 Let us momentarily focus on singular terms. To keep matters simple, consider an impoverished fragment of the language of arithmetic with one constant, \(0\), read: “zero” and one functional symbol, \(S\), read: “the successor of”. The domain of the intended interpretation consists of a set of natural numbers, which are named by singular terms of the form \(0,\) \(SS0,\) \(SSS0,\) …, etc. If we now associate the class of all such terms to the substitutional quantifiers \(\Sigma \alpha\) and \(\Pi \alpha\), then a sentence like \(\Sigma \alpha \ \alpha = \alpha\) is evaluated as true in virtue of the truth of formulas like \(0 = 0\), whereas \(\Pi \alpha \ \alpha &lt; \alpha\) is evaluated as false in virtue of falsity of formulas like \(0 &lt; 0\). In general, a sentence of the form \(\Sigma \alpha \ A\) exhibits the same truth conditions as an infinitary disjunction of the form:  \[A(0/\alpha) \vee A(S0/\alpha) \vee \ldots\vee A(SS \overbrace{\ldots}^n S0/\alpha) \vee \ldots\]  whereas a sentence of the form \(\Pi \alpha \ A\) exhibits the same truth conditions as an infinitary conjunction:  \[A(0/\alpha) \wedge A(S0/\alpha) \wedge \ldots\wedge A(SS \overbrace{\ldots}^n S0/\alpha) \wedge \ldots\]  The case of arithmetic is optimal because we have a name for each member of the intended domain. This means that in general, a quantified sentence of the form \(\exists x A\) will be true if, and only if its substitutional counterpart \(\Sigma \alpha \ A\) is likewise true. But in this respect, however, the language of arithmetic is the exception and not the rule. In real analysis, for example, there are too many objects in the domain to have a name in a countable language. In such a situation, we must confront the risk that a sentence of the form \(\exists x A\) may be true even if \(\Sigma \alpha \ A\) remains false in virtue of the lack of a true substitution instance of the form \(A(t/ \alpha)\). This may happen, for example, if the objects that satisfy the open formula \(A\) are not denoted by any singular term in the language.