Accessibility relation

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A simple Kripke model with only three possible worlds, viz. u, v, w. Since the accessibility relation relates w to v and is true at v, the formula is true at w. Since u is not accessible from w, the fact that is true at u does not lead to be true at w.

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 can depend on what's true at another possible world , but only if the accessibility relation relates to . For instance, if holds at some world such that , the formula will be true at . The fact is crucial. If did not relate to , then would be false at unless also held at some other world such that .[1][2]

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 iff is compatible with the information that's available to the speaker in .

This idea can be extended to different applications of modal logic. In epistemology, one can use an epistemic notion of accessibility where for an individual iff does not know something which would rule out the hypothesis that . In deontic modal logic, one can say that iff is a morally ideal world given the moral standards of . 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 iff running the program can transition the computer from state to state .

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.

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  1. ^ Blackburn, Patrick; de Rijke, Maarten; Venema, Yde (2001). Modal Logic. Cambridge Tracts in Theoretical Computer Science. ISBN 9780521527149.
  2. ^ van Benthem, Johan (2010). Modal Logic for Open Minds (PDF). CSLI. S2CID 62162288. Archived from the original (PDF) on 2020-02-19.