Symbols are encoded and .

Basics

0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on.

For some natural number ''n'', ''n''·''n'' = 25.

This statement is more precise than the original one, as the phrase "and so on" does not necessarily include all natural numbers, and nothing more. Since the domain was not stated explicitly, the phrase could not be interpreted formally. In the quantified statement, on the other hand, the natural numbers are mentioned explicitly.

This particular example is true, because 5 is a natural number, and when we substitute 5 for ''n'', we produce "5·5 = 25", which is true. It does not matter that "''n''·''n'' = 25" is only true for a single natural number, 5; even the existence of a single solution is enough to prove the existential quantification true. In contrast, "For some even number ''n'', ''n''·''n'' = 25" is false, because there are no even solutions.

The ''domain of discourse'', which specifies which values the variable ''n'' is allowed to take, is therefore of critical importance in a statement's trueness or falseness. Logical conjunctions are used to restrict the domain of discourse to fulfill a given predicate. For example: :

For some positive odd number ''n'', ''n''·''n'' = 25"

For some natural number ''n'', ''n'' is odd and ''n''·''n'' = 25".

In symbolic logic, "∃" (a backwards letter "E" in a sans-serif font) is used to indicate existential quantification. Thus, if ''P''(''a'', ''b'', ''c'') is the predicate "''a''·''b'' = c" and $\backslash mathbb\{N\}$ is the set of natural numbers, then : $\backslash exists\{n\}\{\backslash in\}\backslash mathbb\{N\}\backslash ,\; P(n,n,25)$ is the (true) statement :

For some natural number ''n'', ''n''·''n'' = 25.

For some even number ''n'', ''n''·''n'' = 25.

In mathematics, the proof of a "some" statement may be achieved either by a constructive proof, which exhibits an object satisfying the "some" statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one.

Properties

Negation

For example, if P(''x'') is the propositional function "x is between 0 and 1", then, for a domain of discourse ''X'' of all natural numbers, the existential quantification "There exists a natural number ''x'' which is between 0 and 1" is symbolically stated: :$\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$

This can be demonstrated to be irrevocably false. Truthfully, it must be said, "It is not the case that there is a natural number ''x'' that is between 0 and 1", or, symbolically: :$\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$.

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 :$\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$ is logically equivalent to "For any natural number ''x'', x is not between 0 and 1", or: :$\backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically, :$\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

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: :$\backslash lnot\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)\; \backslash not\backslash equiv\backslash \; \backslash lnot\backslash \; \backslash forall\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash equiv\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; \backslash lnot\; P(x)$

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

Rules of Inference

''Existential introduction'' concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,

:$P(a)\; \backslash to\backslash \; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)$

The reasoning behind existential elimination is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name. Symbolically, for an arbitrary ''c'' and for a proposition Q in which ''c'' does not appear:

:$\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash to\backslash \; ((P(c)\; \backslash to\backslash \; Q)\; \backslash to\backslash \; Q)$

$P(c)\; \backslash to\backslash \; Q$ must be true for all values of ''c'' over the same domain ''X''; else, the logic does not follow: If ''c'' is not arbitrary, and is instead a specific element of the domain of discourse, then stating P(''c'') might unjustifiably give more information about that object.

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

$\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash or\; Q(x)\; \backslash to\backslash \; (\backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; P(x)\; \backslash or\; \backslash exists\{x\}\{\backslash in\}\backslash mathbf\{X\}\backslash ,\; Q(x))$

The empty set

Notes

See also

References

Category:Logic Category:Quantification Category:Something