Modal Logic

Abstract

Modal operators, like ‘it is necessary that’ or ‘John knows that’, express an attitude about the proposition to which they are applied. Modal logic studies the reasoning in modal contexts, extending classical logic in which only connectives and quantifiers are taken into account. There are many systems of modal logic, depending on the axioms one wants to accept for the modal operators. The semantics of the modal operators is in terms of possible worlds, where each possible world is supposed to satisfy classical logic. A proposition is necessarily true if it is true in every world accessible or imaginable from the given world. Also tableaux rules are available for the different systems of modal logic. Constructing a tableau-deduction in modal propositional logic of a formula from given premisses, if it exists, is straightforward; and if it does not exist, one easily constructs a counterexample from a failed attempt to construct one. Epistemic logic is about the modal operator ‘knowing that’ and an interesting puzzle in this field is the one of the muddy children. The possible world semantics is useful to understand a number of phenomena in the philosophy of language: rigid designators and the ‘de dicto - de re’ distinction. Also strict implication and counterfactuals may be understood in terms of possible world semantics. In modal predicate logic we study the behavior of modal operators in combination with the quantifiers.We shall see that in order to make sense, modal contexts should be referentially transparent and at the same time extensionally opaque.