universal quantifier

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In
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, a universal quantification is a type of quantifier, a
logical constant In logic, a logical constant of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal. Two important types of logical constants are logical connectives and quantifiers. The equality predicate (us ...
which is interpreted as "given any" or "for all". It expresses that a
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
can be satisfied by every
member Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
of a
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 domain ...
. In other words, it is the predication of a
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
or
relation Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
to every member of the domain. It asserts that a predicate within the
scope Scope or scopes may refer to: People with the surname * Jamie Scope (born 1986), English footballer * John T. Scopes (1900–1970), central figure in the Scopes Trial regarding the teaching of evolution Arts, media, and entertainment * Cinem ...
of a universal quantifier is true of every
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value beyo ...
of a
predicate variable In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...
. It is usually denoted by the
turned A Turned A (capital: Ɐ, lowercase: ɐ, math symbol ∀) is a letter and symbol based upon the letter A. Modern Usage * Lowercase ɐ (in Roman or two story form) is used in the International Phonetic Alphabet to identify the near-open centra ...
(∀)
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 co ...
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 a universal quantifier ("", "", or sometimes by "" alone). Universal quantification is distinct from ''existential'' quantification ("there exists"), which only asserts that the property or relation holds for at least one member of the domain. Quantification in general is covered in the article on
quantification (logic) In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything i ...
. The universal quantifier is encoded as in
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
, and as \forall in
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
and related formula editors.

# Basics

Suppose it is given that
2·0 = 0 + 0, and 2·1 = 1 + 1, and , etc.
This would seem to be a
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 ...
because of the repeated use of "and". However, the "etc." cannot be interpreted as a conjunction 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 premis ...
. Instead, the statement must be rephrased:
For all natural numbers ''n'', one has 2·''n'' = ''n'' + ''n''.
This is a single statement using universal quantification. This statement can be said to be more precise than the original one. While the "etc." informally includes
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 nothing more, this was not rigorously given. In the universal quantification, on the other hand, the natural numbers are mentioned explicitly. This particular example is
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
, because any natural number could be substituted for ''n'' and the statement "2·''n'' = ''n'' + ''n''" would be true. In contrast,
For all natural numbers ''n'', one has 2·''n'' > 2 + ''n''
is 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 A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
is enough to prove the universal quantification false. On the other hand, for all
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s ''n'', one has 2·''n'' > 2 + ''n'' is true, because none of the counterexamples are composite numbers. This indicates the importance of 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 domain ...
'', which specifies which values ''n'' can take.Further information on using domains of discourse with quantified statements can be found in the
Quantification (logic) In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything i ...
article.
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 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 premises ...
. For example,
For all composite numbers ''n'', one has 2·''n'' > 2 + ''n''
is
logically equivalent 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 ...
to
For all natural numbers ''n'', if ''n'' is composite, then 2·''n'' > 2 + ''n''.
Here the "if ... then" construction indicates the logical conditional.

## Notation

In
symbolic 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 properties of formal ...
, the universal quantifier symbol $\forall$ (a turned " A" 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 seri ...
font, Unicode U+2200) is used to indicate universal quantification. It was first used in this way by
Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...
in 1935, by analogy with
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The stand ...
's $\exists$ (turned E) notation for
existential quantification 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, whe ...
and the later use of Peano's notation by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
. For example, if ''P''(''n'') is the predicate "2·''n'' > 2 + ''n''" and N 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 : $\forall n\!\in\!\mathbb\; P\left(n\right)$ is the (false) statement :"for all natural numbers ''n'', one has 2·''n'' > 2 + ''n''". Similarly, if ''Q''(''n'') is the predicate "''n'' is composite", then : $\forall n\!\in\!\mathbb\; \bigl\left( Q\left(n\right) \rightarrow P\left(n\right) \bigr\right)$ is the (true) statement :"for all natural numbers ''n'', if ''n'' is composite, then ". Several variations in the notation for quantification (which apply to all forms) can be found in the '' Quantifier'' article.

# Properties

## Negation

The negation of a universally quantified function is obtained by changing the universal quantifier into an
existential quantifier 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, w ...
and negating the quantified formula. That is, :$\lnot \forall x\; P\left(x\right)\quad\text \quad \exists x\;\lnot P\left(x\right)$ where $\lnot$ denotes
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
. For example, if is the
propositional function In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (''x'') that is not defined or specified (thus bei ...
" is married", then, for 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 all living human beings, the universal quantification
Given any living person , that person is married
is written :$\forall x \in X\, P\left(x\right)$ This statement is false. Truthfully, it is stated that
It is not the case that, given any living person , that person is married
or, symbolically: :$\lnot\ \forall x \in X\, P\left(x\right)$. If the function is not true for ''every'' element of , then there must be at least one element for which the statement is false. That is, the negation of $\forall x \in X\, P\left(x\right)$ is logically equivalent to "There exists a living person who is not married", or: :$\exists x \in X\, \lnot P\left(x\right)$ It is erroneous to confuse "all persons are not married" (i.e. "there exists no person who is married") with "not all persons are married" (i.e. "there exists a person who is not married"): :$\lnot\ \exists x \in X\, P\left(x\right) \equiv\ \forall x \in X\, \lnot P\left(x\right) \not\equiv\ \lnot\ \forall x\in X\, P\left(x\right) \equiv\ \exists x \in X\, \lnot P\left(x\right)$

## Other connectives

The universal (and existential) quantifier moves unchanged across the
logical connective 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 co ...
s
Wedge (∧) is a symbol that looks similar to an in-line caret (^). It is used to represent various operations. In Unicode, the symbol is encoded and by \wedge and \land in TeX. The opposite symbol (∨) is called a vel, or sometimes a (descend ...
,
The descending wedge symbol ∨ may represent: * Logical disjunction in propositional logic * Join in lattice theory * The wedge sum in topology The vertically reflected symbol, ∧, is a wedge, and often denotes related or dual operators. The ...
, , and , as long as the other operand is not affected; that is: :$\begin P\left(x\right) \land \left(\exists\mathbf\, Q\left(y\right)\right) &\equiv\ \exists\mathbf\, \left(P\left(x\right) \land Q\left(y\right)\right) \\ P\left(x\right) \lor \left(\exists\mathbf\, Q\left(y\right)\right) &\equiv\ \exists\mathbf\, \left(P\left(x\right) \lor Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \to \left(\exists\mathbf\, Q\left(y\right)\right) &\equiv\ \exists\mathbf\, \left(P\left(x\right) \to Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \nleftarrow \left(\exists\mathbf\, Q\left(y\right)\right) &\equiv\ \exists\mathbf\, \left(P\left(x\right) \nleftarrow Q\left(y\right)\right) \\ P\left(x\right) \land \left(\forall\mathbf\, Q\left(y\right)\right) &\equiv\ \forall\mathbf\, \left(P\left(x\right) \land Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \lor \left(\forall\mathbf\, Q\left(y\right)\right) &\equiv\ \forall\mathbf\, \left(P\left(x\right) \lor Q\left(y\right)\right) \\ P\left(x\right) \to \left(\forall\mathbf\, Q\left(y\right)\right) &\equiv\ \forall\mathbf\, \left(P\left(x\right) \to Q\left(y\right)\right) \\ P\left(x\right) \nleftarrow \left(\forall\mathbf\, Q\left(y\right)\right) &\equiv\ \forall\mathbf\, \left(P\left(x\right) \nleftarrow Q\left(y\right)\right),& \text \mathbf\neq \emptyset \end$ Conversely, for the logical connectives , , , and , the quantifiers flip: :$\begin P\left(x\right) \uparrow \left(\exists\mathbf\, Q\left(y\right)\right) & \equiv\ \forall\mathbf\, \left(P\left(x\right) \uparrow Q\left(y\right)\right) \\ P\left(x\right) \downarrow \left(\exists\mathbf\, Q\left(y\right)\right) & \equiv\ \forall\mathbf\, \left(P\left(x\right) \downarrow Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \nrightarrow \left(\exists\mathbf\, Q\left(y\right)\right) & \equiv\ \forall\mathbf\, \left(P\left(x\right) \nrightarrow Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \gets \left(\exists\mathbf\, Q\left(y\right)\right) & \equiv\ \forall\mathbf\, \left(P\left(x\right) \gets Q\left(y\right)\right) \\ P\left(x\right) \uparrow \left(\forall\mathbf\, Q\left(y\right)\right) & \equiv\ \exists\mathbf\, \left(P\left(x\right) \uparrow Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ P\left(x\right) \downarrow \left(\forall\mathbf\, Q\left(y\right)\right) & \equiv\ \exists\mathbf\, \left(P\left(x\right) \downarrow Q\left(y\right)\right) \\ P\left(x\right) \nrightarrow \left(\forall\mathbf\, Q\left(y\right)\right) & \equiv\ \exists\mathbf\, \left(P\left(x\right) \nrightarrow Q\left(y\right)\right) \\ P\left(x\right) \gets \left(\forall\mathbf\, Q\left(y\right)\right) & \equiv\ \exists\mathbf\, \left(P\left(x\right) \gets Q\left(y\right)\right),& \text \mathbf\neq \emptyset \\ \end$

## 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 in ...
is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the universal quantifier. ''
Universal instantiation In predicate logic, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with '' dictum de omni'') is a valid rule of inference from a truth about each member of a class of individuals ...
'' 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 :$\forall\mathbf\, P\left(x\right) \to P\left(c\right)$ where ''c'' is a completely arbitrary element of the universe of discourse. ''
Universal generalization In predicate logic, generalization (also universal generalization or universal introduction,Moore and Parker GEN) is a valid inference rule. It states that if \vdash \!P(x) has been derived, then \vdash \!\forall x \, P(x) can be derived. Gener ...
'' 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'', :$P\left(c\right) \to\ \forall\mathbf\, P\left(x\right).$ 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.

## The empty set

By convention, the formula $\forall\emptyset \, P\left(x\right)$ is always true, regardless of the formula ''P''(''x''); see
vacuous truth In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
.

# Universal closure

The universal closure of a formula φ is the formula with no
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
s obtained by adding a universal quantifier for every free variable in φ. For example, the universal closure of :$P\left(y\right) \land \exists x Q\left(x,z\right)$ is :$\forall y \forall z \left( P\left(y\right) \land \exists x Q\left(x,z\right)\right)$.

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 universal quantifier can be understood as the
right 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 kn ...
of a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 po ...
s, 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
existential quantifier 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, w ...
is 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 ...
.
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...
, Ieke Moerdijk, (1992) ''Sheaves in Geometry and Logic'' Springer-Verlag. ''See page 58''
For a set $X$, let $\mathcalX$ denote its
powerset 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 postu ...
. For any function $f:X\to Y$ between sets $X$ and $Y$, there is an
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 $f^*:\mathcalY\to \mathcalX$ 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 $\exists_f$ and the right adjoint is the universal quantifier $\forall_f$. That is, $\exists_f\colon \mathcalX\to \mathcalY$ is a functor that, for each subset $S \subset X$, gives the subset $\exists_f S \subset Y$ given by :$\exists_f S =\,$ those $y$ in the image of $S$ under $f$. Similarly, the universal quantifier $\forall_f\colon \mathcalX\to \mathcalY$ is a functor that, for each subset $S \subset X$, gives the subset $\forall_f S \subset Y$ given by :$\forall_f S =\,$ those $y$ whose preimage under $f$ is contained in $S$. The more familiar form of the quantifiers as used in
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 ...
is obtained by taking the function ''f'' to be the unique function $!:X \to 1$ so that $\mathcal\left(1\right) = \$ is the two-element set holding the values true and false, a subset ''S'' is that subset for which the
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, o ...
$S\left(x\right)$ holds, and :$\begin\mathcal\left(!\right)\colon \mathcal\left(1\right) & \to \mathcal\left(X\right)\\ T &\mapsto X \\ F &\mapsto \\end$ :$\exists_! S = \exists x. S\left(x\right),$ which is true if $S$ is not empty, and :$\forall_! S = \forall x. S\left(x\right),$ which is false if S is not X. The universal and existential quantifiers given above generalize to the
presheaf category In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on ...
.

*
Existential quantification 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, whe ...
*
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 ...
*
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 ∀

* * (ch. 2)