 The basic alethic modal logic contains both a possibility \((\Diamond)\) and a necessity \((\Box)\) modality. Most formal presentations of this system take one of these modalities as the primitive syntactic operator (say, \(\Diamond)\), and then define the other as its modal dual \((\oBox\varphi := \lnot \oDiamond \lnot \varphi)\). This is a seemingly harmless syntactic interdefinability, and comes from the fact that \(\Diamond\) and \(\Box\) are semantically interpreted in terms of the existential and universal  quantifiers,  respectively. It is, in some sense, similar to the interdefinability of Boolean operators in classic propositional logic. Nevertheless, it already reflects important underlying assumptions. From a classic point of view, something is necessary if and only if it is not the case that its negation is possible \((\oBox\varphi \leftrightarrow \lnot \oDiamond \lnot \varphi)\), and something is possible if and only if it is not the case that its negation is necessary \((\oDiamond \varphi \leftrightarrow \lnot \oBox\lnot \varphi)\). However, this may not be the case in all settings. For example, while \(\oDiamond \varphi \rightarrow \lnot \Box \lnot\varphi\) is  intuitionistically acceptable  (the existence of a possibility where \(\varphi\) holds implies that not every possibility makes \(\varphi\) false), its converse \(\lnot \oBox \lnot\varphi \rightarrow \oDiamond \varphi\)  is not  (the fact that not every possibility makes \(\varphi\) false is not enough to guarantee the existence of a possibility where \(\varphi\) is true). Thus, one should always be careful when defining a modality in terms of another.