Accessibility relation explained

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

w can depend on what's true at another possible world v , but only if the accessibility relation R relates w to v . For instance, if P holds at some world v such that wRv , the formula \Diamond P will be true at w . The fact wRv is crucial. If R did not relate w to v , then \Diamond P would be false at w unless P also held at some other world u such that wRu .^{[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

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

This idea can be extended to different applications of modal logic. In epistemology, one can use an epistemic notion of accessibility where

wRv for an individual I iff I does not know something which would rule out the hypothesis that w' = v . In deontic modal logic, one can say that wRv iff v is a morally ideal world given the moral standards of w . 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 wRv iff running the program can transition the computer from state w to state v .

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.

See also

• Modal logic
• Possible worlds
• Propositional attitude
• Modal depth

References

• Gerla, G.; Transformational semantics for first order logic, Logique et Analyse, No. 117–118, pp. 69–79, 1987.
• Fitelson, Brandon; Notes on "Accessibility" and Modality, 2003.
• Brown, Curtis; Propositional Modal Logic: A Few First Steps, 2002.
• Kripke, Saul; Naming and Necessity, Oxford, 1980.
• 2024555. Counterpart Theory and Quantified Modal Logic. Lewis. David K.. The Journal of Philosophy. 1968. 65. 5. 113–126. 10.2307/2024555.
• Book: Gasquet. Olivier. Kripke's Worlds: An Introduction to Modal Logics via Tableaux. 2013. Springer . 978-3764385033. 14–16. 23 July 2020. etal.
• List of Logic Systems List of most of the more popular modal logics.

Notes and References

1. Book: Blackburn . de Rijke . Maarten . Venema. Yde . 2001 . Modal Logic . Cambridge Tracts in Theoretical Computer Science. 9780521527149 .
2. Book: van Benthem, Johan . 2010 . Modal Logic for Open Minds . https://web.archive.org/web/20200219165057/https://pdfs.semanticscholar.org/9bea/866c143326aeb700c20165a933f583b16a46.pdf . dead . 2020-02-19 . CSLI. 62162288 .