Kripke semantics a type of formal semantics for languages with operators A and B for necessity and possibility (‘possible worlds semantics’ and ‘relational semantics’ are sometimes used for the same notion); also, a similar semantics for intuitionistic logic. In a basic version a frame for a sentential language with A and B is a pair (W,R) where W is a non-empty set (the ‘possible worlds’) and R is a binary relation on W – the relation of ‘relative possibility’ or ‘accessibility.’ A model on the frame (W,R) is a triple (W,R,V), where V is a function (the ‘valuation function’) that assigns truth-values to sentence letters at worlds. If w 1 W then a sentence AA is true at world w in the model (W,R,V) if A is true at all worlds v 1 W for which wRv. Informally, AA is true at world w if A is true at all the worlds that would be possible if w were actual. This is a generalization of the doctrine commonly attributed to Leibniz that necessity is truth in all possible worlds. A is valid in the model (W,R,V) if it is true at all worlds w 1 W in that model. It is valid in the frame (W,R) if it is valid in all models on that frame. It is valid if it is valid in all frames. In predicate logic versions, a frame may include another component D, that assigns a non-empty set Dw of objects (the existents at w) to each possible world w. Terms and quantifiers may be treated either as objectual (denoting and ranging over individuals) or conceptual (denoting and ranging over functions from possible worlds to individuals) and either as actualist or possibilist (denoting and ranging over either existents or possible existents). On some of these treatments there may arise further choices about whether and how truth-values should be assigned to sentences that assert relations among non-existents.
The development of Kripke semantics marks a watershed in the modern study of modal systems. In the 1930s, 1940s, and 1950s a number of axiomatizations for necessity and possibility were proposed and investigated. Carnap showed that for the simplest of these systems, C. I. Lewis’s S5, AA can be interpreted as saying that A is true in all ‘state descriptions.’ Answering even the most basic questions about the other systems, however, required effort and ingenuity. In the late fifties and early sixties Stig Kanger, Richard Montague, Saul Kripke, and Jaakko Hintikka each formulated interpretations for such systems that generalized Carnap’s semantics by using something like the accessibility relation described above. Kripke’s semantics was more natural than the others in that accessibility was taken to be a relation among mathematically primitive ‘possible worlds,’ and, in a series of papers, Kripke demonstrated that versions of it provide characteristic interpretations for a number of modal systems. For these reasons Kripke’s formulation has become standard. Relational semantics provided simple solutions to some older problems about the distinctness and relative strength of the various systems. It also opened new areas of investigation, facilitating general results (establishing decidability and other properties for infinite classes of modal systems), incompleteness results (exhibiting systems not determined by any class of frames), and correspondence results (showing that the frames verifying certain modal formulas were exactly the frames meeting certain conditions on R). It suggested parallel interpretations for notions whose patterns of inference were known to be similar to that of necessity and possibility, including obligation and permission, epistemic necessity and possibility, provability and consistency, and, more recently, the notion of a computation’s inevitably or possibly terminating in a particular state. It inspired similar semantics for nonclassical conditionals and the more general neighborhood or functional variety of possible worlds semantics. The philosophical utility of Kripke semantics is more difficult to assess. Since the accessibility relation is often explained in terms of the modal operators, it is difficult to maintain that the semantics provides an explicit analysis of the modalities it interprets. Furthermore, questions about which version of the semantics is correct (particularly for quantified modal systems) are themselves tied to substantive questions about the nature of things and worlds. The semantics does impose important constraints on the meaning of modalities, and it provides a means for many philosophical questions to be posed more clearly and starkly. See also FORMAL SEMANTICS , MODAL LOGIC , NECESSITY , POSSIBLE WORLD. S.T.K.