nLab
normal modal logic

A modal logic is uniform if it is closed under the rule of uniform substitution of ${ℒ}_{\omega }(n)$-formulae for propositional variables and is normal if it also contains the axiom schemata:

(K) ${\diamond }_i(\psi \vee \chi )\to {\diamond }_i(\psi )\vee {\diamond }_i(\chi )$

(N) $\neg {\diamond }_i(\perp )$

and monotonicity (for each $i$):

if $\psi \to \chi \in \Lambda$ then ${\diamond }_i\psi \to {\diamond }_i\chi \in \Lambda$.

The smallest normal modal logic with $m$ ‘agents’ is K(m). (The diamonds correspond to the $M_i$ of that entry.)