1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.

It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form

(1) (Ex1)(x2)(Ex3)(x4)…(xm).

On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove the

Theorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.

2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,

(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

In my American Academy paper of 1937 I gave a mathematical clarification and extension of C. I. Lewis's theory of strict implication using as the base (K, T, ×, ′, ≣). In that paper the element a⊰b (read “a hook b”) was defined by (a⊰b) = (ab′=cc′). The main purpose of the present paper is to show that this definition may be replaced by the simpler definition (a⊰b) = (ab≣a), provided Postulate 15 is replaced by two simpler postulates, 15a and 15b (see below). The paper may also serve to throw further light on the essential difference between Lewis's theory of “strict” implication and the more familiar theory of “material” implication. This paper and the earlier one, when read together, are entirely self-contained, all proofs (including proofs of independence) being given in full and no reference to earlier literature being required.

We begin by reviewing briefly the rôle which logical deduction plays in the writing of any scientific book.

Suppose you sit down to write. What do you actually do? In the last analysis, your activity as an author is a process of voluntary selection, restricted by certain inescapable rules. From an indefinite number of possible propositions which may occur to your imagination, you voluntarily select certain propositions which you accept as being worthy of being written down, one after another, in your book. Every proposition which occurs to you is either accepted, or rejected, or laid aside for further consideration.

In the present paper, the various means used by the author in attempting to find a contradiction in Quine's New foundations are sketched, and the reason why each method failed is indicated. It was not Quine's system itself which was tested, but a stronger one obtained by adding a rule of Kleene's type to Quine's system. Reasons are presented why a contradiction in the stronger system should be as damning to Quine's system as a contradiction in Quine's system itself. Two other features of the system are worthy of note. One is the fact that only two symbols are used in building up the formulas of the system. Other systems have been built using only two symbols, but the particular method used in this paper is very flexible and simple, and is peculiarly adapted to the use of the Gödel technique. It was suggested by the consideration that the formulas of any system can be written by the use of only two symbols by first assigning Gödel numbers to the formulas and then writing those numbers in the binary scale of notation. The second feature is that ι (to be used in ιxp, meaning “the x such that p”) is an explicit and integral part of the system, and the axioms and rules governing its use are such as to make it very simple to handle. By a process similar to “Die Eliminierbarkeit der ι-Symbole” of Hilbert-Bernays, it is shown that the system involving the ι can be reduced to one not involving it. The points of difference with the Hilbert-Bernays technique make possible an especially unhampered use of ι.