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A Kripke model is a poset of worlds with a persistent valuation of atoms. Conjunction and disjunction are forced pointwise, while $w⊩\alpha \to \beta$ exactly when every $v\ge w$ that forces $\alpha$ also forces $\beta$.

The forcing relation is the intuitionistic instance of Kripke semantics; persistence and the order on worlds determine its implication clause.

Kripke semantics interprets logical formulas at worlds related by an accessibility relation or an information order. Modal logics use accessibility to interpret modal operators; intuitionistic logics use persistent valuations and quantify over stronger worlds for implication.