Bases for First-Order Theories and Subtheories

Bases for First-Order Theories and SubtheoriesAuthor(s): William CraigSource: The Journal of Symbolic Logic, Vol. 25, No. 2 (Jun., 1960), pp. 97-142Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964208 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 25, Number 2, June 1960

BASES FOR FIRST-ORDER THEORIES AND SUBTHEORIES

WILLIAM CRAIG

1. Introduction.' The extent to which we can grasp the content of a (non-logical) theory, expressing it economically by means of an axiom system or basis, varies greatly. In this paper we shall investigate what degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories. A useful approach, also of interest in its own right, turns out to be the study of bases for subtheories, where a subtheory of a given theory consists of those theorems from which certain predicate symbols are absent. These predicate symbols might be thought of as the formal counterparts of the "purely theoretical" terms employed by a science, the theory corresponding to the science itself and the subtheory to its "observational consequences". Roughly speaking, the types of operations involving such predicate symbols will be reduced to a minimum, so that their syntactical role in deductions will emerge more clearly.

Some of these notions and results will now be expressed more precisely. PCI= shall be any of the usual systems of applied first-order predicate calculus with identity, and PCI shall be the corresponding system of predicate calculus without the axioms, but in general with the symbol, for identity. Formulas derivable in PCI= or PCI need not be closed, so that our results apply also to formulas expressing concepts or conditions, not only to sentences expressing assertions. A theory shall be any set S of formulas of PCI= or PCI which is closed under deduction in PCI= or PCI, respectively. A basis in PCI= (in PCI) for S shall be any subset B of S whose closure under deduction in PCI= (in PCI) is S. A subtheory of S shall be any set S- such that, for some non-logical predicate symbols R1, R2, . .. (distinct from =, unless stated otherwise), S- consists of those formulas in S from which R1, R2, ... are absent.

Given any basis B in PCI= or in PCI for a theory S, and given any

R1, R2, ... determining a subtheory S- of S, certain bases B- for S- will be constructed. Whereas deduction from B of the formulas in S-

requires various types of operations, deduction from B of the formulas in B-

requires only certain ones of these. Moreover, those that are still required

Received November 9, 1959. 1 Theorem 1 below is the main result of [4]. The present formulation and proof are

somewhat different and perhaps more palatable. A few of the other results, or frag-

ments thereof, also occur in [4]. I am grateful to R. L. Vaught for his painstaking and

penetrating comments on an earlier version. Most helpful were his suggestions concern-

ing exposition and his insistence on an intrinsic description of bases. Work on the

resulting revised and more complete version was partly supported by a grant from

the National Science Foundation.

97



98 WILLIAM CRAIG

can be subjected to certain regularities. The types of operations no longer required and these regularities together constitute or measure the degree of economy achieved by B-. In the case of PCI, this economy is rather slight. In the case of PCI=, it is somewhat better, because one further type of operation can be avoided.

Except for certain trivial cases, each basis B- constructed will be infinite. Also, each B- will be highly redundant in the sense that each formula in S- is implied by a single formula in B-. Now let B;, B', ... be the formulas of B-. Letting B1 = B;, there is a B2 in B- implying B1. B', a B3 in B- implying B2. B', etc. Hence B- has a subset which is also a basis for S- and which is monotonic in the sense that its formulas can be arranged in a sequence B1, B2, ... such that each Bi+1 implies Bi.

If S- has a finite basis, then for some j each formula in B1, B1+1, ... constitutes such a basis. Unfortunately, as Theorem 1 2a (1 2b) below shows, there is no general procedure for proving (disproving) the existence of a finite basis for S- in all and only those cases where there is one (none). Hence our construction can at best suggest some broad sufficient conditions for the existence (non-existence) of a finite basis, or perhaps serve heu- ristically in the discovery of finite bases.

If S- does not have a finite basis, then for each i there is a i > i such that each formula in B1, B,+1, ... implies Bi but not conversely. Then the sequence B1, B2, B3, . . . might be thought of as an approximation (from below) of the content of S-. In a rather elusive sense, this approximation seems in our cases less floundering than, e.g., the approximation by B, B B, B. B . B', ..., where B;, B', B', . . . are all the formulas in B- or in S-.2 At least this seems to be the case for PCI=, where our most economical or most regular bases turn out to be at the same time monotonic, so that the pattern of progression in B1, B2, B3, . . . is rather clearly dis- cernible. (The question of how to approximate (from below) the content of S- efficiently and perspicuously seems closely related to the vexing problem of how to test a scientific theory S rapidly and methodically.)

If B is a basis in PCI= for S and if B is finite, then one of our monotonic bases B- in PCI= for S- can also be described without explicit reference to B in a rather intrinsic way. B- can then be said to consist of the instances of an axiom scheme of a certain kind, a scheme of kind 1 as we shall call it, where the rather vague notion of an axiom scheme is construed in a somewhat broader sense than seems usual.3 Conversely, as will also be shown,

2 An effective dual process of "approximating from above" by forming alternations does not always exist. E.g., if S- has a basis in PCI= consisting of the sentences "There are at least n objects", n = 1, 2, ..., then by [21] the set of those B' which imply every formula in S- is not recursively enumerable.

3 E.g., in contrast to the scheme of induction in number theory or to many other schemes, a scheme of kind 1 does not yield its instances simply by substitution for



BASES FOR FIRST-ORDER THEORIES AND SUBTHEORIES 99

if a set S- has a basis B- in PCI= which consists of the instances of a scheme of kind 1, then S- is the subtheory of some theory S having a finite basis B in PCI=. Thus, for PCI=, finite axiomatizability using additional predicate symbols, as defined in [7], and axiomatizability by a scheme of kind 1 coincide. Then by 3.6 of [7], for PCI= also the semantical notion s./.a.+ of [7] and axiomatizability by a scheme of kind 1 coincide.

If B is a basis in PCI for S and if B is finite, then one of our bases B- in PCI for S- can again be described in a rather intrinsic way. B- will then be said to consist of the instances of a scheme of kind 2. The description of B- however is now quite complex, so that the notion of a scheme of kind 2 is much broader than the usual notion(s) of an axiom scheme or also than the notion of a scheme of kind 1. Nevertheless, it retains some of their features.

The following immediate consequence of a result in Kleene [13] (see also [7]) and of the preceding result about PCI seems important. Consider an arbitrary theory S-, closed under deduction either in PCI or in PCI=, such that the totality of non-logical constants or individual variables occurring free in some formula in S- is finite and such that the G6del numbers of the formulas in S- are recursively enumerable. Then by Kleene [13], there is an S having a finite basis B in PCI such that S- is a subtheory of S.4 Then from B one obtains a B- which axiomatizes S- by a scheme of kind 2.5 Thus for first-order theories6 with only a finite number of non-logical

predicate variables. However, schemes more comparable in complexity to schemes of kind 1 do occur in practice, e.g., schemes admitting all tautologies or schemes of comprehension in set theory where only stratified formulas may be substituted (cf. [17] p. 162). For details concerning schemes of kind 1 see ? 2.

4 Kleene's result is stated explicitly only for the case where S- is closed under deduction in PCI. However, by treating =, which is allowed to be present in PCi, like any of the other predicate symbols occurring in formulas in S-, the result is also seen to apply to the case where S- is closed under deduction in PCI=. Note however that by 4.3 of [7] there does not always exist an S having a finite basis in PCI= such that S- is a subtheory of S (e.g., when S- is the particular theory S-O considered below). Any S which includes a subtheory S- that is closed under deduction in PCI= must of course satisfy the condition: S contains all those identity axioms whose predicate symbol R occurs in formulas in S-. Any S which has a basis in PCI= must satisfy the stronger condition: S contains all those identity axioms whose predicate symbol R occurs in formulas in S.

5 Indeed there is an effective method of obtaining a scheme of kind 2 which yields a basis B- for S-, given a procedure enumerating the formulas in S-. This method can be extracted from Kleene's proof and ours. The B- obtained is always a basis for S- in PCI. A fortiori, it is a basis for S- in PCI=, whenever S- is closed under deduction in PCI=.

6 Whether, or to what extent, similar results hold for higher-order theories is an open problem. Perhaps techniques similar to Kleene's and ours apply to many-sorted theories S-, yielding a basis B- for S- in many-sorted logic. Then perhaps in the special case where the sorts are types, B- is a fortiori a basis for S- in type theory



100 WILLIAM CRAIG

constants or free individual variables, and with or without axioms for identity, recursive enumerability and axiomatizability by a scheme of kind 2 coincide.7

A particular theory S- constructed by Ehrenfeucht [8] is worth considering for a moment. S- is such that, for some set J of positive integers which is recursively enumerable but not recursive, the set of those sentences forms a basis for S- in PCI= which for some n e J express "There are not exactly n objects". Then the content of S- cannot be finitely expressed.8 More precisely each sentence F of first or higher (finite) order fails to have the same models as S-, since it is decidable for any positive integer n whether or not F holds for a domain of n objects. Nor can S- be axiomatized by a scheme of kind 1, since, by the equivalence stated earlier, S- would then be the subtheory of some theory S having a finite basis in PCI=, which is known to be false (see 4.3 of [7]). Thus it is doubtful that there is a much better axiomatization of S- than by a scheme of kind 2.

If S- is that subtheory of a theory having {A} as a basis in PCI= from whose formulas R1, . . ., Rq are absent, then S- is also the set of first- order consequences of the second-order formula 3R1... 3RqA. Thus, as Remark 1 in ? 2 will explain in more detail, our construction of B- solves a part of the elimination problem of second-order predicate calculus, as formulated by Ackermann [1].

One of the uses, and according to some writers on philosophy of science (e.g. [16]) the only use, of the "purely theoretical" terms of a science is the ordering or organizing of "observational assertions". Now this organizational role is clearly syntactical in nature, and therefore can and should be studied by syntactical methods. The present paper constitutes an abstract study of this kind. As mentioned in the beginning, one may think of the predicate symbols R1, R2, . . . present in a first-order theory S and absent from a subtheory S- as formal counterparts of "purely theoretical" terms. In an arbitrary deduction from a basis B for S of a formula C in S-, the syntactical role of R1, R2, . . . is in general far from transparent. We can replace this (total) deduction by one consisting of two parts. In the first part, we deduce from B a formula Bi in our basis B- for S-; in the second part, we deduce from Bi the given C. In the second part, R1, R2, ... no longer occur. In the first part, the types of operations used are reduced to what may be called a minimum and moreover are subjected to certain regularities. These regularities are such that, when B is a basis for S in PCl=, then the first

7This fact, Kleene's result in [13], and in particular Vaught's result 2.1 in [7], may each perhaps fruitfully be regarded as a "cause" of incompletabilities.

8 In contrast, 2.1 of [7] shows that the content of S- can be finitely expressed, and indeed expressed by a second-order formula of a relatively simple kind, whenever S- is a first-order theory which is recursively enumerable, has only a finite number of constants or free individual variables, and has no finite model.




part depends only in one respect on the C to be deduced in the second part, and when B is a basis for S in PCi, then the effect of C on the first part, while apparently much greater, is still limited to only certain kinds of choices to be made.9 The syntactical role of R1, R2, . .. in the first part is thus rather clear.'0 It appears to be quite different from the kind of role (apparently in total deductions) that scientists introducing "purely theoretical" terms seem to envisage. This discrepancy calls for careful further syntactical studies, mainly of actual uses of "purely theoretical" terms in actual theories.

2. Results for PCI=. We shall now state our results for PCI= in detail. For this purpose, certain conventions or abbreviations will be introduced. Derivability in PCI= will be denoted by K=, and in PCI by F. The symbols A, B, .. . shall range over the formulas of PCI= or PCI, K, M, N over the quantifier-free formulas or matrices, (P) and (Q) over the prefixes including the empty prefix, s and t over the individual terms, u, v, . . . over the individual variables, and R over the predicate symbols other than =. Applying subscripts or superscripts to any of these symbols shall leave the range unchanged. The individual terms are formed by means of function symbols, individual constants, and individual variables, the atomic formulas by means of predicate symbols of n > 1 arguments. Vacuous quantification shall be permitted.

Throughout ? 2 we shall consider a theory S having a basis {A} in PCI=, such that A is in prenex normal form. Then A is (Q)0(Q)1... (Q)hN, where h > 0 N is some matrix, (Q)O is either empty or of the form 3vl...3vs, (Q)1, .. ., (Q)h-1 are all of the form vul... vur3vl... 3vs, and, if h # 0, (Q)h is either also of that form or of the form Vul... Vur, r > 1, s _ 1. E.g., if A is 3uVv3w3w'Vx3yN, then (Q)O is 3u, (Q)l is Vv3w3w', (Q)2 is Vx3y, and h = 2.

Each formula in one of our bases B- will be obtained from A in four steps. As first step we transform A into an equivalent formula A', called an expansion of A, by replacing none or more times an occurrence of some (Q)g-1(Q)g . . (Q)hN, where 0 < g < h, by an occurrence of (Q)9-1[(Q)g...

(Q)hN. -- .(Q)9... (Q)hN], where for some i > 2 the expression in brackets is the i-fold conjunction of (Q)g... (Q)hN. Any replacement of

9 One of these choices must be made at the very beginning, both for PCI and for PCI=, so that there is no initial portion which is completely unaffected by C and thus common to all C. Thus "timing" is often important, in addition to the number and kinds of choices.

10 The types of operations used and the role of R1, R2, ... are the same for all formulas in B. (See also the first part of Remark 3 in ? 2.) Proof-theoretically there is no fundamental distinction, sometimes made by philosophers of science, between those formulas in B in which R1, R2, ... are the only non-logical constants and those in which there are also other ("observational") non-logical constants.



102 WILLIAM CRAIG

this kind will be called an (i-/old) reiteration. If h = 3, then one reiteration applied to A yields, e.g., (Q)O(Q)1[(Q)2(Q)3N. (Q)2(Q)3N], which yields by another reiteration, e.g., (Q)O(Q)1[(Q)2(Q)3N. (Q)2[(Q)3N. (Q)3N. (Q)3N]]. Only one further reiteration is then possible, namely replacing (Q)2(Q)3N by (Q)2[(Q)3N. ... . (Q)3N].

As second step we transform AI into an equivalent formula AII in which each occurrence of a quantifier contains a different variable, such that AII is an alphabetic variant of AI in the sense that it differs from AI only in its bound variables and that corresponding occurrences of bound variables are bound by corresponding occurrences of quantifiers (cf. [17] ? 21 and

[12] p. 153). To each (Q)9, 0 < g < h, considered as an occurrence in A, there corre-

spond occurrences of (Q)9 in AI, and to each of these an occurrence of a prefix in AII. Similarly, to the occurrence of N in A there correspond occurrences of N in AI, and to each of these an occurrence of a matrix in AII. Occurrences in AII, AI, and A which thus correspond to each other shall be called images of each other. E.g., suppose A is VuVv3wVx3yVzN and AII is Vu1Vv,3w1[Vx113y11(Vz111N111 . Vz112N112)] . Vu2Vv23w2[Vx2l3y2l Vz211N211 . Vx223y22(Vz22lN221 . Vz222N222)]. Then (Q)1 is VuVv3w, (Q)2 is Vx3y, (Q)3 is Vz, and the images in AII of these occurrences in A are indicated by our choice of letters. Although to an occurrence in A of a proper part of N or of (Q)9, 0 < g _ h, there correspond occurrences in AI and in AII, such occurrences will not be called images. Thus, the occurrence of 3w in A, and of 3w2 in AII, while corresponding to each other, are not images of each other.

As third step we transform AII into an equivalent prenex normal form (P)M, called a development of A, by exporting each prefix image in AII as a unit into the prefix (P). More precisely, (i) M is the conjunction of the images in AII of N and contains these images in the order in which they occur in AII, and (ii) (P) is a string (p)O... (P)t of prefixes such that

(p)O, .. ., (P)t are the images in A"I of the (Q)9 in A, 0 < g ? h, and such that if (P)m has (P)tm within its scope in AII then (P)m also has (P)m' within its scope in (P) so that m < m'. E.g., for the AII described above, M is

N111 . N112 . N211 . N221 . N222 and, disregarding the empty initial prefix, (P) is a string of the prefixes VulVv,3w,, Vxl,3y,,, . . ., VZ222 in some order such that VuiVvi3wi occurs to the left of Vx0j3y0, which occurs to the left of VZi.k-

For the fourth and last step, and for this step only, we must specify the predicate symbols which are present in A and absent from the given subtheory S-. Let R1, . . ., Rq be these predicate symbols and let dl, . . ., dq be their respective degrees. Then as last step we change M into a matrix M- and (P)M into (P)M- by the following process. First, we transform M into some truth-functionally equivalent alternation V A Kj? of

lgmgs O~jgq




s > 1 conjunctions such that, for each m, 1 ? m ? s, (i) KW contains no occurrences of Rl, ..., Rq, (ii) for each 1, 1 < j < q, K7 is a conjunction of formulas Rjsl... Sd, and mRjs'... d,, and (iii) either Ko shall be non- empty or, for at least one j, 1 _ j _ q, at least one Rsl ... Sd, and at least one -,Rjs'. .. s shall be terms of K7. Any alternation of conjunctions

A K7j which satisfies (i) to (iii) shall be in disjunctive (R,, ...,q)- ogisq normal form for PCIl. We shall also require for the present purpose that (iv) no K7 contains individual terms or predicate symbols other than- which do not occur in M. Next, according to some procedure for uniquely ordering terms in a conjunction, we form for each Kin, 1 < j ! q, the conjunction a(K7) of those formulas -n(sl -s ...s . S= S') such that Rsl .. SdJ and ,Rjs'... Sd, are both terms of K7. E.g., let K71 be

Rjsls2. Rjtlt2. -,Rjtls'. Then @(K)) differs at most in the order of the two terms from -(sl= tl .s = s') . (tl = tl . t2 =s'). Finally, in

V A K7 we replace each KX, 1 < j < q, by a(K7j). Letting M- l9 mgs 05jgq be the resulting formula V (Kr. A a(K7n)), we shall say that M

yields M- by (PCI=)-elimination of R1, ..., Rq. If a(K7.) is non-empty, then K= K7 v O(K7). Hence K. M v M-."1 It

follows that K= (P)M v (P)M-, and since F A _ (P)M, that also K= A v (P)M-. Moreover, R1, . .., Rq evidently do not occur in M- and, by (iv), those individual terms or predicate symbols other than = which occur in M- also occur in M. It follows that (P)M- is in S-. Hence the following theorem about PCI=, whose proof will be given in ? 4 to ? 6, implies that the totality of formulas (P)M- obtained from A in four steps as just described constitutes a basis B- in PCI= for S-.

THEOREM 1. If F= A D C, if R1, ...,Rq do not occur in C, and if A is in prenex normal form, then there is a development (P)M of A such that, if M yields M- by (PCI=)-elimination of R1, ..., Rq, then K= (P)M- D C.

Several alternative sets of regularities can be chosen for the first and third step, such that the resulting basis is, or easily becomes, monotonic and such that a large degree of independence from C is achieved. The set which follows is perhaps simplest.

Given any AI or AII, we assign to each image in it a unique sequence of numbers as follows. (i) To the image of (Q)O we assign the empty sequence < >. (ii) If to a certain image of (Q)9-1 the sequence <ml, . .., mgi> has been assigned, 0 < g _ h, then to the mg-th, in the order of occurrence,

11 Note that conversely, by Lemma 2a below, M- D 3R1... 3RqM is also valid. It follows (see also Lemma 2b) that, if {M} is a basis in PCI= for a theory S1, and if Sj is the subtheory of S1 determined by the absence of R1, . Rq, then {M-} is

a basis in PCI= for Sj7. Hence which particular V A Kit is used for forming M- does not matter. i&m:s O0j:q



104 WILLIAM CRAIG

of the images of (Q)9 within its scope we assign <ml, .., mg-,, mg>. (iii) To each image of N, we assign the sequence assigned to that image of (Q)Ih

within whose scope it occurs. E.g., if we construe 221 as <2, 2, 1>, 12 as <1, 2>, etc., then the formula AII used earlier to illustrate the notion of image also illustrates the assignment of sequences to images.

If AI is an expansion of A, and if, for some i _ 1, all those and only those sequences <ml, . . ., mg> are assigned to an image in AI such that 0 < g ! A and ml, .. .., mg E {1, . . ., i}, then AI shall be the regular i-expansion of A. E.g., the regular 1-expansion of A is A itself and the regular 3-expansion of Vv3wVx3yN is

Vv3w(Vx3yN . Vx3yN . Vx3yN) . Vv3w(Vx3yN . Vx3yN . Vx3yN).

Vv3w(Vx3yN . Vx3yN . Vx3yN).

We shall assume from now on some kind of convention which associates with each development (P)M -a unique AII from which it is obtainable in our third step. This also associates with each (P)M a unique AI. (A convention is needed only for those A in which some non-empty (Q)9 contains only vacuous quantifiers. For all other A, the AII for each (P)M is unique without any convention.) Then we can talk about the images in (P)M and the sequences assigned to them. One image in (P)M shall dominate another, which is either a prefix or a matrix image, if and only if in the associated AII the image corresponding to one has the image corresponding to the other within its scope, so that the sequence assigned to one is an initial segment of the sequence assigned to the other.

The rank of an image shall be the highest of the numbers inl, ...,m

in the assigned <ml, . . ., mg> and shall be 0 if < > is assigned. If (i) the associated AI is the regular i-expansion of A, and (ii) any image of lower rank precedes in (P) any image of higher rank, then a development (P)M of A shall be a regular i-development of A. E.g., suppose A is Vv3wVx3yN. Then the prefix of one regular 3-development of A is Vv,3wVxl,3yllVv2 3 W2VX123yl2VX213 y21,x223 y22Vv33 W3VX133 yl3VX233 y23VX313 y31Vx323 y32

Vx333y33, where the corresponding matrix is Nil . N12. N13 . N21 . N22 .

N23. N31 . N32 . N33, and where N yields N23, e.g., by substitution of V2, w2, x23, and y23 for v, w, x, and y, respectively. Other regular 3-developments result, e.g., if we move Vx123y12 to the left of Vv23w2, Vx233y23 to the left of VV33W3, Vx133y,3 to the right of Vx333y33, etc., or if we form an alphabetic variant.

THEOREM 2. Let (P)M be any development of A with < i images in (P). Then:

(a) Each regular i-development (P)iMi has an alphabetic variant (P)* M' from which (P)M is obtainable by first deleting images, deleting with every image all those which are dominated by it, then deleting one conjunction sign next




to every image deleted from Mt, and finally rearranging the non-deleted terms of M,.

(b) It follows that, if Mi yields M- and M yields M- by (PCI=)-elimination of R1, ..., Rq, then K= (P)iM7 v (P)M-.

Theorems 2a and 2b will be proved in ? 7. Theorem 2b supplements Theorem 1. Together they show that {(P)1MT, (P)2M., ...} is a basis in PCl= for S- whenever each (P)iMi is a regular i-development of A. Theorem 2b also yields an evident method of selecting from any such {(P)lMT, (P)2M-, ...} a subset {(P).1M-, (P),2M-, ...} which is monotonic and which is a basis in PCI for a set including {(P)1MT, (P)2M-, ... so that by Theorem 1 it is a basis in PCI= for S-. In some cases {(P)1MT, (P)2M-, ... } is itself monotonic, e.g., when each (P)iMi satisfies the further condition that (iii) among images in (P)i of same rank, those of (Q)g precede those of (Q)g+l, 0 _ g < A, and among images of same rank and of the same (Q)9, those corresponding to an earlier image in the associated AII (and thus assigned a sequence that is lexicographically earlier) precede those corresponding to a later image. Then, except for the choice of i, the operations involving the presence of R1, ..., Rq are unaffected by the particular C in S- to be deduced.

Certain regularities can also be imposed on the fourth step. The relationship of M- to N and the role of R1, ..., Rq will then be more transparent. Roughly speaking, each Rj has the effect that in certain places conditions are conjoined requiring certain d1-tuples to be non-identical.

To express this more precisely below, some notions related to a(Km) will be useful. Consider any formula K such that, if we ignore or remove any parentheses present, then K becomes a conjunction of atomic formulas and of negations of atomic formulas. Then -r(K) shall be the set of those <sI, .., Sd, if any, such that, when parentheses are ignored, some Rs,, ..., sd is a term of K. Likewise -r'(K) shall be the set of those <sl, ,Sd>, if any, such that, when parentheses are ignored, some -Rs'... s is a term of K. For any T(Ki) and -'(K2), let S*[-r(Ki); -r'(K2)] be the conjunction of those formulas ,((s= s= =; s), if any, such that, for some d, <si, . .., Er> e '(Kl) and <s, .., s'> e '(K2). The order of terms in the conjunction shall be chosen such that, if K1 is a conjunction of formulas Rsl... sd, and ,Rjs'... s', then a*[-r(Kj); -r'(K;)] is a(K1).

Assume from now on that the matrix of A is some V A K,. in 1 9mgs 0O5j9q

disjunctive (R1, ..., Rq)-normal form for PCI= satisfying conditions (i) to (iii) for such forms. Let M A Nn be the matrix of a development

of A. Then each No is a matrix V A Knm, where Kj'nm and K,' yield 1; m;s 0< j~q

each other by substitution for individual variables. Then M is



106 WILLIAM CRAIG

A V A K' m. Using distributivity, M can be transformed into 1 gn:5't 1 9m,_s 0< j:9q V A A Kjn v(n) where = {1, ..., s}, t= {1, ..., t} and s is the set

vcs_ 1 srn<t O~gigq of functions q, which map t into s. Each A A K can then be

1gn5t O9j5q

replaced by the equivalent A A K`n vw. If we ignore the parentheses O9j9q 1: n9t

around each K'nvw' then each A Kjn) is a conjunction of formulas

Rjsl... sd, and nRis'... d,, so that V A A K!",(n) is then in dis- qes Ogjgq 1l n9t

junctive (R1, ..., Rq)-normal form for PCI=. Hence, among the formulas M- obtainable from M by (PCI=)-elimination of R1, ..., Rq is V ) A Kn(n). A S*[( A Kjt 9(n)); r'( A K~, (n))

vfsi 1 gnst 0

1 sis~q 1:5'n:'t 2

1 s9n:9t Now consider any S*[T( A Kjn )); 'r'( A K`fv 1 5 j < q, and

1 snst 1 snst suppose that -,(si Si S. = Sd) is one of its terms. Then

<sl, . .,,> E r( A K! v(n)) and <s, .. ., ;I> e r'( A K"nqw). Then, when

parentheses are ignored, Rjsl ... Sd, and nRjs. .. Sd, are terms of A Kj"").

Then Rjsli.. .d, is a term of some K. 9(n) and vRjs'.-. s' is a term of some Kn' ), 1 < n n' < t. Then <sl, . . . > e r(Kj"'(n)) and <s', .., 5,> E

Sr'(Kjn 9(ff))Then -'(s, = si *.... S. , = S) is a term of S*[-r(Kj,(n)); ^,'(K 9w~n))] and hence of A 0*[-r(K!'(fn)); T'(K"'ff"vw))]. Conversely, as

1: n, n' t

can be seen in a similar way, if -i(s= s d =, S',) is a term of A '*[^r(K~n (fl)); '(Kf"'(n ))], then it is also a term of S*[,r( A K!" (n));

1: sn, n':r st 1s '( A KjnvP(fw))]. It follows that among the formulas M- obtainable from

M by (PCI=)-elimination of R1, ..., Rq is

V ( A Kong() . A A a*[T(Kjn9(fl)); T'(K"',%(n') w9i: ln:!.t lsj s-r.q L-5n, n' <t

Each Nn is obtained from N by a 1-1-substitution a for individual variables where a depends on n and M. We can adopt various conventions for choosing bound variables in each AII and (P)M, such that each term Nn of the matrix Mi of a regular i-development of A, and hence the a for Nn, is uniquely determined by i and n, i > 11 < n < ih. For definiteness,

let us suppose that distinct quantifier occurrences in A consist of V or 3 followed by distinct letters, and let us adopt the convention that in each AII and (P)M every variable in a prefix image is obtained from the corresponding variable in the corresponding (Q)g simply by affixing the sequence assigned to the image as subscript. Then Mi can be written as A an (N) or A V A ain(Kjm) and the above M- =M- as lgn<ih

1Vn(ih 1A 5ms AK9 ) ()q V (A din(K9"f)) .A A O* [T (din (Kw(n))) ; r'(cai (KT(n')))]) *= v~~~




Now let crn('r(K7.)) and din(T'(Kj.)) be the set of those <crt(ti), ...,

such that <t1, .. ., to> is in T(K7.) or in -r'(K7.) respectively, where a, (t) is the individual term resulting from t by the substitution car, and where 1 < j < q, 1 < m < s, 1 < n < ih. Then -r(aTQ(K! )) = x(T(K7Z)) and T'(ct(K7m)) = n(-r'(KT)). If we let 17 =r(K7) and T'= -r'(K!), we can then conclude:

THEOREM 3. If the matrix of A is some V A K7. in (R1, ..., Rq)- lames O~j~q

disjunctive normal form for PCI=, and if bound variables in developments are chosen by our convention, then for MV in Theorem 2b one can choose

V ( A cr(K(n)). A A S*[yin(Tj9(fn)); <alf (T' j(n')). qski') 1 gn--ih 1 : Siq 1- Sn, n' < ih

The description of M- in Theorem 3 does not refer explicitly to K7T, 1 i j 5 q, 1 < m < s. Given (Q)0. . . (Q)h and i, which uniquely determine ci, . . ., x), and given Km, T77, and T', 1<j<q, 1 m s, the M, described is uniquely determined, apart from the order in which the

functions f e g(ih) are chosen. Likewise, given (Q)o... (Q)h and i, the prefix (P)i of a regular i-development of any (Q)o... (Q)hN can be made unique by adopting our convention for bound variables and by adopting, e.g., condition (iii) given earlier concerning the order of images in (P)i. A description of this (P)i in terms of (Q)O ... (Q)h and i has thus been given informally. Together with the description of M* in Theorem 3 it constitutes a scheme for generating a set {(P)lM-, (P)2M-, ...} which according to Theorems 1 to 3 is a basis in PCI= for S-. The scheme will be called a scheme of kind 1, in the wider sense.

The scheme becomes simpler in the special case where A is in Skolem normal form for satisfiability. Then the prefix of A is of the form Vul... Vur3vl... 3v8, r > 1, s i 1, so that (Q)O is empty and h = 1. For each i,

Ita then becomes simply the process of affixing <n> as subscript to those individual variables which also occur in the prefix (Q) = (Q)l of A. Thus we can write an in place of of. The resulting description of (P)iMy, which will be called a scheme of kind 1, in the narrower sense, is

a1((Q))af2((Q)).. .cri((Q)) V ( A an(K7(n)). A A veVi 1 9n;5i 1 5. jsgq 1 n, n9 si

a*[yn(TP(n)); an (T 7. (n))]).

If A is not in Skolem normal form for satisfiability, then we can find an A* which is, such that 3R'... 3R",A* and A have the same models. If S* is the theory having {A*} as basis in PCI=, then those formulas in S* from which R1, ..., Rq, R, ..., R". are absent constitute S-. Hence some scheme of kind 1, in the narrower sense, yields a basis B- in PCI= for S-. For PCl=, therefore, finite axiomatizability using additional predicate symbols implies axiomatizability by a scheme of kind 1, in the narrower sense.



108 WILLIAM CRAIG

Conversely, consider any set S- and any scheme of kind 1, in the wider sense, whose instances constitute a basis B- in PCI= for S-. For any pair T7 and T7 of sets of dy-tuples given by the scheme, form a conjunction K7. of formulas Rjsl...sd, and .R~sl...se, such that r(Kw ) = Ti and

= T'7, choosing R1 distinct for different pairs and absent from formulas in S-. Let N be V A Kw and let A be (Q)O... (Q)hN, where

1< m5s 0 5s _q

(Q)O. .. (Q)h and K, ..., Ks are given by the scheme. Then the prefix of the i-th instance of the scheme is the prefix of a regular i-development of A, and the matrix is the M- of Theorem 3. Hence, by Theorems 1 to 3, the set B- of instances of the scheme is a basis in PCI= for the set {C1, C2, . . .} of those C such that F= A v C and R1, ..., Rq do not occur in C. Since B- is also a basis in PCI= for S-, {C1, C2, ... } is S-. To summarize:

THEOREM 4. For first-order theories with axioms for identity, finite axiomatizability using additional predicate symbols, axiomatizability by a scheme of kind 1, in the wider sense, and axiomatizability by scheme of kind 1, in the narrower sense, all coincide.12

REMARK 1. In Theorem 1 one may replace the condition that F= A n C by the condition that 3R1. . .3RA m C is valid, p ! q. Theorems 2 and 3 are then also applicable. We then get a scheme of kind 1, in the wider sense, whose instances form a basis in PCl=, for the set of those first-order consequences of 3R1... .3RpA in which Rv+,, ..., Rq do not occur. For p = q, this solves the following part of Ackermann's form of the elimination problem of second-order predicate calculus.13 Given any 3R1 ... 3RqA, to find a fairly economical and recursive basis for the set {C1, C2, * } of first-order consequences of 3R,... 3RqA.

The remaining part of Ackermann's elimination problem maybe stated as follows. Given any such 3R1.. .3RqA and the set {C1, C2, ... } of its first-order consequences, to show that any model satisfying {C1, C2, *-- } also satisfies 3R1... 3RqA.13 It is now known that there are 3R1... 3RqA for which this is false, e.g., 3R,[VxVy(Rix. Roxy. D Riy) . 3xRlx . 3x ,R1x] in 5 of [3], which is closely related to an example in [1] for which this is not false. In the notation of Tarski [18], if X is the set of models of an

12 Condition (iii) concerning the order of images in (P)i did not enter the proof, so that other conditions, or no condition, could be used in its place in defining schemes of kind 1, in the wider sense. Also, condition (ii) in the definition of regular i-development can be replaced by others, provided there is an analogue of Theorem 2.

13 See [1] and [2], where for certain important special cases all parts of the problem are solved. Some related results occur in [19]. Ackermann's problem is semantical, while our weaker result is primarily syntactical. Note, however, that any finite model satisfying all first-order consequences of 3R, .. . 3RA also satisfies 3R,.. . 3RqA. (Cf. 3.4 of [7]. As Specker has observed in conversation, this holds, of course, for any set of higher-order formulas in place of 3R.... 3RqA.)




3R1... 3RqA for which this is false, then C(X) $ X, which shows that PC $ ACA. Indeed, since X e PCnACL would imply C(X) c X (see 3 of [3]), X ' ACL and hence PC $ ACL. X is so to speak "inaccessible" from AC. Theorem 1 suggests why C(X) $ X is not surprising. 3R1 ... 3RqA is equivalent to 3R1... 3Rq(P)M, while by Lemma 2 below (P)M- is equivalent to (P)3R1...3RqM-l4

Although there are therefore 3R1. . 3RqA whose set {C1, C2, .. .} of first-order consequences has a model <D, Q ... Qv> such that no <D, Q1, . . ., Qp, R1, . . ., Rq> is a model of A, note that if <D, Ql, . . ., Qp> is a model of {C1, C2, . . .}, then it has an arithmetical extension (cf. [20]) which is a model of 3R1... 3RqA. (This was suggested by BUchi. See also Theorem H3 of [15].) A fortiori, if <D, Ql, . . ., Qp> is a model of {C1, C2, . .. }, then it is arithmetically equivalent (cf. [18]) to a model of 3R1 ... 3RqA. Conversely, if <D, Ql, ..., Qp> is arithmetically equivalent to a model of 3R1... 3RqA, then it evidently is a model of {C1, C2, .. .}.

It follows that the set of models of {C1, C2, ... } can be characterized not only as in [ 18], but also as the set of models having an arithmetical extension which is a model of 3R1. . 3RqA, or also as the set of models arithmetically equivalent to a model of 3R1 ... 3RqA (cf. 3 of [3]). To verify Buchi's suggestion, let D = {ao, a,, .. . }, to PCI= adjoin ao, ai, ... as names for ao, a,, . . ., and then let {E1, E2, . . .} be the set of sentences true for <D, Q, . . ., QP, ao, al, . ..>. Since {A}u{El, E2, . . .} is consistent, it has a model <D', Q1, .. .,QR1, .. ., Rq, ao, a,, ...>. Then <D', Q', ...QP> is an arithmetical extension of <D, Q1, ..., Qp> (cf. 1.1 1 (iv) of [20]).

REMARK 2. In place of the condition in Theorem 1 that C does not contain R1, . . ., Rq, consider now the weaker condition that there are only positive occurrences of R1, ..., Rq in C. For this weaker condition Lyndon [15] has recently discovered and used an "interpolation lemma". As will be apparent from their proofs, Theorems 1 to 3 can easily be modified to apply under this condition also. Suppose M has been transformed into a disjunctive (R1, ..., Rq)-normal form V A K. . Then, instead of

1 m<s O 50 j5 replacing each K., 1 j < q, by a(K. ) and thus forming V (Ko.

A O(KX )), we now replace each K.^, 1 < q, by O(K" ) . Kw` and 1gjgq

14 For the Y33-case, Theorem 1 can thus be rather simply stated as follows. If 3R3uYv3wM(u, v, w) D C is valid, then so is 3uVvi3wt ... Vvi3wi3R[M(u, vt, wl) .... M(u, vi, wi)] D C for some i > 1, where 3R, 3u, Vv, and 3w may also be replaced by 3R.. . .3Rq 3ul. . 3UT, Vvi.. . Yvs, and 3w ... .3wt, respectively. The dual form may have some practical use in surveying the first-order sufficient conditions ,C for algebraic properties of the form VRA' such as simpleness of a group. The necessary (sufficient) first-order conditions for VRA' (for 3RA) are, of course, not always recursively enumerable, but even more modest, perhaps nonconstructive, general results would be of great interest.



110 WILLIAM CRAIG

thus form V (Ko . A (O(K? ) . K7 +)), where K?'+ is a conjunction 1 smgs 1 sisq

of the terms Ris,... d, of K?. A similar remark holds if there are only negative occurrences of R,, ..., Rq in C.

REMARK 3. One can generalize to the case where A is a conjunction

Al. ... . A, of prenex normal forms. A development of A shall then be a formula (P)M formed as follows. First, form an expansion AI of each A1, 1 j Y r, then form an alphabetic variant All of AI, such that each occurrence of a quantifier in Al'. .... AII contains a different variable, and finally transform A". ... . All into (P)M by exporting each prefix image in each Al, as a unit into (P). As will be apparent from its proof, Theorem I then also holds for a conjunction A of prenex normal forms. There are also corresponding generalizations of Theorems 2 and 3, since we can embed in a way similar to ? 7, not only a linearly ordered tree, but also a "linearly ordered forest", i.e., a finite set of trees with a single linear ordering that is an extension of all the (partial) tree-orderings. The rather cumbersome statement of these generalizations will be omitted.

Certain further regularities can also be imposed on the sets 17' and T7' in schemes of kind 1. E.g., if N(R, -nR) is in R-disjunctive normal form for PCI=, and if N(R, R') is the result of replacing in N(R, ,R) all occurrences of .,R by occurrences of R', where R' is of the same degree d as R and does not occur in N, then 3R(Q)N(R, ,R) is equivalent to 3R3R'[(Q)N(R, R') . Vx.. . .Vxd(-Rxi... xd V -,R'xl... Xd)]. For (Q)N(R, R') contains only positive occurrences of R', so that if <D, R, R', R,, R2, ...> is a model of (Q)N(R, R') and if R and R' are disjoint, then <D, R, R, R,, R2, ..*>

is a model of (Q)N(R, R') and hence of (Q)N(R, ,R), since R' c R. There are also some variations. E.g., the definition of i-regularity may

be modified. Also, if (Q)h is of the form Vul... Vur, then one can restrict developments to those where each image of (Q)" occurs to the right of all the other prefix images. This will be apparent from the proof, but also follows directly from Theorem 1.

On the other hand, certain variations are not possible. E.g., even for h = 1 one can impose no upper bound on the number of alternations of V- and 3-quantifiers in (P), since, for any k, some C is equivalent to no (P)M with < k such alternations and yet equivalent to some 3R,... 3RqVul... Vur3vl...3vsN. Nor, e.g., can each image of (Q)f-1 be made to precede each image of (Q)9, as can be seen from A = Vv3wVx3yN, where F N =. Rv. -nRy with v and y distinct. (Roughly speaking, the notion of i-regularity must be such that the order of prefix images in (P)i is sufficiently different from their order in AII or from the order of the (Q)9 in A.) Nor, e.g., can reiterations be restricted so that each image of (Q)g, 1 < g < h, dominates only one image of (Q)9+1. This can be seen from A = Vv3wVx3yN, where F N _. Rwx. -nRwy. If (P)M is a development




of A with i images of Vv3w each dominating only one image of Vx3y, then (P)M- is true in any domain of ? i objects.

3. Results for PCI. We now consider the case where {A} is a basis for S in PCI. We shall assume throughout that A also contains a predicate symbol Ro distinct from the predicate symbols R1, .. ., Rq that are absent from S-. Otherwise, by Theorem 5 of [5], we would have either F ,A or else F C for every C in S-. One further type of operation which now plays a role is change from (P)'(P)"M to (P)'Vy(P)"M where y does not occur free in (P)"M. It will be called V-vacuous-introduction (cf. [5] p. 252). Another type of operation is change from (P)'Vy(P)"M(y) to (P)'(P)"M(t) where (i) (P)"M(t) is the result of substituting t for the free occurrences of y in (P)"M(y), and (ii) t is free at the free occurrences of y in (P)"M(y). t will be called the individual term of this operation, and the operation itself V-instantiation (cf. [5] p. 253). Analogous to (PCI=)-elimination of R,, ..., Rq from M there is the following process. First, transform M into some truth-functionally equivalent V A Kay satisfying conditions (i)

1 smus O< j~q and (ii) for disjunctive (R1, ..., Rq)-normal forms for PCI=, and also such that (iii*) no Km is empty, (iv*) no K7' contains individual terms or predicate symbols which do not occur in M, 0 < j ; q, and (v*) no K;7 contains as terms both a formula Rssl. Sdj and its negation ,Rjsl ... sd,, 1 < j < q. If Ro is present in M, all these conditions can be satisfied. Then simply delete each K'y., 1 j? q. The resulting formula V Km will

1fmgs be said to be obtained from M by (PCI)-elimination of R1, ..., Rq. Evidently, - M v V Km.15 Hence, by an argument like that preceding Theorem 1,

the formulas (P)*M*- in the following theorem form a basis B- in PCI for S-. THEOREM 5. Suppose that F A D C, that R1, . . ., Rq do not occur in C,

that A contains a further predicate symbol, and that A is in prenex normal form. Then:

(a) Some development (P)M of A yields some (P)+M by V-vacuous-introductions which yields some (P)*M* by V-instantiations, such that, if M* yields M*- by (PCl)-elimination of R1, . . ., Rq, then F (P)*M*- D C.

(b) Each V-vacuous-introduction can be so chosen that the quantifier is introduced to the right of the image of (Q)O and outside of any image of (Q)9, I < g 5 h, and contains a variable absent from the premiss.

(c) Each V-instantiation can be so chosen that each function symbol in the individual term t also occurs in A or in C and that, if C begins with V, then each individual constant or variable in t either occurs also in the premiss of the V-instantiation or occurs free in C.

15 Lemma 5 in ? 4 is, so to speak, a partial converse and shows that it does not matter which particular V A Mn is used.

15m:s Otigq



112 WILLIAM CRAIG

Theorem 5 will be proved in ? 5 and ? 6. It is related to the proof of Theorem 5 of [5] but goes somewhat beyond it in containing (b) and (c) and in imposing stricter conditions on its (P)M. It is supplemented, as Theorem 1 is supplemented by Theorem 2, by the following theorem, both parts of which will be proved in ? 7.

THEOREM 6. Let A, (P)M, (P)+M, (P)*M* and M*- be as in Theorem 5, and let A contain at least one V. Then:

(a) For each i > io, where io is easily found, every regular i-development

(P)iMi of A has an alphabetic variant (P)4 M' from which (P)+M is obtainable by first partially deleting prefix images and totally deleting prefix images and matrix images, totally deleting with every partially or totally deleted image all those which are dominated by it, then deleting one conjunction sign next to every image deleted from M?, and finally rearranging the non-deleted terms of M?.

(b) It follows that (P)iMi yields some (P)*M! by V-instantiations, such that, if M* yields M*- by (PCI)-elimination of R1..., Rv, then F (P) 'M! D

(P)*M*-. Moreover, the V-instantiations can be chosen as in Theorem 5c. Because of Theorem 6 we shall assume for the rest of ? 3 that A contains

at least one V. Then the set B- of formulas (P)!M!- of Theorem 6 is a basis in PCI for S-. B- is usually not monotonic. However, for any (P) M! - and (P)*M!- in B- one easily finds a formula also in B- implying both, so that one easily finds formulas B1, B2, B3, ... in B- = {B', B', B3, . } implying respectively B', B1 . B', B2.B, ....Suppose (P)iM, =(Q)O(P) M and (P)t2Mt2 yield (P)*M* or (P)*M* respectively by V-instantiations. Then we can find an alphabetic variant (Q)O(P)t M' of (P)t2Mt2, such that

. . Mt) is a development of A yielding by V-instantiations some (Q)O(P)1* (Mr . M 2*), where (Q)0(P)' is (P)*1 and (Q)O(P) 2*M M* is an alphabetic variant of (P)*2v*2, such that (Q)u(P)'*(P) '*(MW1 . Mi*)- is in B- and implies both (P)i*M!- and (P)*M7-.

The condition in Theorem 5c on the V-instantiations used in derivations from A of formulas in B- can be simplified and slightly strengthened, and the dependence of the procedure on C somewhat lessened, by assuming that function symbols, individual constants, and individual variables occurring free in a formula C in S or S- also occur thus in A.16 Then by the definition of development and of V-instantiation, they also occur thus in the premiss of any of the V-instantiations leading from (P)iMi to (P)!M*. Since each C in S- is equivalent to some VxC', it follows that V-instantiations used in derivations from A of formulas in B- can be so restricted that for a given premiss there is only a finite number of function symbols, individual variables, or individual constants which may occur in the individual term t. This still allows for each premiss an infinite number of t's except when there are no function symbols in A.

16 In ? 2 this assumption is not needed.




Certain regularities can also be imposed on the last step in deductions from A of formulas (P)*M*- in B-, namely on (PCI)-elimination. The relationship of M*- to N and the role of R1, . . ., Rz will then become clearer. Roughly speaking each Rj has the effect that in certain places logically false formulas are conjoined, thus ruling out certain alternatives. The use of Rj in PCI is thus included in, but less flexible than, its use in PCIl, since the condition ,(s, = sl . .. s a sI) used in PCI= becomes a logical falsehood when s1, ... , sd, are the same individual terms as s . I..,sd, respectively. This more limited use in PCI explains at least partly why V-instantiation in the presence of R1, ..., Rq cannot always be avoided in PCI.

To express this more precisely below, let y['r(Ki); 'r'(K2)] be Ros... s. Ros..... s whenever -(K1) and -'(K2) have at least one <Si, *.., Sd> in

common, where s is chosen according to some method from the terms occurring in some common <sl, ..., sd>, and let y[T-(Ki); -'(K2)] be empty otherwise.

Assume from now on that the matrix N of A is some V A K!m satis- 1gmgs O~jsq

fying conditions (i), (ii), and (iii*) for (PCI)-elimination of R1, ..., Rq.17 Then, as in ? 2, M can be written as A V A Kjn'm, and is equivalent

1!9n:!gt 1 g5mr.s 0! Y:r.q to V A A Kjn 9(1). We shall write M* ( A Nn)* as A V

v671' l n:5.t 0o9igj q 1:5'n:!t 1!9n:!9t 1:.m:.s A K1, m*, which is equivalent to V A A K!, 9(n)*. Since A Kjz 9(n)*

and y[7-( A Kj 9,()*)',r'( A Kjn9w(n)*)] are both logically false and hence 1 <n_ 1r tn~

equivalent whenever for some si, . sd, both Rjsl ... Sd, and ,Rjsl ... Sd, are terms of A K. ,,,W* (parentheses being disregarded), it follows that

1 5_9n:!.t among the formulas M*- obtainable from M* by (PCI)-elimination is V ( A Kn (n)* A y[r( A K! 5p(n)*), 7'( A Knq(n)*)]). Then by an

gw?F 1 ;5n:!gt 15 rj:q 1;5n:!t 1 5n:!t argument as in ? 2, one also obtains from M* by (PCI)-elimination V( A Ko n(n)* . A A y[r(K`, 9(n)*); '(K *)

96F 1:5n:!gt l 5 q 1-5n,n':5 t Now to allow an explicit notation let us adopt some convention which

gives for each i a unique regular i-development (P)iMi of A and which then orders those (P)*M* which are then obtainable from (P)iMi by V-

17 There is no point in requiring also that (v*) no K"1 contains as terms both a formula R.sl... SdJ and its negation -,Rjsl... Sd1, 1 s j ; q. Even if N and M satisfy this condition, M* may not. E.g., N = M = Rox . Rlx. -nRlt, with x and t distinct, may give rise to M* = Rot. Rt . -nRlt. Hence there often is no M' which is simply related and equivalent to M* and which satisfies (v*). This is why, in place of the previous description of M*- in terms of an M' from which it is obtained by deletions, we now use y and describe M*- in terms of a matrix not satisfying (v*) but more simply related to M*.



114 WILLIAM CRAIG

instantiations. Then for (P)'M* (P),kMk in this ordering, we can write Kim v(n)* as eik(KT ()). As in ? 2, we can then conclude:

THEOREM 7. If the matrix of A is some V A K, satisfying con- 1 5mg8 0O5j~q

ditions (i), (ii) and (iii*) for (PCI)-elimination of R1, ..., Rq, and if unique regular i-developments (P)iMi are chosen and their (P)*M* ordered by a convention, then for each M*, one can choose

V/ ( /A ak(K(n)). A A y[Ti k(T?(n)); cr',(T' 'n)]). tSt)1 n--wi 15 j5Q 1 5n, n':<P

An intrinsic description of the (P),"k associated with one of the above M:k has been given informally. Together with the above description of M*- it constitutes a scheme for generating a basis B- in PCI, called in ? I a scheme of kind 2.

4. Outline of argument. Advancement of elimination. In outline, our proof of the results in ? 2 and ? 3 may be described as follows. Let {A} be a basis for S, either in PCI or in PCI= . According to the Herbrand- Gentzen theorem in the form of [5],18 or its extension in that form to PCI=, contained in Theorem 10 below, there are certain few types of operations such that, for each C in S. there is a sequence or linear deduction A1 = A, A2, ..., Ak= C from A to C, such that each Ai yields At+1 by one of these types of operations and such that F Ai v Ai+1 or F= Ai v Ai+, respectively, 1 < i < k. Moreover, the use of each type of operation is restricted to a certain segment of the deduction. Now consider those linear deductions Al, . . ., Ak from A to C such that C is in S-, and in each Al, . . ., Ak select the first A1 in S-. The set B- of all A1 thus selected evidently forms a basis in PCI or PCI= respectively for S-. Now in these Al, . . ., Ak certain types of operations are used only in the terminal segments As, . . ., Ak and are avoided in the initial segments Al, ..., As which constitute the deductions from A of the formulas in B-.

This method is quite general and can be used for (the deductive closures of) arbitrary subsets S- of S, not only for subtheories in our technical sense.19 The number of types of operations thus avoided and the corresponding gain in economy varies with the kind of set S- considered. E.g., when S- is the set of those C in S in which certain Junction symbols do not occur, then only three types of operations used in the terminal segments A1, . . *, Ak are not used in the initial segments Al, . . ., As (namely V- and 3-importation

18 In [4], the Herbrand-Gentzen theorem was used in one of its customary forms. This made the proof more complicated. Roughly speaking, the Herbrand deduction of A D C from a tautologous matrix was paralleled by a Herbrand deduction of A1 D C.

19 Moreover other forms of linear reasoning, such as those in [6], involving somewhat different types of operations, could be used. However, at least for subtheories in our technical sense, the formulation of [5] seems most advantageous.




and simplication of [5] p. 252). When, instead, S- is the set of those C in S which are of the form Vu1...VurM, then one additional type of operation is thus avoided (namely 3-generalization of [5]). When S- is a subtheory of S, then one further type of operation is thus avoided (namely 3-vacuous-removal and thus all b-operations of [5]). While these economies are slight, it is doubtful that in general substantial improvements are possible.

When S- is a subtheory of S and when S has a basis {A} in PCI=, three improvements are possible. The main improvement consists in advancing, in the linear deduction from A to C, (PCI=)-elimination of Ri, ..., Rq ahead of two other types of operation, namely V-instantiation and then rather trivially V-vacuous-introduction. The possibility of such advancement is shown in Lemma 4 below. A second improvement consists in imposing regularities on that initial segment of a linear deduction which precedes the V-vacuous-introductions, with the result that the last formula of this segment becomes a development of A. This is done in ? 5 and ? 6. Those parts of the work there which are needed here are summarized in Lemma 1 below. Theorem 1 then follows by Lemma 2b below from Lemmas 1 and 4. Theorem 2 then yields a third improvement. It is established in ? 7, by an abstract embedding argument.

When S- is a subtheory of S and when S has a basis {A} in PCI, then in linear deductions from A to C, (PCI)-elimination of Ri, ..., Rq cannot in general be advanced ahead of V-instantiations. However, the same regularities as for PCI= can be imposed on the steps preceding the V-vacuous- introductions. Also some regularities can be imposed on the V-vacuous- introductions and, what seems more important, the V-instantiations. All this is again shown in ? 5 and ? 6. The parts needed were summarized in Theorem 5. Theorem 6, which supplements it and which allows us to avoid V-vacuous-introduction, is proved in ? 7 by arguments related to those for Theorem 2.

For the proof of Theorem 1 we need from ? 5 and ? 6 only the following lemma, which follows immediately from Theorem 1 Oa,, Lemma 18, and the fact that no L-rule other than matrix change diminishes the set of predicate symbols present in a formula.

LEMMA 1. If F= A D C, if R1, .. ., Rq do not occur in C, and if A is in prenex normal form, then some development (P)M of A yields some (P)+M by V-vacuous-introductions which yields some (P)*M* by V-instantiations such that, for some M' not containing R1, ..., Rq, F= M* v M' and F (P)*M' C.

LEMMA 2. Suppose M yields M- by (PCI=)-elimination of R1, ..., Rq. Then:

(a) M- 3R1. ...3 RqM is valid.



116 WILLIAM CRAIG

(b) It follows that, if R1, ..., Rq do not occur in N. and if F. M v N, then F.M- N.20

PROOF. Let M- be V (Kr. A a(K7m)), where V A Kj is in

disjunctive (R1, ..., Rq)-normal form for PCI= and is truth-functionally equivalent to M. Consider any model of M- and hence of some Kin A 1 < m < s. To each R1 assign as value the set of those

dr-tuples which are denoted by some <tl, ..., td,> E 1 < j ? q. Then each Kt, 1 < q, hence A Kjm, and therefore V A K! are satisfied. Q.E.D. O!j5q l m=s O Y!q

LEMMA 3. Suppose M yields a(M) by substitution of an individual term t for an individual variable x, a(M) yields (a(M))- and M yields M- by (PCI=)- elimination of R1, ..., Rq, and M- yields a(M-) by substitution of t for x. Then F= a(M-) D (a(M))-.21

PROOF.22 Let M- be V (KrI. A a(K'm)) where V A K71 is in lM:<s 15 lj~q l m:5 ,s O:< ,jgq

disjunctive (R1, ..., Rq)-normal form for PCI= and is truth-functionally equivalent to M. Then also F a( V A K1 _M), i.e., F( V

1l m :',s 05 Y :5l m q

A a(Km)) _ a(M). Hence F= ( V A a(K71)) : (a(M))-. Then, by O:<jq 1 m5s Oj!q

Lemma 2b, F= ( V (a(Kn). A a(a(K~n))) v (a(M))-. But, disregarding 1 Sm~s 1 <j~q

the order of terms in a conjunction, a(a(K7')) is a(a(K7m)), 1 j q, I < m < s, so that a(M-) is V (a(Kr). A a(a(K7'))). Q.E.D.

LEMMA 4. Suppose M yields a(M) by substitution of an individual term t for an individual variable x, M yields M- and a(M) yields (a(M))- by (PCI=)- elimination of R1, ..., Rq, and (P)'Vx(P)"M yields (P)'(P)"a(M) by V- instantiation. Then F. (P)'VX(P)"M- D (P)'(P)"(a(M))-.21

20 One can prove Lemma 2b directly, using a finitary proof-theoretic argument, so that the non-finitary argument for Lemma 2a is avoided. Let M and N be as stated, and let M be some V A Kim in disjunctive (R1, ..., Rq)-normal form for

1 smss O <jsq PCI=. Then also FK ( A Km) N. 1 m C s, and hence F A W . J .D N

w i o o --q 0J 2L in where J is a conjunction whose terms include all those of the appropriate J1 and JL Lemma 16 below. Now any assignment of truth-values which would make Kr,.

A 8(KrI) . J .D N false can be shown to yield another assignment of truth-values 1j!9q

which makes A KnZ . J .D N false. Hence F Kr. A 8(Krn) . J .3 N, 1 <m Es, Of j!q 1- j: q

and thus F= V (Km. A a(K'7)). D N. Then Theorems I to 4 are proved 1 smss 1 < j!q

entirely by finitary arguments. 21 Also conversely F= (a(M))- D a(M-) and hence F= (P )'(P)"(a(M))- D (P)'(P)"a(Mj. 22 Lemma 3 also follows immediately from the validity of a(M-) 3R... 3 * Rqa(M),

which follows from Lemma 2a. However, the argument is then non-finitary.




PROOF. By Lemma 3, F= (P)'(P)"a(M-) D (P)'(P)"(a(M))-. But also

F (P)'Vx(P)"M- D (P)'(P)"a(M-). Q.E.D. Now suppose A, C, and R1, ..., Rq satisfy the conditions of Lemma 1,

and let (P), (P)+, (P)* and M, M*, and M' be as stated there. By Lemma 2, if M* yields M*- by (PCI=)-elimination of R1, ..., Rq, then F= M*- v M'. Since F (P)*M' v C, therefore F= (P)*M*- v C. By Lemma 4 and induction F= (P)+M- v (P)*M*-, where M yields M- by (PCI=)-elimination of

R,, ..., Rq. Since each variable in M- also occurs in M, all quantifiers occurring in (P)+ but not (P) are vacuous also in (P)+M-, so that F (P)M- O

(P)+M-. It follows that F= (P)M- v C. This establishes Theorem 1.23

For comparison let us briefly consider the situation for PCI. Then, analogous to Lemma 2b for PCI= and playing a similar role, there is the following lemma, which is easily proved (e.g., by the argument of p. 268 of [5], where M* takes the place of the present M-).

LEMMA 5. If M yields M- by (PCI)-elimination of R1, ..., Rq, if R1, ..., Rq do not occur in N, and if F M D N, then F M- D N.

On the other hand, Lemmas 2a, 3, and 4 have no analogues for PCI, as can be seen for M = Rt . Rix . R2y, a(M) = ,Rt . Rit . R2y, and

(a(M))-= R2y . ,R2y, where R1, R2, t, x, and y are distinct.24 REMARK 4. If function or individual symbols are to be eliminated,

instead of predicate symbols, then Lemmas 2a, 3, and 4 have no analogue even for PCI=. E.g., let M be w = g(x, y) . g(y, x) = z with w, x, y, z distinct. If x does not occur in N and F= M v N, then also F= N, since otherwise a model of -N would yield a model of M. ,N by the adjoining of a new object as value for x. If in analogy with Lemma 2b we let w = w be M-, then M- v 3xM is invalid, and also a(M-) v (a(M))- is false when a(M) is w = g(y, y) . g(y, y) = z. (Another example is furnished by

=y= x, y = y, and -y = y as M, M- and, a(M) respectively.) Thus even for those algebraic theories which can be expressed as a conjunction of identities, elimination of individual or function symbols cannot always be moved ahead of V-instantiation.

There is a second reason why Theorem 1 with function symbols fl,.. ., fq in place of R1, ..., Rq is in general false. Usually fl, . . ., fq are still present in the conclusion of the last matrix change with identity (see ? 6) and are removed only by 3-generalization (see [5]). Thus, in practice, it may be

23 In fact, the full proof shows that there is a symmetric L.-deduction (see ? 6) from (P)M- to C without any Y- or 3-exportations or duplications of [5]. Biichi has

suggested in conversation that types of operations not needed for derivations from B- of formulas in S- be used as an indication of the lack of economy of B-. Our own indicator of economy, namely types of operations not needed for derivations of formulas in B-, can be used only if there is some B from which they are to be derived.

24 Note however that, for PCI without a symbol for =, any model of M- can be extended into a model of M by adjoining new objects to the ground domain. (See also end of Remark 1.)



118 WILLIAM CRAIG

best to replace such fl, ..., fq by predicate symbols in the usual way (e.g., [12] Theorem 43), before constructing B-. Unfortunately the prefix of A becomes then more complex, by the introduction of 3 after V. Individual symbols occurring free in A but not in C can, of course, be quantified by an initial 3.

5. Conditions on Herbrand deductions. We shall now show that certain conditions or regularities can be imposed on Herbrand deductions. We shall use the formulation of ? 3 of [5], familiarity with which will be assumed throughout ? 5. ? 5 is practically completely summarized in Theorem 8 below and will be used later on only in the proof of Theorem 9, in ? 6.

We shall consider throughout ? 5 and ? 6 an arbitrary A =A ... . Ar and A* = Ar+i V ... V Art such that each AX is in prenex normal form, 1 < j _ r'. For I < j < r, let As = (Q)9 ... (Q)^;'Nj, where (Q) ...,(

are defined for Aj as (Q)O, ..., (Q) h are defined for A in ? 2. For r < j r', let Nj be again the matrix of AX and let (Q)?, ..., (Q)^' be defined for As in a manner dual to that for 1 < j < r, interchanging V and 3, so that (Q)? is either empty or of the form Vvl.. .Vvn, (Q), ...,

are all of the form 3ul... 3umVvl... Vvn, and, if h # 0, (Q)^' is either also of that form or of the form 3ul...3um. Roughly speaking, we shall show that, if there is an H-deduction of A D A*, then there is an H- deduction of A v A* in which (a) each (Q). is treated as a unit, 1 < j < r', O < g _ A1, (b) certain operations precede (follow) all other operations, and (c) no individual variable or constant is "superfluous".

(A variety of conditions or regularities can be imposed on Herbrand deductions, not all compatible with one another. It is unlikely that one set of conditions will turn out superior to all others. Conditions in the literature can be classified into two kinds, namely (I) conditions on the quantifier- free tautology Bo D Bo from which an H-deduction of A D A* starts, and (II) conditions on the operations whereby A : A* is deduced from Bo D B*. To some conditions of kind (II) there corresponds no condition of kind (I), e.g., to condition (b) below or also to the conditions which are added by Lemmas 7 to 9 below to the condition of Lemma 6. Conversely, some conditions of kind (I) have at least no simple counterpart of kind (II), e.g. the condition that Bo v Bo be related to A D A* as required by Her- brand's properties B or C, [10] pp. 97-108. The more profound conditions seem to be of kind (I). For our purpose conditions of kind (II) are adequate. Also, conditions of kind (I) would be insufficient since the regularities we want to induce in ? 6 in the "transcripts", i.e., the corresponding linear deductions from A to A*, include regularities in the behavior of quantifiers. Condition (a) below apparently has not been stated before,36 although it can be easily derived from results already known. E.g., Lemmas 7 and 12




in Kleene [14] can be applied to Lemma 6 below, which in turn is implied, e.g., by Herbrand's property B or C. Condition (b) is rather trivial but is needed for the proof of Theorems 10, 9c, and 5c. Condition (c) is probably also implied by Herbrand's property B or C, and at any rate is simply a restatement, for the case where function symbols may be present, of a condition in Hintikka [11] and in similar recent treatments on the analogue(s) of VA- and 3C-introduction. To what extent (c) can be strengthened so that in some sense the individual terms t themselves of VA- and 3C- introductions, not only their variables and constants, are "non-superfluous" or of a minimum complexity is not clear. The three conditions together are quite "incomplete" in the sense that different regularities of kinds (I) and (II) may be added.)

As in [5] p. 259, we shall talk about the individual variable of an 3A- or VC-introduction, whether this 3A- or VC-introduction is vacuous or not, and we shall assume that for different 3A- or VC-introductions the variable is different. Moreover, we shall say that the variable of an 3A- or VC- introduction occurs free in all preceding formulas of an H-deduction, even if the 3A- or VC-introduction is vacuous.

Given any H-deduction Bo v B, .B.., B, D B* of A v A*, there corresponds in an obvious manner to each term in Bin-i or B*-1 exactly one term in Bm or B* respectively, such that conversely each term in Bm or B* except perhaps one corresponds to exactly one term in Bmi- or B* 1 ? m ? s. The exceptional term occurs in Bm or B* only when Bm-1 B* yields Bm D B* by an A- or C-simplication and corresponds to two terms in Bi-, or B* -, the term operated on and the term immediately to its right. If 0 < ml < m2 < s, then a term in B., or B*1 shall belong to a term in Bm or B*2 respectively if and only if either ml= m2 and the terms are the same or ml < m2 and the term in B or B*^ belongs to a term in Bm21 or B*_ 1to which the term in B or B* corresponds in the manner just described. (Cf. [14] p. 7.) By induction, any term in Bm or B* belonging to the term A= (Q)?.. (Q)^jNj in B, or B* is a formula (Q)1N' such that, for some (Q)', (Q)' (Q)j is (Q)? ... (Q)^? and Nj yields N' by substitution of individual terms for individual variables which also occur in (Q)1

THEOREM 8. II there is an H-deduction of A D A*, then there is an H-deduction B0 D B*, B1 D B*, ..., B B*=-A D A* having the following three properties:

(a) There is a subsequence BP0 DB* B 0oB*, BP1 D B1, .., BP D Bp A D A* such that each Bpn 2 B* yields Bpn4:1 B*n41 by the replacement for some j, 1 < j < r', either (i) of a term (Q)"+1... (Q)hJN' belonging to Aj in B, or Bs* by a term (Q)q(Q).+1... (Q)hiN;, 0 < g C h1, or (ii) of w0o or more adjacent terms (Q).... (Q)JN'N (and of the appropriate . or V),

suich that to each belongs at most one term ........(Q)fiiN; in any earlier Bm



120 WILLIAM CRAIG

or B*, O < m < Pn, by one term (Q)q7 (Q)j Ni belonging to Aj in B8 or B*, 1 < g < h1.25

(b) If none or more A, ..., Ak have been chosen beforehand such that Q>J- consists only of V- or only of 3-quantifiers according as Aye is a term of A or of A*, 1 < n < k, then all quantifier introductions concerned with forming Quit in a term belonging to A1n in A or A* precede all operations on a term not belonging to one of A,,, . . ., Ain in A or A*, 1 < n < k. Like- wise, if also none or more Ajkxl, .I. *,

Aik' have been chosen beforehand such

that (Q)9 is non-empty and therefore consists only of 3- or only of V-quantifiers according as Aj is a term of A or of A*, k < n < k', then all quantifier introductions concerned with forming (Q)?n in a term belonging to A,1 in A or A* follow all operations on a term not belonging to one of Ajk, . . .,-An'

k < n ? k'. (c) If no variable occurs both bound and free in A v A*, then each individual

variable or constant in the individual term of an VA- or 3C-introduction from Bm D B* to Bm+i D B* 1 also occurs free in Bm+i D B*+1, provided there is an individual variable or constant occurring free in Bm.+ i B*D

Theorem 8 follows from a series of lemmas whose proofs will only be sketched, since the general subject matter is well known. Most of the lemmas state slightly more than is required for the proof of Theorem 8.

LEMMA 6. If there is an H-deduction of A M A*, then there is an H- deduction of A v A* having the property that for any A- or C-simplication the term (Q)N' operated on begins with V or 3 respectively.

PROOF. In an H-deduction Bo D B*, ..., B, D B* of A D A*, consider the first A- or C-simplication, if any, which fails to have the desired property. If it operates on a term without quantifiers, then it can be avoided altogether. If it operates on a term 3x(Q)N'(x) or Vx(Q)N'(x) respectively, then it can (roughly speaking) be advanced ahead of the two 3A- or VC-introductions concerned, so that it will operate on a term (Q)N'(z) instead. The same new individual variable z must replace the individual variable of each of the two 3A- or VC-introductions in all its free occurrences in Boo Bo*, B, v B*. In the resulting H-deduction, only one 3A- or VC-introduction corresponds to the two 3A- or VC-introductions considered. The process is repeated until the desired property holds. Q.E.D.

LEMMA 7. For any H-deduction Bo D B*, B1 D B*, ..., B, D B* =

A v A* having the property of Lemma 6, there is an H-deduction Bo D B*' -

Bo D B*, B' D B*', ..., B D B*' = A D A* having the property that any term on which an A- or C-simplication operates and which belongs to A1 in A or A* has a prefix (Q);. . . (Q)t 1 <j i r', 1 ?!g ? h1.

25 It follows by induction that each term in any Bp. or B*' belonging to AX in B, or B* is a formula (Q)9+1 ... (Q) iN;, where-1 ? g < h1 and where (Q)+1... (Q)+. is empty when g = hj.




PROOF. In an H-deduction Bo D .B.., B8 D B* of A D A* having the property of Lemma 6, the last A- or C-simplication, if any, failing to have the desired property can be moved back of the VA- or 3C-introduction concerned, so that it will operate on a term Vx(Q)N'(x) or 3x(Q)N'(x) respectively, instead of on a term (Q)N'(z). In the resulting H-deduction, Bo v B* remains unchanged and two VA- or 3C-introductions correspond to the given VA- or 3C-introduction. (Inspection of the H-rules should supply needed details. Moreover details of a similar construction are given in the proof of Lemma 12 of [14], first illustration.) The process preserves the property of Lemma 1 and is repeated until the desired property holds. Q.E.D.

LEMMA 8. For any H-deduction Bo D B*, B1 D B*, ..., B8 D B* =

A D A* having the property of Lemma 7, there is an H-deduction B' D B*0= B0 D B*, B1 D B*', . . ., B * B*' = A M A* also having the property of Lemma 7 and having a subsequence Bpo v B0 =B Bo Bpl v B*. B D Bps, = A D A* such that each Bp D B* yields B D B*,, either by one or more A- or C-simplications or else by the replacement for some j, 1 < j ! r', of a term (Q)q+1 ... (Q)&iN;' belonging to A1 in B, or B* by a term

(Q)q(Q)q+1... (Q)^JN, 0 < g ?h1.

PROOF. In an H-deduction Bo D Bo, . . ., B * D B* of A M A* having the property of Lemma 7, consider the last 3A- or VA-introduction, if any, operating on a term 3x(Q)N' and separated by other operations from an earlier 3A-introduction which introduces this 3x into a term 3x(Q)N' belonging to the given term 3x(Q)N'. By the property of Lemma 7, there belongs to the given term 3x(Q)N' at most one term 3x(Q)N' in any earlier formula, so that there is only one such earlier 3A-introduction. This 3A-introduction can be moved back of the separating operations until it immediately precedes the 3A- or VA-introduction operating on 3x(Q)N'. (Inspection of the H-rules shows that this is possible. A similar argument occurs in the proof of Lemma 7 of [14].) The process is repeated as often as needed. It preserves the initial Bo v Bo, the number s of steps, and the property of Lemma 7. Each VC-introduction thus separated from a later VC- or 3C-introduction can be moved backward similarly until there is no such separation. Consider now the first 3A- or VA-introduction, if any, changing the term operated on into a term (Q)N' and separated by other operations from a later VA-introduction which operates on a term (Q)N' to which the given term (Q)N' belongs. Then for some j, 1 < j ? r, and some (Q)N' and Vx, (Q)'Vx(Q) is (Q)?...(Q)> Hence (Q) is distinct from each (Q).... (Q)<9 1 ? g < hj. Hence by the property of Lemma 7, no A-simplication operates on (Q)N'. The VA-introduction operating on (Q)N' can therefore be advanced ahead of the separating operations (as can be seen by inspection of the H-rules or by the proof of Lemma 7 of [14]), until it immediately follows the 3A- or VA-introduction considered. The



122 WILLIAM CRAIG

process is repeated as often as needed. It preserves the initial Bo v Box the number s of steps, the property of Lemma 7, and also the properties obtained by the previous processes. By a similar process, each 3C-introduction thus separated from an earlier 3C- or VC-introduction can be advanced, until finally a deduction with the desired property results. Q.E.D.

LEMMA 9. For any H-deduction Bo D Box B1 D B*, ..., B, D B* A v A* having the property of Lemma 6, there is an H-deduction Bo v B' =

Bo v Bx B' D B*', ..., Bv B*' -A v A* having property (a) of Theorem 8. PROOF. In an H-deduction Bo D B` Bo D Box B' D B*, ... . B,

B*,= A v A* having the properties of Lemmas 7 and 8, only part (ii) of property (a) remains to be satisfied which requires that between certain A- or C-simplications no other operation occurs. Now any earlier A- or C- simplication separated by other operations from later A- or C-simplications from which it should not be separated can be moved back of these separating operations. Eventually, part (ii) of (a) will be satisfied along with the other parts. Q.E.D.

LEMMA 10. For any H-deduction Bo D Box B1 D B*, ..., B, D B* =

A A*, there is an H-deduction B v B` = Bo v Bo B v B I

.

B,' B*' = A : A* having property (b) of Theorem 8, such that property (a), if present, is preserved.

PROOF. In an H-deduction of A D A* the VA- and 3C-introductions concerned can be advanced (as in Lemma 8 above or as in Lemma 7 of [14]), along with any A- and C-simplications concerned that in the absence of (a) may be present, without disturbing (a) if present or the number s of steps, until the first part of (b) is satisfied. Likewise, the 3A- and VC-introductions, along with any A- and C-simplications, concerned can be moved back until the rest of (b) is satisfied. Q.E.D.

LEMMA 11. Given any H-deduction Bo D Bo, B1 D B*, . .., B, D B* A v A*, an H-deduction Bo v B*') B' v Br', ..., B' v B'* A :)A* having property (c) of Theorem 8 can be obtained by changing only free individual variables or constants, preserving properties (a) and (b) if present.

PROOF. Consider any H-deduction of A D A* and assume that no variable occurs both free and bound in A v A*. Then by Lemma 37 of [12], the H-deduction can be transformed by changing individual variables only into a pure variable H-deduction Bo v Bo, ..., B v B,* of A n A*, i.e., an H-deduction in which no variable occurs both free and bound. Now suppose that the individual term of the VA- or 3C-introduction from Bm v B* to Bm+, v B*+1 contains an individual constant or variable s which does not occur free in Bm+, i B* but that some individual variable or constant s' occurs free in Bm+, v B.*+1. Then substitution of s' for s throughout the H-deduction yields another pure variable H-deduction of A =)A*. (Cf. [12] Lemma 36.) Q.E.D.

Theorem 8 follows immediately from Lemmas 6, 9, 10, and 11.




6. Conditions on L- and L=-deductions. We shall now turn to L-deductions, as defined in [5], and shall prove Theorem 9, which somewhat improves Theorem 2 of [5] and allows us to impose on L-deductions certain further conditions or regularities, most of them corresponding to conditions on H-deductions in Theorem 8. We shall then define the notion of L=- deduction, which is an analogue for PCI= of the notion of L-deduction for PCI, and shall show in Theorem 10 that the conditions of Theorem 9 give rise to similar conditions on L=-deductions. Finally, we shall show in Lemma 18 that, in an L- or L=-deduction starting from A and satisfying certain of these conditions, the formula (P)M obtained at the end of the assembling phase ([5] pp. 254, 258) can also be described in a different and perhaps more intuitive language, namely as a development of A. Several corrections needed in [5] and a correction needed throughout [6] are brought out by the proof of Theorem 9 and will be discussed at the end of ? 6.

Those results of ? 6 which are needed elsewhere have been summarized in Lemma 1 of ? 4, which uses Lemma 18 and part of Theorem 10, and in Theorem 5 of ? 3, which uses Lemma 18 and most of Theorem 9. Except for Lemma 15, the proof of Theorem 9 below requires familiarity with Theorem 8 of ? 5 and with the notion of L-transform of [5] pp. 260-261, and thus pretty much with all of [5]. For everything else in ? 6, familiarity with ? 2 of [5] should be sufficient.

A term of the main conjunction in Em will often more briefly be called a term in Em. If E0 = A, E1, . . ., E.1 is that initial segment of a symmetric and Q-balanced L-deduction such that E., is the premiss of the first matrix change, then there corresponds in an obvious manner to each term in E,,,+ exactly one term in Em, such that conversely each term in Em, except perhaps one corresponds to exactly one term in Em+,, 0 _ m < sl. The exceptional term in Em is the term operated on by a duplication and corresponds to two terms in Em+,, the term in the same position in Emil and the term immediately to its right. If sl > ml > m2 > 0, then a term in Emi shall belong to a term in Em2 if and only if either ml = m2 and the terms are the same, or ml > m2 and the term in Em. belongs to a term in

Em2+i to which the term in Em2 corresponds in the manner just described (cf. ? 5). By induction, if Em is in the assembling phase, then any term in Em belonging to the term A= (Q)9... (Q)'JNI in Eo = A is a formula (Q)jN* such that, for some (Q)', (Q)r(Q),N. is an alphabetic variant of (Q)%... (Q)'JNj, and such that, for some (Q)., (Q); (Q)> is (Q)9... (Q)^. A dual notion of belonging can be defined for that terminal segment in which exactly all b-operations occur.

THEOREM 9. If IA D A* and if A Al.. ... Ar and A*=Ar+ V ... V Ar, are such that Al, . . ., Ar, are in prenex normal form, then there is a symmetric and Q-balanced L-deduction E0 = A, E1, ..., Es.= A* having the following properties.



124 WILLIAM CRAIG

(a,) If E81 is the last formula of the assembling phase, then there is a subsequence Ep,, =A, Epps1, ... , EPPO = E.1 such that each Epn+1 either is EPn or else yields EP'n by the replacement for some j, 1 5 j < r, either (i) of a term (Q)N(Q)7+1... (Q)'Nw belonging to Aj in A by a term (Q)q+1... (Q)hJNr*, and of the main prefix (P) of E by (p)(Q),**, such that (Q)J**(Q)J+l* (Q)hJN~* is an alphabetic variant of (Q)q(Q)'+l... (Q)hJN', 0 < g hj, or (ii) of a term (Q)9'N. belonging to A1 in A by two or more adjacent terms (Q).... (Q)h'N' (and the appropriate .) such that to each belongs at most one term (Q)q... (Q)h'N:' in any later Em, p < m C sl, 1 < g ? h1. A dual property holds for the disassembling phase.

(a2) Each V-vacuous-introduction introduces an V-quantifier into the main prefix only before or after, but not into, any part (Q)57**, where (Q)q*1 is exported into the main prefix as a unit in a step from E to E ,s' > n > O. A dual property holds for 3-vacuous-removals.

(b) If none or more A,,, . . ., Ak have been chosen beforehand such that (Q)h'n consists only of V- or only of 3-quantifiers according as A1n is a term of A or of A*, 1 ? n < k, then no chain initiated (terminated) by an V-exportation from (3-importation into) the prefix (Q)h'n of a term belonging to An in A (in A*) occurs to the left of any chain initiated or terminated by an operation on a term not belonging to one of A,,, . . ., Ain in A or A*, 1 < n < k. Likewise, if also none or more A,4+,, . . ., A,k. have been chosen beforehand such that (Q)? is non-empty and therefore consists only of 3- or only of V-quantifiers according as Ain is a term of A or of A*, k < n _ k, then no chain initiated (terminated) by an 3-exportation from (V-importation into) the prefix (Q)9 of a term belonging to Ain in A (in A*) occurs to the right of any chain initiated or terminated by an operation on a term not belonging to one of Ajk+I, . . ., A k,

in A or A*, k < n k'. (c) If at least one Ain, k < n < k', has been selected in (b), then each

individual variable or constant in the individual term of an V-instantiation either occurs also in the remiss of the Y-instantiation or occurs free in A*. A dual property holds for 3-generalizations.

(d) No variable of a chain occurs in a formula of tA'e deduction outside the segment coextensive with the chain.26

Theorem 9 will be proved by means of four lemmas. We shall first show that if an H-deduction has properties (a), (b), or non-trivially (c) of Theorem 8, then any L-transform has respectively properties (al) and (a2), (b), or (c) of Theorem 9. We shall then show that by a suitable change of bound variables in L-deductions these properties, along with (d), can be obtained also in the case where a variable occurs both bound and free in A v A*.

26 Given Q-balance, this property implies Q-pureness but not conversely. For a related but weaker condition on transient chains, which in contrast to (d) could have been imposed on all L-transforms of H-deductions, see [5] p. 264.




LEMMA 12. If an H-deduction of A D A* has property (a) of Theorem 8, then any L-transform has properties (ai) and (a2) of Theorem 9.

PROOF. Let Bo D Boa . ., B, D B* = A D A* be an H-deduction having property (a), and let Eo A, E1, . .., ES* = A* be an L-transform of it. Let sl be the number of A-operations in Bo v B*, ..., Bs v B.* Then by condition (vii) for L-transforms ([5] p. 260), the last formula of the assembling phase is E,,. The subsequence Bp,0 vp B , v1, B: , .B.., y Bp v B* I described in (a) of Theorem 8 induces a sequence EP1.= Eo, . .., Eppl E o = E.,, where p',1 = P2 or pn =t- (Pn+i-Pn) according as

BPn+l does or does not differ from Bpn' By conditions (vii), (viii) and (ix)* for L-transforms the main conjunction of each Epn has the same number of terms as Bpn and moreover corresponding terms have the same prefix. Thus, in case (i) of (a), case (i) of (a,) holds, since (Q)q*'(Q)q+l...(Q)^'N.*

is an alphabetic variant of (Q).(Q),+1... (Q)h?N' by the conditions for V- and 3-exportation. In case (ii) of (a) where two or more terms in Bvn belong to one term BPn+i and give rise to it by A-simplications, it follows from conditions (vii), (viii) and (ix) for L-transforms that the corresponding terms in the main conjunction of Ep,. belong to one term in EPn+1 and are obtained from it by duplications. Moreover to each such term (Q)N* in E I there belongs at most one term (Q)N* in any later Em, p2 < m ? sl, since otherwise, again by (vii), (viii), and (ix), to one of the terms (Q)N' in BPn considered there would belong more than one term in an earlier B., 0 < m < pn. This establishes property (al) for E0, E1, . . ., Es.. Property (a2) is an easy consequence of conditions (xi), (vii), and (viii) for L-transforms. Q.E.D.

LEMMA 13. If an H-deduction of A D A* has property (b) of Theorem 8, then any L-transform has property (b) of Theorem 9.

PROOF. By conditions (vii), (viii), (ix), and (xi) for L-transforms. Q.E.D.

LEMMA 14. If an H-deduction of A D A* has property (c) of Theorem 8, and if no variable occurs both bound and free in A v A*, then every L-transform has property (c) of Theorem 9.

PROOF. Suppose the L-transform is such that at least one Ai.n

k < n < k', has been selected in (b) of Theorem 9, since otherwise there is nothing to prove. Then the last quantifier introduction in the H-deduction is an 3A- or VC-introduction. Then for any VA- or 3C-introduction from

B. O B* to B.+, v B*1 there is an individual variable free in

Bm+l = B* +1. Hence by property (c) of Theorem 8, each individual variable or constant s in the individual term of the VA- or 3C-introduction also occurs free in Bm+l B*M+1 Since this holds for all VA- and 3C-introductions, s either occurs free in Bs v B* = A O A* or else is the variable of an 3A- or VC-introduction in B.+, v B ..., Bs O B*. Now consider any V-instantiation in the L-transform and let t be its individual term,



126 WILLIAM CRAIG

so that by condition (xb) for L-transforms t is also the individual term of the VA-introduction which corresponds to the complementary V- exportation. Consider any individual variable or constant s in t. If s is free in A*, nothing remains to be proved, and if s occurs free in A, then, by inspection of rules L 1a to L5a, it also occurs free in the premiss of the V-instantiation. In the remaining case where s is the variable of an 3A- or VC-introduction in the H-deduction, it is by (xa) the variable of the chain initiated (terminated) by the corresponding 3-exportation (V-importation) and occurs in the prefix of the premiss of the given V-instantiation. Q.E.D.

LEMMA 15. If Eo =A, E1, ..., E8. =A* is a symmetric and Q-balanced L-deduction, and if A' and A*' are alphabetic variants of A or A* respectively, then there is a symmetric and Q-balanced L-deduction E' = A', El,' Eve A*' consisting of alphabetic variants of Eo, E1, ..., Ev. respectively, having property (d), and preserving each of those properties (al) to (c) which hold for Eo, . . ., Es..

PROOF. Let A', A*', and Eo = A, E1, ..., Es*= A* be as stated. In

Eo, E1, . . ., Es* consider any chain whose variable y occurs either in A' or in A*' or also in some other chain, such that y occurs in the given chain further to the right, and replace by z all occurrences of y bound by an occurrence in the chain, where z does not occur in A', A*', or E0, ..., Es.* Let EN = A, EN, ..., E. = A* be the result of repeating this process as often as possible, so that no variable of a chain in Eo, ..., E. occurs in A', A*', or another chain. Next, consider any quantifier occurrence of a variable, say x, in a place in A = E' where A' contains a different variable, say x', and consider all those occurrences of x in A which are bound by this quantifier occurrence. Replace by x' in the main conjunction of each formula in the assembling phase of Eo,..., E". all those occurrences of x, if any, which correspond in a manner which should be clear to one of the occurrences of x considered in E'. Repeat this process for all those quantifier occurrences in A such that a different variable occurs at the corresponding place in A', and also make similar replacements in the disassembling phase for all those quantifier occurrences in A* such that a different variable occurs at the corresponding place in A*'. The sequence E' = A', El, ....

E = A*' which thus finally results has all the desired properties. Q.E.D. Now consider any A and A* as in Theorem 9. Suppose first that no variable

occurs both free and bound in A v A*. Then by the Herbrand-Gentzen theorem there is an H-deduction of A v A*. Then by Theorem 8, there is an H-deduction of A v A* having properties (a), (b) and non-trivially (c), so that by Lemmas 12, 13 ,and 14 any L-transform has the properties (a,) to (c) of Theorem 9. Moreover, by Theorem 1 of [5], this H-deduction has an L-transform which is symmetric and Q-balanced. This proves Theorem 9, except for (d), for the case where no variable occurs both bound and free in A O A*. Property (d), and also the rest of Theorem 9 for the case where




some variable occurs both bound and free in A M A*, then follows by Lemma 15, so that the proof of Theorem 9 is concluded.

Now consider any A and C as in Theorem 5. By first using Theorem 9al, Lemma 18 below, and the fact that no L-rule other than matrix change diminishes the set of predicate symbols present in a formula (cf. the proof of Lemma 1), and then using Lemma 5, one obtains Theorem 5a. By Theorem 9a2 and, if (Q)O is non-empty, the last part of Theorem 9b, one obtains that part of Theorem 5b which concerns the position of quantifiers introduced by V-vacuous-introduction. By Theorem 9d one obtains the remaining part of Theorem 5b concerning the variables of these quantifiers. The part of Theorem 5c concerning the individual constants and variables in t follows from Theorem 9c and the last part of Theorem 9b. If C begins with V, then, if (Q)0 is empty, we choose A,,+= C, and if (Q)0 is not empty, we choose A,,+l A and A,,+s = C. The part of Theorem 5c concerning function symbols follows simply from the fact that the formulas in PCI can be restricted accordingly. This concludes the proof of Theorem 5.

We now turn to the problem of analogues for PCI= of the notion of L-deduction and of Theorem 9 for PCI.

LEMMA 16. Given any matrix L of PCI=, let J1 be the conjunction of those formulas (1) t = t' O f(sl, ..., s-l, t, si+l, ..., sn) = f(sl, ... , Si-l, t', S1+, .I. ., sn) and (2) t t' :. Rs, ... si-ltsi+i ... s. D Rs, ... s.lt'si+l ... Sn with R distinct from , if any, such that each individual term serving as j-th argument of f or R respectively in the consequent, 1 < j ? n, also serves as j-th argument of f or R respectively in L, let J2 be the conjunction of those formulas (3) t = t and (4) t = t' A. t = s : t' = s whose individual terms occur either in JL or in L, and let kL be the number of distinct individual terms in J2. Then either L is invalid and -L is satisfiable in a domain of E kL objects, or else L is valid and JL .J . L is a tautology.

PROOF. Assume that JL . J2 .D L is not a tautology, so that there is a truth-value assignment p which makes JL . J2 and ,L true. For any s and s' occurring in J2, let s s' if and only if p makes s = s' true. Since p makes J2 true, is reflexive, symmetric, and transitive, so that, if for each t in J2 we form the class { of those s such that s t, then t belongs exactly to i. Since p makes, in addition, JL true, if sl ., s' and if f(sl, . . ., so) and f(sl, ..., s) both occur in L, then f(si, ..., sO)

f( si .s). Likewise, if s1 s l, ..., sn so and if Rs.... .s. and Rs....s' both occur in L, then p assigns to Rsl...sn and Rs....s$ the same truth-value. Hence if D is the set consisting of the < kL classes i, then for each f in L there is a function f partially defined on D such that, for each individual term f(s1, . . ., sn) in L, f(sl, . . ., sn) = f(s1, * * *, sn)-

Likewise for each R in L, there is a relation R defined on D such that, for each Rs,... sn in L, <9., . . ., Rn> e K if and only if p makes Rs,.. . sn



128 WILLIAM CRAIG

true. Now let tl, .. ., tp be the individual variables or constants, fl, ..., fq the function symbols, and Rl, . . ., R, the predicate symbols, occurring in L,

and let f,..., f' be some completion of f1, . . ., fq respectively on D. Then <D5; il, . .., p; ll ...,q; R1, ..., R,> is a model of -L, since every atomic formula in ,L, including those whose predicate symbol is receives the same truth-value as under p. Q.E.D.

Lemma 16 shows, and this will be its only use, that the rule which allows us to pass from (P)M to (P)M' if and only if F= M O M' (or, alternatively, if and only if F JM'M . Jm MJ .. M D M') is an effective rule. We shall call this rule matrix change with identity and denote it by L=6. L=-deduction, L=-transform, and for L=-deductions symmetric, Q-balanced, etc., shall then all be defined in the obvious manner as in [5], except for L=6 in place of L6.

Suppose now that F. A v A*. Then also F A . I .v A* where I is a conjunction of identity axioms, each in prenex normal form and without 3 (see, e.g., [12] pp. 399, 403). Then, by the proper choices in Theorem 9b, there is an L-deduction E0 = A. I, El, . . ., Es = A* having all the properties of Theorem 9, such that all V-exportations operating on a term which belongs in E0 to a term occurring in I follow all other assembling operations. Let Es1 = (P)K be the premiss of the first, and Es, = (P)M' be the conclusion of the last, matrix change. Then K can be written as M . J such that those terms in M. J which occur in M or in J respectively belong to a term in Eo which occurs in A or I respectively. Then each term of the conjunction J is obtained from an identity axiom by V-instantiations, so that F= J. Since F M . J . M', also F= M D M', so that (P)M yields (P)M' by matrix change with identity. Now let Eo = A, E', ..., E1 = (P)M be obtained from E0, El, ..., E., respectively by deleting from the main conjunction those terms which belong in Eo to a term occurring in I and by deleting from the main prefix those quantifiers, all innermost and all V, which were exported from one of the deleted terms. Then it is easily verified that (repetitions in E', . . ., E,1 being disregarded) E', . . ., E'1, ES2, . . ., E,. is a symmetric and Q-balanced L=-deduction from A to A* having all the other properties of Theorem 9. To summarize the argument:27

THEOREM 10. If F= A D A*, and if A =A1 .... . A, and A* Ar+l V ... V Ar, are such that A1, . . ., Ar, are in prenex normal form, then there is a symmetric and Q-balanced L=-deduction E0 = A, E1, ..., Es= A* having properties (a,) to (d) of Theorem 9.

We shall now form alphabetic variants Al, of AM in a way different from that described in Remark 3 of ? 2, with different quantifier occurrences no longer necessarily containing different variables, as follows. Letting Ag be (Q) ...(Q)^'N1 as described earlier, choose for each image of NJ

27 Except for Lemma 16, a similar argument can be carried out, and may be useful, for any conjunction of formulas Yxi... YxkM in place of I.




in Aj a component (Q)+1... I (Q)^'N, 1 < g < h1, such that the image occurs within the component in A', 1 < ? r. In forming AMI, let any prefix image occurring in a chosen component be unchanged, and change any prefix image not occurring in a chosen component such that any variable occurs in a changed prefix image at most once and occurs nowhere else in a quantifier in A".. A"r

LEMMA 17. For each Epn in the sequence EP = A, E, ...,E = ESI of Theorem 9ai, there are expansions A,, ..., A, of Al, ., Ar with chosen components and there are alphabetic variants AI,..., AII Of A,.., A with changed prefix images as just described, such that E.,, is obtained from

Al' ... A,, by exporting (without further change of variables) each of the changed prefix images in A`', ..., A`' as a unit into the main prefix. More precisely, (i) the main conjunction of Epn is the conjunction of those (Q)2+1... (Q)jhN' in A" 1 < i < r, which correspond to chosen components (Q)2+ 1... (Q)jhN1 in A, -1 < g < h1, and contains these (Q)+1...( ,N in the order in which they occur in AI' AII, and (ii) the main prefix of EP, is a string (... ... (P)t of the changed prefix images (P)O, ..., (P)t in

AII,. , A'II, such that if (P)m has (P)m' within its scope in Al', 1 < r, then (P)m also has (P)m' within its scope in (P)... . (P)t.

PROOF. We assume as inductive hypothesis that the lemma holds for EP, . , Epn and, in addition, that if a term (Q). . . (Q)hJN* in Ep ~ corresponds to a chosen component (Q) ... (Q)h'Nj occurring in AI within (Q)g-[(Q)g... (Q)h'N1 . - (Q). . (Q)h'N1], then it and another term

....... (Q)^'NP in E belong to the same term ........(Q)^'N; in an earlier Em.,O < m < P+ . For n+I= s', this additional hypothesis holds trivially, since then the terms in Ep'n+= A are (Q)?... (Q)^N, 1 ? j ? r. Consider first case (ii) of (a,) where E Pn+i yields EP, Iby the replacement of a term (Q)q... (Q)hWN? belonging to some Aj in A by two or more adjacent terms (Q).. XX(Q)h'N? (and the appropriate .), 1 < g < hp. Then by the conditions for case (ii), no other term ........(Q)^WN. in E and the given term (Q)-- NW in E can belong to the same term (Q) ...

(Q)&'Nt in an earlier Em. Hence by the additional inductive hypothesis the given term (Q).( N' in E corresponds to a chosen component (Q)g... (Q)^'N1 occurring in AI within (Q)2-(Q)-... (Q)^'Np. We can therefore replace this occurrence of (Q)2-(Q)2... (Q)h'N1 by (Q)q-1[(Q)g...

(Q)h'N. ... ... (Q)(QNI] such that the resulting AI is an expansion

of A>, form an alphabetic variant AJI of A," of the kind described by replacing in AI' the occurrence of .(Q) .... (Q)h'N" concerned by ...

(Q)h'N;". ... .) . . 4(Q)h'N, and then obtain E,,, by exportation of the

changed prefix images from AI" AJ..... A" A . Al AII as E was obtained from AI". ... Ai, . AII. Ai 1. .. . AI,. In case (ii) of (al) the lemma therefore holds for E ..., Ep,,. Moreover, it is seen that the additional inductive hypothesis is preserved. Consider now the remaining



130 WILLIAM CRAIG

case (i) of (al) where E.,,+, yields E,, by the replacement of a term (Q)g(Q)+1.. . (Q)^?N' belonging to some Aj in A by a term (Q) +1... (Q)h'Na *

and the replacement of the main prefix (P) of E by (P)(Q)q, such that

(Q)2**(Q)2 +1...(Q)hiN** is an alphabetic variant of (Q)N(Q)2+1... (Q)^'N7,

0 ? g ? h1. Then replacement of the corresponding occurrence of

(Q)2(Q)2l+l. *(Q)hLNC in AII by (Q)(Q+1 ... (Q)hJN * gives an alphabetic

variant All of AI of the kind described. Then E,,. can be obtained by exportation of the changed prefix images from All' ... AJ I. A il" .

AnI, .... AI' by steps similar to those by which E. was obtained from AII AL, 1 . A'. AnI .+I 1 AII, except that as additional and last step we export (Q)g** into the main prefix. The additional inductive hypothesis is again preserved. Q.E.D.

LEMMA 18. In a symmetric and Q-balanced L- or L.-deduction from A to A* which has property (al) of Theorem 9, the last formula (P)M of the assembling phase is a development of A.

PROOF. Lemma 17 holds in particular for Ep.0 = (P)M. Now all quantifiers occur in the main prefix of (P)M, so that all prefix images in

AII,.. , AI" are changed images. Hence all chosen components in

AI, ...,A are N. Q.E.D. REMARK 5. Lemma 18 together with Theorems 9ai or lOa, yields a

form of deduction in PCI or PCI= respectively which may be useful in studying syntactically how separate assumptions or "pieces of knowledge" fit, or are fitted, together in reasoning. If modus ponens or only super- ficially different substitutes for it are excluded from processes of deriving A* from Al .... A,28 then Lemma 18 with Theorems 9al or lOa, seems to allow us to operate as long as possible on Al, . . ., Ar separately, changing them to AI,..., AP' respectively, before we have to operate on them jointly.

REMARK 6. Corrections to [5] and [6]. In contrast to Theorem 9c, the condition that no variable occurs both bound and free in A v A* cannot be omitted from Theorem 8c. E.g., let x and y be distinct, let

28 If modus ponens is not excluded, then in a trivial way we can operate on

Al, . . ., A, separately for what may be regarded as a longer time. Namely, operating on Al, . .-1, A, separately we can derive trivially, e.g., [(Al . A2 ... . A7) : A*] D [(A2 .. A) D A*], . . , [(AT-1 . Ar) D A*] D [Ar D A*], [(A,. D A*] D A* respectively. From these, and the valid (Al . A2 . . A,,) D A*, we then get A* by r

applications of modus ponens. This illustrates how modus ponens yields certain inferences, so to speak, in one fell swoop, and thereby often glosses over, or completely obliterates, the details of how information is transformed by reasoning. Although perhaps for a different reason, Herbrand [10] p. 117 considers it to be an important corollary of his work that modus ponens can be avoided in proofs of first-order validity,

but notes that his result does not automatically extend to first-order mathematical theories where there are non-logical axioms. This extension is carried out in [5], and indeed may be thought to be its main purpose.




A = VxVy(Rx. Ry), and let A* = Ry, so that there are H-deductions of A v A*. In any such H-deduction the individual term of the last VA-introduction must be distinct from y, since y is not free at the free occurrences of x in Vy(Rx. Ry). Yet y and no other individual term occurs free in the conclusion. An L-deduction from VxVy(Rx. Ry) to Ry satisfying all the conditions of Theorem 9, including (c), is VxVy (Rx. Ry), Vx'[Vy(Rx'. Ry)], Vx'Vy'[Rx'. Ry'], Vy'[Ry. Ry'], Ry. Ry, Ry where x' and y' are distinct from each other and from x and y. The individual term y of the first V-instantiation is "shielded" from the Vy occurring in the first formula, and conflict of variables is avoided, by the change of this Vy to Vy' prior to the V-instantiation.

This difference between Theorems 8c and 9c is closely related to certain corrections needed in [5] and in [6], which will now be described. Basically, there are two independent errors in [5]. First, in the statement of the Herbrand-Gentzen theorem on p. 259 the condition must be added that no variable occurs both free and bound in A v A'. It is easily seen, and in fact has been shown in [12] p. 450, that there is no H-deduction, as defined in [5], of VxVy(Rx . R'y) v Ry where R', R, x, and y are distinct.29 The next to the last sentence on p. 259, purporting to prove the Herbrand- Gentzen theorem without this condition, is false. Apparently the only further effect in [5] of this error is that Theorem 2 of [5] is not in all cases an immediate consequence of Theorem 1 and of the now weaker Herbrand- Gentzen theorem but only in those cases where no variable occurs both free and bound in A O A'. To prove Theorem 2 for the remaining cases, Lemma 15 above can be used as indicated earlier in the proof of Theorem 9. On the other hand, the condition that no variable occur both free and bound in A v A' must be added not only to the Herbrand-Gentzen theorem but also to Theorems 2 to 5 of [6], since Lemma 15 is inapplicable there.

The second basic error in [5], which does not affect [6] but which has rather extensive consequences in [5], is the blatant falsehood of the last sentence of Lemma 3 (p. 254) for each equivalence L-rule other than duplication or simplication. As a consequence, Lemma 4 must be weakened, e.g., by inserting "the traces in Ai and in Ai+, of" before "the given occurrences of t', and its proof modified, e.g., by inserting "by hypothesis" before "t' is free at these traces". Then in the proof of Theorem 1 one must replace on p. 263 line 27 "any occurrence of t in Al" by "any occurrence of t in A1, . . ., Ap". Mainly affected, however, are Theorem 3 and its proof. Let a Q-balanced L-deduction be restricted if and only if the individual term of each V-instantiation (3-generalization) contains no

29 Note that there are L-deductions from VxVy(Rx . R'y) to Ry, e.g., one similar to the above L-deduction from VxVy(Rx . Ry) to Ry.



132 WILLIAM CRAIG

variable which in the premiss (conclusion) of the complementary Y-ex-

portation (3-importation) occurs in a quantifier that is within the scope of

the quantifier being exported (having been imported). E.g., the L-deduction from VxVy(Rx. Ry) to Ry given above is not restricted. To Theorem 3

one must add the condition that the L-deduction also be restricted. In the proof one must insert "restricted" before, say, "and Q-pure" on p. 265

lines 2 and 27 and on p. 266 line 26, and one must also insert "and restrictedness" after "by (fi)" on p. 266 lines 23 and 34. The condition of restrictedness may also be added to the conclusion of Theorem 1, and therefore, for those cases where no variable occurs both bound and free in A O A', to the con-

clusion of Theorem 2. For other A O A', the use of Lemma 15 in proving Theorem 2 may destroy restrictedness. The Herbrand-Gentzen theorem of

p. 259, now limited to A v A' containing no variable both free and bound, is an immediate consequence of the thus augmented Theorem 2 and of

the corrected Theorem 3. (A third independent error in [5], which apparently has no other con-

sequence, is the omission in the proof of Theorem 3 of [5] of another case,

which is like Case 3 except that either the first or the last operation is a

matrix change. The proof for that case is similar to that for Case 3.)

7. Embeddings of linearly ordered trees.30 We shall now prove Theorems 2 and 6, thereby concluding the proof of all results in ? 2 and ? 3.

Theorems 2a and 6a depend on an abstract embedding argument. From the argument it easily follows that in certain not unusual situations one

can, roughly speaking, replace a set of procedures whose elements are not

all comparable by a subset whose elements are. A finite rooted tree, or from now on simply a tree, shall be any finite

set T of elements or points with a relation < such that (i) < is a partial ordering of T, i.e., < is transitive and asymmetric, (ii) for exactly one

point in T, which will be denoted by 0, there is no a e T such that a < 0, and (iii) for any other point b in T there is exactly one immediate pred-

ecessor, i.e., a point a < b such that if a' # a and a' < b then a' < a. Thus for each point b of the tree there is a unique sequence <ao, . .., ag> of one or more points such that ag = b, each anal is the immediate predecessor of as, 1 < i < g, and ao = 0. Then g shall be the height of b.

30 I am grateful to Richard Otter for helpful remark on trees. The notion of linearly

ordered trees was suggested by Herbrand's discussion of his property A and related

topics, [10] pp. 85-97. <T, < > was suggested by his scheme (pp. 87, 92), and <T, <, _ >

by his type dhduit du scheme (p. 89). Whereas these dealt with occurrences of single

quantifiers, our trees deal with larger units, namely, images of (Q)9, 0 < g 5 h.

Belatedly, I should like to point out that several ideas of [5] are present in [10] pp.

85-97, and expressed better or more explicitly.




A point a shall be an end point if and only if, for no b, a < b. The height of a tree shall be the greatest of the heights of its (end) points.

A linearly ordered tree shall be any <T, <, _> such that <T, < > is a tree, (iv) _ is a linear ordering of T, and (v) _ is an extension of <, i.e., a < b implies a _ b. As usually, <T, <, <> shall be an extension of <T', <', _'> if and only if T, <, are extensions of T', <', -', respectively. An embedding of <T', <', <'> in <T, <, _> shall be an iso- morphism between <T', <', _'> and some <T", <", -'> of which <T, <, _> is an extension. For <T', <'> and <T, <>, extension and embedding shall be defined similarly.3'

Given h ?0 and i > 1, let TV be the set of those <ml, .. ., mg> such that 0 < g ? h and ml, mgE E {I, . . ., i}. (Thus TV consists of those sequences which are assigned to the prefix images in the regular i-expansion of (Q).... (Q)hN.) Let <,h be the relation such that a <h b if and only if a, b e TV and a is an initial segment, possibly empty, of b. For any a e TV let max a be the highest of the numbers ml, ..., mg such that a =<ml, ..., mg> when g > 1, and let max a =0 when a= < >. (Thus, if a is the sequence assigned to an image, then max a is the rank of that image.) Evidently, <TV, < > is a tree in which each end point is of height h and in which each other point is the immediate predecessor of exactly i points. Among the linearly ordered trees <Ti, <A, _> to which it gives rise, consider those such that (vi) if max a < max b then a _ b. Any such <TV, <A, <>, and any linearly ordered tree isomorphic to it, shall be a regular i-tree of height h.

THEOREM 11. Any linearly ordered tree with i points and of height hi can be embedded in any regular i-tree of height h2, I < i, 0 < h, < h2, with each point of height g being mapped into a point of height g, 0 < g < h,.

PROOF. For i = 1, the theorem evidently holds. Now assume as inductive hypothesis that it holds for i > 1. Consider any linearly ordered tree <T', <', <'> with i+-I points and of height h'. Let d be the last element of T' in the -'-ordering, let T = T'- {d}, and let < and S be the restrictions of <' and <' respectively to T. Then <T, < , _> is a linearly ordered tree with i points and of height h, < ?h. Also consider any regular (i+ 1)-tree of height h.' > h. For some -<', it is isomorphic to <Th <l, <h2l' _//>, which we shall consider in its place. Let _" be the restriction of _ I' to T^2'. Then <T'^2, <hi2 ?'> is a regular i-tree of height it > h' > hl, having <T h', <v , ~12 > as an extension. By the inductive hypothesis, there is an embedding of <T, <, <> in <Ti2', < 12, %">, such that each point a of height g in <T, <, <> is mapped into a point a* of height g in <Ti2', <;h2', <">, 0 ? g ? h. Now, since d is not 0, it has an

31 In our case below where each end point in <T', <'> and in <T, < > is of the same height A, but not in general, an embedding in our sense also preserves the immediate predecessor relationship.



134 WILLIAM CRAIG

immediate <E-predecessor c in <T, <, _> of height g, < hi h'. Hence, if c*~ is Kin,, .. ., in0>, then in <ml..., ini+I> is an element of Ti,. Then mapping d into <ml, ..., mic, i+ 1> and each a e T into a* gives a desired embedding of <T', <', <'> in <Tih+2, < h'

?m>. Q.E.D. Now consider any development (P)M of A = (Q)O ... (Q)hN, and let A"l

be the associated alphabetic variant of an expansion AI of A. Let T(P)M be the set of prefix images in (P)M. Let < (P)m, be such that (P)r < (P)M (P)r' if and only if (P)r dominates (P)r', i.e., (P)r has (P)r' within its scope in A"I. Let (P)r -(P)M (P)r if and only if (P)r occurs in (P) at the same place as or to the left of (P)r'. Then <T(p)'^, < (P)MY -(P)M> is a linearly ordered tree in which each point of height g is an image in (P)M of (Q)9, 0 ! g < h, and in which each end point is of height h. In particular, if (P)iMi is a regular i-development of A, then <T(P)4M,, < (P)MO C(P)tMt> is a regular i-tree of height h.

Now let (P)M be any development of A with < i images in (P), and let (P)iMi be any regular i-development of A. By Theorem 11, <T(P)MX

<S(P)Mx (P)M> can be embedded in <T(P)-Me, < (P)>M. _(P)tMt>- Since each end point of either tree is of height h, each point of height g in <T(P)M,

<(P)M, _(P)M> is mapped by the embedding into a point of height g in <T(P)tMt) <(P)tM.J <(P)tMt>, so that images of (Q)9 are mapped into images of (Q)9, 0 _ g ? h. Since no two quantifier occurrences in (P) or in (P)i have the same variable, there is an alphabetic variant (P) M' of (P)sMi such that, for any image (Q)9* in (P), if (Q)9* is mapped into (Q)9*, then (P)' contains (Q)9* where (P)i contains (Q)9*. The embedding of <T(P)M,

<(P)M, _(P)M> in <T(P)4M{, < (P).M, _ (P)tMm> induces one in <T(p)1M, , < (P)i'Me', -C(P).M,>. Now from the prefix of (P)' delete all those images into which no (Q)9* of (P) is being mapped, so that the deletion produces (P), since the mapping preserves _. Then along with every deleted image in (P)' all those images in (P)' are deleted which are dominated by it, since the mapping preserves <. Consider now any matrix image N* in (P)M. It is dominated in (P)M by a unique set of images (Q)", ..., (Q)h". They are mapped into images (Q)1, ..., (Q)hl in (P) M'. For these there is a unique matrix image N* in (P)'M' which they all dominate, since (P)'M' is a development of A. Conversely, if the (Q)1, ..., (Q)h' dominating a matrix image N* in (P)'M' are such that (Q)1, ..., (Q)h* in (P)M are mapped into them under the embedding, then N* is a matrix image in (P)M. It follows that deletion from M' of all images which are dominated by a prefix image deleted in the change from (P) to (P), followed by deletion of a conjunction sign next to each deleted image, yields a conjunction differing at most in the order of terms from M. Thus (P) M' satisfies Theorem 2a, whose proof is thus concluded.

LEMMA 19. Let A, (P)M, and (P)+M be as in Theorem 5, (P)+M being formed according to (b), and A containing at least one V. Then some development




(P)"M" of A yields (P)+M by deletion of images and conjunction signs as described in Theorem 6a.

PROOF. Let Vx1, . . ., Vxk be the quantifiers from left to right occurring in (P)+ but not in (P). Let (Q)0 (Q)1 . *(Q)hN*, (Q). (Q)l ... (Q).,Nk

be alphabetic variants of (Q)O(Q)l. . . (Q)hN having except for the variables in (Q)0 no variable in common with each other or with (P)M and such that (Q)0 is the image of (Q)O in (P)M and also such that the first quantifier in (Q)1', .., (Q)1 is Vxi, ..., Vxk respectively. Insert (Q)V... ( . .. . (Q)j in (P) in those places where Vx1, . . ., Vxk respectively occur in (P)+, and conjoin N1, ..., Nk to M. Let (P)"M" be the resulting formula. Since (P)+M is formed as described in Theorem 5b, (P)"M" is a development of A. Evidently, (P)"M" yields (P)+M by deletions as described. Q.E.D.

Now let A, (P)M, (P)+M, and (P)"M" be as just described. Suppose there are ii prefix images in (P) and i2 quantifiers in (P)+ which are not in (P). Then there are io = il+h.i2 prefix images in (P)". Then by Theorem 2a. for each i > io, every regular i-development (P)iMi of A has an alphabetic variant (P) M' from which (P)"M" is obtainable as described there. Then by Lemma 19, (P)+M is obtainable from (P)'M4 as described in Theorem 6a, whose proof is thus concluded.

Now choose those bound variables in (P) M' which do not occur in (P)+M to be distinct from any variable in the individual term of any of the V-instantiations in the sequence leading from (P)+M to (P)*M*. Then to this sequence there corresponds a second sequence of V-instantiations which starts from (P)4 M' such that the quantifier and the individual term of each V-instantiation in the second sequence are the same as those of the corresponding V-instantiation in the first sequence. Let (P) *M"* be the conclusion of the last V-instantiation in this second sequence. Deletion from (P)t of those quantifiers which were deleted in the change from (P)' to (P)+ yields (P)*. Also, deletion from M* of terms and conjunction signs in those places where terms or conjunction signs were deleted in the change from M' to M, followed by rearrangement of the non-deleted terms, yields M*. Also, those variables which occur in (P)t* but not in (P)* also do not occur in M*. For let z be any such variable. Then z also occurs in (P)4 but not in (P)+. Now any image in M' in which z occurs is dominated by the image in (P) in which z occurs, and is therefore deleted in the change from M' to M. Hence the term in M; corresponding to this image in M' is deleted in the change from M; to M*. By our choice of variables occurring in (P) but not in (P)+, z can only occur in such terms in M;*.

Now let M;-- and M*- be obtained from M;* or M* respectively by (PCI)-elimination of R1, ..., Rq. Since M;* yields M* by deletion of terms and adjacent conjunction signs, followed by rearrangement of terms, FM M;* O M*. Hence FM M*' O M*-. Then by Lemma 5, F M>*- O M*-. Now let zi, ..*, Zr be those variables which occur in (P);* but not in (P)*. Then



136 WILLIAM CRAIG

ZP ..., ,Zr do not occur in M*, as has just been shown, and hence also not in M*-, by condition (iv*) for (PCI)-elimination. Hence F 3Z1 ... 3zrM*-

M*-, and thus F (P)*3z ...z3M-- D (P)*M*-. But also F (P)t*M*-: (P)*3zl. . 3zrM`*Th so that F (P)'*M`*- D (P)*M*-. Finally, to the sequence of V-instantiations leading from (P)'M' to (P)'*M*, there corresponds a further sequence of V-instantiations leading from (P)iMi to an alphabetic variant (P)*M* of (P)t*M *, such that each V-instantiation again satisfies Theorem 5c. If M* yields M*- by (PCI)-elimination of R1, ..., Rq, then, for some M*, (P)M*- is an alphabetic variant of (P)'*M'*- and hence F (P)WM*- v (P)*M*-. This concludes the proof of Theorem 6b. The proof of Theorem 2b is similar, using Lemma 2b instead of Lemma 5, and simpler, since V-vacuous-introduction does not enter.

REMARK 7. Theorem 11, or more directly Theorem 2a, may also be used to supplement Theorems 9 and 10. Consider any symmetric and Q-balanced L- or L=-deduction from A to A* such that A and A* are in prenex normal form, such that the last formula (P)M of the assembling phase is a development of A, and such that the first formula (P)*M* of the disassembling phase is dually related to A*. Let (P)iMi be related to (P)M as in Theorem 2a, and such that moreover those prefix images which are deleted from (P)i to obtain (P) contain no variable occurring in the L- or L=-deduction from (P)M to (P)*M*. Then, by first "copying" the V-vacuous-introductions and V-instantiations of the deduction from (P)M to (P)*M* and also applying V-instantiation with arbitrary t to those V-quantifiers which are deleted from (P)i to obtain (P), then deleting from the matrix those terms which correspond to a term which is deleted from Mi to obtain M, and finally "copying" the matrix change, the 3- generalizations, and 3-vacuous-removals of the deduction from (P)M to (P)*M* and also applying 3-vacuous-removal to those 3-quantifiers which are deleted from (P)i to obtain (P), we obtain part of a symmetric and Q-balanced L- or L=-deduction from A to A* such that the first formula of the disassembling phase is still (P)*M* and such that the last formula of the assembling phase is now the given regular i-development (P)iMi of A. A dual result can then be obtained for the disassembling phase, without affecting the assembling phase, so that we can choose i the same for the two phases. By the discussion after Theorem 2, one can then choose for each i 2 1 a unique regular i-development (P)iMi and a unique (P)*M* for the beginning of the disassembling phase, such that, if there is a symmetric and Q-balanced L- or L=-deduction from A to A* with (P)iMi as last formula of the assembling phase and (P)*M* as first formula of the disassembling phase, then the same holds for (P)iMi, and (P)*,M*, in place of (P)iMi or (P)7M* respectively, i' > i.

For a given A, consider now the set of L- or L=-deductions thus chosen. It is complete in the sense that, for any consequence A* of A in prenex




normal form, it contains a deduction from A to A*. Certain features of the initial segments which constitute an assembling phase should be noted. First, assuming certain conventions concerning the order of interchangeable steps, each such segment has a certain deterministic character, being determined by the corresponding i completely and in a relatively simple manner. Second, there is a certain monotonicity in the sense that, if the segment corresponding to i is the initial segment of a deduction from A to A*, then so is the segment corresponding to i' > i. Finally, as a consequence of this monotonicity and completeness, for any finite set A*, ...,A*

of consequences of A in prenex normal form, there is an i such that the segment corresponding to i' > i is, for each A>, an initial segment of a deduction from A to A*, 1 <j <r, so that in deductions from A to

A*, ... I A* we can initially travel a "common road" and thus perhaps save effort.

It is natural to ask how far some or all of these features can be extended beyond the assembling phase to larger initial segments. As long as we restrict ourselves to symmetric and Q-balanced L- or L=-deductions, no extension preserving the last feature is possible. For, evidently, V-vacuous- introductions are needed when A'. contains V, while they cannot be used when A. contains no V, since there is no way of removing the V thus introduced. Among possible modifications of linear deductions, one may therefore wish to consider sacrifice of symmetry and the addition of a rule for removing vacuous V-quantifiers, allowing us to change from (P)'Vy(P)"M to (P)'(P)"M whenever y does not occur free in (P)"M. Then, to the initial segment ending with (P)iMi, one can add as next step the insertion of i vacuous V-quantifiers immediately to the right of every image in (P)j, removing later on in the deduction, after the last matrix change with or without identity, those of the V-quantifiers introduced which are not needed for

A. and are therefore then again vacuous. Most of the desired features then also hold for a certain subset of those

larger initial segments which include all the V-instantiations.32 The rather artificial construction of the subset is easily obtained from the following observation. Suppose, for simplicity, that there are no function symbols in A or in A*, A*, ..., that each individual constant or individual variable occurring free in A*, A*, ... also occurs thus in A, and that A begins with 3. Then, by property (c) of Theorem 9, each (P)jMj yields only a finite number of formulas (P)i, AMi1, . ., (P)ikAMik, by V-vacuous-introductions and V-instantiations as indicated. Then one can find a (P)iMi, which by

32 Theorem 4 of [6] contains a related result. Also, certain related features are

present in the L-transforms of those H-deductions of A D A' where the initial

tautology Bo D B* is obtained from a Herbrand field of order p, p = 1, 2, . . . (see

[10] pp. 97-108). However, for a given A, the number of V-vacuous-introductions is

then not only a function of p but also of A'.



138 WILLIAM CRAIG

V-vacuous-introductions and V-instantiations as indicated yields some (P)',M*', such that an alphabetic variant of each (P)*,M ,1, 1 j i, is obtainable from (P)*,Mt, by deletion from Mt. of terms and of adjacent., and by deletion from (P)', of quantifiers then vacuous.

8. Relationships between bases. Relationships between bases B1 for S1 and B2 for S2 are only partly reflected by relationships between bases BT for S- and B- for S-.

In the special case where B1 yields B2 by a substitution a for function symbols in each formula,33 we shall now show that, roughly speaking, each BT for S- yields a B- for S- by a. More precisely, let t(xi, ...,k)

be an individual term with the individual variables occurring in it given in some order, and for each s1, ..., sk let t(s1, ..., Sk) be the result of substituting sI, . . ., sk for xi, .., xk in t(xI, ..., Xk). Then A shall be said to yield a(A) by substitution of t(x1, ..., Xk) for f(xi, . . . x>,), O < k < k', if and only if A contains no occurrence of f as part of an argument of f and a(A) is obtained from A by replacing simultaneously each occurrence of f(r1, . . ., rk) by an occurrence of t(r1, . . ., rk). Note that, e.g., f(f(x)) = y is equivalent in PCI= to 3z(z = f(x) . f(z) = y), so that any A' which contains f as part of an argument of f can be transformed into an equivalent A which does not. Theorem I then yields the following:

COROLLARY. Suppose A1, A2, . . . yield a(Ai), a(A2), . . . respectively by substitution of t(xl, ..., xk) for f(xi, ..., X S- and S- are subtheories, determined by the absence of the same predicate symbols, of the theories having {A1, A2, .* } or {a(Ai), a(A2), . . .} respectively as basis in PCl=, {B1, B2, .. .} is a finite or infinite basis for S7 in PCI=, and B1, B2, ...

yield o(Bi), a(B2), ... respectively by substitution of t(x1, ..., xk) for

(x) ..., ). Then {a(Bi), a(B2), .} is a basis for S2 in PCI=. PROOF.34 Let or, A1, A2, .. ., S-, S-, and B1, B2, ... be as described.

For convenience assume that A1, A2, ... is a single formula A in prenex normal form. Consider any D in S-. By Theorem 1, there is a development (P)M' of a(A) and an M'-, obtainable from M' by (PCI=)-elimination of the predicate symbols R1, ..., Rv present in a(A) and absent from S-, such that F= (P)M'- v D. Then there is a development (P)M of A such that M' is a(M). Now F. a(M-) (a(M))-, where M- is obtainable from M by (PCI=)-elimination of R1, . . ., Rq, since M- v 3R, . 3RqM, and hence r(M-) v 3R1. *3Rq(M), is valid. Hence F. (P)a(M-) v D. Now (P)M- is

33 The study of this relationship between B1 and B2 was suggested by Buchi. 34 For certain cases there is a trivial proof. Let R1, . . ., Rq be the predicate symbols

present in A1, A2, ... and absent from S-, and for convenience let A1, A2, ... be a single formula A. Then, in those cases where any model of {B1, B2, ... } is also a model of 3R1. . 3RqA, it follows that any model of {a(B1), o(B2), ... } is also a model of 3R1...3RqG(A) and hence of S-.




in Sj, so that FK Bi O (P)M- for some i > 1. Then

F. a(Bi) O (P)a(M-) and hence FK a(Bi) O D. Finally, since FK A O Bi, also F= a(A) O a(Bi), so that a(Bi) is in S-, i > 1. Q.E.D.

If a is a substitution for a predicate symbol, then an argument similar to the one just given can be carried out only for the case where the predicate symbol for which we substitute is distinct from R1, ..., Rq and where the formula being substituted is quantifier-free.34

The proof of the corollary makes use of an interchangeability of (PCI=)- elimination and substitution for function symbols. The sequence A, a(A), (P)oa(M), (P)(a(M))- is replaced by A, (P)M, (P)M-, (P)a(M-).35 A few other operations can be interchanged with (PCI=)-elimination in a related manner. E.g., suppose A is in prenex normal form, A yields A' by an V- instantiation, (P)'M' is a development of A', and M' yields M'- by (PCI =)- elimination of R1, ..., Rq. Then there is a development (P)M of A, obtainable from A by operations closely related to those leading from A' to (P)'M', and an M- obtainable from M by (PCI=)-elimination of R1, ..., Rq, such that (P)M- yields by one or more V-instantiations some (P)'M-' such that F= M-' O M'-. Moreover, the quantifier occurrences associated with the V-instantiations leading from (P)M- to (P)'M-' are those to which the quantifier occurrence associated with the V-instantiation from A to A' gives rise, and the individual terms of these V-instantiations are closely related.

Now let S- and S- be the subtheories, determined by the absence of R1, ...,Rq, of the theories having {A} or {A'} respectively as basis in PCI=, and let B- be a basis for S- in PCI= containing only formulas (P)M- such that (P)M is a development of A. Then this particular basis B- for S- in PCI= yields a basis B- for S- in PCI= by application of V-instantiations to its formulas, so that B- and B- reflect the relationship between A and A'. In contrast to the corollary, however, other bases for S- in PCI= may fail to yield bases for S- in this manner. Results similar to the one just stated also hold when A yields A' by 3-generalization, or by the replacement of all occurrences of a constant term s by a constant term t.

Similar results more likely to apply to an actual situation also hold by Theorem 3 when A' has the same prefix as A and a matrix in (R1, . .., Rq)- disjunctive normal form for PCI= which yields that of A by deletion of terms from the disjunction. Note that by Theorem 3 of [6], some A and A'

35 Model-theoretically, this implies that C(a* (X)) = a* (C(X)) whenever X is the set of models of some 3R1...3RqA, C is the closure operation of Tarski [18], and a* is the model-theoretic counterpart of the substitution a for function symbols. Proof-theoretically, one may regard this as an extension, of the analogue of Lemma 3 concerning substitution for function symbols, from the quantifier-free to the quantificational level.



140 WILLIAM CRAIG

thus related can be formed from the conjunction of the formulas in B1 or in B2 respectively, whenever B1 is finite and B2 is a subset of B1.

9. Non-generabilities. With each subtheory S- of the theory having {A} as basis in PCI= and determined by the absence of R1, ..., Rq, let us now associate the pair <A; R1, . . ., Rq>. Also let us assign G6del numbers in the usual way to symbols, sequences of symbols, and then to pairs <A; R1, *.-, Rq>

THEOREM 12. For pairs <A; R> such that R is a 2-place predicate symbol, there is no recursive enumeration of (a) the set of Godel numbers of those pairs which are associated with an S- that has a finite basis in PCI=, nor of (b) the set of Gddel numbers of those pairs which are associated with an S- that has no finite basis in PCI

PROOF. (a) Let A1 be satisfiable in some infinite but in no finite domain, and contain the 2-place R as only predicate symbol. For any A2 containing neither R nor =, consider the S- with which <A1 . A2; R> is associated. In the case where A2 is satisfiable in no finite domain, A2 . 3RA1 is equivalent to A2, so that {A2} is a basis for So in PCI=. Consider now the case where A2 is satisfiable in some finite domain of m objects and hence in any domain of > m objects, and consider any B in S-. A2 O B is valid in every infinite domain and hence, for some k > 1, in every finite domain of > k objects (see [9] p. 166). Hence B is satisfiable in any domain of n objects, where n ? k and n > m. Now let C be a sentence containing only - as predicate symbol and asserting that there are more than n objects. Then F= Al O C so that C is in S-. Yet B O C is not universally valid. It follows that in this case S- has no finite basis in PCI =. Hence S- has a finite basis in PCI= if and only if A2 is satisfiable in no finite domain. But by [21], the set of Gddel numbers of such A2 is not recursively enumerable.

(b) Let A contain the 2-place R as only predicate symbol, and consider the S- with which <A; R> is associated. If (i) A is satisfiable in exactly the domain of > n objects, then any sentence containing only = as predicate symbol and asserting that there are at least n objects constitutes a basis for S- in PCI=. If (ii) A is satisfiable in no domain, then {Vx ox = x} is a basis for S- in PCI=. If (iii) A is satisfiable in some infinite but no finite domain, then S- has no finite basis in PCI=. Since for case (i), and also for case (ii), the G6del numbers can be recursively enumerated, they cannot be recursively enumerated for case (iii). Q.E.D.

Results for PCI similar to those just proved for PCI= are obtained by similar arguments. In place of A we consider A. I, where I is a suitable conjunction of identity axioms: Then the subtheory, determined by the absence of R, of the theory having {A. I} as basis in PCI coincides with the S- considered earlier.




Concerning the elimination problem of second-order predicate calculus, non-generability can also be proved for each of the four sets most relevant. More precisely, one easily proves, for an arbitrary sentence A containing the 2-place R as only predicate symbol, that: (c) If 3R'A' is not implied by its first-order consequences, then A is satisfiable if and only if any model satisfying all first-order consequences of 3R'A' V 3RA also satisfies 3R'A' V BRA. (d) If 3R'A' is not implied by its first-order consequences, then A is satisfiable if and only if some model satisfying all first-order consequences of 3R'A'. 3RA does not satisfy 3R'A'. BRA. (e) A is satisfiable in some infinite but in no finite domain if and only if there is no B having the same models as BRA. (f) If A' is satisfiable but in no finite domain, and if R' is the only predicate symbol in A', then A is satisfiable in no finite domain if and only if there is some B having the same models as A. 3R'A'.

BIBLIOGRAPHY

[1] W. ACKERMANN, Untersuchungen iber das Eliminationsproblem der mathematischen Logik, Mathematische Annalen, vol. 110 (1934), pp. 390-413.

[2] W. ACKERMANN, Zum Eliminationsproblem der mathematischen Logik, Mathe- matische Annalen, vol. 111 (1935), pp. 61-63.

[3] J. R. BtUCHi and W. CRAIG, Notes on the family PCA of sets of models, abstract, this JOURNAL, vol. 21 (1956), pp. 222-223.

[4] W. CRAIG, A theorem about first order functional calculus with identity, and two applications. PhD. thesis, Harvard University, 1951.

[5] W. CRAIG, Linear reasoning. A new form of the Herbrand-Gentzen theorem, this JOURNAL, vol. 22 (1957), pp. 250-268.

[6] W. CRAIG, Analysis of first-order implications, Summaries of talks presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University, mimeographed, 1957, pp. 175-180.

[7] W. CRAIG and R. L. VAUGHT, Finite axiomatizability using additional predicates, this JOURNAL, vol. 23 (1958), pp. 289-308.

[8] A. EHRENFEUCHT, Two theories with axioms built by means of pleonasms, this JOURNAL, vol. 22 (1957), pp. 36-38.

[9] L. HENKIN, The completeness of the first-order functional calculus, this JOURNAL, vol. 14 (1949), pp. 159-166.

[10] J. HERBRAND, Recherches sur la thedorie de la demonstration, Travaux de la Socidtd des Sciences et des Lettres de Varsovie, Classe III sciences mathdmati- ques et physiques, no. 33 (1930).

[11] K. J. J. HINTIKKA, Form and content in quantification theory, Two papers on symbolic logic, Acta philosophica Fennica, no. 8, Helsinki 1955, pp. 7-55.

[12] S. C. KLEENE, Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952.

[13] S. C. KLEENE, Finite axiomatizability of theories in the predicate calculus using additional predicate symbols, Memoirs of the American Mathematical Society, no. 10 (Two papers on the predicate calculus), Providence, 1952, pp. 27-68.



142 WILLIAM CRAIG

[14] S. C. KLEENE, Permutability of inferences in Gentzen's Calculi LK and Lj, ibid., pp. 1-26.

[15] R. C. LYNDON, Properties preserved under algebraic constructions, Bulletin of the American Mathematical Society, vol. 65 (1959), pp. 287-299.

[16] E. MACH, The science of mechanics, translation of the second German edition, Chapter IV, Section IV (The economy of science), pp. 481-494, LaSalle, Ill. (Open Court), 1893.

[17] W. V. QUINE, Mathematical Logic, revised edition, Cambridge (Harvard University Press), 1951.

[18] A. TARSKI, Some notions and methods on the borderline of algebra and metamathematics, Proceedings of the International Congress of Mathematicians (Cambridge, Mass. U.S.A., Aug. 30-Sept. 6, 1950), 1952, vol. 1, pp. 705-720.

[19] A. TARSKI, Contributions to the theory of models, I and II, Koninklijke Neder- landse Akademie van Wetenschappen, Proceedings, ser. A, vol. 57 (also Inda- gationes mathematicae, vol. 16) (1954), pp. 572-588.

[20] A. TARSKI and R. L. VAUGHT, Arithmetical extensions of relational systems, Compositio mathematica, vol. 13 (1957), pp. 81-102.

[21] B. A. TRAHTtNBROT, Ndvozmoznost' algorifma dld problem razrlfimosti na kondenyh hlassah (Impossibility of an algorithm for the decision problem in finite classes), Doklady Akaddmii Nauk SSSR, n.s., vol. 70 (1950), pp. 569-572.

THE PENNSYLVANIA STATE UNIVERSITY

36 (Note to page 1 8, added January 25, 1960.) Cf., however, the construction in pp. 149-155 of D. HILBERT and P. BERNAYS, Grundlagen der mathematik, vol. 2, Berlin (Springer), 1939.



Bases for First-Order Theories and Subtheories

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