The logic and mathematics of occasion sentences

The prime purpose of this paper is, first, to restore to discourse-bound oc casion sentences their rightful central place in semantics and secondly, ta king these as the basic propositional elements in the logical analysis of l anguage, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus pr eparing the ground for more adequate semantic, logical and mathematical fou ndations of the study of natural language. Some of the insights elaborated in this paper have appeared in the literature over the past thirty years, a nd a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development i n semantics. In Section I it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a syste m of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and co ntext and thus belong to cognitive psychology rather than to logic. For sen tences to express a proposition and thus be interpretable and informative, they must first be properly keyed in the world, i.e. is about a situation i n the world. Section 2 deals with the logical properties of natural languag e. It argues that presuppositional phenomena require trivalence and present s the trivalent logic PPC3, withy two kinds of falsity and two negations. I t introduces the notion of Sigma -space for a sentence A (or /A/, the set o f situations in which A is true), as the basis of logical model theory, and notion of /P-A/ (the Sigma -space of the presuppositions of A), functionin g as a 'private' subuniverse for /A/. The trivalent Kleene calculus is rein terpreted as a logical account of vagueness, rather than of presupposition PPC3 as a truth-functional model of presupposition is considered more close ly and given a Boolean foundation. In a noncompositional extended Boolean a lgebra, three operators are defined: 1(a) for the conjoined presuppositions of a, a similar to for the complement of a within 1(a), and ($) over cap f or the complement of 1(a) within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possi ble models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of us ing the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and re fined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned.