The semantics of normal modal systems

Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 12)

The problem of endowing modal systems with appropriate semantics has been considered to be a difficult, or even impossible task, for decades. In their Symbolic Logic, Lewis and Langford did not formulate truth conditions for modal propositions, nor did they propose any decision procedure for their five systems. They were able to prove, however, in spite of S1 to S5 being ordered by an inclusion relation, that these systems are all distinct. The strategy they used to establish this factwas thematrixmethod.Amatrix M, as defined below (see Definitions 3.1.1 and 3.1.2), consists of a set of objects (usually sets, natural or rational numbers) insidewhich some special objects are selected as “distinguished” objects (which play the role of the value “true” in the class of truth-values).

A logical system S can be interpreted in amatrixM if we take the propositional variables of the wffs from this logic as ranging over the elements of the matrix and interpret the connectives and modal operators of S as operations in M. A wff α is then said to be verified by M if, for all interpretations of the atomic variables in α (also called valuations), the value of α is a distinguished value ofM. Otherwise, we say that α is falsified by the matrix. We also say that a system S is verified by a matrix M (or that M is a model for S) when all the theorems of S are verified by M. We say that S is characterized by M if the set of wffs of S verified by the matrix coincides with the set of theorems of S.


Modal Logic Relational Model Accessibility Relation Correspondence Theory Modal Formula 
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