The flavor of modal logic called $K$ is propositional logic equipped with a single modality usually written ”$\Box$” subject to the rules that for all propositions $p, q \colon Prop$ we have
Often one adds to this the following further axioms
$\Box T \colon \Box p \to p$ (T modal logic)
$\Box 4 \colon \Box p \to \Box \Box p$. (S4 modal logic).
$K$ is the basic epistemic logic.
(Taut) All (instances of ) propositional tautologies.
For each $i = 1,\ldots, m$, the axiom, ($K_i$):
(i.e. modus ponens);
The second deduction rule corresponds to the idea that if a statement has been proved, then it is known to all ‘agents’.
This logic is the smallest normal modal logic.
The semantics of $K_{(m)}$ is just the Kripke semantics of this context, so a frame, $\mathfrak{F}$ is just a set, $W$ of possible worlds with $m$ relations $R_i$. A model, $\mathfrak{M} = (\mathfrak{F},V)$, is a frame in that sense together with a valuation, $V: Prop \to \mathcal{P}(W)$, and the satisfaction relation is as described in geometric models for modal logics with just the difference implied by the fact that that page correspond to the use of $\Diamond_i = M_i$ whilst this uses $K_i$. This means that