MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 31M Q x M ′ iff D = D ′ , R = R ′ and f and f ′ agree except on xgives ✷ the interpretation “for all x”. Thus quantification could be handled byassociating a modality with each variable, and Montague suggests that this shoulddispel Quine’s uneasiness about combining modality with quantification.The relationM LM ′ iff D = D ′ and f = f ′gives ✷α the interpretation “it is logically necessary that α”, meaning that α holdsno matter what its individual constants and predicates denote.To interpret physical necessity, Montague uses the idea that a statement isphysically necessary if it is deducible from some set of physical laws specified inadvance. This is formalised by fixing a set K of first-order ✷-free sentences andspecifying a relation P byM P M ′ iff D = D ′ , f = f ′ and M ′ is a model of K.Similarly, “it is obligatory that α” is taken to mean that α is deducible from someset of ethical laws specified in advance. This is formalised by fixing a class I ofideal models, those in which the constants and predicates mean what they oughtto according to these laws. Montague suggests as an example that I could bethe class of models which, in Tarski’s sense, satisfy the ten commandmentsformulated as declarative, rather than imperative, sentences.The deontic modality then corresponds to the model-relation E such thatM EM ′ iff D = D ′ , f = f ′ and M ′ belongs to I.If a model-relation X fulfills the conditionsfor all M there exists M ′ with M XM ′ ,M XM ′ and M ′ XM ′′ implies M XM ′′ ,M XM ′ and M XM ′′ implies M ′ XM ′′ ,(the last two mirror Kanger’s conditions) then every S5-theorem is valid, i.e. satisfiedby every model. Montague states that the converse is true, and that thereis a decision method for the class of formulas valid in this sense.4.7 HintikkaIf M is a model for predicate logic, of the kind used by Montague, let µ M be the setof all formulas that it satisfies. In Jaakko Hintikka’s approach to semantics, suchmodels M are in effect replaced by the sets µ M . These sets can be characterisedby their syntactic closure properties, obtained by replacing “M satisfies α” by“α ∈ µ M ” in the clauses of the inductive definition of satisfaction of formulas. Amodel set is defined as a set µ of formulas that has certain closure properties, suchas