The initial event, in the series which led to the establishing of the Journal of Symbolic Logic and of the Association for Symbolic Logic, was a conversation early in 1934 between C. J. Ducasse and C. A. Baylis at Brown University in room 112, Rhode Island Hall. Ducasse, in the Graduate School at Harvard in 1911, had had work in Symbolic logic with Josiah Royce, and had become much impressed with the potential importance of the subject. Baylis, years before at the University of Washington, had been the outstanding student in the logic course Ducasse was conducting there, and eventually joined Ducasse on the staff of the Philosophy Department at Brown.

The theories considered here are based on the classical functional calculus (possibly of higher order) together with a set A of non-logical axioms; they are also assumed to contain classical first-order number theory. In foundational investigations it is customary to further restrict attention to the case that A is recursive, or at least recursively enumerable (an equivalent restriction, by [1]). For such axiomatic theories we have the well-known incompleteness phenomena discovered by Godei [6]. Quite far removed from such theories are those based on non-constructive sets of axioms, for example the set of all true sentences of first-order number theory. According to Tarski's theorem, there is not even an arithmetically definable set of axioms A which will give the same result (cf. [18] for exposition).

Let ℒ be the set of all formulas of a given first order predicate logic (with or without identity). For each positive integer n, let ℒn be the set of all formulas φ in ℒ logically equivalent to a formula of the form

where Q is a (possibly empty) string of quantifiers, m is a positive integer, and each αij is either an atomic formula or the negation of an atomic formula.

In this paper a formalized logic of propositions, PA1, is presented. It is proven consistent and its relationships to traditional logic, to PM ([15]), to subjunctive (including contrary-to-fact) implication and to the “paradoxes” of material and strict implication are developed. Apart from any intrinsic merit it possesses, its chief significance lies in demonstrating the feasibility of a general logic containing the principle of subjunctive contrariety, i.e., the principle that ‘If p were true then q would be true’ and ‘If p were true then q would be false’ are incompatible.

By a theory is meant an applied first-order predicate calculus with at least one predicate symbol and perhaps some individual constants and function symbols and a specified set of axioms. In addition to the terms defined by means of the individual variables, constants, and function symbols a theory may also include among its terms those constructed by means of operators such as the epsilon or iota operators; that is, expressions like (εχΡ) or (οχΡ), where P is a well formed formula (wff) of the theory, may also be terms. A constant term of a theory F is then a term in which no variable occurs free. We are interested only in theories which have at least one constant term so that if a theory doesn't have any individual constants it must necessarily admit as terms expressions constructed by means of operators. A sentence of a theory F is a closed wff.