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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.
^Blackburn, Patrick; de Rijke, Maarten; Venema, Yde (2001). Modal Logic. Cambridge Tracts in Theoretical Computer Science. ISBN 9780521527149.
^van Benthem, Johan (2010). Modal Logic for Open Minds(PDF). CSLI. S2CID 62162288. Archived from the original (PDF) on 2020-02-19.
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set, and R is a binary relation on W. Elements of W are called nodes or worlds, and R is known as the accessibilityrelation. A Kripke model is a triple...
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A} does not contain an isolated point Kripke semantics — a dense accessibilityrelation corresponds to the axiom ◻ ◻ A → ◻ A {\displaystyle \Box \Box A\rightarrow...
mathematical logic. Examples include in modal logic as an incestual accessibilityrelation,[clarification needed] and in lambda calculus as a manner of reduction...
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about their formal properties and their patterns of interaction. Accessibilityrelation Affect (linguistics) Attitude Belief Disposition Embedded clause...
accessible worlds where its antecedent holds. Warmbrōd's idea was that speakers will switch to a model with a more permissive accessibilityrelation in...
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Fx\to \Diamond \exists xFx} . If a frame is based on a symmetric accessibilityrelation, then the Barcan formula will be valid in the frame if, and only...
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underlying logic (which, in the case of SDL, is classical). The accessibilityrelation between possible world is interpreted as acceptability relations:...
believe p {\displaystyle p} "). This corresponds to having a dense accessibilityrelation in Kripke semantics, and any accurate reasoner is always stable...
role-bounded quantifier to a modal operator with that role as its accessibilityrelation. Operations on roles (such as composition, inversion, etc.) correspond...
refers to the accessibilityrelation that should be used: normally, ◻ {\displaystyle \Box } and ◊ {\displaystyle \Diamond } refer to a relation R {\displaystyle...
{\displaystyle W} and an accessibilityrelation ≤ {\displaystyle \leq } on W × W {\displaystyle W\times W} to the frames. The accessibilityrelation is required to...
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particular, the correspondence of modal axioms to properties on the accessibilityrelation is lost. This can be remedied by imposing additional conditions...
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