1 The System K : A Foundation for Modal Logic

Strictly speaking,modal logic studies reasoning that involves theuseof the expressions ‘necessarily’ and ‘possibly’. The main idea is to introduce the symbols∫ (necessarily) and∂ (possibly) to a system of logic so that it is able todistinguish threedifferentmodesof assertion:∫A(Aisnecessary), A (A is true), and ∂A (A is possible). Introducing these symbols (or operators) would seem to be essential if logic is to be applied to judging the accuracy of philosophical reasoning, for the concepts of necessity and possibility are ubiquitous in philosophical discourse. However, at the very dawn of the invention of modal logics, it was recognized that necessity and possibility have kinships with many other philosophically important expressions. So the term ‘modal logic’ is also usedmore broadly to cover a whole family of logics with similar rules and a rich variety of different operators. To distinguish the narrow sense, some people use the term ‘alethic logic’ for logics of necessity and possibility. A list describing some of the better known of these logics follows.