of mixed proofs in which our two methods are employed together. The method by means of which the completeness theorem was proved is now seen to be tied to the second type of disproof (introduction of new free individual symbols) much more closely than with the first (adding to the depth of constituents) although it is the first one which was initially considered in the completeness proof. In fact, in the completeness proof we first transformed the increase in depth into an increase in the number of free individual symbols. This may seem a rather roundabout way of proving the completeness of the first main method of disproof. In fact, we could have avoided this complication, but only at the expense of complicating the proof considerably in other respects. For this reason, the "direct" proof will not be attempted here. It would be of interest to construct a model for a sequence So of deeper and deeper constituents which satisfy (A)-(E) and which are subordinate to their predecessors directly without going by way of the auxiliary sequence St, for such a construction promises us a survey of the different kinds of model which So (essentially, an arbitrary consistent and complete theory) can have. An interesting line of development seems to open here, a line in which the
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important results of Vaught's on models of complete theories (which e.g. entail the Ryll-Nardzewski No·categoricitytheorem) appear to assume a natural place.')
1) For Vaught's results see Denumerable Models of Complete Theories, in Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 2-9 September 1959, pp. 303-321 (Pergamon Press, Oxford and London 1961).
SEMANTICAL ANALYSIS OF INTUITIONISTIC LOGIC I SAUL A. KRIPKE Harvard University, Cambridge, Mass., USA
The present paper gives a semantical model theory for Heyting's intuitionist predicate logic, and proves the completeness of that system relative to the modelling. The model theory and completeness theorem were announced in [1]. The semantics for modal logic which we announced in [1] and developed in [2], [3], together with the known mappings of intuitionistic logic into the modal system S4, inspired the present semantics for intuitionist logic. It would in fact be possible to derive the completeness of Heyting's predicate logic in our semantics by using the mappings into S4 together with the results of [2], [3]. We prefer, however, to develop the semantics of intuitionistic logic independently of that of S4; this procedure will enable us, we believe, to obtain somewhat more information about intuitionistic logic, including the mapping into S4 as a consequence thereof"), Further, a fairly recently worked-out development, not contained in the announcement of [1], is included: an exposition of Cohen's notion of "forcing" [5] in terms of the present semantics. In addition to giving a simple decision procedure for Heyting's propositional calculus, Part II will present a result not announced in [1] but mentioned in [4]-the undecidability of monadic intuitionistic quantification theory. The proof is based on the semantics previously developed. It should be mentioned that, for the pure implicational intuitionistic propositional logic, Beth [6] has announced the rediscovery of essentially the present modelling; also that, for all of intuitionist propositional logic, 1) The reader who wishes to understand thoroughly the deeper motivation of the present paper, however, is strongly urged to consult [2], [3], and [16], which give the underlying analysis of modal logic.
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a modelling equivalent to ours can be extracted from the results of Lemmon and Dummett [7].1) The results of this paper, though devoted to intuitionistic logic, are proved only classically, except as mentioned below. Intuitionistically, the situation is essentially the same as that for Beth's completeness theorem [8], as analysed by Dyson and Kreisel in [9]; a reader who is interested in intuitionistically valid proofs can consult [9] and apply a similar analysis to the present results. We will give indications below which (we believe) will be sufficient for a reader familiar with [9] to make such an analysis. In the course of these indications, we will prove some results about Kreisel's system Fe which are parenthetical to the main theme of this paper. In particular, we will show that Kuroda's conjecture and Markov's principle are both refutable in Fe. Some notations that will be used throughout the paper are the following: P", Qn, R" (n ~ 0) are n-adic predicate letters; a O-adic predicate letter is usually called a "proposition(al) letter." Occasionally the superscript on a predicate letter will be omitted if this does not sacrifice clarity. We use letters x, y, Z, . . . , with or without subscripts, as (individual) variables. The formulae of the intuitionistic propositional calculus are to be built out of the usual connectives A, V, ::::>, -', starting with the propositional letters as atomic formulae. In the predicate calculus, not only propositional letters but also formulae pn(x u . . . , x n) are taken as atomic; thence formulae are built up from these in the usual manner, using the connectives just given and the quantifiers (x) and (3x). We use A, B, C, . . . , for arbitrary formulae of propositional or predicate calculus, depending on the context; if we wish to call attention to certain free variables in a formula, we use such notations as A(x 1 , ... , x n ) . We assume, finally, that the reader is familiar with standard presentations of Heyting's formalized intuitionistic propositional and predicate calculus, say the presentation in [10].
1) Kreisel's conjectured "reinterpretation of the (intuitionistic) logical constants" in [171 is also, if his conjectures prove correct, related to the present model theory.
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1. The model theory
We define an (intuitionistic) model structure (m. s.) to be an ordered triple (G, K, R) where K is a set, G is an element of K, and R is a reflexive and transitive relation on K. An (intuitionistic) model on a m. s. (G, K, R) is a binary function ¢(P, H), where P ranges over arbitrary proposition letters") and H ranges over elements of K, whose range is the set {T, F}, and which satisfies the following condition: if ¢(P, H) = T and HRH' (H, H'cK), then ¢(P, H') = T. Given a model ¢(P, H), we can define a value ¢(A, H) (=T or F) for an arbitrary formula A of propositional calculus by induction on the number of connectives in A .If A has no connectives, then it is a proposition letter and ¢(A, H) = T or F has already been defined for each H. Assume that ¢(A, H) and ¢(B, H) have already been defined. Then we stipulate: a) ¢(A A B, H)= T iff ¢(A, H) = ¢(B, H) = T; otherwise, ¢(A A B, H) = F. b) ¢(A v B, H) = T iff ¢(A, H) = T or ¢(B, H) = T; otherwise, ¢(A v B, H) = F.
c) ¢(A ::J B, H) = T iff for all H' E K such that HRH', ¢(A, H') = F or ¢(B, H') = T; otherwise, ¢(A ::J B, H') = F. d) ¢(,A, H) = T iff for all H' E K such that HRH', ¢(A, H') = F; otherwise, ¢(,A, H) = F.
Notice that the conditions on A and v are exact analogues of the corresponding conditions on classical conjunction and disjunction; but the conditions on :::> and, are not analogous to the classical conditions. It is easy to show by induction, for any H, H' E K such that HRH', that if ¢(A, H) = T, then ¢(A, H') = T. This property has been stipulated 1) In [2), we let ¢(P, H) range over H E K and atomic subformulae of a fixed formula A. We called this a model of A. We could equally well have adopted this orientation
here; conversely (2) could have adopted, mutatis mutandis, the present definition. The viewpoint of (2) is exploited in the analysis of Cohen's "forcing", where we consider models assigning values only to formulae built out of a fixed atomic formula P(x). We should also remark that, although in this section we have taken the atomic formulae to be proposition letters and formulae pn(x" ... , x n), the definitions would equally well go through if formulae were built out of an arbitrary fixed class of atomic formulae; this fact is exploited in the "provability interpretation," section 1.3, below.
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for a propositional letter, and it follows for more complex formulae using the clauses (a) - (d). Notice that, intuitionisticaIly, the inductive definition here given does not work, since it clearly appeals to the law of excluded middle in clause (c) and (d) (e.g., in (d), either for all H', ¢(A, H') = F, or not). Thus intuitionistically, it would be best to define a model ¢ as a mapping ¢(A, H) in {T, F}, where A ranges over arbitrary formulae of propositional calculus, and which happens to satisfy the clauses (a) - (d) as well as the condition that ¢(P, H) = T and HRH' implies ¢(P, H') = T. Clearly, from the classical viewpoint, this modification leaves the notion of a model essentially unchanged. We call a formula A of propositional calculus valid iff ¢(A, G) = T for every model ¢ on a model structure (G, K, R). A model ¢ on a m. s. (G, K, R), such that ¢(A, G) = F, is called a countermodel for A. To extend the modelling to quantification theory, we define a quantificational model structure (q. m. s.) to be a model structure (G, K, R), together with a function IjJ (the "domain function"), defined on K, such that IjJ(H) is a non-empty set for all H E K, and IjJ(H) s:; IjJ(H') if HRH' (H,H' EK). (Intuitionistically, we require that IjJ(H) not only be non-empty, but that it contains at least one element; of course, a species may be known not to be empty without any particular element thereof being known.) We define a quantificational model ¢ on a q. m. s. (G, K, R) to be a function ¢(p n , H), where P" ranges over l1-adic predicate letters (for all n), and H ranges over elements of K. If 11 = 0, ¢(pn, H) = T or F, and if n ~ 1, ¢(r, H) is a subset of the Cartesian product [1jJ(H)r. We again require for n = 0, that ifHRH', and ¢(pn, H) = T, ¢(r, H') = T; for n ~ 1, analogously we require that if HRH ', ¢(r, H) s:; ¢(r, H'). Let U
= U IjJ(H). H£K
Given a quantificational model ¢, we can define, for each formula A of intuitionistic quantification theory, a value ¢(A, H) = T or F, for each HE K, relative to a fixed assignment of elements ofU to the free individual variables of A. If A is an atomic formula, it is either a propositional letter P, in which case ¢(P, H) = TorFis given, or itis a formula P'(x., ... , x n)
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(n ~ 1). In this latter case, let elements aI' ... , an of U be assigned to Xl' ... , Xn; then we can define, relative to this assignment, ¢(pn(XI' ... , x n), H) = Tiff (a l, ... , an) E ¢(r, H), and ¢(r(Xl' ... , x n), H) = F iff (a l, ... , an) ¢: ¢(pn, H). Given this assignment to atomic formulae, we
can build up the assignment to more complex formulae by induction. Suppose A(Xl' ... , Xm y) is a formula, where at most the variables 'X m y). Assume, that relative to each assignindicated occur free in A(x l' , Xm y, a truth-value ¢(A(x l , . . . , Xm y), ment of elements of U to Xl' H) has been defined for each H. We can then obtain values for ¢«y)A (Xl' ... , Xn, y), H) and ¢«3y)A(Xl' ... , Xn, y), H) as follows. Let the elements al' ... , an of U be assigned to the variables Xl' ... , x.; Then:
e) We say ¢«3y)A(Xl' ... , Xm y), H) = T iff there is e b e l/f(H) such that ¢(A(x l, ... , Xn, y), H) = T when Xl' ... , Xn are assigned al' ... , am respectively, and y is assigned b; otherwise ¢«3y)A(x l , . . . ,X n, y), H) = F.
f) We say ¢«y)A(x l, ... , Xm y), H) = T iff for each H' E K such that HRH' ¢(A(x l, ... , Xm y), H') = T when Xl> ... , x, are assigned al' , an, and y is assigned any element b of l/f(H'); otherwise, ¢«y)A(x l, , X n , y), H) = F.
Finally, we stipulate that if truth-values ¢(A, H) and ¢(B, H) (for all E K), are given relative to an assignment to the free variables of A and B, then corresponding values ¢(A A B, H), ¢(A v B, H), ¢(A :::J B, H), and ¢(,A, H) are to be defined according to the prescriptions (a) - (d). To get a proper intuitionistic definition of model, we should again modify the given conditions and stipulate that a model ¢ is a function ¢(r, H) as above, together with a function ¢(A, H), assigning T or F to ¢(A, H) relative to a given assignment of elements of U to the free variables of A, and satisfying the previously stated conditions (e.g., that ¢(r(x l, ... x n ) , H) = T when Xi is assigned aD .:0;; i .:0;; n) iff (a l , . . . , an) E ¢(pn, H)). Again, this definition is classically substantially equivalent to the old one. We note that all of the results to follow would remain valid if we allowed
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1. 1. Intuitive interpretation
A triple (G, K, S), with K a set, G E K, and S a relation defined on K, is called a tree (and G is called its origin) iff: (1) There is no HE K such that HSG; (2) for every HE K except G, there is a unique H' E K such that H'SH; (3) for every HE K, GS*H, where S* is the ancestral of the relation S(i.e., H 1S*H z iff H 1 =H z or H 1SnH z for some power n of S). If HSH', wecallH the predecessorofH', and H' a successor of H; the tree is finitary iff every H has only finitely many successors. An element H without successors is called an endpoint. Note that K can be characterized in terms of S as its field, and G can then be characterized as the unique element of K without a predecessor. (This definition of tree is adopted from [2]. Intuitionistically, we must further require that the elements H be natural numbers and that S be decidable.) A m. s. (G, K, R) is called a tree m. s. iff there exists a relation S such that (G, K, S) is a tree and R is the smallest reflexive and transitive relation containing S (i. e., R = S*). In our remarks on intuitive interpretation, we will primarily be concerned with tree models (i. e., models defined on a tree m. s. (G, K, In fact, we will show below in section 1. 2 that any model can be replaced by an "equivalent" tree model. The rest of this section will consist of an informally stated intuitive interpretation of the modelling, together with indications how to state the interpretation more formally in terms of Kreisel's theory [11] of absolutely free choice sequences.") The reader unfamiliar with [11] (or uninterested in these details is advised to omit the remarks relating to [11] but to read the rest of the section. The interpretation proceeds as follows. Suppose we are given a model ¢ for a formula A of propositional calculus whose sole atomic subformulae are P, Q, R. For example, suppose we have a tree model ¢ on a m. s. (G, K, R) diagrammed as follows:
R».
1) We are informed that Godel (unpublished) has proposed that such sequences be called "absolutely lawless," presumably on the ground that they are not completely free, being governed by the "higher order" requirement that no restrictions, other than those defining the spread in question, ever be placed on choices later by a free decision. Since Godel's suggestion has not yet been adopted in print, we hesitate to make this change ourselves.
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P
G
Figure 1.
The elements of K are G, Hi' H 2 , H 3 , H 4 . We have written an atomic formula above a node G or Hi if ¢ assigns it the value T on this node; we omit it if ¢ assigns it the value F. Thus, e.g., ¢(P, G) = T while ¢(Q, G) = ¢(R, G) = F. We intend the nodes H to represent points in time (or "evidential situations"), at which we may have various pieces of information. If, at a particular point H in time, we have enough information to prove a proposition A, we say that ¢(A, H) = T; if we lack such information, we say that ¢(A, H) = F. If ¢(A, H) = T we can say that A has been verified at the point H in time; if ¢(A, H) = F, then A has not been verified at H. Notice, then, that T and F do not denote intuitionistic truth and falsity; if ¢(A, H) = T, then A has been verified to be true at the time H; but ¢(A, H) = F does not mean that A has been proved false at H. It simply is not (yet) proved at H, but may be established later. Now given a point in time G, there are various possibilities open for gaining further information about the propositions. One situation is diagrammed in Figure 1. At the point G (representing our present information), we have proved P. For all we know, we may remain "stuck" at G for an arbitrarily long time, without gaining any new information. But it is possible that we will gain enough information to "jump" to point Hi (in which case we have a proof of R in addition to P), or to the point H 2 (where we get a proof of Q in addition to P), or even to the points H 3 or H 4 • If we have "jumped" to the point H 2 , so that we have proved both P and Q, then as far as we know, we may remain "stuck"
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for an arbitrarily long time at HZ; but we may advance to H 3 or H 4 • Notice that if we jump to the "situation" H 3 , we still have proved no more than P and Q; but this does not mean that the situation H 3 is exactly like Hz. In fact, as long as we remain at Hz, the possibility is still open to us that we will some time or other be able to advance to H 4 and prove R; but, if we are at the situation H 3 , we have gained enough information to exclude the option that R will ever be proved. Now, in general, in a model structure (G, K, R), we interpret G as the present "evidential situation." If H is any situation, we say HRH' if, as far as we know, at the time H, we may later get enough information to advance to H'. Thus, since the information we have at H may be all the knowledge we have for an arbitrarily long time, we stipulate that HRH; and the transitivity property of R is intuitively obvious. The requirement that, for any A, if ¢(A, H) = T and HRH', then ¢(A, H') = T, simply means that if we already have a proof of A in the situation H, then we can accept A as proved in any later situation H'-we don't forget. Finally, the inductive clauses for propositional calculus are in consonance with the intuitionistic interpretations of these notions. Thus A A B [A v B] is proved when both A and B have been proved [either A has been proved or B has been proved]; so ¢(A A B, H) = Tiff ¢(A, H) = ¢(B, H) = T [¢(A v B, H) = T iff ¢(A, H) = Tor ¢(B, H) = T]. Notice that disjunction and conjunction behave, in a given situation H, as if they were classical truth-functions. Negation and implication, on the other hand, are not so treated. To assert-cs intuitionistically in the situation H, we need to know at H not only that A has not been verified at H, but that it cannot possibly be verified at any later time, no matter how much more information is gained; so we say that ¢(.,A, H) = T iff ¢(A, H') = F for every H' E K S.t. HRH'. Again, to assert A => B in a situation H, we need to know that in any later situation H' where we get a proof of A, we also get a proof of B; the inductive definition of ¢(A => B, H) formalizes this requirement. Consider the following two point countermodel to the law of excluded middle: p
H
G
Figure 2.
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We have ¢(P,H) = T, ¢(P, G) = F. Since ¢(P, H) = T, ¢(,P, G) = F, and hence ¢(P v ,P, G) = F. Intuitively, at the present situation G, we have not yet proved P; nor can we assert ,P, since the possibility remains that we will get enough information later to advance to H and assert P. Thus, at the point G, we are not in a position to assert P v v]', These considerations can readily be formulated in terms of Kreisel's theory FC of absolutely free choice sequences. Intuitively, an absolutely free choice sequence (a.f.c.s.) is a free choice sequence a, chosen from a given spread S, in which it is stipulated from the beginning that no restrictions, other than the conditions defining the spread S, can ever be placed on the choices. Figure 2, then, for example, can be interpreted in terms of the present theory as follows: Consider a.f.c.s.'s from the spread S consisting of free choices ofO's and 1's, in which, however, 1 can be followed only by 1. Intuitively, we interpret the situation G as a choice of 0 and H as a choice of 1. Since, starting with G, we can remain "stuck" at G as long as we like, we permit 0 to be followed by an arbitrary number of O's as well as by 1; but, since H is followed only by itself, we permit 1 to be followed only by 1. Then P{a) is the assertion "a 1 occurs on the a.f.c.s. a" (i.e., (3n) (a{n) = 1), where n ranges over natural numbers). As long as we have chosen only O's in a, we have not established P{a); but on the other hand, since a is chosen with no restrictions other than being in S, we cannot exclude the possibility of the choice of a 1 later, so we cannot establish ,P{a). These considerations can be formalized easily in Kreisel's FC so as to yield a proof of ,(a t S) (P{a) v,P{a)), where at S ranges over a.f.c.s.'s in S. More generally, given any (intuitionistically defined) countable tree model ¢ of A on a m.s. (G, K, R), suppose we identify the nodes (elements of K) with natural numbers, identifying G in particular with 0. Define in terms of (G, K, R) a spread S consisting of all free choice sequences in which the initial choice is 0, and the choice of any natural number m must be followed either by a further choice of m or by a choice of some successor ofm on the tree. To any atomic subformulaP of A, and a.f.c.s. a in S, associate a formula P{a) abbreviating (3x) (3m) (a{x) = m and ¢(P, m) = T). Given B, C, and associated formulae B{a) and C{a), associate with B A C, B{a) A C{a); with B v C, B{a) v C{a), etc. Then, it is easily seen by induction that, for any subformula B of A,
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if¢(B, m) = T,then(a~S)«3x)(a(x) = m) =:> B(a)),andif¢(B,m) = F, , (et ~ S) «3x) (et(x) = m) =:> B(et)). In particular, if ¢(B, G) = T[ =F], then since every a.f.c.s. in S contains = G), we have (et ~ S) B( «) [,(a ~ S) B(et)]. If the m.s. (G, K, R) and model ¢ can be formally described in
°(
Kreisel's FC, the preceding reasoning can be formalized in FC, and thus in particular, if ¢(A, G) = F, f- ,(et ~ S)A(et) in FC, giving a counterexample to the validity of A. To extend this treatment to quantifiers, consider first the following countermodel to (x) (P(x) v Q). =:> • (x)P(x) v Q:
{a}
{a, b}
pea)
P(a),Q
G
H
Figure 3.
We have ¢(P(x), G) = ¢(P(x), H) = T, when x is assigned a, but ¢(P(x), G) = ¢(P(x), H) = F when x is assigned b. Further, ¢(Q, G) = F, ¢(Q, H) = T, GRH but not HRG, and ljJ(G) = {a}, ljJ(H) = {a, b}. All this information is included in the diagram. It is easily verified that ¢«x) (P(x) v Q), G) = T, but ¢«x)P(x) v Q, G) = F. Intuitively, we can interpret the situation as follows: Identify the elements a and b with the integers and 1, respectively. Let R be Fermat's last theorem, and let
°
°
Q be R v ,R. Let V be the species containing 0, and containing 1 if Q is true (i.e., V = {mlm = v (m = 1 A Q)}), and let x be a variable ranging over V. Let P(x) be the statement x = 0. Then, already at the present situation G, we can assert V s; {a, l}, and 1 E ViffQ; so we can assert (x) (P(x) v Q). But so long as we have not advanced to the situation H, where Fermat's last theorem has been decided, so that we can assert Q, we cannot assert (x)P(x) v Q. N.B. It should be remarked that (x) (P(x) v Q). =:> • (x)P(x) v Q holds in any quantificational model such that ljJ(H) is constant. Thus, in general, if the variables in a formula A range over a domain D, then for each situation H, ljJ(H) is the species of all individuals known to be in D on the basis of the information available at H. (So, in the case of the paragraph above, at the present situation G, ljJ(G) = {O}; but when at H, Q has been proved, ljJeH) = {a, I}. Since D is to contain an
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SAUL A. KRIPKE
element, we must know at least one element ofD from the outset, so that I/J(G) must contain at least one element. The restriction that HRH' is to imply I/J(H) C;; I/J(H') should now be obvious on the intended interpretation. Notice that, to assert in a situation H that for every element x of D, P(x) is true, we must know not only that P(x) is true for every x in I/J(H), but also that it is true for every x which may later be proved to be in D; i.e., for every x in I/J(H'), where HRH'; and this is exactly the inductive clause for universal quantification. On the other hand, to assert the existence of an x in Dsuch that P(x) is true, we need to find an element x which has already been proved to be in D (i.e., which is in I/J(H)), and such that P(x) is true; and this is exactly what the condition on existential quantification requires. These facts can again be stated more formally in terms of the theory of absolutely free choice sequences. Suppose we are given an (intuitionistically defined) countable tree m. s. (G, K, R) in which D, and hence I/J(H} for each H is countable. Then, we can identify both the elements of K and the elements of D with natural numbers, identifying G in particular with O. We then associate with (G, K, R) a spread S of absolutely free choice sequences, defined just as above. Further, for any a.f.c.s. a in S, let D" be the species of all natural numbers n such that there is a natural number x such that n E I/J(a(x)). Let x, be a variable ranging over D" (i.e., (x,,) ( ... ) isto be interpreted as (x)(x E D" ::l . . . ) and similarly for (3x,,)). Then since «(O) = 0 = G, and I/J(G) contains a natural number, D" has an element for all a. Let ¢ be an (intuitionistically defined) q. model on (G, K, R)for some formula A. Given any atomic subformulaP'(x., ... , x n) , and an a.f.c.s. a of S, we associate with these two an assertion pea, x I, , . . . , x n), where the variables XI~' ... , xn~ range over D", and where pea, XI~' ... , xnJsays that ¢(pn(x l , •.• , x n), m) = T for some m on a, when Xi~ is assigned to the variable Xi (i = 1, ... , n; note that Xi~ E D" C;; D). Given formulae A( ct, x I~' •.. , xnJ and B( a, Y I~' ... , ymJ associated with A(x l , . . . , x n) and B(y I' ... , Ym), respectively, associate A(a, XI~' ... , xnJ /I B(a, YI~' ... , YmJ with A(x l , . . . , x n) /I B(YI' ... , Ym), and similarly for the other connectives. Further, associate (xi)A(a, XI~' ... , xnJ with (xi)A(x l , . . . , x n), and similarly for the existential quantifier. Then, we prove, by induction, that, for any mE K, if A(x l , . . . , x n ) contains only the free variables listed and XI' ... , Xn are assigned ai' ... , an E I/J(m), then if ¢(A(xI' ... ,
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x n) , m) = T[ =F] relative to this assignment, we have in FC (a t S) ((3x) (a(x) = m) ::::> A(a, aI' ... , an» [.,(a t S)((3x) (a(x) = m) ::::> A (a, aI' ... , an))]. In particular, if m = 0 = G, since (a t S) (3 x) (a(x) = 0), we get (a tS)A(a,a l, ... ,an) [.,(atS)A(a,al, ... , an)]. Thus, if A does not contain free variables, and ¢(A, G) = F, we get a proof in FC that A is not generally valid.
To translate, then, the example given above into FC, notice that, where B is the full binary spread, (a)
(a
t B)(x)((3y) (a(y)
= x)
::::>
(x = 0 V (3y) (a(y) =
1»),
but also (b)
,(a t B)((x)((3y) (a(y) = x)
::::> X
= 0) v (3y) (a(y) = 1»).
Thus we have refuted the "law" (x) (P(x) v Q). ::::> • (x)P(x) v Q; for if it held, it would hold for any free choice sequence a, with x ranging over the species of all z such that (3y) (a(y) = z), contrary to (a) and (b). Notice that, since (a) is a triviality and (b) follows from the fan theorem, we could simply have used the ordinary theory of free choice sequences instead of Fe. We remark that, following Dyson and Kreisel [9], the countermodels in FC that we have described, assigning certain infinite sequences of natural numbers to formulae, can classically be interpreted as countermodels in Baire space (the space of all sequences of natural numbers, with the usual topology). In fact, by examination of the countermodels actually produced below, it follows that every unprovable formula has a countermodel in the Cantor set, as Dyson and Kreisel assert. The following remarks on the uses of absolutely free choice sequences are not relevant to the main point of the present paper, but will be added here: REMARK.
1. All the theorems which are proved in the last chapter of Heyting
[12], using Brouwer's method of free choice sequences depending on the
solving of problems, can be carried out in Fe. To take the first example given by Heyting: to show that it is absurd that, for every real number a, a i= 0 should imply a # O. For if this were true, then for any free choice sequence a in the binary spread, by associating with a the real number
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SAUL A. KRIPKE 00
~ rx(x)/2 x=o
x
,
we could show that -,(x)(rx(x) = 0) ~ (3x) (rx(x) = 1); hence, in particular, this would hold for absolutely free choice sequences. But it is easy to show, in FC, that (rx ~ B) -,(x)(rx(x) = 0). Hence we need only show in FC that -,(rx ~ B)(3x)(rx(x) = 1); but this easily follows from the fan theorem, since (rx ~ B)(3x)(rx(x) = 1) would imply (3m)(rx ~ B)(3x S m) (C«(x) = 1), which is absurd. Similar treatments are possible for all the refutations of classical theorems treated by Heyting by this method in [12].
I think it probable that such treatments in FC will extend to all the counterexamples to classical theorems which Brouwer gives by his method; but I have not made a survey of the literature. A careful reader of the present section on the interpretation of our models will find it plausible that, conversely, a good deal of the interpretation, at least for propositional calculus, that has just been carried out in Fe, could be carried out using Brouwer's method of ips depending on the solving of problems. 2. The following example, which refutes both Kuroda's conjecture (cf. [13]) and Markov's principle (cf. [14]) in FC, was inspired by applying the methods of the present section to obtain a countermodel to (x) -, -, A(x) ~ -, -, (x)A(x). Let S be the finitary spread consisting of all free choice sequences o: such that «(x + 1) = «(x) or «(x + 1) = «(x) + I for every x. We show in FC (a)
(rx ~ S)(m)-'-'(3n) (rx(n) ~ m)
(b)
(rx
~
S)-,(m) (3n) (rx(n)
~
m).
To prove (a), let o: be an a.f.c.s. in S, let m be an integer and suppose for reductio ad absurdum that -'(3n)(rx(n) ~ m). Then, since o. is absolutely free, by axiom 5. 1 of FC, there is an initial segment iX(x) of a such that (*) (13 t S) ((j(x) = iX(x) ~ -,(3n) (p(n) ~ m». Now rx(x) < m, for otherwise (3n) (rx(n) ~ m). Hence, since every ips on S is non-decreasing, for all y < x, rx(y) < m. Now (*) asserts that, if we have chosen the first x components of 13 so that P(x) = iX(x), we can never choose pen) ~ m for any n. But by axiom 5.3 of FC, there are a.f.c.s.'s 13 in S, satisfying the conditions P(x) = iX(x) and f3(x + i) = «(x) + i (0 s i s m - rx(x»,
SEMAN TICAL ANALYSIS OF INTUITIONISTIC LOGIC I
105
since this finite sequence of choices accords with the spread law of S. But then if n = x + m - a(x), {len) = m, contrary to (*). To prove (b), let a be an a.f.c.s. in S, and for reductio ad absurdum assume (m) (3n) (a(n) ~ m). Then, again by axiom 5.1 of FC, there is and x such that (**) ({l ~ S) (~(x) = ii(x) ::) (m) (3n)(~(n) ~ m». Given any a.f.c.s, {l in S, assign a value f({l) as follows: If ~(x) ¥ ii(x), let f({l) = 0; if ~(x) = ii(x), let f({l) be the least n such that {len) ~ a(x) + 1. By (**), f is well defined for all such {l, so by the fan theorem there is some finite integer p such thatf({l) is wholly determined by ~(p). We can thus write f({l) as f(~(p». Clearly, by the definition of J, p ~ x. Now, again using axiom 5.3 of FC, determine {l by requiring ~(x) = ii(x), {l(x + i) = a(x) (0 ::::; i ::::; P - x). Then (**) asserts that {l(f(~(p»)., ~ o(x) + 1. But this is clearly absurd, since again by 5.3 we are perfectly free to continue the choices by {l(p + j) = rx(x) (0 ::::; j ::::; f(~(p» -- p), so that, takingj =f(~(p» -p, we would get {l(f(~(p») = «(x) < «(x) + 1. So (b) is proved. We will now use (a) and (b) to refute Kuroda's conjecture [13] and Markov's principle [14]. Kuroda's conjecture asserts that for m a number variable, (m) -, -,A(m) implies -, -, (m)A(m). Using Kuroda's conjecture, we could derive from (a) the assertion (rx ~ S) -, -, (m)(3n)(rx(n) ~ m), which directly contradicts (b); so Kuroda's conjecture is refutable in FC. Similarly Markov's principle asserts that, for a decidable predicate A(x) and number variable n, -, -, (3n)A(n) implies (3n)A(n). But, if we take A(n) to be rx(n) ~ m, then A(n) is primitive recursive and hence decidable. Then Markov's principle would allow us to derive (rx ~ S) (m) (3n) (rx(n) ~ m) from (a), again contradicting (b). In spite of the proofs by G6del and Kreisel that strong completeness of Heyting's predicate calculus implies certain forms of Markov's principle, I am unable to see how to convert these results into a proof in FC that Heyting's predicate calculus is not strongly complete, and I doubt that such a conversion is in fact possible. If S' is the spread consisting of all tx such that there is a {l in S such that (x) (rx(x + 1) = {lex»~, it is easy to conclude from the present results that (rx ~ S') -'(3n)(rx(n) ~ «(O) and that -,(rx ~ S') (3n)(rx(n) ~ rx(O»; but, since rL here ranges over absolutely free choice sequences of S' and not ordinary free choice sequences, we are unable to apply Theorem 1 of Kreisel [15] to conclude that Heyting's predicate calculus is not strongly complete.
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SAUL A. KRIPKE
1.2. Relationship to the Beth models
In this section, we discuss the relationship of the present model theory to that of Beth [8]. We will show that the present models can be "translated," in a natural way, into Beth models. Using an intuitive interpretation of the Beth modelling, we will also show that the mapping leads to an interpretation of our own quantificational models which is alternative to that of the previous section; in this interpretation, the variables always range over the species of natural numbers. This section can be omitted, if desired, without loss of continuity. First, we present the notion of Beth model in our terminology as follows: Let (G, K, S) be a tree, and let R = S*, so that (G, K, R) is a tree m.s. By a path in the tree (G, K, S) we mean a sequence {H;} of elements of K, indexed on either the sequence of natural numbers or on some finite initial segment thereof, satisfying the conditions: (a) H o = G; (b) for i > 0, H;_ISH;; (c) if {HJ has a last element H m H, is an endpoint of (G, K, S). If some H; = H, we say the path is through H. Let B be a subset of K. If every path through H intersects B, we say that H is barred by B. Thus, for example, H is barred by {H}. By a Beth model on (G, K, S), we mean a binary function IJ(P, H) satisfying the following conditions: (a) rJ(P, H) = T or F, where P is atomic and HE K. (b) If HSH' and rJ(P, H) = T, then rJ(P, H') = T. (c) IfH is barred by Band rJ(P, H') = T for every H' E B, then rJ(P, H) = T. Given a Beth model n, we define by induction values rJ(A, H) for an arbitrary formula A of the propositional calculus. Suppose rJ(A, H) and rJ(B, H) have already been defined. Define rJ(,A, H), rJ(A A B, H), and rJ(A :::> B, H) exactly as was done above for a model ¢; simply replace "¢" by "IJ" throughout. Finally define rJ(A v B, H) = T iff there is a subset B ofK such that H is barred by Band rJ(A, H') = Tor rJ(B, H') = T for every H' E B; otherwise, I1(A v B, H) = F. Notice that if I1(A, H) = T or I1(B, H) = T, I1(A v B, H) = T; we can take {H} as the set B barring H. Notice that condition (b) above actually implies the strengthened condition (b'): If IJ(P, H) = T and HRH', then I1(P, H') = T. Using this fact, it is easy to prove by induction that the properties (a) - (c) actually hold not only for an atomic formula P, but also for an arbitrary formula A. As in the case of models ¢, the inductive definition of I1(A, H) just
SEMANTICAL ANALYSIS OF INTUITIONISTIC LOGIC I
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given depends on the law of the excluded middle. Again as in the case of a model, we can correct the situation by modifying the definition of a Beth model. We leave the modification to the reader. A Beth model '1 on a tree (G, K, S) is called finitary if (G, K, S) is finitary. Beth's own version of his models in [8] is actually equivalent to our notion of a finitary Beth model. We call a Beth model '1 a strong Beth model iff for all HE K and formulae A and B, '1(A v B, H) = T implies '1(A, H) = T or '1(B, H) = T. Notice that, on account of the validity of condition (b') above, a Beth model '1 is also a model in our sense. However, since the inductive clause for disjunction in a Beth model differs from the inductive clause for our sense of model non-atomic formulae may be given different values according as '1 is considered as a model in our sense or as a Beth model. A strong Beth model is precisely a Beth model in which this eventuality never happens. The intuitive rationale behind the Beth models is simple: Again the elements ("nodes") of the tree model (G, K, S) are points in time, or evidential situations; but we no longer suppose that we are allowed to remain at a given point H as long as we please. On the contrary, if H is a node of the tree (G, K, S), we are forced, unless H is an endpoint, to "jump" within a fixed, finite time to one of the successors of H in the tree. (Paradigmatic of such a game, of course, are free choices in an (absolutely) free choice sequence (J(: after each choice we are forced to make another, within a finite length of time, unless the spread-law states that the choice we have just made is terminal.) '1(P, H) = T[ =F] means that P has been established [has not yet been established] at the time H, so the conditions (a) and (b) on '1 are clear. If H is barred by B ~ K, condition (c) says if we know that P will be established at any H' E B, then we already know at H that P is true; for, once we are at H, we must get to some H' E B in a finite time. Similarly, the inductive clause which defines '1(A v B, H) observes that to establish A v B at H it is sufficient to know that, in a finite number of "moves," we must either establish A or establish B; that is to say, it suffices to know that there is a B which bars H such that every H' E B either establishes A or establishes B. The inductive clauses for the other connectives are as before. As in section 1.1, we can give a more precise justification of the
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SAUL A. KRIPKE
definition in terms of absolutely free choice sequences. As before, we identify the elements of the (countable) tree (G, K, S) with natural numbers, associating 0 with G. We then consider the spread S of all absolutely free choice sequences of elements of K whose first term is 0 and which satisfy the condition that a(n)Sa(n + 1), unless a(n) is an endpoint of (G, K, S), in which case a(n) = a(n + 1). For any atomic P, associate a formulaP(a) which says (3n) (3x) (a(x) = n A I/(P, n) = T). We then define inductively a formula A(a) associated with an arbitrary formula A, exactly as in section 1.1. Again as in 1.1, if IJ(A, 11) = T [=F] we can derive (a ~ S) «3x)(a(x) = n) ~ A(a)) [.(a ~ S)«3x) (a(x) = n) ~ A(a))] in FC We now show how the ideas of section 1.1 can be modified so as to show how every model can be transformed into at. "equivalent" strong Beth model. Let ¢ be a model on a m.s. (G, K, R). Define a tree (G', K', S') as follows: Let K' be the set of all finite non-empty sequences {H;}7= l' where Hi E K(1 sis n), H 1 = G, and HiRH i + 1(1 s i < n). Let G' be the sequence whose sole term is G. We say, for H~, H~ E K', that H~ S'H; iff H~ is the initial segment of H~ formed by omitting the last term of H;. (Then, if R' = S'*, H~ R'H; iff H~ is an initial segment of H~. ) For any H' E K', let l(H') be the last term of H', then l(H') E K. Define IJ(P, H') (P atomic, H' E K') by IJ(P, H') = ¢(P, l(H')). Let H' E K', H' = {HJ~= l' We define an associated path P(H') = {Hj}.f:=o as follows: For 0 s j < n, let H, be the unique initial segment of H' with j + 1 terms. For j :2: n, let Hi be the j + I-termed sequence whose first n terms are H 1 , . . . , H, and whose other terms are all equal to H n • So for j :2: n, let l(Hi) = Hn- Clearly P(H') is a path through H'; further, for any Hi on this path, l(Hj)Rl(H'). We now assert: THEOREM 1 (First part): IJ is a strong Beth model. Further IJ is equivalent to ¢ in the sense that IJ(A, H') = ¢(A, l(H')) for any H' E K' and formula A. In particular IJ(A, G') = ¢(A, G) for any A. PROOF. First we show that IJ is a Beth model. Condition (a) is clear. For (b), if IJ(P, H~) = T and H~ S'H;, then l(H~)Rl(H;). Since ¢(P, l(H~)) = IJ(P, H~) = T, and since ¢ is a model, ¢(P, l(H~) = T, hence IJ(P, H;) = T. For (c) let H' be barred by B ~ K'. Then the path P(H') intersects B. Let H" be some point of the intersection. To establish (c)
SEMAN TICAL ANALYSIS OF INTUITIONISTIC LOGIC I
109
it is sufficient to show that if 'l(P, H") = T, 'l(P, HI) = T. Since H" is on the path P(H '), l(H")Rl(H '). Since 17(P, H") = T,
pI---~1
1
p
l l
p p
i
l
p
1" ..
p~
I
.l
p l
I
i
.l
pl p! I
Figure 4a.
It is clear that the model would not essentially change if the infinite vertical branches were reduced to a single point:
pI I
pI
II
p[ I I
I
Figure 4b.
i
p[oo, I
i
Figure 4b is exactly Beth's countermodel in [8] to P v-P.
110
SAUL A. KRIPKE
Since 11 is a strong Beth model, it is also a model; thus the method allows us to transform a model
Let (G, K, R) be any q.m.s., with domain function Ijt(H). For HRH' and H'RH. Let K be the set of all such Hfor every H E K. For H, H' E K, let HRH' ijfHRH', and let $(H) = Ijt(H). Then (G, K, R) is a q.m.s. with domain function l/i. Moreover, if
H
E
K, let
H be the set of H' E K such that
The proof of the lemma is straightforward and is left to the reader.
Theorem I (Second part): Let (G, K, R) be a m.s. such that R is a partial ordering. Let S be any irreflexive relation such that R = S*. Let K be the set of all finite non-empty sequences {H i }7= 1 such that H 1 = G, and H,SH i + 1 for every i(l :5: i < n). Let G be the sequence whose sole term is G. For any H 1, Hz E K, let H 1SHz ijfH 1 is the initial segment of Hz formed by omitting the last term of Hz· Let R = S*. Then (G, K, R) is a tree m.s. Moreover, if
B,
SEMAN TICAL ANALYSIS OF INTUITIONISTIC LOGIC I
111
H) = T. Let HI be any member ofK such that HRH I , where H = 1(0). Then either H = HI' or HSnH I for some n > 0: in either case there exists HI E K such that HRH I and HI = l(H I ). By assumption, either '1(A, HI) = F or '1(B,Ol) = T, whence, by the induction hypothesis, either q,(A, HI) = F or q,(B, HI) = T. Since HI was arbitrary (subject to HRH I , q,(A => B, H) = T. Conversely, suppose l1(A => B, H) = F. Then for some HI such that HRH I , '1(A, 01) = T and l1(B, HI) = F. By the induction hypothesis, q,(A, l(H I ) ) = T and q,(B, 1(H 1 ) ) = F; since l(O)Rl(H I ) , q,(A => B, l(H)) = F, as desired. The case of' is quite
similar. Q.E.D.
Notice that the situation contrasts with that in S4, where it is often impossible to replace an arbitrary finite model by an equivalent finite tree model (cf. [2]). The third part of Theorem 1 extends the procedure for finding a tree model equivalent to an arbitrary model to quantificational models. Here we cannot use the same construction as a tree q. model and as a Beth q. model, as will be seen when we define the latter, in preparation for the fourth part of theorem 1. THEOREM 1 (Third part): Let (G, K, R) be a q.m.s. with domain function ljJ(H). (R need not be anti-symmetric.) Let S be any relation (not necessarily irreflexive) such that R = S*. Let q, be a quantificational model on (G, K, R). Let (G, K, R) be defined as in the second part of the theorem, and let ilI(H) = ljJ(l(O)). Let '1(r, H) = cp(r, l(A)) for each predicate letter P" and each H E K. Then '1 is a quantificational model on the q.m.s. (G, K, R) with domain function ill. Further, relative to a given assignment to the free variables of A, '1(A, H) = q,(A, l(H)): in particular, '1(A, G) = q,(A, G).
The proof is left to the reader. Notice that, since S is not required to be irreflexive, it may in particular be R itself: thus (G, K, R) may be as in the second part of Theorem 1, or may be identical with the Beth model (G', K', R') of the first part. As a quantificationa1 model, however, '1 will not be a Beth quantificationa1 model, to the definition of which we now turn. Unlike our own models, with their variable domains (a feature we have noted to be essential), the Beth quantificationa1 models are based on a fixed domain D. We define a Beth q.m.s. to be a Beth m.s. (G, K, R),
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SAUL A. KRIPKE
together with a domain D with at least one element. A Beth q, model Yf is a binary function Yf(pn, H), whose value is T or F when n = 0, and is a subset of D" for n ~ 1. We require, in addition to the conditions (b) and (e) above on n, the analogues for n ~ I: (bn). If HRH', Yf(r, H) S; H'); (en) if H is barred by B S; K, then
«r:
n
H', B
Yf(r, H')
S;
Yf(pn, H).
For an atomic formula r(x l , . . . , x n), define Yf(r(x I , . . . , x n), H) = T, relative to an assignment of aI' .. , anEDtox l , . . . ,xmiff(a I , . . . ,an)E Yf(r, H); otherwise, =F. We then define the values for more complex formulae by induction. The inductive clauses for the propositional connectives are as above. Let the formula A(x I , ••• , X m y) contain only the free variables listed. We define Yf«y)(A(x I , . . . , x m y), H) = T, relative to an assignment of a, ED to Xi (1 ::::; i ::::; n), iff Yf(A(x I , ••• , (xn,y), H) = T relative to any assignment of an element bED to y and a, to Xi; otherwise, = F. Again Yf«3y)A(x I , ••• , X n, y), H) = T when a, is assigned to Xi iff there is a B S; K such that H is barred by B and for any H' E B there is a bED such that Yf(A(x I , .•• , X m y), H') = T when ai is assigned to Xi and y is assigned b; otherwise, = F. Using the inductive clauses and the conditions on atomic formulae, we can prove the analogues of (b) and (e) for an arbitrary formula A, relative to a fixed assignment to its free variables in a Beth quantificational model n, If Yf(A, H) = T and HRH', Yf(A, H') = T. If H is barred by B and 1'f(A, H') = T for any H' E B, then Yf(A, H) = T. Suppose we are given a quantificational model ¢ on a m.s. (G, K, R) such that u = U l/I(H) H,K
is countable. We will transform ¢ into a Beth quantificational model whose domain D is the set N of non-negative integers. Let (G/, K /, S') be as above, and R' = S'*. Notice that N is a countable union of disjoint countable sets; call these NJi = 0, ... ). We have a procedure, which, for each H' E K', generates certain elements of N at H'; the set of elements generated at H' will be identical with n
UN;
i=O
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for some n. Further, ifP is any path in K', every pEN will be generated at some H' E P. Further, the procedure will satisfy the condition that if H' R'H", every element generated at H' is also generated at H". An element generated at H', but not at its predecessor (if any exists), is said to be introduced at H'. Further, any natural number n generated at H' is assigned a unique element of t/J(l(H')); this element is called v(n, H'). The v-function will satisfy the condition that if n is generated at H', and H'R'H", then v(n, H') = v(n, H"). We give an inductive definition on the tree (G', K', S') of a procedure with these properties; at any stage, satisfaction of these properties will be taken to be part of the inductive hypothesis. First, consider the origin G' of the tree. We generate exactly the elements of No at G', and we define v(n, G '), for n E No, in such a way that No is mapped onto t/J(G). (This is possible since t/J(G) is at most countable. All arbitrary choices can be made precise, if desired, using well-orderings of the denumerable sets Nand Ll.) Suppose we have defined the set of all integers generated at H' it is, say, m
(M = U N;) i~O
and have defined v(n, H') for each n E M. Let H'S'H". Then introduce all elements of N n + l' so that the set of elements generated at H" is M v N m + l' Define v(n, H") for n E M v N m + 1 by v(n, H") = v(n, H') for n E M, and such that v(n, H") maps N n + 1 onto t/J(1(H")). Then the inductive definition is complete. We now define a Beth quantificational model Yf whose domain is N on the Beth m.s. (G', K ' S') as follows: If P is a propositional letter, define Yf(P, H') = ¢(P, l(H')). For an n-adic predicate letter n define Yf(r, H') to be the set of n-tuples (ml' , m n ) of natural numbers such , m; are all generated at H" and that, for every H" E K' such that m l , H'R'H", (v(m l , H"), ... , v(mm H")) E ¢(r, l(H")). THEOREM I: (Fourth part): Yf is a Beth quantificational model on (G', K ', S') whose domain is N. For any H' E K' and formula A(x l , . . . , x n ) , whose free variables are exactly those listed, and natural numbers m l' . . . , m.; which have been generated at H', Yf(A(x1 , ••. , x n) , H') = T when Xl' ••• , X n are assigned m l , . . . , m i; respectively, if and only if ¢(A(x 1 , ... , x n) , l(H')) = T when Xl' ••. , X n are assigned v(m 1 , H'), ... , v(m n ,
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H'), respectively. In particular (n = 0), ifA is a closedformula, rJ(A, H') = cf>(A, l(H'». PROOF. We show first that rJ is a Beth quantificational model. Conditions (b) and (b n ) are obvious. Condition (c) is proved as in the first part of the theorem. Condition (en) (n ~ 1) is proved as follows: Suppose H' E K' is barred by B £; K', and suppose {m I' ... , m n) is not in rJ(pn, H'). We show that there is an H" E B such that (m l , ... , m n ) is not in rJ(pn, H"). Since (m I' ... , m n) is not in I/(pn, H'), there is an H~ E K' such that H' R'H~, m I' . . . , m n are all generated at H~, and (v(m I' H~), ... , v(mn> H~» is not in cf>(pn, l(H~)). As in the first part of this theorem, let P be the path P(H~) through H~, with the property that, for H" on the path and H~R'H", l(H~) = l(H"). Then P intersects B in an element H". If H" R'H~, then since clearly (m l , ... , mn) is not in rJ(P", H~), by condition (b"), it is not in rJ(pn, H"). If H~R'H", then since l(H") = l(H~), and v(mi' H~) = v(mi' H"), we have «v(m l , H"), ... , v(m n, H"» tt ¢(pn, l(H"», so that (m l , . . . , mn) ¢ rJ(P", H"), the desired conclusion.
We now prove the assertion in the second sentence of the present Fourth part by induction; the third sentence is a special case. Let A(x l , . . . , nX) be atomic. If n = 0, see the proof of the first part of this theorem. If n > 0, write A(x l , . . • , xn) as P"(x l , . . . , xn) . Suppose m l , . . . , mn are all generated at H' E K '. Let H = l(H'), and a, = v(mi' H'). If c/>(P"(XI' ... , x n) , H) = T, when Xi is assigned a, (l ~ i ~ n), then (a I ' . . . , an) E cf>(P", H). If H' R'H~ (H~ E K'), let H o = l(H~). Then HRH o, hence a I' , an E ",(H o) . Also a, = v(mi' H') = v(mi , H~). This shows that , Inn) E rJ(pn, H'), hence rJ(pn(x l , . . . , x n) , H') = T, relative to the. (m l , assignment of m, to Xi' as desired. On the other hand, if cf>(P"(x I' •.. , x n ) , H) = F relative to this assignment, and hence (ai' ... , an) ¢ cf>(pn, H), we clearly have (m l , . . . , mn ) ¢ rJ(P", H'), again as desired. The inductive clauses for the propositional connectives are as in the first part of this theorem. Suppose the result proved for A(x l , . . . , X n , y). Again let m, be assigned to Xi' let H = l(H'), and let a, = v(mi' H') (i = 1, ... , n). Let cf>«3y)A(xl , ••• , X n , y), H) = T when Xi is assigned a.. Then there is e b e ",(H) such that cf>(A(x I' ... , X n , y), H) = T when in addition y is assigned b. v(p, H') maps the elements generated at H' onto ",(H), so let v(p, H') = b, where p is generated at H'. Then, by inductive hypothesis rJ(A(xl , • . . , X n , y), H') = T when Xi is assigned m, (i = 1, ... , n) and
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X n, y), H') = T when x, is assigned m.; On the other hand, suppose
y is assigned p; hence '7«3y)A(x I' ... ,
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SAUL A. KRIPKE
1](A(Xl' ... , x n , y), Hi) = F when Xi is assigned m, and y is assigned p. Hence 1]«y)A(x 1 , ••• , x n ' Y), H~) = F when Xi is assigned m.; but since H'R'H'b 1]«y)A(x 1 , ••. , Xm y), H') = F relative to this same assignment.
This concludes the proof of the theorem. Q.E.D.
The fourth part of Theorem I shows how any quantificational model 1]. Essentially if we have arrived at a certain position H' E K' and if H = 1(H'), the numbers introduced at H' are "identified" with certain elements 0 f I/I(H) by v (n, H'). An example, following the spirit though not the letter of the proof of Theorem I, fourth part, converts the countermodel of section I . I, Figure 3, for (x) (P(x) v Q). ~ . (x)P(x) v Q into a corresponding Beth quantificational countermodel in the natural numbers. In Figure 3, there are two evidential situations, G and H; I/I(G) = {a}, t/J(H) = {a, b}. As natural numbers are generated, as long as we remain at the evidential situation G, we must "identify" each natural number with a (and therefore give it all properties assigned to a in Figure 3), but if we pass to H, we must "identify" some natural number with b. These considerations lead to the following figure:
¢ can be transformed into a Beth quantificational model
P(O)
P(1)
i
P(2)
Q i H. I
i---r
Q i H.
Figure 5.
P(3)
I
Q i H, I
This is exactly Beth's counterrnodel to (x) (P(x) v Q). ~ . (x)P(x) v Q. As long as we remain on the horizontal branch but are uncertain that we will continue thereon, we have not established (x)P(x) v Q; but on the other hand, for each natural number x, either P(x) or Q is eventually established. We have not mechanically applied the proof of Theorem I to obtain this model, but instead have reproduced its spirit; in particular, we have introduced a simplification analogous to that required to obtain Figure 4b from Figure 4a. Notice that Figure 5 can be interpreted in terms of absolutely free choice sequences as follows: Let a be an absolutely free choice sequence on the binary spread. Let P(x) abbreviate a(x) = 0, and let Q be (3x) (a(x) = I). Then, if x ranges over the natural numbers, clearly (a ~ B) (x) (P(x) v Q), but ,(a ~ B) «x)P(x) v Q). And, analogously, as Kreisel
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and Dyson (Kreisel [II] and Dyson & Kreisel [9]) have observed, countable Beth quantificational models can always be interpreted thus. So Theorem I gives a new intuitive interpretation of our models, in which all quantifiers range over the natural numbers. Since below we will obtain a completeness theorem for countable quantificational models, and since such models can always be transformed into Beth (q.) models, our completeness results include those of Beth. (Beth required his models to be finitary, but we will show in part II how to obtain finitary Beth models.)
1.3. Other interpretations ofthe models Sections 1. I and 1.2 gave interpretations of our models which were intended to accord with the interpretations intuitionists customarily assign to their logical constants. In this section we will give two formal interpretations of the modelling which do not claim any direct intuitionistic content. (Both interpretations are actually direct special cases of the modelling; they simply consider a restricted class of models.) One interpretation is based on provability in formal systems; it was described briefly in [3]. The other is based on Paul Cohen's notion of forcing [5]. The two interpretations are intimately related to each other. This section may be omitted without loss of continuity.
I. Provability interpretation. Let Eo be a formal system, and let E be an arbitrary extension thereof. Let K be the set of all such E, and let ERE' iff E' is an extension of E. We define an atomic formula P to be a closed wff of Eo. (Note that P need not be an atomic formula of Ea.) We can then build non-atomic formulae out of the P's using the connectives A, ::J, " v. If we define l/>(P, E) = T iff P is provable in E and F otherwise, then l/>(P, E} is a model on the m.s. (Eo, K, R). Thus for any complex formula A which is a theorem of the intuitionist propositional calculus, l/>(A, Eo} = T. If Eo is elementary number theory Z, and P is Godel's undecidable formula, then l/>(P v ,P, Eo} = F; for P is not provable in Eo, but it is provable in certain extensions E. The larger problem, whether Heyting's propositional calculus is complete with respect to this particular choice of Eo, remains open. To interpret intuitionistic quantification theory in this manner, we must assume that the system Eo and its extensions have notions of free
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variables and of constants, and that Eo contains at least one constant. For any E E K, let I/I(E) be the set of all constants of E. Then if ERE', I/I(E) s I/I(E'). For every n, define an n-adic atomic predicate P" to be a formula of Eo with n free variables, together with a I-I function from the integers I, ... , n to the free variables of P", The variable assigned by this function to m(1 ~ m ~ n) is called the mth free variable of P". Define, for n ~ 1, the set ¢(pn, E) s [I/I(E)]n as follows: An n-tuple (a l ' . . . , an) of constants in I/I(E) is in ¢(P", E) iff the result of the simultaneous substitution of aj(1 ~ i ~ n) for the ith free variable of P" is a theorem of E. Out of the atomic n-adic predicates (which play the role of the n-adic predicate letters above), we can build more complex formulae using the propositional connectives and the quantifiers. ¢(P", E) then becomes an intuitionistic quantificational model. It is clear that in the preceding K can be replaced by any subset K' thereof (e.g., the finitely axiomatizable extensions of Eo). Further, restrictions, such as recursive enumerability, on the notion of formal system, can be removed at will. There is also a more "model-theoretic" variant of the present interpretation of Heyting's predicate calculus, which eliminates the assumption that E must-contain constants. Further, the interpretations can be extended in other directions so as to yield new interpretations of larger parts of intuitionistic mathematics; in particular, we can give an interpretation of FC which leads to a proof that FC is an inessential extension of Heyting's arithmetic"). For more on provability interpretations of intuitionistic and modal logics, cf. [3]. 2. Cohen's notion of "forcing," Let D be an arbitrary countable infinite set. Let 9 = (9 0 , ( 1) be a pair of finite, disjoint subsets of D, and let K be the set of all such pairs. If 9 = (.0/'0' !!J I) and 9' = (9~, 9'1) are in K, theqdefine 9 R9' (or, f!}' is an extension of 9) iff 9"0 s 9~ and 9 1 S 9~. Further, let I/I(g» = 9 0 u:3' 1. Now consider a single monadic predicate letter P. For any g; E K, define ¢(P,9) = 9 0 . Let K' be the set of all 9 E K such that 1/1(.9) is non-empty. Then for any g; E K', (g>, K', R) is a q.m.s., with the associated domain function 1/1. (If we had modified Heyting's predicate calculus so as to admit the empty domain and thus permit I/I(Y') to be empty, the rather artificial use of K' in place 1) Kreisel has independently obtained this result using an elimination of free choice sequences by contextual definition.
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of K could be dropped.) Then ¢ is a model on (g'J, K', R), and for any formula A built from P using propositional connectives and quantifiers, the inductive definitions we have given define a truth-value ¢(A, g'J'), for any g'J' E K', relative to a fixed assignment of elements of D to the free variables of A. If this value is T, we say that g'J' forces A relative to the assignment. (Notice that the value of ¢(A, g'J') is clearly independent of the choice of the "designated" element g'J of (g'J, K', R).) If D' is a subset of D, we say that g'J , agrees with D' iff g'J~ s D' and g'J~ s D - D'. We can say that D' forces A (relative to a given assignment to the free variables) iff there is a g'J' E K' which agrees with D' and forces A. Notice that if g'J' and g'J" agree with D', they have a common extension which agrees with D'; thence it easily follows that D' cannot force a statement together with its negation. Call D' generic iff for every A, and fixed assignment to the fret: variables thereof, D' forces either A or ..,A. Cohen proves that generic sets exist: Let {An} be an enumeration ofall the ordered couples Ai = BIB. If D' is generic and forces ..,..,A, it clearly must force A; hence a nonempty, generic D' forces every classically valid formula not containing universal quantifiers. Cohen has proved an even stronger fact: If D' is generic and A has no universal quantifiers, then (relative to an assignment to free variables), A is forced by D' if and only if it is true when the existential quantifiers (taken as ranging over D) and the propositional connectives are interpreted classically, and "P(x)" is interpreted as "x ED'," The
e::
e-:
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SAUL A. KRIPKE
assertion is readily proved by induction on the complexity of A. Since, classically speaking, a (x) can always be replaced by ,(3xh the restriction that universal quantifiers be absent is not important. The definition we have given differs from Cohen's in inessential respects. (It may be closer to a definition given by Feferman, which we have not seen 1). It is clear that the notion can be extended. For example, we need not deal with a single predicate P(x); we can deal with several such, not all of which need be monadic. The modifications needed for this more general situation should be obvious. Further, we can replace the countable set D by a set of regular cardinality N,,; K will consist of disjoint pairs of sets of cardinality less than ~". Cohen's motivation was radically different from ours, but it is clear that his notion is intimately related to our model theory. The "deeper" reasons for this relation may yet be unknown. It should be noted that Dana Scott had already observed that Cohen's idea was similar to an interpretation conjectured by Kreisel [17]. And indeed, if Kreisel's conjectures prove correct, his interpretation of intuitionism will be closely related to ours.
2. Semantic tableaux In this section we develop Beth semantic tableaux for intuitionistic logic. The notion developed here is similar to those of [2], [11], which can be read as background if desired. We deal at each stage of the construction with a system of alternative sets of tableaux; each alternative set is ordered in the form of a tree, and the origin of the tree is called the main tableau of the set. We call the tree ordering relation on an alternative set "S"; the smallest reflexive and transitive relation containing "S" is called "R". We can assume, at a given stage of the construction, that each alternative set is diagrammed on a piece of paper; corresponding to the system of all the alternative sets of the stage, we have a leaflet of which the separate sheets of paper are pages. Given a formula A of Heyting's predicate calculus, to see whether it is valid we attempt to find a countermodel to A. If A has the form Al A •.. Am. :=> • B I V ... B n, then what we need is a model ¢, such that relative to some assignment to the free variables of A, ¢(A j, G) = T and 1) See note at end of paper.
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¢(B j ' G) = F, 1 :0; i :0; m, 1 :0; j :0; n. We represent the situation by putting A l , ... , Am on the left, and B l , . . . , B; on the right of the main tableau of a construction. We continue the construction, which gives a systematic attempt to find a tree countermodel to A, by the following rules, which apply to any tableau of any alternative set of the construction:
NI. If..,A appears in the left column of a tableau, put A in the right column of that tableau. Nr. If..,A appears in the right column of a tableau t, start out a new tableau t l , with tSt", by putting A on the left of t l • AI. If A left of t.
A
B appears on the left of a tableau t, put A and B on the
Ar. If A A B appears in the right column of a tableau t, there are two alternatives; extend the tableau t either by putting A in the right column or by putting B in the right column. If the tableau t is in an ordered set Y, it is clear that at the next stage we have two alternative sets, depending on which extension of the tableau t is adopted. Informally speaking, if the original ordered set is diagrammed structurally on a sheet of paper, we copy over the entire diagram twice, in one case putting in addition A in the right column of the tableau and in the other case putting B; the two new sheets correspond to the two new alternative sets. The formal statement is rather messy: Given a tableau t in an alternative set Y, if t has A A B on the right, we replace Y by two alternative sets Y 1 and Y 2, where !f'l = Y - {t} u {tl} and Y 2 = Y - {t} u {t 2}, and t l [t2 ] is like t except that in addition it contains A [B] on the right. The tree ordering S 1 of the new set Y 1 is precisely the same as S, save that t l replaces t throughout; and similarly for the tree ordering S2 of Y 2 • (Formally, Sl agrees with S on !f' - {t}, and, if t' is the predecessor [a successor] of t, then t'Sltl[tlSlt'].) We say!f' splits into Y l and !f'2' Similar remarks apply to the rule VI and PI below. VI. If A v B appears on the left of t, put either A on the left of t or B on the left of t. (As in the case of Ar, this splits the set !f' containing t into two alternative sets.) Vr. If A v B appears on the right of t, put A and B on the right of t. PI. If A
=:>
B appears on the left of t, either put A on the right of t
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or put B on the left. (Thus again the set g> containing t is replaced by two alternative sets.) Pro If A ::> B appears on the right of t, start out a new tableau t l, with A on the left of t 1 and B on the right, such that tSt l • For a construction involving quantifiers, we associate, at a given stage of a construction, a set I/1(t) of variables with each tableau t. We start out the definition of I/1(t) by assuming that, at the initial stage of the construction, which starts out with a single tableau to, I/1(to) consists of a single variable x. At later stages I/1(t) is to be enlarged only as required by the rules Ilr and II below and the stipulation that tSt l is to imply that I/1(t) S I/1(t 1 ) . We are now in a position to state the rules for quantifiers as follows: Ill. If (x)A(x) appears on the left of t and y is any variable in I/1(t), put A(y) on the left of t. Ilr. If (x)A(x) appears on the right of t start out a new tableau t 1 with tSt l • If y is the alphabetically earliest variable which has not yet occurred in any tableau of any alternative set at this stage, put y E I/1(t 1) and put A(y) on the right of t': El. If (3x)A(x) appears on the left of a tableau t, and y is the alphabetically earliest variable which has not yet appeared in any tableau of any alternative set at this stage, put y E I/1(t) and put A(y) of the left of t. Et, If (3x)A(x) appears on the right of a tableau t, and y is a variable in I/1(t), put A(y) on the right of t.
In addition to the rules we have stated, the following stipulation holds throughout the construction: if t and t 1 are tableaux of some one alternative set, at any given stage, such that tSt! , and A appears on the left of t, then put A on the left of t 1. Notice that, since the stipulation is to be iterated an arbitrary number of times, it also applies when A is on the left of t and tRt l • The relation tSt l is to hold in a construction only as required by the rules listed above. The rules may be applied in any order, as long as the order stipulated is such that every applicable rule is eventually applied. A tableau t is called closed iff some formula occurs in it on both the left and the right. A set or tree of tableaux is closed iff some tableau in
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the set is closed. A system of alternative sets is closed iff every set of the system is closed. A construction started out by putting A on the right of the main tableau of the construction is called the construction for A. We can place the following restrictions on constructions: A rule is not to be applied to a tableau of a closed set; nor is it to be applied if it is "superfluous" (e.g., Al is not to be applied if A and B already appear on the left of the tableau t in question). Let us call an alternative set at any stage of a construction terminal iff it is not replaced at any stage of the construction by another set or pair of sets; thus, in particular, every closed set is terminal. In any construction, let a be some fixed sequence Y'1' Y'2' ... of alternative sets such that Y'1 is a set at the first stage of the construction and Y'i+ 1 is the set or one of the two sets, which, at the (i + 1)-th stage, replaces Y'i; a terminates at Y' n iff Y' n is terminal. (If the construction does not terminate there is at least one infinite such sequence a.) Any tableau t in Y'I or in Y'i + 1 which is not an immediate descendant of any tableau in Y'i is called an initial tableau. Let K be the set of all sequences. of tableaux tl' ' 2, ... such that 11 is an initialtableau andr, + I isan immediate descendant of t, and r terminates at iff belongs to a terminal set Y'm. Let be that member of K whose first term II is in Y'1' Let tpx', for r, t ' in K, iff for some Y'i in a there are terms t, I' of t, t' in Y'j such that IRt' (R the ancestral of the tree ordering S). Then, intuitively, (.0' K, p) forms a q.m.s. with domain function
'n 'n
.0
'in
If a quantificational model ¢ is defined so that, for any sentence letter P, ¢(P, r) = Tiff P appears on the left of some I in r, and, for any predicate letter P", ¢( P", r) is the set of n-tuples (x l ' . . . , x n) of variables such that pn(x 1 , .•• , x n) appears on the left of some I in r, then, for every formula B, if B appears on the left of some I in r, ¢(B,.) = T (relative to the
assignment of each free variable in B to itself). Further, the dual law that, for every B. if B appears on the right of some I in r, then ¢(B, r) = F, holds iff z does not terminate in a closed set Y' n' Hence, if the construction was a construction for A, this is just the condition under which a provides a countermodel for A.
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2: The construction for A is closed if and only if A is valid.
The proof, which follows the lines sketched intuitively above, and in addition shows that the alternative sets of the construction for A exhaust the possibilities of finding a countermodel for it, is omitted because it is a routine variation on the proofs of the corresponding theorems of [2] and [16jl). 3. Completeness theorem 3. 1. Consistency property THEOREM
3: If A is provable in Heyting's predicate calculus, then A is
valid.
This theorem is almost trivial; we need only verify that, in a standard formalization of Heyting's predicate calculus, the axioms are all valid, and the rules preserve validity. Such a verification is left to the reader. It follows that if A is provable, the construction for A is closed. 3 . 2. Completeness property
We show that every valid formula A is provable by showing that if the construction for A is closed, then A is provable. As in [2] and [16], we do this using a notion of "characteristic formula." As in [2], define the rank of a tableau in a finite tree of tableaux (or, indeed, of a node in any finite tree), as follows: An endpoint of the tree has rank O. If tis not an endpoint, let t 1 , ..• , t n be its successors; then Rank(t) = Max {Rank (ti)}+l. It is easy to verify that, for any finite tree of tableaux, a unique rank is defined for each tableau of the tree. 1) Define A to be tree valid iff '" (A, G) = T for every model rp on a tree q.m.s,
(G, K, R). Then what really is readily proved is that the construction is closed iff A is tree valid. But, by section 1.2 above, validity coincides with tree validity. Alternatively, we can argue as follows without use of section 1.2: Clearly validity implies tree validity, and provability implies validity. The completeness result below shows that tree validity implies provability, so the three notions coincide. We could have defined a tableau procedure, based on a relation R, which would have been more appropriate to models than to tree models; a reader familiar with [21 will know how this could be carried out. Notice that, as observed in analogous cases in [21 and [161, the countermodels for non-valid formulae obtained by Theorem 2 from tableaux are always on a countable tree q.m.s. (G, K, R) with a countable set U of individuals involved. This "LowenheimSkolem" result will be used in part II to show that the present completeness results include those of Beth [81.
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Given any tableau t in a tree of tableaux, define the following sequence {til: to = t, t j + l = the predecessor of t j , if such a predecessor exists, and undefined otherwise. The sequence is clearly finite, and its last term is the origin of the tree We call it the "path from t back to the origin." The terms of the sequence other than t "come before t" on the tree. For any t on a tree, let X(t) be the set of all variables occurring free in t but not in any tableau coming before it. At any stage of a construction, the tableaux of an alternative set form a finite tree. We define the characteristic formula of a tableau t in the set at a given stage by induction on its rank in the set. Given a tableau t, let AI' ... , Am[Bl , . . . , B n] be the formulae occurring on the left [right] of t. Further, let Xl' ... , x q be the elements of X(t). (Possibly q = 0.) If Rank (t) = 0, then the characteristic formula of t is defined as (x.). .. (Xq ) (AI A .. . A m.::::> .B l V .. • B n ) ; or, if there are no formulae on the left [right] of t, as (Xl)' .. (Xq) (B I V ... B n ) [(Xl)' .. (xqHA 1 A Am)]' If Rank (t) > 0, let t l , . . . , t p be the successors of t, and let C l , , Cp be the corresponding characteristic formulae. Then the characteristic formula of t is (Xl)' .. (x q) (AI A •.. Am.::::> .B; V ... B; V C, V ... C p ) ; or, if there are no formulae on the left [right] of t, the characteristic formula is (Xl)" . (Xq ) (B 1 V .. . B; V C l V " .Cp ) [(Xl)" . (Xq ) (A 1 A .. • A m.::::> ,C 1 V ... C p ) ] . The characteristic formula of an alternative set (tree) of tableaux is defined as the characteristic formula of the main tableau of the set. The characteristic formula of the entire system of alternative sets at a given stage of a construction is defined as the conjunction of the characteristic formulae of the alternative sets of the system. In a natural sense, the present notion of characteristic formula is "dual" to that of [2] and [16]. It may facilitate the reader's comprehension of the notion of characteristic formula if he consults the corresponding treatment of characteristic formulae in [2], [16]. LEMMA: If A o is the characteristic formula of the initial stage of a construction, and B o is the characteristic formula of any stage of the construction, then I- B o ::::> A o.
PROOF. It suffices to show that the characteristic formula of any stage of the construction implies the characteristic formula of the preceding stage. But the characteristic formula of the mth stage has in general the orm D 1 A .. . Dj A .. . Dn , where the Di(l :::; i :::; 11) are the characteristic
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SAUL A. KRIPKE
formulae of the alternative sets of the stage. The rule which is applied and changes the mth stage into the m + lth affects only one alternative set, say with characteristic formula D i: If the rule is PI, Ar, or VI, it will change this set into two distinct alternative sets, with characteristic formulae D', and Dj; we wish to prove, then, J- D 1 A .. . D', A DjA ... D n . =:> • D 1 A ... D j A ... D w To do this, it suffices to prove D', A Dj. =:> • D i: Similarly, if the rule applied is other than PI, Ar, or VI, then D j is transformed into Dj; to prove that J- D 1 A ... D', A ... Dn' =:> • V 1 A ... D j A ... D n , it suffices to prove J- D', =:> D i: So, when a rule is applied transforming the mth stage of a construction into the m + 1th, we need only consider the characteristic formula of the set to which the rule is actually applied. Suppose, then, a rule (other than PI, or Ar, or VI) transforms a set Y with characteristic formula D j into one with characteristic formula Dj; we wish to prove J- Dj =:> D i: Let t be the tableau to which the rule is actually applied, and let C be its characteristic formula. Further, let C' be the characteristic formula of the tableau t' into which t is transformed by the given rule. (The rules Nr, Pr and IIr leave t unchanged, appending a new tableau t': In this case t' will be identical with t, but the new characteristic formula C' of t will not be identical with the old one C.) Suppose we can show J- C' =:> C. Then if t is the main tableau of the set Y, we have shown J- Dj =:> D j • Otherwise, let t 1 be the predecessor at stage m of t, let t~ be the predecessor at stage m+ Lof r', and let Cl[C~J be the characteristic formula oft 1 [ t a Then C 1 is a universal quantification (u.q.) of a formula of the form X. =:> • Yv C, and C~ is a u.q. of X. =:> • Yv C'. Since J- C' =:> C, clearly J- (X. =:> • Yv C') =:> (X. =:> • Y V C). Applying universal generalization to this last statement, and distributing universal quantifiers across the implication sign, we obtain J- C~ =:> C iIf t 1 is the main tableau of g, then C~ =:> C 1 is D', =:> D ; Otherwise, let t 2[t;J be the predecessor of tl[t~J, and apply the same reasoning as before. Eventually we will obtain D', =:> D j • Thus in the case ofany rule other than PI, VI, or Ar, we need only consider the tableau t to which the rule is actually applied, and prove the formula C' =:> C stated above. Notice that in general C, the characteristic formula of t, is a u.q. of a certain formula B, and C' is a u.q. of a certain formula B'. If we prove J- B' =:> B, then by universal generalization and distribution of the quantifiers across the implication sign, we can obtain C' =:> C.
SEMANTICAL ANALYSIS OF INTUITIONISTIC LOGIC I
127
Bearing these remarks in mind, we break down the proof into the following cases, depending on the rule applied to obtain the m + 1th stage from the mth. We can say a case is "justified," if we have shown, for the case, that f- D / ::::J D j' which usually reduces to f- B' ::::J B. The reader is advised to consult the similar treatments in [2] and [16]. In considering a rule, we will in general assume that the tableau t to which it is applied contains formulae both on the left and the right, and that its characteristic formula is therefore an implication. The cases where the left or right side is empty will be left to the reader. Case NI. The characteristic formula of t is a u.q. of X A ,A. ::::J. Y; after A has been put on the right, its characteristic formula becomes a u.q. of X A ,A. ::::J • Y v A. The case is justified by f- X A ,A. ::::J • Yv A: ::::J : XA,A.::::J .Y. Case Nr. The characteristic formula of t is a u.q. of X. ::::J • ,A v Y. When we start out a new tableau t 1 with A on the left, and t St", the characteristic formula of r ' is ,A (since X(t 1 ) is empty because any free variable of A already occurs in t), and that of t becomes a u.q. of X.::::J .,A v Yv ,A. The case is justified by f- X.:::::>. ,A v Yv,A::::::> :X. :::::>. ,A v Y. Case A 1. Justified by f- X
A
A
A
BAA
A
B. ::::J . Y: ::::J :X A A
A
B. ::::J . Y.
Case Ar. Let the characteristic formula of t, call it C, be a u.q. of X.:::::> . Yv (A A B). The rule Ar "splits" t into two alternative tableaux, t' and t", whose characteristic formulae C' and C" are u.q.'s of X . :::::> • Y v (A A B) v A and X.::::J . Yv (A A B) v B, respectively. Using f- (X.:::::>. Y v (A A B) v A) A (X. ::::J . Yv (A A B) v B): ::::J: X. ::::J . Yv (A A B), and generalizing, and distributing quantifiers, we obtain f- C' A C". :::::> • e. If t is the main tableau of the set, this is the desired result f- Dj A Dj. ::::J .Dj • Otherwise, let t 1 be the predecessor of t. The characteristic formula C 1 of t 1 is a u.q. of Xl' ::::J. Y1 V C; it is transformed by Ar into two alternative characteristic formulae C~ and e~, which are u.q.'s, respectively, of Xl' ::::J . Y1 V C' and Xl' ::::J . Y 1 V C", Using f- C' A C", ::::J • C, we easily obtain f- C~ A e~. ::::J . C 1 . Continuing this process along the path from t back to the origin, in a finite number of steps we obtain f- Dj A Dj. ::::J . Di: Case PI. Like Ar, using f- (X A (A ::::J B). ::::J . Y v A) B.:::::>. y)::::::>:XA (A :::::> B).:::::>. Y.
A
(X A (A ::::J B)
A
128
SAUL A. KRIPKE
Case Pro Let the characteristic formula of t be a u.q. of X. ::::> • Y v (A ::::> B). Pr instructs us to start out a tableau t 1 , with A on the left and B on the right, whose characteristic formula is thus A ::::> B (X(t 1 ) being empty). Then the characteristic formula of t is transformed into a u.q. of X.::::>. Yv (A ::::> B)v (A::::> B), and I- X.::::>. Yv(A ::::> B) v (A::::> B):::::>: X. ::::> • Yv (A ::::> B) justifies the case. Case VI. Like Ar, using I- (X A (A v B) B.::::>. Y):::::> :(X A (A v B).::::>. Y).
A
A.
::::> • Y) A
(X A (A v B)
A
Case Vr. Justified by I- X.::::>. Yv (A v B)v A vB:::::> :X.::::>. Yv (A v B). Case 171. If t has as characteristic formula C, a u.q. of X A (3x)A(x). ::::> • Y, after application of 171, t is transformed into t 1 , whose characteristic formula C' is a u.q. of X A (3x)A(x) A A(a). ::::> • Y. Since a is a new variable not previously introduced, a E X(t 1 ) . Thus, we can take C' to be a U.q. of (a) (X A (3x)A(x) A A(a). ::::> • Y). So I- (a) (X A (3 x)A (x) A A(a). ::::> • Y):::::> :X A (3x)A(x). ::::> • Y justifies the caseCase 17r. Justified by I- X.::::>. Y v (3x)A(x) (3x)A (x). Case III. Justified by I- X
A
(x)A(x)
A
A(a).
A
A(a):::::> :X.::::>. Y v
::::> • Y: ::::>
:X A (x)A(x).
::::> • Y.
Case IIr. The characteristic formula of t is a u.q. of X. ::::> • Y v (x)A(x). IIr instructs us to start out a new tableau t\ with tSt\ and with A(a) on the right, where a has not previously been used. Then X(t 1 ) = {a}, since a is the only free variable of t 1 which does not occur in t, Hence the characteristic formula of t 1 is (a)A(a), and the characteristic formula of t is transformed into a u.q. of X.::::>. Yv (x)A(x) v (a)A(a). So I- X. ::::> • Y v (x)A(x) v (a)A(a): ::::> :X. ::::> • Y v (x)A(x) justifies the case.
Finally, we must justify the rule stipulating that if a formula A appears on the left of a tableau t, and t St", we must put A on the left of t 1 • This is justified by XAA.::::>.Yv«X' AA)::::> Y'):::::>:XAA.::::>.Yv (X' ::::> Y'). The lemma is proved. THEOREM 4: If A is valid, then A is provable in Heyting's predicate calculus. PROOF. We can assume A has no free variables. Since A is valid, the
SEMANTICAL ANALYSIS OF INTUITIONISTIC LOGIC l
129
construction for A is closed. Then there is a stage at which each alternative set is closed; let the characteristic formula of that stage be D 1 /\ • • • D., where the D /s are the characteristic formulae of the alternative sets of the stage. By the lemma, D 1 A ..• D n • =:l.A (since A is the characteristic formula of the initial stage). So it suffices to show D j for each j. The alternative set whose characteristic formula is D j' being closed, contains a closed tableau t. Then t contains a formula B on both sides, so its characteristic formula C is a u.q. of X /\ B. =:l • Y v B. Clearly I- C. If t is the main tableau of the set, this is D i: Otherwise, let t 1 be the predecessor of t. Then the characteristic formula C 1 of t 1 is a u.q. of X'. =:l • Y ' V C. Clearly I- C 1 • Continuing in this manner, we are driven back along the path from t to the origin until we obtain I- D I: Q.E.D. REMARK. The theorem gives a finitary proof that if the construction for A is closed, I- A. We could have proved it alternatively by showing that the tableau procedure is equivalent to a standard Gentzen formulation of Heyting's system. Of course the theorem and proof apply to the propositional calculus, even though the proof was carried out for the predicate calculus.
References [I] Saul A. Kripke, Semantical Analysis of Modal Logic (abstract). The Journal of Symbolic Logic 24 (1959) 323-324. [2] Saul A. Kripke, Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi. Zeitschrift fur Mathematische Logik und Grundlagen der Mathemathik 9 (1963) 67-96. [3] Saul A. Kripke, Semantical Considerations on Modal and Intuitionistic Logic. Acta Philosophica Fennica 16 (1963) 83-94. [4] Saul A. Kripke, The Undecidability of Monadic Modal Quantification Theory. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 8 (1962) 113-116. [5] Paul J. Cohen, The Independence of the Continuum Hypothesis. Proceedings of the National Academy of Sciences, U.S.A. 50 (1963) 1143-1148. [6] E. W. Beth, Observations on an Independence Proof for Peirce's Law (abstract). The Journal of Symbolic Logic 25 (1960; published, 1962) 389. [7] M. A. E. Dummett and E. J. Lemmon, Modal Logics between S4 and S5. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 4 (1958) 250-264. [8] E. W. Beth, Semantic Construction of Intuitionistic Logic. Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afd, Letterkunde, Nieuwe Reeks, Deel 19, No. 11. [9] V. H. Dyson and G. Kreisel, Analysis of Beth's Semantic Construction of In-
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tuitionistic Logic. Technical Report no. 3, Stanford University Applied Mathematics and Statistics Laboratories, Stanford, California. [l0] S. C. Kleene, Introduction to Metamathematics. (Van Nostrand, New York; North-Holland Publishing Co., Amsterdam and P. Noordhoff Ltd., Groningen). 1952. [ll] G. Kreisel, A Remark on Free Choice Sequences and the Topological Completeness Proofs. The Journal of Symbolic Logic 23 (1958) 369-388. [l2] A. Heyting, Intuitionism: An Introduction. (North-Holland Publishing Co., Amsterdam 1956). [l3] S. Kuroda, Intuitionistische Untersuchungen der formalistischen Logik. Nagoya Mathematical Journal 2 (1951) 35--47. Known only from references. [14] A. A. Markov, 0 nepreryvnosti konstruktivnyh funkcij (On the continuity of constructive functions). Uspehi Matern. Nauk 9 (1954) 226-230. Known only from references. [15] G. Kreisel, On Weak Completeness of Intuitionistic Predicate Logic. The Journal of Symbolic Logic 27 (1962) 139-158. [l6] Saul A. Kripke, A Completeness Theorem in Modal Logic. The Journal of Symbolic Logic 24 (1959) 1-14. [l7] G. Kreisel, Set Theoretic Problems suggested by the Notion of Potential Totality in: Infinitistic Methods (Warsaw 1961). Note (added in proof, August 9,1964). We have since seen Feferman's paper, and his version of forcing is indeed virtually identical with ours, although he, of course, does not base it on any model theory for or connection with intuitionistic logic. He credits his version to Dana Scott. Note (added in proof, October 28,1964). In connection with the "Remark" at the end of section 1.1, it should be pointed out that the example in part I of the Remark already refutes Markov's principle. For we observed there that, in FC, (a) (IX I B) , (x) (IX (x) = 0), but also (b) , (IX I B) (3x) (IX x) = 1). By (b), noting that since B is the binary spread, (IX I B) (x) (IX (x) =F 0 :::> IX (x) = I), we have (c) , (IX I B) (3x) (IX (x) =F 0). But (a) and (c) jointly contradict Markov's principle. The example in part 2 of the Remark is of interest in showing that a single counterexample can refute both Markov's principle and Kuroda's conjecture. It should be noted that Markov's principle would imply, for IX on the full binary spread, that , (x) (IX (x) = 0) :::> (3x) (IX (x) = 1). From this it is easy to derive, for a real number a, that a =F 0 implies a # 0 (similarly to part 1 of the Remark). Hence if Brouwer's disproof (using ips depending on the solving of problems) of the latter is accepted, Brouwer has already refuted Markov's principle. I wish to thank M. A. E. Dummett and John Crossley for their help in editing this paper, and in particular, M. A. E. Dummett for an important correction in section 1.2.
SET THEORY AND HIGHER-ORDER LOGICl) RICHARD MONTAGUE University of California, Los Angeles, Calif., USA
Several mutual applications of set theory and higher-order logic are developed. Second-order logic is used to discover the standard models of Zermelo-Fraenkel set theory, consideration of standard models leads to the introduction of new systems of set theory, one of these systems is applied in finding a definition of truth for higher-order sentences, and finally Zerrnelo-Fraenkel set theory with individuals is given a philosophical justification as logically true within higher-order logic. 1. Standard models
Let us consider three well-known first-order theories. The first, called Peano's arithmetic, has the non-logical constants 0, S, +, " and the following axioms"): ,0= Sx,
Sx = Sy
--+
x = y,
x+O = x, x+Sy = S(x+y), l) I am indebted to the United States' National Science Foundation, which supported the preparation of most of this paper under grant number NSF GP 1603 (Montague). 2) It is convenient for the purposes of the present paper to regard a first-order theory as determined by a sequence of non-logical constants and a set of axioms. I use the logical constants " A, Y, -->-,"-', A, Y, =, which are the respective symbols of negation, conjunction, disjunction, implication, equivalence, universal quantification, existential quantification, and identity. (I use ",", "A", etc. as names of certain symbols of the object language, and I indicate concatenation by juxtaposition.)
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RICHARD MONTAGUE
x :0
= 0,
x'Sy
= (x'y)+x,
prO] A Ax[P[x]
P[Sx]]
-+
-+
AxP[x].
The last principle is regarded as a schema, called the Induction Schema; we take as axioms all formulas of Peano's arithmetic obtainable without clash of variables by substituting a formula for P in the schema. The second theory, called (at least in an alternative formulation 1» the theory of real closed fields has the non-logical constants 0, 1, +, " -, -1, ~, and the following axioms: x+(y+z) x+y x+O
= (x+y)+z,
= y+x, = x,
x+( -x) x . (y . z)
= 0, = (x . y) . z,
x : y = y' x,
= x, = 0 -+ x
x·l ., x
. x- 1 = 1,
x· (y+z) = (x : y)+(x' z),
.,0
o~ o~ o~ x
=
~
0- 1
1,
x v0 A 0 xA0
x
~ ~ ~
-x, - x -+ x = 0, Y -+ 0 ~ x +Y A 0
~ X •
y.
y ...... O ~ y+(-x),
= 0,
Vx P[x]
A
VyAx[P[x]
Az[Ax[P[x]
-+
-+
x ~ y]
x ~ z]
-+
-+
Vy(Ax[p[x]
-+
x ~ y]
A
y ~ z)).
The last principle is called the Continuity Schema and plays a role analogous to that of the Induction Schema; that is, we consider as axioms all formulas of the present theory obtainable (again without clash of vari1) The usual formulation involves fewer primitive symbols, and hence has somewhat more complicated axioms. It is clear that several of our symbols, for instance -1 could be defined in terms of the others.
133
SET THEORY AND HIGHER-ORDER LOGIC
abIes) by substituting a formula for P in the schema. As a final example, consider Zermelo-Fraenkel set theory, whose only non-logical constant is e and whose axioms are the following: Au[u e a
+-+ u
Vu u e a
-+
s b]
-+
a
= b,
Vu[u e a It. ., Vv(ve u It. v e a)],
VaAu[uea+-+u
= xvu = y],
VbAu[u s b +-+ Vv(u e v It. v s a)], VbAu[u e b +-+ Ax(x s u Va[Vu u s a
It.
Au(u s a
-+ -+
x e a)], Vv[u s v It. v e a])],
AxAyAzAaAbAcAqAr[Au(u e a +-+ u
Au(u e b +-+ u Au(u s q +-+ U It.
P[q] It. P[r]
=
x)
It.
= x v u = y) It. Au(u s c +-+ u = x v u = z) = a v u = b) It. Au(u s r +-+ u = a v u = c) -+
y = z]
-+
AsVtAy[y e t
= x] It. Au[u e b +-+ u = x Au[ueq+-+u = avu = b] It. P[q])]. Au[u s a
+-+ u
+-+ VxVaVbVq(x
vu
It.
e s It.
= y] It.
The last principal is called the Replacement Schema; as with the schemata above, we take as axioms all formulas of Zermelo-Fraenkel set theory obtainable without clash of variables by substituting a formula for P in this schema. 1) A possible model of Peano's arithmetic is a structure
AxAyAz[P[(x, y)]
A
P[(x, z)]
AsVtAy[y e t <--> Vx(x
£ S A
->-
y
= z] ->-
P[(x, y)])].
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RICHARD MONTAGUE
quantifiers on those variables. 1) The concepts of possible model and model may of course be used also in connection with other theories. For instance, a possible model of the theory of real closed fields will have the form
°
1) For an exact definition of the notion of truth in a possible model, see Tarski and Vaught [12]. 2) For the general notion of isomorphism of structures, see, for instance, Tarski [11].
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SET THEORY AND HIGHER-ORDER LOGIC
ventionally identifying 1/0 with 0), and R is the usual relation of magnitude among real numbers. For set theory we first introduce a transfinite hierarchy of types. Intuitively, if a is any ordinal, T(a) is to be the a~ type in a Russellian hierarchy which begins with an empty set of individuals; the recursive definition is the following. T(O) = the empty set A. T(a+ 1) = the set of all subsets of T(a).
If A is a limit number, T(A) is the union of all sets
T(~)
for
~
< A.
(Notice that the types are according to this definition cumulative and hence not pairwise disjoint. For instance, the empty set is a member of both T(l) and T(2). This is in no way incompatible with Russell's theory. Though that theory requires distinct symbols for the empty set of type o and the empty set of type 1, nothing prevents the two symbols from designating the same object. The identity between the two symbols is not a sentence of the theory and can be neither asserted nor denied.) By a standard model of Zermelo-Fraenkel set theory is understood a structure isomorphic to
~
P[Sx]]
~
AxP[x]).
(Here P now serves as a genuine predicate variable.) Similar the secondorder theory of real numbers and second-order Zermelo-Fraenkel set 1) For several equivalent definitions of the notion of a strongly inaccessible ordinal, see Montague and Vaught [8], which contains investigations of certain properties of the structures (T(rx), E;(T(rx))).
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RICHARD MONTAGUE
theory are to be exactly like the theory of real closed fields and ZerrneloFraenkel set theory respectively, except that the Continuity Schema and the Replacement Schema are to be replaced by the second-order axioms obtainable from them by prefixing universal quantifiers on P. The possible models of any of these three theories coincide with the possible models of the corresponding first-order theory. (Thus as always the collection of possible models depends only on the sequence of nonlogical constants of the theory in question.) A possible model of a second order theory will be considered a model of that theory if all axioms of the theory are true under the interpretation supplied by the possible model; individual variables are regarded as ranging over the universe of the possible model (that is, the set which is the first constituent of the possible model), and (one-place) predicate variables as ranging over the set of all subsets of that universe. Now it is well known that the models of second-order Peano's arithmetic coincide exactly with those structures which were described earlier as standard models of (first-order) Peano's arithmetic, and it is rather easily shown that the models of the second-order theory of real numbers and those of second-order Zermelo-Fraenkel set theory coincide respectively with the standard models of the theory of real closed fields, and the standard models of (first-order) Zermelo-Fraenkel set theory. These facts suggest a unification of the divergent special notions: whenever we speak of the standard models of a first-order theory T we have in mind a related second-order theory U; the standard models of Tare then identified with the models of U. In any context in which standard models are considered the theory of basic interest seems always to be a second-order theory (or, in some cases which will not arise in this paper, a theory of higher than second order). We may, of course, consider as well various first-order subtheories of the second-order theory.') For some purposes, indeed, it is essential to do so in view of certain properties - compactness and the Lowenheim-Skolern property, for example - which are possessed by all firstorder theories but not by any interesting second-order theory. Suppose that we have selected a certain second-order theory as our 1) One theory is called a sub theory of another if all theorems of the first are theorems of the second. (A theorem of a first- or second-order theory T is a first- or second-order formula of T which is true in every model of T.)
SET THEORY AND HIGHER-ORDER LOGIC
137
basic object of attention. How are we to select a subtheory that might be called "the corresponding first-order theory" ? Let us consider a procedure applicable to those second-order theories T whose axioms, like those in our examples, all have the form
where Po, ... , P n-l are predicate variables and ¢ is a formula without second-order quantifiers. 1) By a first-order instance within T of the formula displayed above, we understand a first-order formula obtainable (without clash of bound variables) by substituting formulas of T for the predicate variables Po' ... , P n - 1 in ¢. The first-order theory corresponding to T might then be identified with that theory whose non-logical constants are those of T and whose axioms are the first-order instances within T of axioms of T. It is according to this notion that the three second-order theories considered above can be said to have Peano's arithmetic, the theory of real closed fields, and Zermelo-Fraenkel set theory as their corresponding first-order theories. This way of selecting a first-order theory is, however, unnatural: we can easily find two equivalent second-order theories") whose first-order counterparts in the present sense are not equivalent. A much more natural procedure, and the one we shall adopt, is to identify the firstorder counterpart of a second-order theory T with that theory whose nonlogical constants are those of T and whose axioms consist of those firstorder sentences which are theorems of T. In the light of this analysis two of the first-order theories considered above, Peano's arithmetic and Zermelo-Fraenkel set theory, lose interest. Neither is equivalent to what we now regard as the first-order counterpart of second-order Peano's arithmetic or second-order ZermeloFraenkel set theory. The first-order counterparts of these two theories are, to be sure, not equivalent to any recursively axiomatized theories (or even to theories with arithmetical axiom sets, in the sense of Kleene [3]), and we may for some purposes wish to consider recursively axiomatized first-order subtheories. But Peano's arithmetic and Zermelon
1) The first-order axioms of our examples can be regarded as having this form with
= O.
2) Two first- or second-order theories are said to be equivalent if they have the same models.
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RICHARD MONTAGUE
Fraenkel set theory, though they satisfy this description, seem in no tangible sense pre-eminent among a wide range of other, non-equivalent theories which also qualify. The situation changes when we consider the second-order theory of real numbers. It is a result of Tarski [10] that what we now call the first-order counterpart of this theory is equivalent to the theory of real closed fields. 2. Rank-free set theory There are interesting structures of the form
