 Given a standard first-order language \(\scrL\) with the truth functional operators \(\neg,\,\to,\) a distinguished identity predicate \(=,\) and the universal quantifier \(\forall,\) a Tarskian interpretation \(\calI\) for \(\scrL\) specifies a nonempty set \(D\)—the universe of \(\calI\)—for the quantifiers of \(\scrL\) to range over and assigns appropriate semantic values to the terms (i.e., the (individual) constants and variables) and predicates of \(\scrL\). Specifically, to each term \(\tau\) of \(\scrL,\) \(\calI\) assigns a referent \(\tau^\calI \in D\) and, to each n-place predicate \(\pi\) of \(\scrL,\) \(\calI\) assigns a set \(\pi^\calI \subseteq D^n\) of n-tuples of members of \(D,\) often referred to as the extension of \(\pi\) in  \(\calI\).[30]  The extension \(=^\calI\) assigned to the identity predicate, of course, is always stipulated to be the “real” identity relation for \(D,\) i.e., the set \(\lbrace\langle a,a\rangle:a \in D\rbrace\). Those assignments to the terms and predicates of \(\scrL,\) in turn, completely determine the truth values of all the formulas of \(\scrL\) by means of a familiar set of recursive clauses. To facilitate the quantificational clause, let \(\calI[\frac{\nu}{a}]\) be the interpretation that assigns the individual \(a\) to the variable \(\nu\) and is otherwise exactly like \(\calI\). This apparatus then delivers a  compositional  theory of meaning for \(\scrL,\) that is, an account on which the meaning (in this case, the truth value) of a complex formula is determined by its grammatical structure and the meanings of its semantically significant parts and hence, ultimately, in the Tarskian case, by the semantic values assigned to the terms and predicates of the language: