A Philosophical Conception of Propositional Modal Logic
Edward N. Zalta · Philosophical Topics · 1993
The definitions of propositional modal logic are traditionally formulated in the following way. 1 First, a formal language is defined, usually with atomic formulas p,q,...and complex formulas involving the connectives ¬, → and the ✷ operator. Models M for this language are then defined as triples 〈W, R, V 〉 in which W is a nonempty set of worlds, R is an accessibility relation, and V is a valuation function that maps each atomic sentence of the formal language to a set of worlds. Truth (at a world, in a model), validity, and logical consequence are then defined as semantic properties of, or relations among, sentences of the language. Finally, a proof theory is developed so that the consequences of (sets of) sentences may be derived. In recent developments of this proof theory, rules of inference are conceived as relations between sentences, and a logic Σ is defined to be a set of sentences closed under certain rules. With such a framework of definitions, modal logicians then investigate metatheoretic questions, such as whether sentences valid in certain models are theorems