By a logical measure (for a given language) we mean a syntactically defined function which associates some value with each well-formed formula of the language. Various such logical measures have played a fundamental role in the development of Logic and the Philosophy of Science. The purpose of this paper is to define a logical measure which has much wider applications than measures so far studied.

The new measure has two fundamental advantages, which will be referred to throughout the paper. First of all it can be applied to more (and richer) languages than the older measures. Most of the measures now in use are restricted to the first-order functional calculus, and frequently even to a first-order calculus with one-place predicates only. The measure to be defined will be applicable to richer languages as well, e.g., to functional calculi of all finite orders. But even as far as the first-order functional calculus is concerned, the new measure has a great advantage: We do not have to require that the atomic sentences of the calculus be independent. The previous measures depended in their construction on the requirement that “the basic statements of the language express independent facts.”

By an atomic sentence we mean a well-formed formula formed by applying an h-place primitive predicate to h individuals (a well-formed formula no part of which is well-formed); by a permissible conjunction we mean a conjunction of atomic sentences and negations of other atomic sentences; and the requirement of independence is that an atomic sentence (or its negation) is logically implied by a permissible conjunction only if it (its negation) is one of the components of the conjunction.

Axiomatizations of two systems of modal logic are presented in this paper. The first consists of six axiom schemata and one rule of inference; this axiomatization is proved equivalent to Lewis' S3. The addition of a seventh schema, the analogue of C10. 1, yields an axiomatization equivalent to S4. Our axiom schemata for S3 are proved mutually independent, as are our schemata for S4.

The program of “basic logic” can be summarized as follows:

I. To treat every syntactical system as a subclass of a certain fixed infinite classUof “U-expressions.” This can always be done by modifying in trivial ways the notation of each syntactical system which is not already such a subclass. As a result all syntactical systems become comparable with each other in the sense that they are merely different subclasses of a single class of expressions. The class U can be chosen in such a way as to be inductively definable thus in terms of a fixed symbol ‘σ’: (1) The symbol ‘σ’ is a U-expression. (2) The result of placing two U-expressions (or two occurrences of the same U-expression) next to each other, and enclosing this total expression within a pair of parentheses, is a U-expression.

II. To formulate a particular syntactical systemKwithin which every syntactical system (and indeedKitself) is “represented.” Such a system is here said to be a “basic system,” and an appropriate interpretation of it is said to be a “basic logic.” Within such a logic every finitary logic is definable, as well as the basic logic itself. Such a logic should be of fundamental importance, especially if it is so constructed as to be the weakest such logic and so contain no theorems that are not essential to its being basic.

With the above considerations in view, the system K has been defined in such a way that it is a subclass of U and is a basic syntactical system. A simpler definition of K will be given than heretofore in previous papers, and the minimum character of K will be made more clear.