Is modal logic the logic of necessity, possibility, and impossibility alone, or the logic of truth, falsity, necessity, possibility, and impossibility?
In other words, is modal logic concerned with modal statements only, or concerned with both modal statements and truth-functional statements?
See What is Modal Logic?:
Narrowly construed, modal logic studies reasoning that involves the use of the expressions ‘necessarily’ and ‘possibly’. However, the term ‘modal logic’ is used more broadly to cover a family of logics with similar rules and a variety of different symbols [including logics for belief, for tense and other temporal expressions, for the deontic (moral) expressions such as ‘it is obligatory that’ and ‘it is permitted that’, and many others.]
Usually, modal propositional logic is built "on top" of classical (i.e. truth-functional) propositional logic.
See e.g. George Boolos & John Burgess & Richard Jeffrey, Computability and Logic (4th ed - 2002): 27.1 Modal Logic, page 327-on:
Modal sentential logic adds to the apparatus of ordinary or ‘classical’ sentential logic one more logical operator, the box $\square$, read ‘necessarily’ or ‘it must be the case that’. One more clause is added to the definition of sentence: if $A$ is a sentence, so is $\square A$. [...]
A modal sentence is said to be a tautology if it can be obtained from a valid sentence of nonmodal sentential logic by substituting modal sentences for sentence letters. Thus, since $p ∨ \lnot p$ is valid for any sentence letter $p$, $A ∨ \lnot A$ is a tautology for any modal sentence $A$.
