MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

30 Robert Goldblattwhere α ′ is any formula differing from α only in having free occurrences of y insome places where α has free occurrences of x. Taking α to be the valid ✷(x ≈ x),this allows derivation ofx ≈ y → ✷(x ≈ y),which is arguably invalid. For example, it is an astronomical fact that the MorningStar and the Evening Star are the same object (Venus), but this equality is not anecessary truth.Kanger pointed out that his new semantics for quantification and modality madeit possible to “recognize and explain the error in the Morning Star paradox”: theprinciple of substitutivity of equals is not valid without restriction, but only in theweaker form✷(x ≈ y) → (α → α ′ ).Jaakko Hintikka [1969] later expressed the opinion that this discussion by Kangerof the Morning Star paradox willremain a historical landmark as the first philosophical application of an explicitsemantical theory of quantified modal logic.4.6 MontagueKanger’s quaternary relation R i might equally well be viewed as a binary relation(r ′ , V ′ ) R i (r, V ) between systems. Such a notion appears in a paper by RichardMontague [1960] which was originally presented to a philosophy conference at theUniversity of California, Los Angeles, in May of 1955. Montague did not initiallyplan to publish the paper because “it contains no results of any great technicalinterest”, but eventually changed his mind after the appearance of Kanger’s andKripke’s ideas.The aim of the paper is to interpret logical and physical necessity, and the deonticmodality “it is obligatory that”, and to relate these to the use of quantifiers.Tarski’s model theory for first-order languages is employed for this purpose: amodel is taken to be a structure M = (D, R, f) where D is a domain of individuals,R a function fixing an interpretation of individual constants and finitarypredicates in D in the now-familiar way, and f is an assignment of values inD to individual variables. Montague uses these models to provide a semanticsfor formulas that are constructible from atomic first-order formulas by using thepropositional connectives and ✷, but not quantifiers. 30 His approach is to take arelation X between such models, and then inductively defineM satisfies ✷α iff for every model M ′ such that M XM ′ , M ′ satisfies α.His first example shows that the Tarskian semantics for ∀ fits this definition.Taking X to be the relation Q x specified by30 Montague uses several symbols for various kinds of modality, but ✷ will suffice here.