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| {{redirect|Universal quantifier|the symbol conventionally used for this quantifier|Turned A}}
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| In [[predicate logic]], a '''universal quantification''' is a type of [[quantification|quantifier]], a [[logical constant]] which is [[interpretation (logic)|interpreted]] as "given any" or "for all". It expresses that a [[propositional function]] can be [[satisfiability|satisfied]] by every [[element (mathematics)|member]] of a [[domain of discourse]]. In other terms, it is the [[Predicate (mathematical logic)|predication]] of a [[property (philosophy)|property]] or [[binary relation|relation]] to every member of the domain. It [[logical assertion|asserts]] that a predicate within the [[free variables and bound variables|scope]] of a universal quantifier is true of every [[Valuation (logic)|value]] of a [[predicate variable]].
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| It is usually denoted by the [[turned A]] (∀) [[logical connective|logical operator]] [[Symbol (formal)|symbol]], which, when used together with a predicate variable, is called a '''universal quantifier''' ("∀x", "∀(x)", or sometimes by "(x)" alone). Universal quantification is distinct from [[existential quantification|''existential'' quantification]] ("there exists"), which asserts that the property or relation holds only for at least one member of the domain.
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| Quantification in general is covered in the article on [[quantification]]. Symbols are encoded {{unichar|2200|FOR ALL|note=as a mathematical symbol|html=|ulink=}}.
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| == Basics ==
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| Suppose it is given that
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| <blockquote>2·0 = 0 + 0, and 2·1 = 1 + 1, and 2·2 = 2 + 2, etc.</blockquote> | |
| This would seem to be a [[logical conjunction]] because of the repeated use of "and." However, the "etc." cannot be interpreted as a conjunction in [[formal logic]]. Instead, the statement must be rephrased:
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| <blockquote>For all natural numbers ''n'', 2·''n'' = ''n'' + ''n''.</blockquote> | |
| This is a single statement using universal quantification.
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| This statement can be said to be more precise than the original one. While the "etc." informally includes [[natural number]]s, and nothing more, this was not rigorously given. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly.
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| This particular example is [[true (logic)|true]], because any natural number could be substituted for ''n'' and the statement "2·''n'' = ''n'' + ''n''" would be true. In contrast,
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| <blockquote>For all natural numbers ''n'', 2·''n'' > 2 + ''n''</blockquote>
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| is [[false (logic)|false]], because if ''n'' is substituted with, for instance, 1, the statement "2·1 > 2 + 1" is false. It is immaterial that "2·''n'' > 2 + ''n''" is true for ''most'' natural numbers ''n'': even the existence of a single [[counterexample]] is enough to prove the universal quantification false.
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| On the other hand,
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| for all [[composite number]]s ''n'', 2·''n'' > 2 + ''n''
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| is true, because none of the counterexamples are composite numbers. This indicates the importance of the ''[[domain of discourse]]'', which specifies which values ''n'' can take.<ref group="note">Further information on using domains of discourse with quantified statements can be found in the [[Quantification]] article.</ref> In particular, note that if the domain of discourse is restricted to consist only of those objects that satisfy a certain predicate, then for universal quantification this requires a [[logical conditional]]. For example,
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| <blockquote>For all composite numbers ''n'', 2·''n'' > 2 + ''n''</blockquote>
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| is [[logically equivalent]] to
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| <blockquote>For all natural numbers ''n'', if ''n'' is composite, then 2·''n'' > 2 + ''n''.</blockquote>
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| Here the "if ... then" construction indicates the logical conditional.
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| === Notation ===
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| In [[First-order logic|symbolic logic]], the universal quantifier symbol <math> \forall </math> (an [[turned A|inverted]] "[[A]]" in a [[sans-serif]] font, Unicode 0x2200) is used to indicate universal quantification.<ref>The [[turned A|inverted "A"]] was used in the 19th century by [[Charles Sanders Peirce]] as a logical symbol for 'un-American' ("unamerican").
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| Page 320 in Randall Dipert, "[http://books.google.com/books?id=3suPBY5qh-cC&pg=PR7&dq=Cheryl+Misak,+unamerican&source=gbs_selected_pages&cad=3#v=onepage&q=unAmerican&f=false Peirce's deductive logic]". In Cheryl Misak, ed. ''The Cambridge Companion to Peirce''. 2004</ref>
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| For example, if ''P''(''n'') is the predicate "2·''n'' > 2 + ''n''" and '''N''' is the [[Set (mathematics)|set]] of natural numbers, then:
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| : <math> \forall n\!\in\!\mathbb{N}\; P(n) </math>
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| is the (false) statement:
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| <blockquote>For all natural numbers ''n'', 2·''n'' > 2 + ''n''.</blockquote>
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| Similarly, if ''Q''(''n'') is the predicate "''n'' is composite", then
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| : <math> \forall n\!\in\!\mathbb{N}\; \bigl( Q(n) \rightarrow P(n) \bigr) </math>
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| is the (true) statement:
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| <blockquote>For all natural numbers ''n'', if ''n'' is composite, then 2·''n'' > 2 + n</blockquote>
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| and since "''n'' is composite" implies that ''n'' must already be a natural number, we can shorten this statement to the equivalent:
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| : <math> \forall n\; \bigl( Q(n) \rightarrow P(n) \bigr) </math>
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| <blockquote>For all composite numbers ''n'', 2·''n'' > 2 + ''n''.</blockquote>
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| Several variations in the notation for quantification (which apply to all forms) can be found in the [[quantification]] article. There is a special notation used only for universal quantification, which is given:
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| : <math> (n{\in}\mathbb{N})\, P(n) </math> | |
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| The parentheses indicate universal quantification by default.
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| == Properties ==
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| <!-- ''We need a list of algebraic properties of universal quantification, such as distributivity over conjunction, and so on. Also rules of inference.'' -->
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| ===Negation===
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| Note that a quantified [[propositional function]] is a statement; thus, like statements, quantified functions can be negated. The notation most mathematicians and logicians utilize to denote negation is: <math>\lnot\ </math>. However, some (such as [[Douglas Hofstadter]]) use the [[tilde]] (~).
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| For example, if P(''x'') is the propositional function "x is married", then, for a [[Universe of discourse|Universe of Discourse]] X of all living human beings, the universal quantification
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| <blockquote>Given any living person ''x'', that person is married</blockquote>
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| is given:
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| :<math>\forall{x}{\in}\mathbf{X}\, P(x)</math>
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| It can be seen that this is irrevocably false. Truthfully, it is stated that
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| <blockquote>It is not the case that, given any living person ''x'', that person is married</blockquote>
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| or, symbolically:
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| :<math>\lnot\ \forall{x}{\in}\mathbf{X}\, P(x)</math>.
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| If the statement is not true for ''every'' element of the Universe of Discourse, then, presuming the universe of discourse is non-empty, there must be at least one element for which the statement is false. That is, the negation of <math>\forall{x}{\in}\mathbf{X}\, P(x)</math> is logically equivalent to "There exists a living person ''x'' such that he is not married", or:
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| :<math>\exists{x}{\in}\mathbf{X}\, \lnot P(x)</math>
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| Generally, then, the negation of a propositional function's universal quantification is an [[existential quantification]] of that propositional function's negation; symbolically,
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| :<math>\lnot\ \forall{x}{\in}\mathbf{X}\, P(x) \equiv\ \exists{x}{\in}\mathbf{X}\, \lnot P(x)</math>
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| It is erroneous to state "all persons are not married" (i.e. "there exists no person who is married") when it is meant that "not all persons are married" (i.e. "there exists a person who is not married"):
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| :<math>\lnot\ \exists{x}{\in}\mathbf{X}\, P(x) \equiv\ \forall{x}{\in}\mathbf{X}\, \lnot P(x) \not\equiv\ \lnot\ \forall{x}{\in}\mathbf{X}\, P(x) \equiv\ \exists{x}{\in}\mathbf{X}\, \lnot P(x)</math> | |
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| ===Other connectives===
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| The universal (and existential) quantifier moves unchanged across the [[logical connective]]s [[logical conjunction|∧]], [[logical disjunction|∨]], [[material conditional|→]], and [[converse nonimplication|<math>\nleftarrow</math>]], as long as the other operand is not affected; that is:
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| :<math>P(x) \land (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \land Q(y))</math>
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| :<math>P(x) \lor (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \to (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \to Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \nleftarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y))</math>
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| :<math>P(x) \land (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \land Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \lor (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y))</math>
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| :<math>P(x) \to (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \to Q(y))</math>
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| :<math>P(x) \nleftarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| Conversely, for the logical connectives [[Sheffer stroke|↑]], [[Logical NOR|↓]], [[Material nonimplication|<math>\nrightarrow</math>]], and [[converse implication|←]], the quantifiers flip:
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| :<math>P(x) \uparrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y))</math>
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| :<math>P(x) \downarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \nrightarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \gets (\exists{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y))</math>
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| :<math>P(x) \uparrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| :<math>P(x) \downarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y))</math>
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| :<math>P(x) \nrightarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y))</math>
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| :<math>P(x) \gets (\forall{y}{\in}\mathbf{Y}\, Q(y)) \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y)),~\mathrm{provided~that}~\mathbf{Y}\neq \emptyset</math>
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| <!-- What about:
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| *[[logical biconditional|Biconditional (if and only if) (xnor)]] (<math>\leftrightarrow</math>, <math>\equiv</math>, or <math>=</math>)
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| *[[Exclusive or|Exclusive disjunction (xor)]] (<math>\not\leftrightarrow</math>)
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| -->
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| === Rules of inference ===
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| A [[rule of inference]] is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the universal quantifier.
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| ''[[Universal instantiation]]'' concludes that, if the propositional function is known to be universally true, then it must be true for any arbitrary element of the Universe of Discourse. Symbolically, this is represented as
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| :<math> \forall{x}{\in}\mathbf{X}\, P(x) \to\ P(c)</math>
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| where ''c'' is a completely arbitrary element of the Universe of Discourse.
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| ''[[Generalization (logic)|Universal generalization]]'' concludes the propositional function must be universally true if it is true for any arbitrary element of the Universe of Discourse. Symbolically, for an arbitrary ''c'',
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| :<math> P(c) \to\ \forall{x}{\in}\mathbf{X}\, P(x).</math>
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| The element ''c'' must be completely arbitrary; else, the logic does not follow: if ''c'' is not arbitrary, and is instead a specific element of the Universe of Discourse, then P(''c'') only implies an existential quantification of the propositional function.
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| <!-- ''Discuss universally quantified types in [[type theory]].'' -->
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| === The empty set ===
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| By convention, the formula <math>\forall{x}{\in}\emptyset \, P(x)</math> is always true, regardless of the formula ''P''(''x''); see [[vacuous truth]].
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| == Universal closure ==
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| The '''universal closure''' of a formula φ is the formula with no [[free variable]]s obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of
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| :<math>P(y) \land \exists x Q(x,z)</math>
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| is
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| :<math>\forall y \forall z ( P(y) \land \exists x Q(x,z))</math>.
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| == As adjoint ==
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| In [[category theory]] and the theory of [[elementary topos|elementary topoi]], the universal quantifier can be understood as the [[right adjoint]] of a [[functor]] between [[power set]]s, the [[inverse image]] functor of a function between sets; likewise, the [[existential quantifier]] is the [[left adjoint]].<ref>Saunders Mac Lane, Ieke Moerdijk, (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. ISBN 0-387-97710-4 ''See page 58''</ref>
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| For a set <math>X</math>, let <math>\mathcal{P}X</math> denote its [[powerset]]. For any function <math>f:X\to Y</math> between sets <math>X</math> and <math>Y</math>, there is an [[inverse image]] functor <math>f^*:\mathcal{P}Y\to \mathcal{P}X</math> between powersets, that takes subsets of the codomain of ''f'' back to subsets of its domain. The left adjoint of this functor is the existential quantifier <math>\exists_f</math> and the right adjoint is the universal quantifier <math>\forall_f</math>.
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| That is, <math>\exists_f\colon \mathcal{P}X\to \mathcal{P}Y</math> is a functor that, for each subset <math>S \subset X</math>, gives the subset <math>\exists_f S \subset Y</math> given by
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| :<math>\exists_f S =\{ y\in Y | \mbox{ there exists } x\in X \mbox{ s.t. } f(x)=y \}</math>.
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| Likewise, the universal quantifier <math>\forall_f\colon \mathcal{P}X\to \mathcal{P}Y</math> is given by
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| :<math>\forall_f S =\{ y\in Y | f(x)=y \mbox{ for all } x\in X \}</math>.
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| The more familiar form of the quantifiers as used in [[first-order logic]] is obtained by taking the function ''f'' to be the [[projection operator]] <math>\pi:X \times \{T,F\}\to \{T,F\}</math> where <math>\{T,F\}</math> is the two-element set holding the values true, false, and subsets ''S'' to be [[predicate (mathematical logic)|predicates]] <math>S\subset X\times \{T,F\}</math>, so that
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| :<math>\exists_\pi S = \{y\,|\,\exists x\, S(x,y)\}</math>
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| which is either a one-element set (false) or a two-element set (true).
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| The universal and existential quantifiers given above generalize to the [[presheaf category]].
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| == See also ==
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| {{Wiktionary|every}}
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| * [[Existential quantification]]
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| * [[Quantifier]]s
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| * [[First-order logic]]
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| * [[List of logic symbols]] - for the unicode symbol ∀
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| == Notes ==
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| <references group="note" />
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| ==References==
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| {{Reflist}}
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| *{{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = [[A K Peters]] | year = 2005 | isbn = 1-56881-262-0}}
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| *{{cite book | author = [[James Franklin (philosopher)|Franklin, J.]] and Daoud, A. | title = Proof in Mathematics: An Introduction | url = http://www.maths.unsw.edu.au/~jim/proofs.html | publisher = Kew Books | year = 2011 | isbn = 978-0-646-54509-7}} (ch. 2)
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| [[Category:Quantification]]
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| [[Category:Logic symbols]]
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| [[Category:Logical expressions]]
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