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 quanti ...
, 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
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 conc ...
∃, 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. 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 ...
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 ...
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 formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
.
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 ...
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 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 \\
4 ...
''n'', ''n''·''n'' = 25" is false, because there are no even solutions.
The ''
domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The dom ...
'', 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 thi ...
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 font) is used to indicate existential quantification. Thus, if ''P''(''a'', ''b'', ''c'') is the predicate "''a''·''b'' = c", and
is the
set of natural numbers, then
:
is the (true) statement
:
For some natural number ''n'', ''n''·''n'' = 25.
Similarly, if ''Q''(''n'') is the predicate "''n'' is even", then
:
is the (false) statement
:
For some natural number ''n'', ''n'' is even and ''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
A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The
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:
:
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:
:
.
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
:
is logically equivalent to "For any natural number ''x'', ''x'' is not greater than 0 and less than 1", or:
:
Generally, then, the negation of a
propositional function's existential quantification is a
universal quantification of that propositional function's negation; symbolically,
:
(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:
:
Negation is also expressible through a statement of "for no", as opposed to "for some":
:
Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:
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, and returns a conclusion (or conclusions). For example, the rule of ...
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,
:
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''. Th ...
, 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:
:
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
is always false, regardless of ''P''(''x''). This is because
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 oth ...
, 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 and the theory of
elementary topoi, the existential quantifier can be understood as the
left adjoint of a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
between
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is ...
s, the
inverse image 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
*
Existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
*
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 quanti ...
*
Lindström quantifier
*
List of logic symbols – for the unicode symbol ∃
*
Quantifier variance
*
Uniqueness quantification
Notes
References
*
{{Mathematical logic
Logic symbols
Quantifier (logic)