We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

In teaching symbolic logic to beginners, it is a not infrequent experience to encounter difficulties in the transition from the informal “if …, then …” construction of common discourse to its analysis in terms of material implication and its truth-table. There is no trouble with the idea that “if p, then q” must be so understood that “if p/T1, then we must have (p⊃q)T when q/T, and (P⊃q)/F when q/F. There is thus no problem in leading the explanation to the point.

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 fundamental role of the restricted calculus of predicates in applications of symbolic logic, and particularly in Hubert's Beweistheorie as summed up by Hilbert and Bernays, makes it important that this logical calculus should be accurately defined. The first standard formulation of the calculus was that of Hilbert and Ackermann's Grundzüge der theoretischen Logik. This employed (in the first three editions) a finite set of axioms and rules of derivation, with rules of substitution included. A reaction by Hilbert and Ackermann's successors to persistent difficulty encountered with the rules of substitution has been to omit these rules, and instead enlarge the set of axioms and the other rules of derivation so as to encompass all possible substitutions. Such an enlargement seems to me to be undesirable. As an alternative, this note is designed to put the original approach of Hilbert and Ackermann for once and for all on a sound basis.

Let Pp, Pd, and N be the intuitionistic formal systems of prepositional calculus, predicate calculus, and elementary number theory, respectively.1 Consider the following six propositions.8

(1) ├A V B only if ├A or ├B.

(2) ├∋xA(x) only if ├Ã(t) for some formula Ã(x) congruent to A(x) and some term t free for x in Ã(x).

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).

A standard set of axioms for the intuitionist propositional calculus is the following (we associate to the right, and use the convention that ⊃ binds less strongly than the other connectives).

Suppose the ri-placed relation symbols Pi, 1 ≦ i ≦ k, are all the non-logical constants occurring in the closed formula , also written as , of Heyting's predicate calculus (HPC). Then HPC is called complete for provided , i.e.

Here D ranges over arbitrary species, and over arbitrary (possibly incompletely defined) subspecies of ;

The “implicational paradoxes” are treated by most contemporary logicians somewhat as follows:

“The two-valued prepositional calculus sanctions as valid many of the obvious and satisfactory inferences which we recognize intuitively as valid, such as

(A→.B→C)→.A→B→.A→C,2

and

A→B→.B→C→.A→C;

it consequently suggests itself as a candidate for a formal analysis of implication.

In dieser Arbeit soll eine unendlichwertige Logik untersucht werden, deren Aussagenkalkül von Łukasiewicz und Tarski [3] erstmals diskutiert wurde. Eine ausführliche Darstellung des Prädikatenkalküls findet man in J. D. Rutledge [6] und B. Rosser [5]. Von A. Rose und Rosser wurde die Vollständigkeit des Aussagenkalküls bewiesen [4]. Später wurde von Rutledge die Vollständigkeit des einstelligen Prädikatenkalküls bewiesen, und zwar mit algebraischen Methoden, den sogenannten MV-Algebras, die von C. C. Chang [1] eingeführt worden sind. Hier soll bewiesen werden, dass der volle unendlichwertige Prädikatenkalkül nicht axiomatisierbar ist. Genauer: es soll gezeigt werden, dass die Menge der wahren Formeln, d.h. derjenigen Formeln, die immer den Wert Eins annehmen, nicht rekursiv auf zählbar ist.

In [1] McKinsey gives a method of defining possibility in terms of “substitutions”, or transformations of one sentence into another. Roughly, the motivation is that a sentence be possible if a sentence of “like form”, i.e. a transformation of it, is true.

The purpose of this paper is to show that the two systems given in [1], shown by McKinsey to be at least as strong as Lewis' systems S4 and S5, do exactly characterize S4 and S5, and that the system given by omitting rule A4 from the first system exactly characterizes von Wright's system M.1

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.

For the m-valued (2<w<∞) logics of Łukasiewicz as described in [2] using the values 1,2, …, m and based on the functions f(p, q) = max(1,q—p+1) and g(p) = m—p+1, Evans and Schwartz have essentially2 proved in [1] that the addition of a constant function i, 1<i<m, yields functional completeness if and only if (m—1, i—1)= 1.3 In this note it will be shown that if n constant functions i1, …, in, 1<ik<m and 1≦k≦n, are added, functional completeness obtains if and only if (m—1, i1—1, …, in—1) = 1.

It is shown in Sobociński [1] that the number of irreducible modalities in T is infinite. The infinity of modalities in T collapses if the axiom CLpL2p is added to T, since T plus CLpL2p is equivalent to S4 and the number of irreducible modalities in S4 is finite as proved in Parry [2]. But the addition of the axiom CL2pL3p does not collapse the infinity of modalities in T. The aim of the present paper is to prove this theorem and its corollaries.

The systems of predicate logic considered here are extensions of ordinary first-order predicate logic. They were obtained by first admitting infinite conjunctions, disjunctions, and quantifications into formulas, and then adjusting the axiom schemata and rules of inference to accommodate them. In this paper, only systems with formulas of denumerable length and “formal proofs” with denumerably many steps will be discussed, though such restrictions are hardly essential. The systems with denumerable conjunctions and disjunctions, but finite quantifications, are semantically complete; i.e., valid formulas are provable. However, systems in which denumerably infinite quantifications are also allowed, are semantically incomplete.