We begin our study of explicit numerical claims with existential claims or claims of exemplification. The unrestricted existential quantifier says that the predicate it applies to is exemplified—i.e., it has a non-empty extension, an extension with at least one member. The restricted existential quantifier says that its quantified predicate is exemplified within the extension of its restricting predicate—i.e., the intersection of their extensions is non-empty. Both use the sign ∃ (there exists) and we will refer to sentences formed with either as existentials. An unrestricted existential can be restated as a restricted existential whose restricting predicate is universal, and a restricted existential can be restated by applying an unrestricted existential to a predicate formed from the restricting and quantified predicates using conjunction (note: not using the conditional). Although English existentials can appear with either singular or plural quantifier phrases, this does not seem to affect the proposition expressed and the difference will not be captured in our analyses.

To deny a generalization is to claim the existence of a counterexample, and this suggests that the negation of a universal should be equivalent to an existential with a negative quantified predicate. This is so, and the negation of an existential is also equivalent to a negative generalization. We extend the traditional term obversion to both principles.

Another traditional principle is conversion, which tells us that we can interchange the restricting and quantified predicates of a restricted existential. This suggests that we could regard the single predicate in an unrestricted existential as either a restricting or a quantified predicate. That provides some explanation of English there-is existentials, which can have class indicators without quantified predicates. A rule of thumb for handling the simpler examples of such sentences is to replace there by something (or someone).

English sentences that claim the existence of an example can vary widely in the way they distribute the properties of this example between the quantifier phrase and quantified predicate. The logical equivalence of different ways of distributing this information explains why the difference between restrictive and non-restrictive relative clauses does not affect what is said in cases where they modify the class indicator of an existential quantifier phrase. Other forms of equivalent restatement are the result of confining the scope of an existential to a formula in which all its bound variables appear. Confinement principles sometimes require a change between universal and existential quantifiers, and this explains why a quantifier phrase stated using any can sometimes be treated either by a universal with wide scope or an existential with narrow scope.

Any existential sentence—indeed any sentences that entail an existential—can be said to involve an existential commitment, but the examples whose existence make existentials true can be any reference values, even the nil value. This may seem to conflict with the substantive existential commitment, to objects rather than mere reference values, that many find in English existentials. This commitment might be traced to the logical properties of non-logical vocabulary; but, if that account is rejected, it is possible to introduce a logical predicate that carries the commitment (through a stipulation that its extension includes only non-nil values).