A logical measure function

Abstract

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.

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