Kripke Semantics - History and Terminology

History and Terminology

Kripke semantics does not originate with Kripke, but instead the idea of giving semantics in the style given above, that is based on valuations made that are relative to nodes, predates Kripke by a long margin:

• Rudolf Carnap seems to have been the first to have the idea that one can give a possible world semantics for the modalities of necessity and possibility by means of giving the valuation function a parameter that ranges over Leibnizian possible worlds. Bayart develops this idea further, but neither gave recursive definitions of satisfaction in the style introduced by Tarski;
• J.C.C. McKinsey and Alfred Tarski developed an approach to modeling modal logics that is still influential in modern research, namely the algebraic approach, in which Boolean algebras with operators are used as models. Bjarni Jónsson and Tarski established the representability of Boolean algebras with operators in terms of frames. If the two ideas had been put together, the result would have been precisely frame models, which is to say Kripke models, years before Kripke. But no one (not even Tarski) saw the connection at the time.
• Arthur Prior, building on unpublished work of C. A. Meredith, developed a translation of sentential modal logic into classical predicate logic that, if he had combined it with the usual model theory for the latter, would have produced a model theory equivalent to Kripke models for the former. But his approach was resolutely syntactic and anti-model-theoretic.
• Stig Kanger gave a rather more complex approach to the interpretation of modal logic, but one that contains many of the key ideas of Kripke's approach. He first noted the relationship between conditions on accessibility relations and Lewis-style axioms for modal logic. Kanger failed, however, to give a completeness proof for his system;
• Jaakko Hintikka gave a semantics in his papers introducing epistemic logic that is a simple variation of Kripke's semantics, equivalent to the characterisation of valuations by means of maximal consistent sets. He doesn't give inference rules for epistemic logic, and so cannot give a completeness proof;
• Richard Montague had many of the key ideas contained in Kripke's work, but he did not regard them as significant, because he had no completeness proof, and so did not publish until after Kripke's papers had created a sensation in the logic community;
• Evert Willem Beth presented a semantics of intuitionistic logic based on trees, which closely resembles Kripke semantics, except for using a more cumbersome definition of satisfaction.

Though the essential ideas of Kripke semantics were very much in the air by the time Kripke first published, Saul Kripke's work on modal logic is rightly regarded as ground-breaking. Most importantly, it was Kripke who proved the completeness theorems for modal logic, and Kripke who identified the weakest normal modal logic.

Despite the seminal contribution of Kripke's work, many modal logicians deprecate the term Kripke semantics as disrespectful of the important contributions these other pioneers made. The other most widely used term possible world semantics is deprecated as inappropriate when applied to modalities other than possibility and necessity, such as in epistemic or deontic logic. Instead they prefer the terms relational semantics or frame semantics. The use of "semantics" for "model theory" has been objected to as well, on the grounds that it invites confusion with linguistic semantics: whether the apparatus of "possible worlds" that appears in models has anything to do with the linguistic meaning of modal constructions in natural language is a contentious issue.