Die Unabhängigkeit der Elementaren prädikatenlogischen schlussregeln. by Karl SchröterReview by: Perry SmithThe Journal of Symbolic Logic, Vol. 37, No. 1 (Mar., 1972), p. 194Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272606 .

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194 REVIEWS

Several other results are also proved in the paper. We mention only a few of them: (1) Each nth-order formula is equivalent with respect to satisfiability to a second-order formula (the author is apparently unaware that this result was established already by Hintikka (XXXI 660)). (2) There exists a class of models characterized by an (n + 1)st-order axiom but not by any nth- order axiom. (3) Results which generalize for higher orders a theorem of Zykov (XXII 360) on a form of the prefix of a second-order formula. (4) The set of Gbdel numbers of those second- order formulas which are true in every finite domain is not definable by any arithmetical for- mula of whatever finite order.

Proofs of all the theorems are merely sketched and, moreover, only the cases of formulas whose orders are _3 are discussed; generalizations for higher orders are stated without proofs.

Errata: Page 94, line 23, read S ? R for S C R. Page 95, replace "y" by " Y" in the first displayed formula. ANDRZEJ MOSTOWSKI

KARL SCHROTER. Die Unabhangigkeit der elementaren pradikatenlogischen Schlussregeln. Zeitschriftfir mathematische Logik wnd Grwndlagen der mathematik, vol. 2 (1956), pp. 218-227.

The following complete set of axioms and inference rules for first-order logic (with Vx 4' and 3x 4 taken to be well-formed formulas only if x occurs free but not bound in 4) was given in the author's XXXII 418 (we use '4'(x)' to mean that x occurs free but not bound in 4'). All tautologies are axioms, and the inference rules are modus ponens; f(x) -#4' .. Vx +(x) -+4'; 4' --# (x) .. 4 -* Vx 54(x) (provided x does not occur in 4); 4'(x) --).4 .'. 3x +(x) -. 4 (provided x does not occur in @); 4 -+ +(x) . . 4 -* 3x +(x); 4 .'. 4' (provided 4' comes from 4 by replace- ment of all occurrences of a variable x in a subformula Vx +(x) or 3x +(x) of 4 by a variable y which does not occur in 4+(x)); and +(x, y) .'. +(x, x). In this paper it is shown syntactically that removing any one of the inference rules destroys completeness. For example, the last inference rule is needed to prove VxVyRxy -+ Rxx. PERRY SMITH

A. DuMimRru. The antinomy of the theory of types. International logic review-Rassegna internazionale di logica (Bologna), vol. 2 no. 3 (1971), pp. 51-54.

The author claims to show, by constructing an antinomy, that the simple theory of types is inconsistent. This is done without using an axiom of infinity or other added axioms, but only functional calculus. Indeed only singulary predicates of types m and m - 1 appear in the con- struction of the antinomy, with no indication that m must be large-so that ostensibly an incon- sistency has been found in singulary functional calculus of some low order.

In view of the elementary consistency proof for simple type theory which is due originally to Herbrand (3825) and is now well known (see an especially detailed version by Gentzen I 119), it is certain in advance that there is a fallacy. The assertion (p. 52) that there are only a finite number of singulary predicates of type m may be ignored as not essential to the argument. It seems to the reviewer that the critical fallacy is in the passage that begins at line 14 from the bottom of page 53, where it is overlooked that a given predicate of type m - 1 may be both incongruent and congruent because the sets Sp -1 may have elements in common.

Dumitriu relies on a quotation from Wang and McNaughton (XIX 64) for the idea that the consistency of simple type theory is not certain. But Wang and McNaughton must not be under- stood as referring to type theory without axiom of infinity. ALONZO CHURCH

GEROLD STAHL. A paratheory of type theory. Zeitschrift fur mathematische Logik and Grundlagen der Mathematik, vol. 9 (1963), pp. 169-171.

By a " paratheory of type theory," the author intends a theory based on the simple theory of types in which the individuals are the objects of all types in some model of the simple theory of types. It is not made clear how this semantic feature is expressed syntactically. The consistency of the paratheory is said to reduce trivially to that of simple type theory; and an "amplified" paratheory is introduced containing constants for the individuals corresponding to the universal classes of each type and a predicate for the e-relation. These facts might suggest that the basic paratheory is just the original theory with the above interpretation. But in the absence of a definite formulation of the basic paratheory, the reader cannot be sure.

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