K Modal Logic

Origin. Kripke (1959, 1963). Minimal normal modal. No frame conditions. Weakest normal system. Foundation of modal semantics.

Models. Arbitrary accessibility. No special conditions. Kripke frames. Distribution and necessitation only.

Formalism.

Language: Propositional + □, ◇. □A: necessarily A. ◇A := ¬□¬A: possibly A.

Axioms: All propositional tautologies. K: □(A → B) → (□A → □B). Distribution axiom.

Rules: Modus ponens. Necessitation: from ⊢ A, infer ⊢ □A. Theorems are necessary.

Semantics: Frame F = ⟨W, R⟩. W: worlds. R: accessibility (arbitrary). No conditions on R.

Truth: M, w ⊨ □A iff for all v: wRv implies M, v ⊨ A. M, w ⊨ ◇A iff for some v: wRv and M, v ⊨ A.

What K lacks: Not □A → A (T). Not □A → □□A (4). Not ◇A → □◇A (5). Minimal assumptions.

Theorem examples: □(A ∧ B) ↔ □A ∧ □B. □A → ◇A invalid. ◇(A ∨ B) ↔ ◇A ∨ ◇B.

Extensions: KT: + reflexive. K4: + transitive. KB: + symmetric. S5 = KT45.

Symbols.

Metatheory. Minimal normal. Arbitrary frames. Completeness. Decidable.

Applies to. Modal logic foundation. Provability. Knowledge. Temporal reasoning.

Limitations. Too weak alone. Needs extensions. Abstract accessibility. No intuitive reading fixed.