Accessibility relation

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" formula_1 can depend on what's true at another possible world formula_2, but only if the accessibility relation formula_3 relates formula_1 to formula_2. For instance, if formula_6 holds at some world formula_2 such that formula_8, the formula formula_9 will be true at formula_1. The fact formula_8 is crucial. If formula_3 did not relate formula_1 to formula_2, then formula_9 would be false at formula_1 unless formula_6 also held at some other world formula_18 such that formula_19.
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 was 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 formula_8 iff formula_2 is compatible with the information that's available to the speaker in formula_1.
This idea can be extended to different applications of modal logic. In epistemology, one can use an epistemic notion of accessibility where formula_8 for an individual formula_24 iff formula_24 does not know something which would rule out the hypothesis that formula_26. In deontic modal logic, one can say that formula_8 iff formula_2 is a morally ideal world given the moral standards of formula_1. In application of modal logic to computer science, the so-called possible worlds can be understood as representing possible states and the accessibility relation can be understood as a program. Then formula_8 iff running the program can transition the computer from state formula_1 to state formula_2.
Different applications of modal logic can suggest different restrictions on admissible accessibility relations, which can in turn lead to different validities. The mathematical study of how validities are tied to conditions on accessibility relations is known as "modal correspondence theory".