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Accessibility Relation

Defines Accessibility Relation, accessibility relations

Let WW be a set (of possible worlds).

Definition. An accessibility relation on WW is a binary relation R⊆W×WR \subseteq W \times W. We write wRvwRv to mean (w,v)∈R(w, v) \in R, read as “vv is accessible from ww.”

The accessibility relation determines the semantics of the modal operators. In a Kripke model (W,R,V)(W, R, V):

  • Necessity: □φ\Box\varphi holds at ww iff φ\varphi holds at every vv with wRvwRv.
  • Possibility: ◊φ\Diamond\varphi holds at ww iff φ\varphi holds at some vv with wRvwRv.

Proposition (Frame correspondence). The properties of RR determine which modal system the Kripke frame (W,R)(W, R) validates:

Property of RR Axiom validated
Reflexive (∀w. wRw\forall w.\, wRw) T: □φ→φ\Box\varphi \to \varphi
Transitive (wRv∧vRu⇒wRuwRv \wedge vRu \Rightarrow wRu) 4: □φ→□□φ\Box\varphi \to \Box\Box\varphi
Symmetric (wRv⇒vRwwRv \Rightarrow vRw) B: φ→□◊φ\varphi \to \Box\Diamond\varphi
Euclidean (wRv∧wRu⇒vRuwRv \wedge wRu \Rightarrow vRu) 5: ◊φ→□◊φ\Diamond\varphi \to \Box\Diamond\varphi

Proof sketch. Each direction is a direct semantic argument. For example, if RR is reflexive and □φ\Box\varphi holds at ww, then wRwwRw gives φ\varphi at ww. Conversely, if RR fails reflexivity at some ww (i.e. ¬wRw\neg wRw), choose VV making φ\varphi true everywhere except ww to falsify T at ww. □\square

Proposition. Combining properties gives the standard systems: reflexive + transitive (preorder) gives S4; equivalence relation gives S5.

Examples.

  • The total relation R=W×WR = W \times W makes every world accessible from every other; the resulting frame validates S5.
  • The identity relation R={(w,w):w∈W}R = \{(w, w) : w \in W\} collapses □φ\Box\varphi to φ\varphi; the resulting modal logic is non-modal.
  • A strict partial order (irreflexive, transitive) models temporal “earlier-than” in tense logic.

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Cite

@misc{emsenn2025-accessibility-relation,
  author    = {emsenn},
  title     = {Accessibility Relation},
  year      = {2025},
  url       = {https://emsenn.net/library/math/domains/logic/terms/accessibility-relation/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}