4 - Valuation and satisfaction

Summary

Introduction

For each signature I we have now defined two quite different entities; the polymodal language of signature I, and the class of structures (labelled transition structures) of signature I. These two entities must now be made to interact. Thus the structures will be made to support a semantics for the language, or, equivalently, the language will be used to describe properties of structures.

The polymodal language has the usual propositional facilities together with a family of new 1-ary connectives [i], one for each label i. We now wish to evaluate each formula of this language, i.e. determine whether or not a formula φ is True or False. Of course, this can not be done in isolation, we need to work in an appropriate context. To determine the truth value of φ we need three pieces of information together with an agreed procedure for using the information.

1. We need to know the truth values of the variables appearing in φ. As in the propositional case this information will be conveyed by a valuation, however, these modal valuations are more complicated than the propositional versions.

2. We need to know how to handle the propositional connectives. This will be done in exactly the same way as the propositional language (i.e. using the defining truth tables of the connectives). In this sense, modal logic subsumes propositional logic.

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