# Existential quantification

Consider a formula that states that some natural number multiplied by itself is 25.

This statement is more precise than the original one, since the phrase "and so on" does not necessarily include all natural numbers and exclude everything else. And since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, however, the natural numbers are mentioned explicitly.

This can be demonstrated to be false. Truthfully, it must be said, "It is not the case that there is a natural number *x* that is greater than 0 and less than 1", or, symbolically:

If there is no element of the domain of discourse for which the statement is true, then it must be false for all of those elements. That is, the negation of

A common error is stating "all persons are not married" (i.e., "there exists no person who is married"), when "not all persons are married" (i.e., "there exists a person who is not married") is intended:

Negation is also expressible through a statement of "for no", as opposed to "for some":

Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:

A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.