RULES FOR SUBATOMIC DERIVATION

In proof-theoretic semantics the meaning of an atomic sentence is usually determined by a set of derivations in an atomic system which contain that sentence as a conclusion (see, in particular, Prawitz, 1971, 1973). The paper critically discusses this standard approach and suggests an alternative account which proceeds in terms of subatomic introduction and elimination rules for atomic sentences. A simple subatomic normal form theorem by which this account of the semantics of atomic sentences and the terms from which they are composed is underpinned, shows moreover that the proof-theoretic analysis of first-order logic can be pursued also beneath the atomic level.

On supervaluations in free logic

The concepts “classical valuation” and “supervaluation” were introduced by van Fraassen around 1966, to provide a semantic analysis of the then extant axiomatic systems of free logic. Consider an atomic sentenceand a “partial” model which fails to interpret c. Then (1) has no truth value in , nor doesWhile the valuelessness of (1) was found intuitively acceptable, that of (2) was not. Indeed, (2) and all other tautologies are theorems of free logic.Van Fraassen found a way to accommodate both intuitions. He interprets the unproblematic atomic sentences as usual, while “interpreting” those like (1) by simply assigning them a truth-value in arbitrary fashion. Then a truth-value for every sentence can be defined in the usual way; the result van Fraassen calls a “classical valuation” of the language. The arbitrary element in any given classical valuation is then eliminated by passage to the “supervaluation” over , which agrees with the classical valuations where they agree among themselves, and otherwise is undefined. In the supervaluation over , (1) is valueless but (2) true (since true on all classical valuations), as was required.There is a slight, but crucial oversimplification in the preceding account. Evaluation of the sentencerequires prior evaluation of the open formulaBut here mere assignment of truth-value is not enough; a whole set must be arbitrarily assigned as extension. The quantification over classical valuations involved in passage to the supervaluation thus involves an implicit quantification over subsets of the domain of : supervaluations are second order.

A logical measure function

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.