Editing Existential quantification (section)

=== Negation ===
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated.  The <math>\lnot\ </math> symbol is used to denote negation.

For example, if ''P''(''x'') is the predicate "''x'' is greater than 0 and less than 1", then, for a domain of discourse ''X'' of all natural numbers, the existential quantification "There exists a natural number ''x'' which is greater than 0 and less than 1" can be symbolically stated as:
:<math>\exists{x}{\in}\mathbf{X}\, P(x)</math>

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:
:<math>\lnot\ \exists{x}{\in}\mathbf{X}\, P(x)</math>.

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
:<math>\exists{x}{\in}\mathbf{X}\, P(x)</math>
is logically equivalent to "For any natural number ''x'', ''x'' is not greater than 0 and less than 1", or:
:<math>\forall{x}{\in}\mathbf{X}\, \lnot P(x)</math>

Generally, then, the negation of a [[propositional function]]'s existential quantification is a [[universal quantification]] of that propositional function's negation; symbolically,
:<math>\lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x)</math>
(This is a generalization of [[De Morgan's laws]] to predicate logic.)

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:
:<math>\lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x) \not\equiv\ \lnot\ \forall{x}{\in}\mathbf{X}\, P(x) \equiv\ \exists{x}{\in}\mathbf{X}\, \lnot P(x)</math>

Negation is also expressible through a statement of "for no", as opposed to "for some":
:<math>\nexists{x}{\in}\mathbf{X}\, P(x) \equiv \lnot\ \exists{x}{\in}\mathbf{X}\, P(x)</math>

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

<math> \exists{x}{\in}\mathbf{X}\, P(x) \lor Q(x) \to\ (\exists{x}{\in}\mathbf{X}\, P(x) \lor \exists{x}{\in}\mathbf{X}\, Q(x))</math>