This is the first in a series of articles outlining a new approach to Semantics. The novelty in the approach is that the concept of an interpretation of a logical system is taken as the central concept of Semantics. I hope to show that by means of this approach a satisfactory definition can be given for such controversial concepts as analyticity, and at the same time the approach leads to a unified foundation for formalized Semantics.

As the possibility of such definitions has been questioned in recent years, it is an important task to try to give precise definitions. Clearly, this is the task of those philosophers who believe that concepts like analyticity should play a fundamental role in Semantics. On the other hand, the philosophers who have criticized these concepts will now be able to tell just exactly why these definitions are unacceptable to them — instead of being forced to talk in generalities. It seems to me that no further progress is possible until we have precise definitions available for discussion.

The fundamental semantic concepts fall into two classes: those, like truth, for which Tarski has offered definitions; and those additional ones, like analytic truth, for which we have Carnap's proposed definitions.

In this note we point out that two concepts which have been introduced separately into the theory of models are extensionally equivalent. The basic ideas can be expressed conveniently in the terminology of Tarski's theory of arithmetical classes. It will be recalled that an arithmetical class is a class consisting of all algebraic structures which satisfy some fixed sentence of the first-order functional calculus. A more general concept is that of a quasi-arithmetical class, whose elements are all algebraic structures which satisfy some fixed set of first-order sentences.

Now where H and K are any two classes of algebraic structures, Abraham Robinson has defined the relation H is persistent with respect to K to mean that (i) H∩K is non-empty, and (ii) whenever M ϵ H∩K, M′ is an extension of M, and M′ ϵ K, then M′ ϵ H. Concerning this notion Robinson has established the following theorem: If K is the class of abelian groups (or of commutative fields) and H is an arithmetical class which is persistent with respect to K, then H∩K contains a group (or field) which is finite. This leads us to the following definition.

A statement X in the lower predicate calculus is said to be persistent with respect to the set of statements K ([2], [3]), if whenever X holds in a model M of K then X holds also in all extensions of M which are models of K. If X is persistent with respect to the empty set, then it may be said to be absolutely persistent.

A statement X is called existential, if it is in prenex normal form and does not contain any universal quantifiers. This includes the possibility that X does not contain any quantifiers at all.

Let E be the class of all existential statements. Then it is not difficult to see that E is quasi-disjunctive. That is to say, given statements Y1, Y2 in E, there exists a statement Y in E such that

is provable.

L. Henkin [1] has raised the question how to characterise the statements X which are persistent with respect to a given set K (e.g. a set of axioms for a field or a group) by a syntactical condition. He has shown that, in order that a statement X be absolutely persistent, it is necessary and sufficient that there exist a statement Y ϵ E such that

is provable.

In part I of this paper it is shown that if the simple theory of types (with an axiom of infinity) is consistent, then so is the system obtained by adjoining axioms of extensionality; in part II a similar metatheorem for Gödel-Bernays set theory will be proved. The first of these results is of particular interest because type theory without the axioms of extensionality is fundamentally rather a simple system, and it should, I believe, be possible to prove that it is consistent.

Let us consider — in some unspecified formal system — a typical expression of the axiom of extensionality; for example:

where A(h) is a formula, and A(f), A(g) are the results of substituting in it the predicate variagles f, g for the free variable h. Evidently, if the system considered contains the predicate calculus, and if h occurs in A(h) only in parts of the form h(t) where t is a term which lies within the range of the quantifier (x), then 1.1 will be provable. But this will not be so in general; indeed, by introducing into the system an intensional predicate of predicates we can make 1.1 false. For example, Myhill introduces a constant S, where ‘Sϕψχω’ means that (the expression) ϕ is the result of substituting ψ for χ in ω.

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?

We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for which

We now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we define

The only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formula

T is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

Since 1934 various different techniques for natural deduction have been developed by Gentzen, Jaśkowski, Rosser, Quine, and others (see [1], pp. 147–167; [2], especially footnotes 1, 3, and 4; and [3], pp. 75-83, 96-107, and 289-294). It has been pointed out to me by Professor Donald Kalish of U.C.L.A. that the restrictions placed upon Universal Generalization (UG) and Existential Instantiation (EI) in [3] force one to construct a less natural proof than seems desirable for such arguments as

(cf. [3], p. 139, and [1], pp. 175f.) It is my purpose in this note to formulate an alternative restriction on UG which will permit a more natural proof for such arguments, and to prove the consistency of the altered rule.

In the notation of [3], “The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added proviso that when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.” ([3], p. 100.) The statement of EI is relatively unrestricted; being

provided that ν is a variable which occurs free in no earlier step.” ([3], p. 104.)

It has been shown by Quine that an interpreted theory Θ, formulated in the notation of quantification theory, is translatable into a theory Θ′ in which the only primitive predicate is a dyadic F. In establishing this result Quine takes for the universe of Θ′ a set which comprehends the universe of Θ and has the property that if x and y are members then so is {x, y}. For this fragmentary set theory the distinctness of x from {x, y} and {{x}} is assumed. It will be shown here that, if a pair of somewhat more stringent restrictions are assumed for the set theory, then there is a symmetric dyadic predicate G definable within Θ′, i.e., in terms of F alone, in terms of which F is in turn definable. It follows from this extended result that the theory Θ is translatable into a theory in which the only primitive predicate is symmetric and dyadic.

In the first of her papers on functional calculi based on strict implication, Ruth Barcan Marcus takes as her starting point the Lewis systems S2 and S4, supplemented by one of the normal bases for quantification theory, and by one special axiom for the mixture, asserting that if possibly something φ's then something possibly φ's. In the symbolism of Łukasiewicz, which will be used here, this axiom is expressible as CMΣxφxΣxMφx. In the present note I propose to show that if S5 had been taken as a startingpoint rather than S2 or S4, this formula need not have been laid down as an axiom but could have been deduced as a theorem.

It has been shown by Gödel that a system equivalent to S5 may be obtained if we add to any complete basis for the classical propositional calculus a pair of symbols for ‘Necessarily’ and ‘Possibly,’ which here will be ‘L’ and ‘M’; the axioms

the rule

RL: If α is a thesis, so is Lα;

and the definition

Df. M: M = NLN.

In the present paper we investigate in some detail two systems closely related to those contained in [4]. The system K1 of [4] is simplified by eliminating conjunction and disjunction, giving rise to a system Σ whose primitives are two individuals, concatenation, identity, existential and limited universal quantification. Any relation expressible in Σ may be expressed in the form (Eα1)(β)α1(Eα2)…(Eαn)(γ = δ). A further reduction of primitives leads to a system Σμ, which differs from Σ by containing the operator μ (i.e. “the first string, such that” with respect to a certain ordering) in place of the quantifiers. Reasons are given for restricting Myhill's definition of constructive ideas to those expressible in Σμ.

Acquaintance with [4], [5], [6] is assumed. We deviate from [4] by dropping concatenation-signs and parentheses in chain matrices from the official notation, and, furthermore, by using Greek and German letters as metalinguistic variables standing for chain matrices and chains, respectively, while retaining small Roman letters (excluding a, b) for use as variables of the systems.