Kripke model

English

Etymology

Named after Saul Kripke.

Noun

Kripke model (plural Kripke models)

(logic) A Kripke frame together with either one of the following: (1) a function associating each of the frame's worlds to a set of prime formulae which are "true" for the given world, (2) a function associating each prime formula to a set of worlds for which the prime formula is "true", (3) a forcing relation between worlds and prime formulae. Additionally, there is a set of rules for deducing (from the given function or relation) what formulae are forced to be true by a given world. (The set of rules depends on which logic the Kripke model is being applied to, whether one of several modal logics or intuitionistic logic).
A formula is intuitionistically valid iff it is forced true by every world of every Kripke model.
A terminal world of a Kripke model (for intuitionistic logic) has a forcing relation equivalent to a classical model/interpretation.
If a given world (in a Kripke model (for intuitionistic logic)) forces neither ${\\displaystyle A}$ nor ${\\displaystyle \eg A}$ then there is some possible "future" world (accessible from the "present" one) in which ${\\displaystyle A}$ is forced true. In particular, if ${\\displaystyle A}$ eventually becomes forced somewhere along any possible time thread (towards the "future"), then the present world would force ${\\displaystyle \eg \eg A}$ to be true, but if this does not happen there there exists some possible future world in which ${\\displaystyle \eg A}$ becomes true.

Usage notes

For intuitionistic logic the forcing relation satisfies a persistence relation, namely, if world w forces proposition p, then all worlds accessible from w also force p.

单词	Kripke model
释义	Kripke model English Etymology Named after Saul Kripke. Noun Kripke model (plural Kripke models) (logic) A Kripke frame together with either one of the following: (1) a function associating each of the frame's worlds to a set of prime formulae which are "true" for the given world, (2) a function associating each prime formula to a set of worlds for which the prime formula is "true", (3) a forcing relation between worlds and prime formulae. Additionally, there is a set of rules for deducing (from the given function or relation) what formulae are forced to be true by a given world. (The set of rules depends on which logic the Kripke model is being applied to, whether one of several modal logics or intuitionistic logic). A formula is intuitionistically valid iff it is forced true by every world of every Kripke model. A terminal world of a Kripke model (for intuitionistic logic) has a forcing relation equivalent to a classical model/interpretation. If a given world (in a Kripke model (for intuitionistic logic)) forces neither ${\\displaystyle A}$ nor ${\\displaystyle \eg A}$ then there is some possible "future" world (accessible from the "present" one) in which ${\\displaystyle A}$ is forced true. In particular, if ${\\displaystyle A}$ eventually becomes forced somewhere along any possible time thread (towards the "future"), then the present world would force ${\\displaystyle \eg \eg A}$ to be true, but if this does not happen there there exists some possible future world in which ${\\displaystyle \eg A}$ becomes true. Usage notes For intuitionistic logic the forcing relation satisfies a persistence relation, namely, if world w forces proposition p, then all worlds accessible from w also force p.
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