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.

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 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.