The relational semantic framework that assigns truth values to formulas by evaluating them at worlds in a Kripke frame together with a valuation; modalities are interpreted via the accessibility relation connecting worlds.

Principle

Truth is world-relative and defined inductively on the syntax of formulas: propositional atoms are decided by the valuation at a world, Boolean connectives follow standard truth tables, and modal operators quantify over accessible worlds according to R (e.g., □φ is true at w iff φ is true at every v with wRv).

Demonstration

In modal propositional logic, a formula □(p→q)→(□p→□q) is evaluated at a world w by checking, for each v with wRv, whether the implication p→q holds at v; the frame property of transitivity or reflexivity may determine the formula's global validity across models based on that frame class.

Misapplication

Applying Kripke truth conditions unchanged to non-normal modal logics or to semantics with different structural primitives (e.g., neighborhood semantics), or reading the accessibility relation as a metaphysical relation of 'real possibility' without checking formal constraints.

Consequence

Kripke semantics yields clear notions of local truth, global validity, frame correspondence and completeness theorems; it provides a concrete method to relate syntactic axioms to semantic constraints on frames.

Reversal

Algebraic semantics (e.g., Boolean algebras with operators) or proof-theoretic semantics invert the emphasis by treating formulas via algebraic operations or inferential rules rather than explicit world-based evaluations.

Boundary

Covers modal, temporal and intuitionistic logics and variants where the world/ accessibility picture applies; it does not directly capture some substructural or paraconsistent logics without adaptation, nor does it replace algebraic or proof-theoretic perspectives though they are often equivalent in expressive power under translation.

Semantic Tension

Competes with algebraic and proof-theoretic accounts: Kripke semantics emphasizes concrete relational models and local satisfaction, while competing accounts emphasize algebraic structure or inferential roles; the tension centers on whether modality is best seen as relational accessibility or as algebraic/operational structure.

Synthesis

Kripke semantics is the practice of interpreting modal and related logics by evaluating formulas at worlds in frames equipped with an accessibility relation and valuation, thereby connecting syntactic axioms to semantic frame properties and enabling proofs of validity and correspondence.