Open Futures in the Foundations of Propositional Logic

Abstract

AbstractThis chapter weaves together two themes in the work of Nuel Belnap. The earlier theme was to propose conditions (such as conservativity and uniqueness) under which logical rules determine the meanings of the connectives they regulate. The later theme was the employment of semantics for the open future in the foundations of logics of agency. This chapter shows that on the reasonable criterion for fixing meaning of a connective by its rule governed deductive behavior, the natural deduction rules for classical propositional logic do not fix the interpretation embodied in the standard truth tables, but instead express an open future semantics related to Kripke’s possible worlds semantics for intuitionistic logic, called natural semantics. The basis for this connection has already been published, but this chapter reports new results on disjunction, and explores the relationships between natural semantics and supervaluations. A possible complaint against natural semantics is that its models may disobey the requirement that there be no branching in the past. It is shown, however, that the condition may be met by using a plausible reindividuation of temporal moments. The chapter also explains how natural semantics may be used to locate what is wrong with fatalistic arguments that purport to close the door on a open future. The upshot is that the open future is not just essential to our idea of agency, it is already built right into the foundations of classical logic.KeywordsNatural deduction Propositional logicLogical rulesMeaningSupervaluationNatural semanticsFatalismInferentialism