Deontic, Epistemic, and Temporal Modal Logics

Abstract

Modal ConceptsModalities (necessity, possibility, and related concepts and expressions) can be interpreted in different ways: for example, the possibility of a proposition or a state of affairs can be taken to mean that it is not ruled out by what is known (an epistemic interpretation) or believed (a doxastic interpretation), or that it is not ruled out by the accepted legal or moral requirements (a deontic interpretation), or that it has not always been or will not always be false (a temporal interpretation). These interpretations are sometimes contrasted with alethic modalities, which are thought to express the ways ('modes') in which a proposition can be true or false. For example, logical possibility and physical (real or substantive) possibility are alethic modalities.The basic modal concepts are represented in systems of modal logic as propositional operators; thus they are regarded as syntactically analogous to the concept of negation and other propositional connectives. The main difference between modal operators and other connectives is that the former are not truth-functional; the truth-value (truth or falsity) of a modal sentence is not determined by the truth-values of its subsentences. The concept of possibility ('it is possible that' or 'possibly') is usually symbolized by ‡ and the concept of necessity ('it is necessary that' or 'necessarily') by ᮀ; thus the modal formula ‡p represents the sentence form 'it is possible that p' or 'possibly p,' and ᮀp should be read 'it is necessary that p.' Modal operators can be defined in terms of each other: 'it is possible that p' means the same as 'it is not necessary that not-p'; thus ‡p can be regarded as an abbreviation of ÿᮀÿp, where ÿ is the sign of negation, and ᮀp is logically equivalent to ÿ ‡ÿp. Systems modal propositional logic or quantification theory (predicate logic) are obtained by adding the symbols ‡ and ᮀ (and possibly other modal signs), together with appropriate rules of sentence formation (e.g. if A is a formula, ‡A and ᮀA are formulas), to a system of (non-modal) propositional logic or quantification theory.