The purpose of this article is to give a survey of some results I have found in investigations concerning logical modalities. The results refer: (1) to semantical systems, i.e., symbolic language systems for which semantical rules of interpretation are laid down; (2) to corresponding calculi, i.e., syntactical systems with primitive sentences and a rule of inference; (3) to relations between a semantical system and the corresponding calculus.

The semantical systems to be dealt with are the following: pro positional logic (PL), functional logic (FL), and the corresponding modal systems, viz. modal propositional logic (MPL) and modal functional logic (MFL).

General syntax, the formal part of the general theory of signs, has as its basic operation the operation of concatenation, expressed by the connective ‘⌢’ and understood as follows : where x and y are any expressions, x⌢y is the expression formed by writing the expression x immediately followed by the expression y. E.g., where ‘alpha’ and ‘beta’ are understood as names of the respective signs ‘α’ and ‘β’, the syntactical expression ‘alpha⌢beta’ is a name of the expression ‘αβ’.

Tarski and Hermes have presented axioms for concatenation, and definitions of derivative syntactical concepts. Hermes has also related concatenation theory to the arithmetic of natural numbers, constructing a model of the latter within the former. Conversely, Gödel's proof of the impossibility of a complete consistent systematization of arithmetic depended on constructing a model of concatenation theory within arithmetic.

A fundamental problem of symbolic logic is to define logical calculi sufficient to comprise important parts of mathematics, and to develop systematic methods of calculation therein.

The possibility of progress in this direction has been severely limited by Gödel's proof that a consistent system sufficient to comprise arithmetic must contain propositions whose truth-value cannot be decided within the system, and by Church's extension of Gödel's method to the result that even in the first order logical function calculus the general decision problem cannot be solved.

The following system is an extension of the Lewis calculus S2 to include quantification.

In a previous paper, a functional calculus based on strict implication was developed. That system will be referred to as S2. The system resulting from the addition of Becker's axiom will be referred to as S4. In the present paper we will shw that a restricted deduction theorem is provable in S4 or more precisely in a system equivalent to S4. We will also show that such a deduction theorem is not provable in S2.

The following theorems not derived in Symbolic logic will be required for the fundamental theorems XXVIII* and XXIX* of this paper. We will state most of them without proofs.

In a previous note I showed a new way to define the ordered pair. I made use of the notations ‘Nn’ (for class of natural numbers) and ‘Sv’ (for successor of v), remarking that they are readily defined without appeal to ordered pairs or relations. Adopting the auxiliary abbreviation:

I defined the ordered pair thus:

In this note I give an account of the relations between different notions of weakened quantification, which can be defined in intuitionistic logic. The reason for publishing it is the fact that the last gap in our knowledge of these relations has been filled in an article by Kleene [5].

For the intuitionistic calculus of propositions I refer to my article [2]; besides the rules of substitution and of inference I need only a few formulas from this calculus, which I shall repeat below. The simplest way of formalizing the intuitionistic predicate calculus is obtained by adjoining to the calculus of propositions the symbols, rules, and axioms as given by Hilbert and Ackermann [4], Chapter 3, §§4-5, pp. 53-57. (Reference is made to this book as HA.) I repeat only the axioms (e) and (f) and the rule γ.

In his excellent review of four notes of Skolem on recursive functions of natural numbers Bernays states: “The question whether every relation y = f(x1,…, xn) with a recursive function ƒ is primitive recursive remains undecided.” Actually, the question is easily answered in the negative by a form of the familiar diagonal argument.

We start with the ternary recursive relation R, referred to in the review, such that R(x, y, 0), R(x, y, 1), … is an enumeration of all binary primitive recursive relations.