Haskell B. Curry

The Inconsistency of Certain Formal Logics

A proof that certain systems of formal logic are inconsistent, in the sense that every formula expressible in them is provable, was published by Kleene and Rosser under the above title in 1935. For the case where the underlying system satisfies an additional condition—viz. the possession of an operator analogous to Schonfinkels K —I gave a simpler derivation of this condition in a previous paper. That argument, like the original one of Kleene and Rosser, was a refinement of the Richard paradox. The object of the present paper is to show that, if we use other paradoxes, an inconsistency will result from a very much simpler argument and on much less restrictive hypotheses. The contradiction can no longer be called “the paradox of Kleene and Rosser,” because it is based on an entirely different principle; but, in deference to the work of the original discoverers of the inconsistency, the paper is given the same title as that which their paper bears. The central idea of the new derivation was suggested by some work of R. Carnap. This paper is based on the one above cited, which will be referred to as PKR. However, acquaintance with that paper will be presupposed only through 3.4, i.e., through the statement of the basic hypotheses and conventions for a combinatorially complete system—except that for the second (alternative) method of construction given below the substance of PKR through 9.6 is needed.

Some Philosophical Aspects of Combinatory Logic

This is a discussion of some philosophical criticisms of combinatory logic, preceded by a brief survey to give background.

On the Use of Dots as Brackets in Logical Expressions

Such chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine,1 is at variance with our ordinary algebraic usage, in which we write a-b+c-d for ((a-b)+c)-d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.2 Such a modification is easily obtained by generalizing somewhat a procedure of Church.3 The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation. Thus the chain implication (1) can be symbolized a la Church by

REMARKS ON INFERENTIAL DEDUCTION

Publisher Summary The term inferential deduction is used to describe deduction based on inferential rules of G. Gentzen type. This chapter discusses three loosely connected topics. M. Anderson and H. Johnstone presented the Gentzen natural rules in a linear form that was used by S. Jaskowski and B. Fitch. This formulation is, in principle, equivalent to the formulation by proof trees that G. Gentzen used. However, for certain rather technical purposes, it is desirable to have an explicit process or algorithm for transforming either of these forms of proof into the other. The chapter describes this technicality. It also discusses the semantics of Gentzen L-rules. The natural rules of Gentzen have the peculiarity that some of the rules involve discharge of assumptions.