This tag is for questions relating to "Kripke’s models" for modal logic (or variants thereof) are the basis for many modern approaches to reasoning about knowledge and belief. For philosophers, by far the most important examples are ‘Kripke models’, which have been adopted as the standard type of models for modal and related nonclassical logics.
Kripke model ( or Kripke semantics or relational semantics or frame semantics):
Given a nonempty set $~\mathcal P~$ of propositional letters and a finite nonempty set $~\mathcal A~$ of agents, a Kripke model is a structure $$M=(W,R,V)$$ consisting of
- a nonempty set $~W~$ of worlds identifying the possible states of affairs that might obtain,
- a function $~R:A→℘(W×W)~$ that assigns to each agent $~a~$ a binary possibility relation $~R_a⊆W×W~$ with $~wR_av~$ indicating that agent $~a~$ will entertain world $~v~$ as a candidate for the actual world whenever we assume that $~w~$ is in fact actual), and
- a propositional valuation $~V:\mathcal P→℘(W)~$ mapping each propositional letter $~p∈\mathcal P ~$to the set $~V(p)⊆W~$ of worlds at which that letter is true.
Notes:
- The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').
- The main defect of Kripke semantics is the existence of Kripke incomplete logics, and logics which are complete but not compact. It can be remedied by equipping Kripke frames with extra structure which restricts the set of possible valuations, using ideas from algebraic semantics. This gives rise to the general frame semantics.
References:
https://en.wikipedia.org/wiki/Kripke_semantics
