Modal logics

In this chapter we will consider the way in which possible worlds came to the aid of logicians working with modal logic. Modal logic is generally seen as the logic of possibility and necessity. Possible worlds have made formal modal logic quite clear and precise. In order to see possible worlds and modal logic in clearer perspective we will consider a little of the historical context.

Modern formal logic began with Frege's first-order logic. First-order logic is now seen in philosophy, mathematics, linguistics and computer science as the stepping-off point for virtually all work in logic. First-order logic includes classical propositional logic, which we looked at in Chapter 1, together with predicate logic. We have already said that we take first-order logic to be an artificial language that gives a precise and unambiguous account of logical concepts that are very like, but not exactly the same as, the logical concepts expressed in ordinary language.

We have seen that some of the logical concepts expressed in ordinary language are negation (standardly expressed with “not”), conjunction (“and”), disjunction (“or”) and implication (“if … then …”). We now turn to possibility and necessity. I have introduced special font expressions for the propositional logic operators used to translate these ordinary language logical operations: not for negation, and for conjunction, or for disjunction and imp for implication. Standard modal logic adds to these the diamond, ◊, for possibility and the box, □, for necessity.