# Kripke semantics

**Kripke semantics**, also called possible-worlds semantics or relational semantics, is a formal semantics for modal logic in which formulas are evaluated at points called possible worlds, and a necessity operator is true at a world exactly when the embedded formula holds at every world accessible from it. A Kripke model consists of a non-empty set of worlds W, a binary accessibility relation R on W, and a valuation V assigning truth values to propositional variables at each world.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup> Since the 1960s, Kripke models have served as the basis of the most widely used semantics for modal logic.<sup>[2](https://plato.stanford.edu/entries/logic-epistemic/index.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A Kripke model is a triple M = ⟨W, R, V⟩: a non-empty set of possible worlds, a binary accessibility relation R on W, and a valuation V assigning a truth value to each propositional variable at each world; the pair F = ⟨W, R⟩ is the frame.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup> |
| Truth clause for necessity | M, w ⊨ □A iff for every w′ with wRw′, M, w′ ⊨ A; possibility ◇A is truth at some accessible world.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup><sup> • </sup><sup>[3](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)</sup> |
| Founding papers | "A Completeness Theorem in Modal Logic" (Journal of Symbolic Logic 24(1): 1–14, 1959) proved completeness for S5 with quantifiers and equality; "Semantical Analysis of Modal Logic I" (1963) treated normal modal propositional calculi.<sup>[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/completeness-theorem-in-modal-logic1/2656899380CA2607B27A2E90DFEF7CFE)</sup><sup> • </sup><sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup> |
| Correspondence example | The axiom □A→A corresponds to reflexivity of R: its second-order translation ∀P∀x[∀y(Rxy→Py)→Px] collapses to ∀xRxx.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup> |
| Beyond modal logic | Kripke's 1965 possible-world semantics for intuitionistic predicate logic is sound and complete, and Kripke models for modal logic predated those for intuitionistic logic.<sup>[7](https://plato.stanford.edu/entries/logic-intuitionistic/)</sup> |
| Computing applications | Kripke structures are the standard model of systems in model checking with temporal logics such as CTL and LTL.<sup>[8](https://www.cs.cmu.edu/~15414/lectures/19-ctl.pdf)</sup> |

## Frames, models, and truth

The semantics separates two components. A **Kripke frame** is a pair ⟨W, R⟩ where W is a non-empty set of possible worlds and R is a binary accessibility relation; the relation records which worlds are "relevant" or possible relative to a given world.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup><sup> • </sup><sup>[3](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)</sup> A **Kripke model** adds a valuation V, a function from atomic propositions and worlds to truth values, so that V(p, w) = T means p is true at world w.<sup>[3](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)</sup>

Truth of a formula at a world is defined recursively. The propositional connectives behave as in ordinary propositional logic, evaluated at the current world. The modal clauses are:<sup>[3](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)</sup>

\[ M,w \models \Box\varphi \text{ iff for all } w' \text{ such that } wRw',\ M,w' \models \varphi \]

\[ M,w \models \Diamond\varphi \text{ iff there is } w' \text{ such that } wRw' \text{ and } M,w' \models \varphi \]

This definition makes the modal operators non-truth-functional: the truth value of □φ at w depends not only on φ's value at w but on its values at other worlds.<sup>[3](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)</sup> The accessibility relation is what lets a semantics restrict the range of worlds relevant to a necessity claim. In ordinary speech, saying that something is necessary does not require its truth in all possible worlds, only in a certain class of worlds the speaker has in mind; the relation R formalizes that restriction.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup> Different readings of the box operator supply different relations: in [deontic logic](https://www.edgepedia.org/deontic-logic), where □ reads "it is obligatory that", wRw′ holds just in case w′ is a morally acceptable variant of w.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup>

## Origin

[Saul Kripke](https://www.edgepedia.org/saul-kripke), an American philosopher and logician, did much of the technical work on modal logic in the late 1950s, partly as a high-school student.<sup>[9](https://collaborate.princeton.edu/en/publications/saul-kripke-naming-and-necessity/)</sup> His 1959 paper stated and proved a completeness theorem for the system S5 supplemented by first-order quantifiers and the sign of equality.<sup>[4](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/completeness-theorem-in-modal-logic1/2656899380CA2607B27A2E90DFEF7CFE)</sup> The paper "Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi" (1963) treated the normal propositional systems, and "Semantical Considerations on Modal Logic" (1963) extended the framework.<sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup>

Kripke was not working alone in spirit. Kanger (1957), Montague (1960, originally presented in 1955), Hintikka (1961), and Prior (1957) all thought of a relation between worlds, and Hintikka, like Kripke in 1959, adopted a notion of validity requiring truth in all arbitrary sets of worlds.<sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup> What distinguished Kripke's version was that he characterized the worlds as simple points of evaluation, whereas other logicians still treated worlds fundamentally as models of first-order logic; this abstract characterization proved crucial in linking the model-theoretic semantics to the algebra of modal logic.<sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup> A key technical innovation was defining validity as truth across all subsets of a maximal structure of worlds rather than only in the maximal structure itself, which disconnected validity from necessity and made completeness proofs possible.<sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup> Kripke's completeness proof used Beth's method of semantic tableaux.<sup>[5](https://plato.stanford.edu/entries/logic-modal-origins/)</sup>

## Correspondence theory

Kripke semantics turns modal axioms into conditions on frames. Kripke's semantics provides a basis for translating modal axioms into sentences of a second-order language with quantification over one-place predicate letters: replace the metavariable A with an open sentence Px, translate □Px to ∀y(Rxy → Py), and close the free variables and predicate letters with universal quantifiers.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup> For the axiom schema □A→A this yields ∀P∀x[∀y(Rxy→Py) → Px]; instantiating P to the predicate Rx gives ∀x[∀y(Rxy→Rxy) → Rxx], which reduces to ∀xRxx, the condition that R is reflexive.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup> Similar results hold for many other axioms and frame conditions, and the collapse of second-order axiom conditions to first-order frame conditions underlies Sahlqvist's 1975 completeness results.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup>

In the 1970s, Johan van Benthem characterized exactly when such a collapse happens: the second-order translation of an axiom reduces to a first-order condition on R if and only if the translation's holding in a model entails its holding in any bisimular model, where two models are bisimular when there is a bisimulation between them. A bisimulation is a weaker notion than isomorphism, since it need not be one-to-one, but it suffices to preserve the truth values of modal formulas.<sup>[6](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)</sup>

## Completeness and its limits

The semantics supports completeness theorems of a uniform shape: a modal logic is sound and complete for a class of frames when its theorems are exactly the formulas valid on all frames in the class. The basic logic K is valid in all Kripke models.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup> Stronger logics correspond to restricted frame classes. In provability logic, the axiom (GL) corresponds to the condition that R is transitive and conversely well-founded, meaning there are no infinite ascending sequences w₁Rw₂Rw₃R…; K. Segerberg proved in 1971 that GL is complete with respect to transitive conversely well-founded frames, with D. de Jongh and S. Kripke proving the result independently, and Segerberg showed completeness even for finite transitive irreflexive trees. These theorems yield a decision procedure, and GL, like K, T, and S4, is decidable in the complexity class PSPACE and PSPACE-complete.<sup>[1](https://plato.stanford.edu/entries/logic-provability/)</sup>

Completeness does not hold everywhere. Work beginning with Makinson's 1969 example of a normal logic lacking the finite model property, extended by Dov Gabbay in 1972, led to the construction of a decidable modal logic that is finitely axiomatizable but incomplete with respect to Kripke frames.<sup>[10](https://www.sciencedirect.com/science/article/pii/S1570868303000089)</sup> Such Kripke-incomplete logics show that the relational semantics does not capture every normal modal system.

## Applications

The framework extends well past alethic modality, since any binary relation can serve as accessibility.

**Epistemic logic.** In the representation of knowledge, a widespread interpretation in theoretical economics and theoretical computer science understands knowledge in terms of informational indistinguishability between possible worlds, with an accessibility relation per agent; pointed Kripke models with indistinguishability relations are used to determine what an agent does or does not know.<sup>[2](https://plato.stanford.edu/entries/logic-epistemic/index.html)</sup>

Of all the interpretations given to intuitionistic systems, the one coming closest to classical model theory is the 1965 possible-world semantics of Kripke, for which intuitionistic predicate logic has been shown both sound and complete. A Kripke structure there consists of a partially ordered set of nodes with a domain function D assigning each node an inhabited set, monotone in the ordering, plus a forcing relation.<sup>[7](https://plato.stanford.edu/entries/logic-intuitionistic/)</sup>

**Quantified modal logic.** Kripke's 1963 paper proposed, as one strategy for retaining classical quantification, that all worlds share the same non-empty domain of individuals; many modal logicians instead loosen these fixed-domain strictures and adopt [free logics](https://www.edgepedia.org/free-logic), since fixed-domain semantics validates the Barcan formula and the claim that everything exists necessarily.<sup>[11](https://plato.stanford.edu/entries/logic-free/)</sup>

**Temporal logic and computer science.** Branching time frames were initially introduced and developed by Saul Kripke and Arthur Prior, and branching time semantics for tense-modal logics have been independently developed in philosophical logic and computer science.<sup>[12](https://plato.sydney.edu.au/entries/branching-time/index.html)</sup> In verification, the Kripke structure, a set of states with a transition relation and labels, is the standard model of a finite-state system against which CTL and LTL specifications are checked.<sup>[8](https://www.cs.cmu.edu/~15414/lectures/19-ctl.pdf)</sup> Interval temporal logics likewise receive Kripke-style semantics over Allen's interval relations.<sup>[13](https://plato.stanford.edu/entries/logic-temporal/)</sup>

## References

1. [Provability Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-provability/)
2. [Epistemic Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-epistemic/index.html)
3. [Logic textbook, chapter 11: Kripke frames and models (S. Luckelman, Durham University)](https://sluckelman.webspace.durham.ac.uk/wp-content/uploads/sites/218/2026/01/logicbook20260123.pdf)
4. [Saul A. Kripke, "A completeness theorem in modal logic", Journal of Symbolic Logic 24(1): 1–14, 1959](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/completeness-theorem-in-modal-logic1/2656899380CA2607B27A2E90DFEF7CFE)
5. [Modern Origins of Modal Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-modal-origins/)
6. [Modal Logic, Stanford Encyclopedia of Philosophy (Spring 2013 archive)](https://plato.stanford.edu/archives/spr2013/entries/logic-modal/)
7. [Intuitionistic Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-intuitionistic/)
8. [Lecture Notes on Computation Tree Logic, Carnegie Mellon University 15-414](https://www.cs.cmu.edu/~15414/lectures/19-ctl.pdf)
9. [John P. Burgess, "Saul Kripke: Naming and Necessity", in Central Works of Philosophy Volume 5, Acumen, 2010](https://collaborate.princeton.edu/en/publications/saul-kripke-naming-and-necessity/)
10. [Mathematical modal logic: A view of its evolution, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S1570868303000089)
11. [Free Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-free/)
12. [Branching Time, Stanford Encyclopedia of Philosophy](https://plato.sydney.edu.au/entries/branching-time/index.html)
13. [Temporal Logic, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/logic-temporal/)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Logical calculi and logical syntax › Modal and temporal logic › Modal correspondence and frame theory*

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