“Kripke semantics”的概念、定义、翻译、参考文献-科学参考

Kripke semantics

Logic

A kind of semantics developed by Saul Kripke (1940– ) for modal and intuitionist logic. The distinctive feature of the semantics is that it deploys a set of possible worlds, and formulae are assigned truth values at each world. For modal logics, there is a binary accessibility relation, $R$ , which is deployed in the truth conditions for the modal operators, e.g.:
- • $□ φ$ is true at world $w$ iff for all worlds, $w^{'}$ , such that $w R w^{'}$ , $φ$ is true at $w^{'}$ .
Placing various constraints on $R$ generates a wide variety of modal logics. The machinery can be augmented by a collection of non-normal worlds to deliver semantics for an even wider range of modal logics. In intuitionist logic, $R$ is required to be transitive and reflexive, and is used in giving the truth conditions of the conditional, thus:
- • $φ \to ψ$ is true at world $w$ iff for all worlds $w^{'}$ such that $w R w^{'}$ where $φ$ is true at $w^{'}$ , so is $ψ$ .

Philosophy

The standard semantic treatment for modal languages with symbols for necessity and possibility, due to Saul Kripke. The model is a set of possible worlds and a relation on them, corresponding to the idea of one world being ‘accessible’ from another. A valuation function then evaluates sentences as true or false at worlds. Using these models, Kripke was able to bring new treatments to issues such as the decidability, independence, and completeness of different systems of modal logic, where different conditions on the accessibility relation correspond to different strengths of system.

单词	Kripke semantics
释义	Kripke semantics Logic A kind of semantics developed by Saul Kripke (1940– ) for modal and intuitionist logic. The distinctive feature of the semantics is that it deploys a set of possible worlds, and formulae are assigned truth values at each world. For modal logics, there is a binary accessibility relation, $R$ , which is deployed in the truth conditions for the modal operators, e.g.: • $□ φ$ is true at world $w$ iff for all worlds, $w^{'}$ , such that $w R w^{'}$ , $φ$ is true at $w^{'}$ . Placing various constraints on $R$ generates a wide variety of modal logics. The machinery can be augmented by a collection of non-normal worlds to deliver semantics for an even wider range of modal logics. In intuitionist logic, $R$ is required to be transitive and reflexive, and is used in giving the truth conditions of the conditional, thus: • $φ \to ψ$ is true at world $w$ iff for all worlds $w^{'}$ such that $w R w^{'}$ where $φ$ is true at $w^{'}$ , so is $ψ$ . Philosophy The standard semantic treatment for modal languages with symbols for necessity and possibility, due to Saul Kripke. The model is a set of possible worlds and a relation on them, corresponding to the idea of one world being ‘accessible’ from another. A valuation function then evaluates sentences as true or false at worlds. Using these models, Kripke was able to bring new treatments to issues such as the decidability, independence, and completeness of different systems of modal logic, where different conditions on the accessibility relation correspond to different strengths of system.
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