Some Advances in the Combinatory Theory of Quantification by Haskell B. CurryReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 8, No. 1 (Mar., 1943), p. 52Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268007 .

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52 REVIEWS

of types), can be reduced to the decision problem for S, and on the other hand that the de- cision problem for S is unsolvable. Hence the same results now follow for S"' in place of S. A fortiori, the general decision problem for systems in normal form is unsolvable. This pro- vides an example of an unsolvable decision problem which is of very simple form, and which may for that reason be found useful in obtaining proofs of unsolvability of decision problems arising in various special branches of mathematics. Moreover, as'the author points out, if we define a normal set of sequences on the letters a,, a2, . * *, a, as the set of assertions contain- ing those letters only, in any system in normal form whose primitive symbols include those letters, and a binormal set as a normal set whose complement is normal, then on this basis a new and independent approach to the question of effective calculability is possible, the notion of binormality taking t1e place of that of recursiveness or of X-definability, and the notion of normality taking the place of that of recursive enumerability.

The paper contains also a discussion of the "problem of tag," a decision problem closely related to the decision problem for systems in Post's normal form; and a statement, without proof, of an alternative reduction of systems in canonical form (in place of the reduction to normal form).

According to the historical footnote above referred to, the author had in 1921 at least an outline of a proof (using the diagonal method) of the unsolvability of the general decision problem for systems in normal form, and in consequence of this an anticipation of G6del's incompleteness theorem (4183). These results were not published. ALONZO CHURCH

HASKELL B. CURRY. Some advances in the combinatory theory of quantification. Pro- ceedings of the National Academy of Sciences of the United States of America, vol. 28 (1942), pp. 564-569.

This paper is concerned with the problem of uniting in a single system the author's "com- binatory logic"-or, what is known to be essentially equivalent in consequence of results of the author and of Rosser, one of the forms-of the reviewer's "calculus of X-conversion"-and a theory of quantifiers and implication.

Such familiar systems as the theory of types or the Zermelo set theory impose restrictions upon the ranges of variables or upon the operation of forming a class or a function by abstrac- tion which prevent their containing combinatory logic as an integral part of the system (i.e., in such a way that the system as a whole is a combinatory logic with added axioms). And on the other hand, early attempts at a combined theory of quantification-implication and com- binatory logic (or X-conversion) led either to inconsistent systems-cf. 5451-or, as in the case of the theory of quantifiers proposed in the reviewer's II 39, to systems probably not adequate for the whole of classical mathematics.

The present paper is based on the proposal of the last section of the author's VIII 31(2), under which consistency is to be secured by distinguishing as "canoniqal" a certain class of terms (not including combinators or quantifiers in isolation) and restricting certain of the syntactical variables in the rules and axiom schemata to canonical terms. Various systems are considered, some of them requiring a notion of relative canonicalness, or a hierarchy of levels of canonicalness. The central problem is that of consistency proofs for these systems.

The paper has the character of an abstract, and is at the same time a report of unfinished research. For some of the consistency theorems involved the author has proofs, which are here outlined but are not given in detail. However, a final consistency theorem, which may turn out to be the crucial one, is offered only as a conjecture, or as an unsolved problem. Adequacy of the various systems to the ordinary uses of quantifiers, and in particular to mathematics, is not indicated, except for an important negative indication in the case of the system which the author calls W2; some suggestions in regard to the expected adequacy for arithmetic may, however, be found in the above-mentioned last section of VIII 31(2).

The following errata were pointed out, or have been verified, by the author: On page 564, line 8 of text, for 22 read ft. On page 564, line 12 from the bottom, for "The system F1" read "The system i." On page 565, line 6, for .nX1... nx read Xn.... xnX. On page 565, line 19, for , (yw) read ,3w (yw). On page 567, line 8 should read "canonicalness and verifiability are invariant with respect to conversion." On page 568, line 7 from the bottom, insert a semicolon after the displayed formula. On page 569, in footnote 3, for 8X2 read j2-

ALONZO CHURCH

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