Necessity and Possibility

Abstract

We give a basic introduction to modal logic. This includes possible world semantics, axiom systems, and quantification. Ideas and formal machinery are discussed, but all proofs (and meta-proofs) are omitted. Recommendations are given for those who want more. 15.1 Introduction Modal operators qualify truth in some way: necessary truth, knowable truth, provable truth, eventual truth, and so on. All these have many formal properties in common while, of course, differing on others. One can abstract these properties and study them for their own sake just as elementary algebra abstracts algebraic equations from natural language problems about weights, measures, distances, and ages. The idea in all cases is that abstraction should provide us with a simple setting in which the formal manipulation of symbols according to precise rules will lead us to results that can be applied back to the complex 'real' world in which the problems arose. If modal operators are many, what then formally constitutes a modal operator? We do not want to get into the infinite regress of philosophical debate here. A good working definition is, a modal operator is one we can investigate using the formal tools that have been developed for this purpose. Of course this is a time-dependent characterization-tools are human artifacts after all. Here we just consider the core of the subject, normal modal logics. These are the best understood using the simplest tools. They do not exhaust the subject. Modal operators come in dual pairs. Dual to necessity is possibility: X is possibly true if it is not necessary that not-X is true, and X is necessarily true if it is not M. Fitting () Professor emeritus,