Propositional Modal Logic

Abstract

We witnessed in the previous Chapter a system of basic modal logic constructed in the framework of a 4—valued logic. In addition to propositional connectives, studied by us in Chapter 2 on propositional logic, some other statements common in everyday speech and reasoning were studied there. These statements called also modal statements encompass such phrases as it is necessary that.., it is possible that.... These statements were interpreted semantically by means of truth functions of modal functors L (necessity) and M (possibility). We have seen that basic modal system have accomodated some basic postulates about modalities while rejecting some less natural postulates e.g. (S5): CMpLMp (what is possible is necessarily possible.) It is desirable to look for other systems with eventually distinct semantics which would accept some of expressions rejected by the basic modal system.

This brings us to the Spindle—destiny, spun according to the ancient by the Fates. [...]; the Fates, with Necessity, Mother of the Fates, manipulate it and spin at the birth of every being so that all comes into existence through Necessity

Plotinus, the Enneads, II.3.9