Modal operators

Modal logic is principally concerned with the alethic modalities of necessity and possibility, although this branch of logic is applied to a wide range of linguistic and conceptual phenomena, including natural language semantics, proof theory, theoretical computer science and the formal characterization of knowledge and belief. This wide range of application stems from the basic form of modal assertions, such as ‘it is necessarily the case that φ’, where an entire statement φ is embedded within a context possessing rich logical structure.

When constructing a formal representation of these embedding contexts, there are several choices concerning their specific symbolic form. The most standard approach symbolizes modal contexts as operators, which combine directly with formulas of the object language to yield new formulas. The primary alternative to this approach is to treat modal contexts as predicates, which attach not to formulas directly, but to names of formulas, and thereby attribute a metalinguistic property to a syntactic object. A variation on the operator approach, which assumes the interpretive framework of possible worlds semantics, is to treat modal contexts as quantifications over possible worlds. Finally, a variation on the predicate approach is to analyse modal contexts as predicates of propositions rather than as predicates of syntactic objects.