In a previous number of this Journal, Dr. Arnold F. Emch argues that the mathematical properties of the system of strict implication preclude the interpretation of p⊰q as synonymous with “q is deducible from p” and of ◊(pq), or p o q, as “p and q are consistent.” He proposes a new calculus of propositions in which the primitive ideas of the system of strict implication are retained but an additional idea, “logical consistency,” Op, is introduced, and two new relations, “logical implication,” p∾q, and “logical equivalence,” p = q, are denned. He believes that the properties of this new system are in accord with the facts about deducibility, consistency, and independence of propositions where those of p⊰q and ◊(pq) fail of such accord.

Not all of the objections which Dr. Emch advances against strict implication are given due consideration in what follows. Concerning one especially important point, something will be said at the end of this paper. For the rest, discussion will here be limited to certain formal properties of the two systems, and certain consequences for the interpretation of them.

For brevity, the system of strict implication will be referred to, in what follows, as the “system S,” or merely as S; and Dr. Emch's calculus as the “system L,” or as L.

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.”

First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).

The system of formal logic to be presented in this paper has the following main properties:

(1). The Mengenlehre paradoxes do not arise in it.

(2). It seems to be free from other contradictions.

(3). It does not resort to a theory of types.

(4). It uses negation and material implication and does not weaken the principle of excluded middle.

(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.

(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….

(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.

(8). A theory of integers and real numbers cannot, apparently, be deduced from it.

(9). A form of the axiom of choice is provable in it.

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkül. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables).

The additional axioms of L other than x=y→[F(x)→F(y)] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x]. Moreover the axiom x=y→[F(x)→F(y)] may be replaced by the following set of axioms:

and similar axioms for each of the functions b1, b2, …, bk.

The present formulation should prove significant in the development of symbolic logic along the lines of Gödel's theorem on the incompleteness of symbolic logics and Church's results concerning absolutely unsolvable problems.

We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem.

In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.

In the present formulation the symbol space is to consist of a two way infinite sequence of spaces or boxes, i.e., ordinally similar to the series of integers …, − 3, − 2, − 1, 0, 1, 2, 3, …. The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.