Kripke semantics

Remarks

Then $\mathfrak{M}, w \models \lozenge\varphi$ if and only if some $v$ with $w R v$ has $\mathfrak{M}, v \models \varphi$.

Example: on any frame, the axiom $\square(p \to q) \to (\square p \to \square q)$ is valid. At a world $w$ where $\square(p \to q)$ and $\square p$ hold, every $v$ with $w R v$ satisfies $p \to q$ and $p$, hence $q$.

Non-example: $\square p \to p$ is not valid on the frame $(\set{w}, \emptyset)$. With $V(p) = \emptyset$, the formula $\square p$ holds at $w$ because no world is accessible, while $p$ fails at $w$.

The formulas valid on all frames form the logic $\mathrm{K}$. The formulas valid on all reflexive frames form $\mathrm{T}$, and those valid on all reflexive and transitive frames form $\mathrm{S4}$.

For intuitionistic logic, the relation is a preorder, $V(p)$ is closed upward, and $\varphi \to \psi$ holds at $w$ when $\psi$ holds at every $v \ge w$ where $\varphi$ holds.