Kripke semantics in the context of "Modal logic"

Saul Aaron Kripke (/ˈkrɪpki/; November 13, 1940 – September 15, 2022) was an American analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. From the 1960s until his death, he was a central figure in a number of fields related to mathematical and modal logic, philosophy of language and mathematics, metaphysics, epistemology, and recursion theory.

Kripke made influential and original contributions to logic, especially modal logic. His principal contribution is a semantics for modal logic involving possible worlds, now called Kripke semantics. He received the 2001 Schock Prize in Logic and Philosophy.

An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a possible world ${\displaystyle w}$ can depend on what is true at another possible world ${\displaystyle v}$, but only if the accessibility relation ${\displaystyle R}$ relates ${\displaystyle w}$ to ${\displaystyle v}$. For instance, if ${\displaystyle P}$ holds at some world ${\displaystyle v}$ such that ${\displaystyle wRv}$, the formula ${\displaystyle \Diamond P}$ will be true at ${\displaystyle w}$. The fact ${\displaystyle wRv}$ is crucial. If ${\displaystyle R}$ did not relate ${\displaystyle w}$ to ${\displaystyle v}$, then ${\displaystyle \Diamond P}$ would be false at ${\displaystyle w}$ unless ${\displaystyle P}$ also held at some other world ${\displaystyle u}$ such that ${\displaystyle wRu}$.

Accessibility relations are motivated conceptually by the fact that natural language modal statements depend on some, but not all, alternative scenarios. For instance, the sentence "It might be raining" is not generally judged true simply because one can imagine a scenario where it is raining. Rather, its truth depends on whether such a scenario is ruled out by available information. This fact can be formalized in modal logic by choosing an accessibility relation such that ${\displaystyle wRv}$ if ${\displaystyle v}$ is compatible with the information that is available to the speaker in ${\displaystyle w}$.