 Modal Logic: background notions: A modal logic is an extension of the language of propositional logic with a set of modal operators \(\Box_1,\ldots , \Box_n, \ldots\), defined on a countable set of atomic propositions \(\texttt{Atoms}=\{p_1,p_2, \ldots \}\), over which the set of well-formed formulas is inductively built (for a mathematical treatment of logic and induction see for instance Dalen 1980). Each well-formed formula \(\varphi\) of a modal language \(\mathcal{L}\), henceforth simply called formula, is constructed using the following grammar: