Introduction to Philosophy/Logic/Modal Logic

Introduction to Philosophy > Logic > Modal Logic

Modality
"Still holding the copy of The Grasshopper Lies Heavy, Robert said, 'What sort of alternate present does this book describe?' Betty, after a moment, said, 'One in which Germany and Japan lost the war.' They were all silent. (Philip K. Dick, The Man in the High Castle)" This is the basic premise of modal logic: some things could have been otherwise. In our everyday reasoning, we distinguish between necessary truths and statements that just happen to be true but could have been false. We say that the latter are contingently true, or that their negation is possible. The concepts of necessary, contingent, possible and impossible are very closely related: something is necessary if it is impossible for it to be false, and something is possible if its falsehood is not necessary. Contingency is the negation of necessity, and impossibility the negation of possibility. Modal logic attempts to include the notions of modality (necessary, contingent, possible and impossible, among others) into the structure of classical logic (propositional logic and predicate calculus) and is therefore an extension of classical logic.

Syntax
Modal logic introduces three new symbols into classical logic: $$\Box$$(necessary),$$\Diamond$$ (possible) and $$\Rightarrow$$ (if, then). Formulae are constructed in the usual way, with the following rules added before the closure clause (see the rules for Well-Formed Formulae in propositional calculus):  If $$\alpha$$ is a well-formed formula, then $$\Box \alpha$$ and $$\Diamond \alpha$$ are also well-formed formulae. If $$\alpha$$ and $$\beta$$ are well-formed formulae, then $$\alpha \Rightarrow \beta$$ is also a well formed formula.  Actually, one can define the modal propositional calculus axiomatically using just $$\Box$$ or $$\Diamond$$. For example, if we wish to define $$\Diamond$$ and $$\Rightarrow$$ in terms of $$\Box$$, $$\Diamond \alpha$$ can be defined simply as a abbreviated notation for $$\neg \Box \neg \alpha$$ and $$\alpha \Rightarrow \beta$$ as another way of writing $$\Box (\alpha \rightarrow \beta)$$.