| Existential quantification | |
| Type: | Quantifier |
| Field: | Mathematical logic |
| Statement: | \existsxP(x) P(x) x |
| Symbolic Statement: | \existsxP(x) |
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("" or "" or "). Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain.[1] [2] Some sources use the term existentialization to refer to existential quantification.[3]
Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as in Unicode, and as \exists in LaTeX and related formula editors.
Consider the formal sentence
For some natural number
n
n x n=25
This is a single statement using existential quantification. It is roughly analogous to the informal sentence "Either
0 x 0=25
1 x 1=25
2 x 2=25
This particular example is true, because 5 is a natural number, and when we substitute 5 for n, we produce the true statement
5 x 5=25
n x n=25
n
n x n=25
For some positive odd number
n
n x n=25
is logically equivalent to the sentence
For some natural number
n
n
n x n=25
The mathematical proof of an existential statement about "some" object 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 without concretely exhibiting one.
In symbolic logic, "∃" (a turned letter "E" in a sans-serif font, Unicode U+2203) is used to indicate existential quantification. For example, the notation
\exists{n}{\in}N:n x n=25
There exists some
n
n x n=25
The symbol's first usage is thought to be by Giuseppe Peano in Formulario mathematico (1896). Afterwards, Bertrand Russell popularised its use as the existential quantifier. Through his research in set theory, Peano also introduced the symbols
\cap
\cup
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The
lnot
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:
\exists{x}{\in}XP(x)
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:
lnot \exists{x}{\in}XP(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
\exists{x}{\in}XP(x)
\forall{x}{\in}XlnotP(x)
Generally, then, the negation of a propositional function's existential quantification is a universal quantification of that propositional function's negation; symbolically,
lnot \exists{x}{\in}XP(x)\equiv \forall{x}{\in}XlnotP(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:
lnot \exists{x}{\in}XP(x)\equiv \forall{x}{\in}XlnotP(x)\not\equiv lnot \forall{x}{\in}XP(x)\equiv \exists{x}{\in}XlnotP(x)
Negation is also expressible through a statement of "for no", as opposed to "for some":
\nexists{x}{\in}XP(x)\equivlnot \exists{x}{\in}XP(x)
Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:
\exists{x}{\in}XP(x)\lorQ(x)\to (\exists{x}{\in}XP(x)\lor\exists{x}{\in}XQ(x))
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.
Existential introduction (∃I) 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)\to \exists{x}{\in}XP(x)
Existential instantiation, when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation while substituting an existentially quantified variable for a subject—which does not appear within any active sub-derivation. If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then one can exit that sub-derivation with that conclusion. The reasoning behind existential elimination (∃E) 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:
\exists{x}{\in}XP(x)\to ((P(c)\to Q)\to Q)
P(c)\to Q
The formula
\exists{x}{\in}\varnothingP(x)
\varnothing
In category theory and the theory of elementary topoi, the existential quantifier can be understood as the left adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the universal quantifier is the right adjoint.[5]