Eighteenth Meeting of the Association for Symbolic Logic

Eighteenth Meeting of the Association for Symbolic LogicAuthor(s): William CraigSource: The Journal of Symbolic Logic, Vol. 20, No. 2 (Jun., 1955), pp. 200-206Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266965 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 20, Number 2, June 1955

EIGHTEENTH MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC

The annual meeting of the Association for Symbolic Logic was held at the University of Pittsburgh, Pennsylvania, on December 29, 1954, in conjunction with the annual meetings of the American Mathematical Society and of the Mathematical Association of America. A banquet in the evening was open to members of all Societies.

Professor G. H. von Wright gave an invited hour address at the morning session on The meaning of probability. All other papers were contributed twenty minute papers. Abstracts of all papers are appended. Part II of J. C. E. Dekker's paper was presented by title only. Part I of J. W. Addison's paper was presented at an earlier meeting of the American Mathematical Society, and an abstract will appear in their Bulletin.

Professors W. V. Quine and H. G. Rice presided at the morning and afternoon sessions respectively.

The Council of the Association met between the morning and afternoon sessions and again after the afternoon session.

Professor J. C. Knipp of the University of Pittsburgh and his colleagues deserve the gratitude of the Association for their valuable help in making arrangements.

WILLIAM CRAIG

PAUL C. GILMORE. On the relative irreducibility of polynomials. Let K be an infinite field such that for any p(t, x) e K t, x], t transcendental, with

no root in K(t) there is a t' e K for which p(t', x) has no root in K. Let a be the first order predicate calculus to which has been adjoined a set of individual parameters, one assigned to t and to each member of K. Further let there be assigned one predicate of a to each subset of K, to each subset of the set of pairs of K and so forth. RI is defined to be the set of statements of a for which K is a model. Then there is an extension S' of S= K(t) which is a model for ft and for which every element of S'-S is transcendental with respect to S. The proof consists in showing that every finite subset of the set of statements t U G1 U 62 from a has K as a model, where G, is the set of statements expressing that t is transcendental and G2 is the set of statements expressing that each irreducible polynomial has no root. From this theorem can immediately be proven that for any irreducible p(t, x) there are infinitely many t' e K for which p(t', x) is irreducible (Hilbert's irreducibility theorem for K). If in addition it is assumed that K is ordered then for any k, m e K with k < m, there exists a t' e K such that k < t' < m. For the case that K is of the form K'(cx), a transcendental, see a forthcoming paper "Metamathematical considerations on the relative irreducibility of polynomials" by P. C. Gilmore and A. Robinson, which contains an algebraic proof of the assumption for K as well as a further application of the main theorem.

ABRAHAM ROBINSON. On predicates in model-complete systems. A set of statements K in the lower predicate calculus (l.p.c.) is called complete if

for every statement X in the l.p.c. which includes only constants and relations contained in K, either X or -X is deducible from K. The diagram N of a given structure M is the set of statements R(a1, . . ., a") for all relations and constants for which the statement in question holds in M, together with the statements ~R(a,, .. ., a") for the relations and constants of M for which R(a,, ..., a") does not hold in M. A consistent set of statements H in the l.p.c. is called model-complete if for every model M of H, the set H U N is complete, where N is the diagram of M. For example, the systems of axioms for the concepts of an algebraically closed field, or of a real-closed ordered field, are model-complete.

200



ABSTRACTS OF PAPERS 201

For a given model-complete set H, let Q(xL, . . ., x,") be a predicate in the l.p.c. whose relations and constants are included in H. Then the following theorem can be proved:

There exists a predicate

P(xL, - - * . n) = (3w1) .... (3Wm)X(W1, Wm. Xi, ...,X") such that X contains only constants and relations of H, but does not contain any quantifiers and such that the equivalence

(Xi) ***(Xn) [P(Xi, * n) --Q (xi .., Xn)] is deducible from H.

Various mathematical illustrations of the theorem can be given.

WILLIAM W. BooNE. An unsolvable problem about inner automorphisms of a certain finitely presented group.

One can explicitly exhibit: (1) a particular group, ?2, with a finite number of defining relations on a finite number of generators, g,, g2, . ., gn; (2) a fixed subset of these generators, g2, g3, . ., gm, 1 < m < n, such that it is recursively unsolvable to determine for an arbitrary positive word W of AT (i.e., a word in which all exponents are positive) whether or not there exists a positive word U made up only of g2, g3, .., g, such that W = UggU-L.

The construction of the group Q$T as described in UPGT (Certain simple, unsolvable problems of group theory, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, series A, vol. 57 (Cf. the abstract, this JOURNAL, vol. 16, An extension of a result of Post)) can be altered so as to yield 0dT Let 2 be &82 with KL, KiR KsRX

qN+i, qN+2, adjoined. Let 12' be the operations obtained from U. by adding the operations qo20S2 -- qN+1S2, qN+la` qN+l, qN+1KiR qN.+2K,, aaqN+2 -- qN+2 where a. is any S-, X-, or E-symbol. (Cf. Post, XII90.)

Let - n 3, 4, 5, 6, be the system obtained from in by replacing T2 by Z2' in the hierarchy of construction in UPGT, but letting a also be K in 113.

Let ', be the system obtained from Air by adjoining la r, --a l., and their converses for each r. and la of 8,.. Let L,, L2, be variables for words consisting solely of I-symbols. Let L* be the word obtained from L, by replacing Ix by r2 and reversing order. It may then be shown: There is an L, such that KLEKIR F'5 L1KLqN+2K2RL* if and only if there is an L2 such that KLIKiR F7' L2KLqN+2K2RL-2*

The demonstration, similar to that of the primary technical result of UPGT, page 235, so alters a given proof of KLYKiR/7'L1KLqN+2K2RL* that if 0q - del(qiLqiL) (if Ng = 0) no i-markers occur between SL and qiL (left of q0).

GEORG HENRIK VON WRIGHT. The meaning of probability. A notion P(a/h), "the probability of a given h", was introduced. It signifies a non-

negative real number, subject to the following rules: I. P(a/h) , P(a/h) = 1. 2. P(a & b/h) = P(a/h) x P(b/h & a). 3. If P(h/h) > 0, then P(h/h) 1. These rules suffice, with principles of logic, for "classical" probability theory.

With this axiom system was compared a system of relative modalities. We introduce a notion M(a/h), "a is possible given h", and the following axioms: I. M(a/h) v M(-a/h). 2. M(a & b/h) -M(a/h) & M(b/h & a). 3. M(h/h) D -M(-hlh).

~.M(a/lh) expresses that h is a necessary condition of a. The logic of the relative modalities can thus be viewed as a logic of necessary and sufficient conditions, i.e. as a logic of nomic (causal) determination. Probability theory may be regarded as a metric extension of this modal logic.

It may be shown that, subject to appropriate conditions, P(alh) can be interpreted either as a proportion in a (finite or infinite) "population" or a ratio of measures of ranges (of propositions or attributes). As proposed analyses of the "meaning of probability", however, both interpretations encounter grave difficulties.



202 ABSTRACTS OF PAPERS

The finite frequency interpretation must be rejected as semantically inadequate to the concept of probability, because it does not reconcile the assumed constancy in the probability-values with variations in the actual frequencies. The limiting-frequency interpretation again makes probability-propositions undecidable and thus fails to illuminate the applicability of the calculus to experiential situations. Another anomaly of the limiting-frequency view is that it makes probability dependent upon a way of ordering the populations.

On the range-interpretation a probability-value is relative to a choice a) of a measure (-function) and b) of a "language" in which to develop the ranges. The problem arises, how these two choices should be mutually adjusted so as to make the calculations of probabilities adequate. The only criterion of adequacy, which seems to be forthcoming, is agreement with intuitively grasped probability-values. This threatens the theory with circularity. (The threat is analogous to the difficulty caused to the classical range-theory by the notion of "equal possibility".)

As an answer to the question, how abstract probability is linked with the world of experience, the following suggestion was followed up:

Bernoulli's theorem and other principles of asymptotic probabilities may be used to let us proceed from initial assumptions about (usually mediocre) probabilities to the prediction of frequencies with a high probability. We adopt the methodological rule that highly probable frequency-distributions should be regarded as "practically certain". If our predictions are not fulfilled we therefore conclude, by virtue of this Principle of Practical ("Moral") Certitude, that our initial probability-hypothcses have to be revised. This seems to be in good accord with the actual practice of sta- tisticians and actuaries. It explains, why it is possible- without detriment to the usefulness of the calculus- to dismiss as irrelevant the question whether probability "means" a relative frequency or a ratio of range-measures.

The "meaning-pattern" of probability thus seems to be different from the semantic patterns of the traditional theories, - the patterns of "models" for probability. It was suggested that the same may, mnutatis mutandis, hold true for other scientific notions, the semantics of which has been causing the logicians and philosophers difficulties.

J. R. SHOENFIELD. The independence of the axiom of choice. The axiom of choice is not provable from the following axioms for set theory:

(a) axioms A-C of G6del's The consistency of the continuum hypothesis; (b) the axiom that every set may be simple ordered; (c) for each it, the axiom

E(X1 C X2 X X3 . * * . . Xn-1 C Xn - Xn E x1).

This is stronger than a result of Mostowski in that the axiom of extensionality is included; it is weaker in that Gddel's axiom D is not included (although the weaker set of axioms (c) is included.) The proof is a modification of Mostowski's proof. The principal difference is that in the model constructed, the membership relation is not represented by the membership relation.

J. W. ADDISON. Analogies in the Borel, Lusin and Kleene hierarchies. II. The effective Lusin hierarchy (c1 levels) on N (natural numbers) and on NN is defined

(using images of recursive transforms from NN, complements, and recursively enumerable unions). Recursive sieves and recursive operations (A) are applied in the theory. On N and NN the finite effective Lusin (effective projective) hierarchy is equivalent to the Kleene function-quantifier (VT) hierarchy. Parallel theorems about the projective hierarchy on NN and 7 hierarchies on N and NN are demonstrated by essentially identical proofs. A set of recursive predicates which are 'notations' for 'effective ordinals' is defined; it is an I1V but not an 31-set of NN; i.e., it is expressible in

(P)(Ey)R(0, Vt, y) but not in O(EVp)(y)R(o, Vp, y) form, where k, tV e NN, y e N,




and R is recursive. Using this, any two disjoint 31-sets of N are separable by an effective Borel set (and hence by an 31 n VI-set). Kleene's theorem (any 31 n Y1-set is an effective Borel set) follows as a special case. There exist two disjoint IV-subsets of N not separable by any 31 n VI-set. The analogy of the projective with the i

hierarchy thus lacks the imperfections of the earlier analogy with the Kleene (number- quantifier) hierarchy (cf. Kleene, Konink. Ned. Akad. 53(1950) pp. 800-802).

STEVEN OREY. co-consistency and related properties. Let I, be the collection of all systems L such that (1) L is a first-order system con-

taining the classical predicate calculus and possibly additional axioms, but no additional rules, (2) L contains a set of "numerals", Z0, Z1, . (3) L contains a statement N(x) such that FN(Zk), k = 0, 1, ...

A special model for a system of 4, is a model in which N(x) is satisfied by the images of the numerals and nothing else. It is shown that L will have a special model if and only if there is a consistent set of formulas, C, with the following property: if ylp(x), p2(X), ... is an enumeration of all statements containing x free, C contains 'pk(Zik) or (x).N (x) D Vyk(x) for k = 1, 2, .. ., and ik a non-negative integer.

The following construction leads to a logic L. in I, which is co-consistent but has no special model. Let Lo have as terms the variables Xk and numerals Zk, k = 0, 1,. Lo has three predicates: =, N(x), x(x). The axioms of Lo are those of the predicate calculus with equality, Zk * Z, for k different from j, N(Zk) for k = 0, 1, . .. , and _(x).N(x) D (). Ln,1 is like Ln except that it contains a new predicate, q~n+1 and the additional axioms 9Pn+l(Zk), A = 0, 1, ... , and -(x).N(x) D 9n+1(*) v :X(Zn). L. is the union of the L., n 0, 1, . Clearly each Ln has a special model, but L. does not. One can show that for each n Ln+1 is a conservative extension of Ln. If an co-contradiction were derivable in L. it would therefore also be derivable in some Ln; but this is impossible since Ln has a special model.

If L is in 4. and FZk * Z. for k different from j, and if L is consistent, L will have models in which there are elements satisfying N(x) and x * Zk for k = 0, 1,.

If P is in I and P. stands for the logic obtained from P by allowing a uses of Carnap's rule, it can be shown that Po is consistent if and only if P has a special model.

JOHN G. KEMENY. Unified foundations for semantics. A number of concepts in formalized semantics, notably the concept of analyticity,

have been the source of considerable discussion lately. It has even been maintained that no precise definition of these concepts is possible. It is my purpose to present a new approach to semantics leading to such precise definitions, and to a unification of these controversial concepts with those defined by Tarski, like truth.

In XIII 154(3) I proposed a general definition of models of formalized languages. The only reason that these models do not suffice for the foundation of semantics is that they include some unintended models if the system is incomplete. And due to G6del's incompleteness theorem, this is the usual situation. But the author of a system can compensate for an incomplete formulation by telling us which models were in- tended to be models - these I call interpretations.

An interpretation is a description of a "possible world", as far as it is describable in our language. Corresponding to truth in a possible world we have the semantic concept of validity in the interpretation. Thus I characterize analytic truth by validity in all interpretations, which corresponds to Leibnitz's "true in all possible worlds".

One of the interpretations describes the "actual world". [It can be shown that this is the interpretation determined by the translation of the object language in the

meta-language.] Tarski's definition of truth is equivalent to validity in this interpretation. And all the other semantic concepts related to truth and analytic truth can be defined with respect to all interpretations or this particular interpretation.




I can further show that the interpretations are determined if we are given (1) a translation from our object language to the meta-language, and (2) a division of the primitive constants into logical and extra-logical constants. Thus all my definitions are forthcoming if these two weak requirements are fulfilled.

CLIFFORD SPECTOR. Recursive well-orderings. Let W be the set of Gbdel numbers / of general recursive two-place functions /(x, y)

such that f(x, y) = 0, as a 5 relation, well-orders the set x (Ey)[f(x, y) O0 v /(y, x) = 0], and let I/I be the ordinal of this ordering. Various decision problems of this set are investigated, and are related to the Davis-Kleene hierarchy Ha, for a in the set 0 of Church-Kleene ordinal notations. Theorems: (1) The predicate / e W is a complete predicate of the form (m)A (I, a) with a a function variable and A arithmetical. (2) It P(x) is expressible in both one-function-quantifice forms with arithmetical scope, then P is recursive in the set TV, = f[f e TV & If/ < y] for some constructive ordinal y. (3) For each a e 0, Wia. is recursive in H... Theorems (2) and (3) provide a new proof of Kleene's theorem that every predicate expressible in both one-function-quantifier forms is recursive in Ha for some a e 0. Theorem (3) remains valid if Wila is replaced by the set b[b e 0 & Jbi < la!], which fact is used to settle the question of uniqueness ordinals left open by Martin Davis in his thesis (Princeton University, 1950): (4) For each a e 0, the degree of unsolvability of Ha depends only on the ordinal lal. As a consequence of results obtained concerning well-orderings for non-constructive ordinals, the function- quantifier analog of Post's problem is solved.

J. C. E. DEKKER. A no-n-constructive extension Qf the number system. Part. I. Consider the system [e, + ,-] consisting of the set e of all non-negative integers

(numbers) and the arithmetic operations of addition and multiplication. Collections of numbers are called sets, collections of sets classes. The sets a and ft arc recursively equivalent (x cat fi) if there is a partial recursive 1-1 function at least defined on a which maps x on P. An infinite set without an infinite recursively enumerable subset is immune, a set which is finite or immune is isolated. A set is not recursively equivalent to. a proper subset if and only if it is isolated. This fact forms the germ of our extension of the number system. The equivalence classes into which the class of all isolated sets is decomposed by the relation are called isols. An isol is finite if it contains a finite set, infinite if it contains an immune set. Since two finite sets are recursively equivalent if and only if they are equivalent, we identify every finite isol with the number by which it is characterized. A denotes the collection of all isols; it has cardinality c.

The operations of addition and multiplication defined for numbers can now be extended in a "natural" manner to isols such that: (1) + and are associative and commutative, (2) - is distributive over +, (3) A + B = A * B = 0, (4) A - B = 0 (A =0vB= 0), (5) A 0-(A -B=A -+B= 1).

The definitions are as follows. Let Is(a) = {flBa}. A + B = Is(c + fi), where m A, #e B, a and P recursively separable. A B = Is(j(ot x P)), where c A, P B, c x f the Cartesian product of a and fi and j(x, y) = j(x + y)(x + y + 1) + x.

J. C. E. DEKKER. A non-constructive extension of the number system. Part II. A + C = B + CO--A = B. Thus we can: (1) extend the operation of subtraction

from e to A, (2) imbed [A,+,-] in a ring [A*,+,-] in the same way as [?,+,-] can be imbedded in the ring of all integers. A* is the ring of all isolic integers. A o0? B if A + U=B forsome U, A o<B if A + U= B forsome U# O.Theo? relation in A is an extension of the < relation in e. It is reflexive, antisymmetric and transitive, but there exist isols which are incomparable relative to o0 . Moreover, A + C o< B + C * -A o< B. There are infinite descending chains in A, e.g., A, A-1, A-2, ... for any infinite isol A.




As yet we have only established (A * C = B * C A C 0) -A = B, in case C is finite. Division by finite isols different from 0 can therefore be introduced in A. It can be shown that: (1) if 0 < m < k the relations (3B)(A = kB) and (3C)(A = kC + m) exclude each other (in particular, no isol is both even and odd), (2) A (A - 1) . . . . - (A - k + 1) is divisible by k!, whenever 1 k of? A, (3) there exist infinite isols without any finite divisor > 1.

A function /(x) from ? into ? is called finite if {xlf(x)} # 0 is finite. Every finite function is clearly recursive. All finite functions can be effectively generated without repetitions in a sequence {rn(x)}. Using finite functons AB can be defined for A, B e A. Exponen- tiation in A has the following properties: A B - A C = A B+ C, (AB)C = A B-C (A *.B)c C_ AC BC, (A,B,C 0 A Ao< B) TACo< BC, (1 o< A A0o< BABo< C)ABo< AC.

JOHN MYHILL. A fixed-point theorem in recursion theory. It was observed by Fitch (Proceedings of the XIth Congress of Philosophy,

vol. XIV) that there exists a combinatorial operator Z such that Zxy conv xy(Zx). Combining this with the result of Church that a set of integers is recursively enumerable (r.e.) just in case it is lambda-definable, we obtain the following result which we call the fixed-point theorem. Let qn(x) = q(n, x) be a partial recursive function which enumerates all partial recursive functions of one variable, and let (On = {qn(x)}. Then given any recursively enumerable relation 'D(x, y) we can effectively find infinitely many solutions n of the equivalence x e (On ?ID(x, n). Thus if we call n the index of onw, there exist recursively enumerable sets containing just those numbers which are greater than their index, just those numbers which are factors of their index etc. The same fact can be proved by purely recursion-theoretic methods, or alternatively by the use of canonical systems. Under appropriate definitions, we may express the theorem succinctly as: Every effective operation on r.e. sets has a fixed-point.

We have the following corollaries. (1) The Kleene recursion theorem (in a slightly weakened form); (2) Any two creative sets (Post) are one-one reducible to one another; (3) Let a class A of r.e. sets be called completely recursively enumerable (c.r.e.; Rice) if (3 (0o ((On e A '-+ n e Ok). Let a collection { i} of finite sets be called an array if the set of all prime-power representations of ci's is r.e. Let T(fl) be the collection of all r.e. supersets of i. Rice proved that the class A = ZT(m,) is c.r.e. for each array {mi} and conjectured that conversely every c.r.e. class can be written in the form ZT(ox). We can use the fixed-point theorem to prove this conjecture (which was also proved in another way by MacNaughton and Shapiro).

The ASSOCIATION FOR SYMBOLIC LOGIC announces the following elections, each for a period of three years from January 1, 1955.

As members of the Executive Committee, Dr. Ilse Novak Gal, of Cornell University, and Professor Hao Wang of Harvard University.

As members of the Council, Professor W. Ackermann of the University of Miinster, Germany, and Professor Georg H. von Wright, of Helsingfors University.

The ASSOCIATION FOR SYMBOLIC LOGIC will hold a joint meeting with the American Philosophical Association at Boston University, Boston, Massachusetts, on December 29, 1955. Members desiring to present papers will please by October 1 submit abstracts,




in duplicate and not longer than three hundred words each, to George Berry, De- partment of Philosophy, Boston University, College of Liberal Arts, Boston, Mass- achusetts. Every effort will be made to accommodate papers whose abstracts are submitted after October 1.

INSTITUTIONAL CONTRIBUTING SUBSCRIBERS to the JOURNAL for the year 1954 are the following:

HARVARD UNIVERSITY

UNIVERSITY OF ILLINOIS

LEHIGH UNIVERSITY

RUTGERS UNIVERSITY

SMITH COLLEGE

EDWARD C. HEGELER TRUST FUND

INSTITUTE FOR ADVANCED STUDY

UNIVERSITY OF MICHIGAN

PENNSYLVANIA STATE UNIVERSITY

UNIVERSITY OF WISCONSIN



Eighteenth Meeting of the Association for Symbolic Logic

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