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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 In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different ...
∃, which, when used together with a predicate variable, is called an existential quantifier ("" or "" or "). Existential quantification is distinct from
universal quantification In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
("for all"), which asserts that the property or relation holds for ''all'' members of the domain. Some sources use the term existentialization to refer to existential quantification.


Basics

Consider a formula that states that some
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
multiplied by itself is 25. :
0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, ...
This would seem to be a
logical disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
because of the repeated use of "or". However, the ellipses make this impossible to integrate and to interpret it as a disjunction in
formal logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
. Instead, the statement could be rephrased more formally as :
For some natural number ''n'', ''n''·''n'' = 25.
This is a single statement using existential quantification. This statement is more precise than the original one, since the phrase "and so on" does not necessarily include all
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s 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 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 Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solut ...
is enough to prove this existential quantification as being true. In contrast, "For some
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
''n'', ''n''·''n'' = 25" is false, because there are no even solutions. The '' domain of discourse'', which specifies the values the variable ''n'' is allowed to take, is therefore critical to a statement's trueness or falseness.
Logical conjunction In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...
s are used to restrict the domain of discourse to fulfill a given predicate. For example: :
For some positive odd number ''n'', ''n''·''n'' = 25
is logically equivalent to :
For some natural number ''n'', ''n'' is odd and ''n''·''n'' = 25.
Here, "and" is the logical conjunction. In symbolic logic, "∃" (a rotated letter " E", in a
sans-serif In typography and lettering, a sans-serif, sans serif, gothic, or simply sans letterform is one that does not have extending features called "serifs" at the end of strokes. Sans-serif typefaces tend to have less stroke width variation than ser ...
font) is used to indicate existential quantification. Thus, if ''P''(''a'', ''b'', ''c'') is the predicate "''a''·''b'' = c", and \mathbb is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of natural numbers, then : \exists\mathbb\, P(n,n,25) is the (true) statement :
For some natural number ''n'', ''n''·''n'' = 25.
Similarly, if ''Q''(''n'') is the predicate "''n'' is even", then : \exists\mathbb\, \big(Q(n)\;\!\;\! \;\!\;\! P(n,n,25)\big) is the (false) statement :
For some natural number ''n'', ''n'' is even and ''n''·''n'' = 25.
In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the proof of a "some" statement may be achieved either by a
constructive proof In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existe ...
, 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

A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The \lnot\ 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: :\exists\mathbf\, P(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\mathbf\, 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 :\exists\mathbf\, P(x) is logically equivalent to "For any natural number ''x'', ''x'' is not greater than 0 and less than 1", or: :\forall\mathbf\, \lnot P(x) Generally, then, the negation of a propositional function's existential quantification is a
universal quantification In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
of that propositional function's negation; symbolically, :\lnot\ \exists\mathbf\, P(x) \equiv\ \forall\mathbf\, \lnot P(x) (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: :\lnot\ \exists\mathbf\, P(x) \equiv\ \forall\mathbf\, \lnot P(x) \not\equiv\ \lnot\ \forall\mathbf\, P(x) \equiv\ \exists\mathbf\, \lnot P(x) Negation is also expressible through a statement of "for no", as opposed to "for some": :\nexists\mathbf\, P(x) \equiv \lnot\ \exists\mathbf\, P(x) Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions: \exists\mathbf\, P(x) \lor Q(x) \to\ (\exists\mathbf\, P(x) \lor \exists\mathbf\, Q(x))


Rules of Inference

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\mathbf\, P(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\mathbf\, P(x) \to\ ((P(c) \to\ Q) \to\ Q) P(c) \to\ 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.


The empty set

The formula \exists \varnothing \, P(x) is always false, regardless of ''P''(''x''). This is because \varnothing denotes the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
, and no ''x'' of any description – let alone an ''x'' fulfilling a given predicate ''P''(''x'') – exist in the empty set. See also Vacuous truth for more information.


As adjoint

In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
and the theory of elementary topoi, the existential quantifier can be understood as the
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
of a functor between power sets, the
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
functor of a function between sets; likewise, the universal quantifier is the right adjoint. Saunders Mac Lane, Ieke Moerdijk, (1992): ''Sheaves in Geometry and Logic'' Springer-Verlag . ''See p. 58''.


Encoding

In Unicode and HTML, symbols are encoded and . In TeX, the symbol is produced with "\exists".


See also

*
Existential clause An existential clause is a clause that refers to the existence or presence of something, such as "There is a God" and "There are boys in the yard". The use of such clauses can be considered analogous to existential quantification in predicate lo ...
* Existence theorem *
First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
* Lindström quantifier *
List of logic symbols In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subs ...
– for the unicode symbol ∃ * Quantifier variance * Uniqueness quantification


Notes


References

* {{Mathematical logic Logic symbols Quantifier (logic)