 Intuitively, a Tarskian interpretation of an applied non-modal language represents a possible world, a way in which the properties and relations expressed by the predicates of the language might be exemplified by the things in the universe of the interpretation. The idea underlying possible world semantics is simply to interpret a modal language \(\scrL_\Box\) by bringing a collection of Tarskian interpretations together to represent a modal space of many possible worlds in a single interpretation of  \(\scrL_\Box\).[33]  So let \(\scrL_\Box\) be the result of adding the modal operator \(\Box\) to some standard first-order language \(\scrL\). As with a Tarskian interpretation of \(\scrL,\) an SQML interpretation \(\calM\) for \(\scrL_\Box\) specifies a nonempty set \(D\) to serve as its universe. Also as in Tarskian semantics, \(\calM\) assigns each term \(\tau\) of \(\scrL_\Box\) a semantic value \(\tau^\calM \in D\). Additionally, however, \(\calM\) specifies a nonempty set \(W\)—these are typically called the set of “possible worlds” of \(\calM\) but can be any nonempty set. One member \(w^\ast\) of \(W\) is designated as the “actual world” of \(\calM\). To give substance and structure to these “worlds”, \(\calM\) then assigns extensions to the predicates of \(\scrL_\Box\) relative to each world—that is, for every n-place predicate \(\pi\) of \(\scrL\) and each world \(w \in W,\) \(\calM\) assigns a set \(\pi_w^\calM \subseteq D^n\) of n-tuples of members of \(D,\) the extension of \(\pi\) at  \(w\);[34]  in particular, the extension \(=_w^\calM\) of the identity predicate at all worlds \(w\) is stipulated to be \(\lbrace\langle a,a\rangle:a \in D\rbrace\). In this way \(\calM\) represents the different ways that the properties and relations expressed by those predicates can change (or not) from world to world.