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In
predicate 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 Quantifica ...
, an existential quantification is a type of quantifier, a
logical constantIn logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=n ...
which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the
logical operator Logic (from Greek: grc, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason ...
symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...
∃, which, when used together with a predicate variable, is called an existential quantifier ("" or ""). Existential quantification is distinct from
universal quantification In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
("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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
multiplied by itself is 25. :
0·0 = 25, or 1·1 = 25, or 2·2 = 25, or 3·3 = 25, and so on.
This would seem to be a
logical disjunction In logic, disjunction is a logical connective typically notated \lor whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics of logic, seman ...
because of the repeated use of "or". However, the "and so on" makes this impossible to integrate and to interpret it as a disjunction in
formal logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
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) Image:SaltInWaterSolutionLiquid.jpg, upMaking a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent. In chemistry ...
is enough to prove this existential quantification as being true. In contrast, "For some
even number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
''n'', ''n''·''n'' = 25" is false, because there are no even solutions. The ''
domain of discourse In the formal sciences Formal science is a branch of science studying formal language disciplines concerned with formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used f ...
'', 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 Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...
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 Logic (from Greek: grc, λογική, label=none, lit=possessed of reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=n ...
to :
For some natural number ''n'', ''n'' is odd and ''n''·''n'' = 25.
Here, "and" is the logical conjunction. In
symbolic logic Mathematical logic, also called formal logic, is a subfield of mathematics exploring the formal applications of logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. ...
, "∃" (a rotated letter "
E
E
", in a
sans-serif In typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by ...
font) is used to indicate existential quantification. Thus, if ''P''(''a'', ''b'', ''c'') is the predicate "''a''·''b'' = c", and \mathbb is the set 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the proof of a "some" statement may be achieved either by a
constructive proofIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, which exhibits an object satisfying the "some" statement, or by a
nonconstructive proofIn mathematics, a constructive proof is a method of mathematical proof, 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 kn ...
, 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 In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of logical truth, true or false (logic), false, except that within the sentence there is a Variable (mathematics), variab ...
's existential quantification is a
universal quantification In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
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 In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...
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 In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their Syntax (logic), syntax, and returns a conclusion (or multiple-conclusion logic, ...
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 In predicate logic, existential instantiation (also called existential elimination)Moore and Parker is a rule of inference which says that, given a formula of the form (\exists x) \phi(x), one may infer \phi(c) for a new constant symbol ''c''. The ...
, 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 Logical truth is one of the most fundamental concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts and belief ...
, 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 \emptyset \, P(x) is always false, regardless of ''P''(''x''). This is because \emptyset denotes the
empty set #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

empty set
, and no ''x'' of any description – let alone an ''x'' fulfilling a given predicate ''P''(''x'') – exist in the empty set. See also
Vacuous truthIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
for more information.


As adjoint

In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
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 have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
of a
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

functor
between
power set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, the
inverse image In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

inverse image
functor of a function between sets; likewise, the
universal quantifier In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
is the
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
.
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
, Ieke Moerdijk, (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. ''See page 58''


Encoding

In Unicode and HTML, symbols are encoded and . In
TeX TeX (, see below), stylized within the system as TeX, is a typesetting system which was designed and mostly written by Donald Knuth and released in 1978. TeX is a popular means of typesetting complex mathematical formulae; it has been noted ...
, the symbol is produced with "\exists".


See also

* Existential clause *
Existence theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
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 Quantificat ...
* Lindström quantifier *
List of logic symbols In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents state ...
– for the unicode symbol ∃ * Quantifier variance *
Quantifiers Quantifier may refer to: * Quantifier (linguistics), an indicator of quantity * Quantifier (logic) * Quantification (science) See also

*Quantification (disambiguation) {{disambiguation ...
*
Uniqueness quantification In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
*
Universal quantification In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...


Notes


References

* {{Mathematical logic Logic symbols Quantification (science)