In [1] Anderson gave what he claimed was a decision procedure for Lewis's calculus S4. One of the proofs of [1] was faulty, however, and he sought to remedy the situation in [2] by modifying slightly one of the conditions of the decision procedure of [1]. Poliferno subsequently reported in [5] that, according to Anderson, the procedure of [2] was not adequate either, but tha t a correct decision procedure for S4 could be obtained by combining the procedures of [1] and [2]. The purpose of the present note is to show tha t none of the three procedures referred to above are correct decision procedures for S4.

In this paper we continue the investigations begun in [2] of equivalence classes of (denumerable) well-orderings under one-one partial recursive, order-preserving maps (recursive isotonisms). The numbering and notation are continued from [2] except that we now use lower case Greek letters for classical ordinals and upper case Roman letters for sets and we slightly modify the definition of (see §IX.3).

In section IX, after proving some lemmata which we use repeatedly in this paper, we establish the existence of Cantor Normal Forms for all co-ordinals less than a co-ordinal of the form WA (a fortiori for all co-ordinals less than a principal number for exponentiation).

In their original paper on degrees [3], Kleene and Post showed that there is a degree between 0 and 0′. Later, Friedberg [1] and Muchnik [4] showed that there is a recursively enumerable degree between 0 and 0′. Since then, this phenomenon has been repeated several times: a result has been proved for degrees, and then, after considerable additional effort, it has been proved for recursively enumerable degrees.

There are some obvious respects in which that set of all degrees differs from the set of recursively enumerable degrees; e.g., the former is uncountable and has no largest member.

In [1] (p. 131 et seq) Professor A. N. Prior suggests a calculus of functions of propositions in which the range of the function variables is not restricted to truth functions.1 If f, g, … etc. represent such variables and we have quantification over all variables we can introduce propositional identity by definition as (ƒ)(ƒp ⊃ ƒq) Alternatively identity may be primitive with the usual axioms and schemata (v.e.g. [4] p. 190 et seq). We shall refer to such systems as ‘functorial calculi’2 (abbreviated as FC).

In [1], page 85, Quine states: “… ponentials of logically true statements are logically true.” It is perhaps of some interest that this assertion is not completely precise as can be seen from counterexamples. One simple example is the following:

Let L be some language in which ‘c’ and ‘b’ are atomic sentential constants. Let ‘c’ be false and ‘b’ true. Suppose that L includes parentheses and the material conditional ‘⊃’ as logical notation, but that ‘b’, at least, counts as extra-logical.

An implication A ⊃ B is said to be null in case A is false in all instances. This note describes several formal systems for first-order predicate logic and arithmetic in which null implications do not occur. In fact, the most restrictive systems avoid the occurrences of any formulas which represent null predicates. The precise sense of this avoidance is explained below.

An intuitionistic negation is a statement of the form A ⊃ F where F is a contradiction. The absence of null implications among true statements entails the absence of intuitionistic negations, and hence systems described here might be called negationless intuitionistic systems.

In this note is proved the following:

Theorem. Iƒ A × B is universal and one oƒ A, B is r.e. then one of A, B is universal.

Let α, τ be 1-argument recursive functions such that x goes to (σ(χ), τ(χ)) is a (1–1) map of the natural numbers onto all ordered pairs of natural numbers. A set A of natural numbers is called universal if every r.e. set is (many-one) reducible to A; A × B is called universal if the set

In [1], McCall and I offered an axiomatization of the implicational fragment of the 3-valued Łukasiewiczian system defined in [2]. The following axiom schemata sufficed, together with the rule of modus ponens.1

By inequation is meant an expression of the form,

where a, b are terms taken from any first-order formal language. An inequation is consistent iff it cannot be brought to the form,

by substitution.

Briefly, the following things are done in this paper. An informal exposition of the significance of partial truth tables in modal logic is presented. The intuitive concept of a truth tabular sentence connective is examined and made precise. From among a vast assortment of truth tabular connectives, the semantically well-behaved ones (called regular connectives) are singled out for close investigation. The familiar concept of functional completeness is generalized to all sets of regular connectives, and the functional completeness (or incompleteness) of selected sets of connectives is established. Some modal analogues of Sheffer's stroke are presented, i.e., single connectives are introduced which serve to define not only the truth functional connectives but all the regular modal connectives as well. The notion of duality is extended to all regular connectives and a general duality theorem is proved. Lastly, simplified proofs are given of several metatheorems about the system S5.

In the course of a recent paper on modal' extensions of the intuitionist propositional calculus, [1], I made some suggestions as to the relationships between the system MIPC, the intuitionist predicate calculus, and the question of producing a genuine intuitionist concept of modality. This paper may be regarded as a clarification of those rather inaccurate ideas in the light of Kripke's outstanding analysis of the intuitionist predicate calculus, [2]. (I use Kripke's notation and terminology here without explanation — this work is intended to be read in conjunction with [2].) In particular, I shall adapt his interpretation of his modelling to give an account of MIPC in terms of differing mathematical intuitions.

In the first-order language, quantifiers (∀x) and (∃#) are understood “to say” that “for all elements” and “there is at least one element such that …”, respectively. We are interested in changing the interpretation to “for all elements with fewer than κ exceptions” and “there are at least κ elements such that”, respectively, where κ is a cardinal. We call this the κ-interpretation of the quantifiers.1 The first question which presents itself is “What is the relationship between the κ-interpretation and the λ-interpretation?” For instance, is a formula valid under one interpretation also valid in all other interpretations? In the second section, it will be shown that as far as infinite interpretations, i.e. κ-interpretations for infinite cardinals κ, are concerned, the validity of a formula is preserved. Actually, a more general result is obtained there by model theoretic methods.

The equi-cardinality quantifier1 to be used in this article, written as Qx, is characterised by the following semantical rule: A formula QxA(x) is true in a relational system exactly when the cardinality of the set consisting of these elements which make A(x) true is the same as that of the universe. For instance, QxN(x) is true in 〈Rt, N〉 but false in 〈Rl, N〉 where Rt, Rl, and N are the sets of rational numbers, real numbers, and natural numbers, respectively. We notice that in finite domains the equi-cardinality quantifier is the same as the universal quantifier. For this reason, all relational systems considered in the following are assumed infinite.