S5 Modal Logic

Origin. Lewis (1932), Kripke (1963). Equivalence relations. Maximal normal modal. Metaphysical necessity. Foundation of possible worlds metaphysics.

Models. Universal accessibility within clusters. Equivalence frames. Modal realism. Possible worlds.

Formalism.

Axioms: K: □(A → B) → (□A → □B). T: □A → A. 5: ◇A → □◇A.

Alternative axiomatizations: KT5 = KTB4 = KT4 + B. B: A → □◇A. Multiple equivalent formulations.

Frame condition: R is equivalence relation. Reflexive, symmetric, transitive. Or: universal within components.

Characteristic formulas: ◇A → □◇A (5). ◇□A → □A. Reduces modality iterations.

Reduction laws: □□A ↔ □A. ◇◇A ↔ ◇A. □◇A ↔ ◇A. ◇□A ↔ □A. One modal operator suffices.

Metaphysical reading: □A: A is metaphysically necessary. ◇A: A is metaphysically possible. Leibniz: true in all possible worlds.

Semantic simplicity: All worlds accessible to all (in component). No accessibility distinctions. Just: some world vs all worlds.

Model theory: Propositional S5 = monadic predicate logic. One-variable fragment. Decidable, PSPACE-complete.

First-order S5: Complications with quantifiers. Barcan formula debates. Possibilism vs actualism.

Symbols.

Symbol	Unicode	Meaning
5	—	axiom ◇A → □◇A
B	—	axiom A → □◇A
S5	—	KT5
E	—	equivalence frame

Metatheory. Equivalence relations. Reduction. Metaphysical modality. Decidable.

Applies to. Metaphysics. Possible worlds. Essentialism. Logical necessity.

Limitations. Too strong for some readings. First-order complications. Accessibility distinctions lost. B axiom debated.