MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 35of the logics. To prove them would require showing in each case that a deductivelyconsistent formula is a member of some model set that belongs to a modelsystem of the appropriate kind, but again the issue of axioms and proof theoryis not taken up. The paper is mainly devoted to a discussion of the problem ofcombining modalities with quantifiers, and proposes various modifications on theclosure properties of Ω depending on whether it is required that whatever existsin a particular state of affairs should do so necessarily.4.8 The Place of KripkeThe earlier efforts to develop the seminal ideas of Kripke semantics have inevitablyraised questions of priority. In fact, as the above material is intended to show,the idea of using a binary relation to model modality occurred independently to anumber of people, and for different reasons, with Hintikka being the first to explainit in terms of conceivable alternatives to a given state of affairs. Kanger was thefirst to recognise the relevance of [Jónsson and Tarski, 1951] to modal logic, 33 andthe first to apply this kind of semantical theory to the resolution of philosophicalquestions about existence and identity.But it is only in Kripke’s writings that we see such seminal ideas developed intoan attractive model theory of sufficent power to fully resolve the long-standing issueof a satisfactory semantics for modality and of sufficient generality to advance thefield further. A fundamental point (mentioned in section 4.1) is that he was the firstto propose, and make effective use of, arbitrary set-theoretic structures as models.The methods of Hintikka, Kanger and Montague are all variations on the themeof a binary relation between models of the non-modal fragment of the predicatelanguages they use. Also, they did not present complete axiomatisations of theirsemantics. Kripke was the first to do this, and by allowing R to be any relationon any set K, he opened the door to all kinds of model constructions, which wererapidly provided by himself and then others. (His models for non-normal logicsappear to lack any historical antecedents.) It is due to his innovation that we nowhave a model theory for intensional logics.As already noted in section 4.2, Kripke developed his ideas independently. Hisanalysis of S5 was inititiated in 1956 when he was still at high-school (he turned16 years old on November 13th of that year). From the paper [Prior, 1956] helearned of the axioms for S5, and began to think of modelling that system bytruth tables with missing rows (see section 4.1). Early in 1957 E. W. Beth senthim his papers on the method of semantic tableaux, which provided Kripke witha technique for proving completeness theorems. By 1958 Kripke had worked outhis relational semantics for modal and intuitionistic systems, as announced in hisabstract [1959b] which was received by the editors on 25 August 1958. It wasthrough exploring different conditions connecting tableaux in order to model thedifferent subsystems of S5 that Kripke came to the idea of using a binary relation33 As Føllesdal [1994] emphasis.