Does it hold that if $\models_cA$, then $\models_K\Box A$?

Question

Does it hold that if $\models_cA$, then $\models_K\Box A$? (Where $\models_c$ and $\models_K$ refer to classical propositional logic and Kripke modal logic, respectively.)

Answer 1

Yes, this is an immediate consequence of the Kripke semantics. Suppose $\varphi$ is a classical tautology. Then if $K$ is a Kripke frame, $\varphi$ holds at every world of $K$ (since these worlds are models of classical propositional logic) - in particular, no matter what the "marked" world of $K$ is, $\varphi$ will hold in every model that that world "sees". So we will have $\Box \varphi$ true in $K$. Since $\Box\varphi$ is true in every Kripke frame, by the completeness theorem for Kripke frames this means $\Box\varphi$ is a modal tautology.

Incidentally, note that in general you need to specify which modal logic you're referring to. There are lots which are frequently assumed as "the right" modal logic - e.g. $S_4$, $S_5$, and $K$, just to name a few I've seen used as the background modal theory within the last week or so. In this case it really doesn't depend on what modal theory we use, but this is only because you're consider a very restricted class of modal formulas (those of the form "$\Box \varphi$" for $\varphi$ classical).