Quantifier (logic)
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In
The most commonly used quantifiers are and . These quantifiers are standardly defined as
In a first-order logic statement, quantifications in the same type (either universal quantifications or existential quantifications) can be exchanged without changing the meaning of the statement, while the exchange of quantifications in different types changes the meaning. As an example, the only difference in the definition of uniform continuity and (ordinary) continuity is the order of quantifications.
First order quantifiers approximate the meanings of some natural language quantifiers such as "some" and "all". However, many natural language quantifiers can only be analyzed in terms of generalized quantifiers.
Relations to logical conjunction and disjunction
For a finite domain of discourse , the universally quantified formula is equivalent to the logical conjunction . Dually, the existentially quantified formula is equivalent to the logical disjunction . For example, if is the set of
Infinite domain of discourse
Consider the following statement (using dot notation for multiplication):
- 1 · 2 = 1 + 1, and 2 · 2 = 2 + 2, and 3 · 2 = 3 + 3, ..., and 100 · 2 = 100 + 100, and ..., etc.
This has the appearance of an infinite conjunction of propositions. From the point of view of formal languages, this is immediately a problem, since syntax rules are expected to generate finite words.
The example above is fortunate in that there is a procedure to generate all the conjuncts. However, if an assertion were to be made about every irrational number, there would be no way to enumerate all the conjuncts, since irrationals cannot be enumerated. A succinct, equivalent formulation which avoids these problems uses universal quantification:
- For each natural number n, n · 2 = n + n.
A similar analysis applies to the
- 1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc.
which can be rephrased using existential quantification:
- For some natural number n, n is equal to 5+5.
Algebraic approaches to quantification
It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow^{[clarification needed]} and interest in such algebra has been limited. Three approaches have been devised to date:
- Peano arithmetic;
- Cylindric algebra, devised by Alfred Tarski, Leon Henkin, and others;
- The polyadic algebra of Paul Halmos.
Notation
The two most common quantifiers are the universal quantifier and the existential quantifier. The traditional symbol for the universal quantifier is "
An example of translating a quantified statement in a natural language such as English would be as follows. Given the statement, "Each of Peter's friends either likes to dance or likes to go to the beach (or both)", key aspects can be identified and rewritten using symbols including quantifiers. So, let X be the set of all Peter's friends, P(x) the predicate "x likes to dance", and Q(x) the predicate "x likes to go to the beach". Then the above sentence can be written in formal notation as , which is read, "for every x that is a member of X, P applies to x
Some other quantified expressions are constructed as follows,
- ^{[4]}
for a formula P. These two expressions (using the definitions above) are read as "there exists a friend of Peter who likes to dance" and "all friends of Peter like to dance", respectively. Variant notations include, for set X and set members x:
- ^{[5]} ^{[6]}
All of these variations also apply to universal quantification. Other variations for the universal quantifier are
- ^{[citation needed]} ^{[7]} ^{[8]}
Some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified; for a given mathematical theory, this can be done in several ways:
- Assume a fixed domain of discourse for every quantification, as is done in Zermelo–Fraenkel set theory,
- Fix several domains of discourse in advance and require that each variable have a declared domain, which is the type of that variable. This is analogous to the situation in statically typed computer programming languages, where variables have declared types.
- Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain (or the typeof the objects in that domain).
One can use any variable as a quantified variable in place of any other, under certain restrictions in which variable capture does not occur. Even if the notation uses typed variables, variables of that type may be used.
Informally or in natural language, the "∀x" or "∃x" might appear after or in the middle of P(x). Formally, however, the phrase that introduces the dummy variable is placed in front.
Mathematical formulas mix symbolic expressions for quantifiers with natural language quantifiers such as,
- For every natural number x, ...
- There exists an x such that ...
- For at least one x, ....
Keywords for uniqueness quantification include:
- For exactly one natural number x, ...
- There is one and only one x such that ....
Further, x may be replaced by a pronoun. For example,
- For every natural number, its product with 2 equals to its sum with itself.
- Some natural number is prime.
Order of quantifiers (nesting)
The order of quantifiers is critical to meaning, as is illustrated by the following two propositions:
- For every natural number n, there exists a natural number s such that s = n^{2}.
This is clearly true; it just asserts that every natural number has a square. The meaning of the assertion in which the order of quantifiers is reversed is different:
- There exists a natural number s such that for every natural number n, s = n^{2}.
This is clearly false; it asserts that there is a single natural number s that is the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables.
A less trivial example from
- Pointwise continuous if
- Uniformly continuous if
In the former case, the particular value chosen for δ can be a function of both ε and x, the variables that precede it. In the latter case, δ can be a function only of ε (i.e., it has to be chosen independent of x). For example, f(x) = x^{2} satisfies pointwise, but not uniform continuity (its slope is unbound). In contrast, interchanging the two initial universal quantifiers in the definition of pointwise continuity does not change the meaning.
As a general rule, swapping two adjacent universal quantifiers with the same scope (or swapping two adjacent existential quantifiers with the same scope) doesn't change the meaning of the formula (see Example here), but swapping an existential quantifier and an adjacent universal quantifier may change its meaning.
The maximum depth of nesting of quantifiers in a formula is called its "quantifier rank".
Equivalent expressions
If D is a domain of x and P(x) is a predicate dependent on object variable x, then the universal proposition can be expressed as
This notation is known as restricted or relativized or bounded quantification. Equivalently one can write,
The existential proposition can be expressed with bounded quantification as
or equivalently
Together with negation, only one of either the universal or existential quantifier is needed to perform both tasks:
which shows that to disprove a "for all x" proposition, one needs no more than to find an x for which the predicate is false. Similarly,
to disprove a "there exists an x" proposition, one needs to show that the predicate is false for all x.
In
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement " such that " can be viewed as a question "When is there an such that ?", and the statement without quantifiers can be viewed as the answer to that question.^{[9]}
One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest.
ARange of quantification
Every quantification involves one specific variable and a domain of discourse or range of quantification of that variable. The range of quantification specifies the set of values that the variable takes. In the examples above, the range of quantification is the set of natural numbers. Specification of the range of quantification allows us to express the difference between, say, asserting that a predicate holds for some natural number or for some real number. Expository conventions often reserve some variable names such as "n" for natural numbers, and "x" for real numbers, although relying exclusively on naming conventions cannot work in general, since ranges of variables can change in the course of a mathematical argument.
A universally quantified formula over an empty range (like ) is always
A more natural way to restrict the domain of discourse uses guarded quantification. For example, the guarded quantification
- For some natural number n, n is even and n is prime
means
- For some even numbern, n is prime.
In some
- For every natural number n, n·2 = n + n
in Zermelo–Fraenkel set theory, one would write
- For every n, if n belongs to N, then n·2 = n + n,
where N is the set of all natural numbers.
Formal semantics
Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: a mathematical specification of a class of objects via syntax, a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted. The syntax of a formula can be given by a syntax tree. A quantifier has a
the occurrence of both x and y in C(y, x) is free, while the occurrence of x and y in B(y, x) is bound (i.e. non-free).
An
is the function G of n-1 arguments such that G(v_{1}, ..., v_{n-1}) = T if and only if F(v_{1}, ..., v_{n-1}, w) = T for every w in X. If F(v_{1}, ..., v_{n-1}, w) = F for at least one value of w, then G(v_{1}, ..., v_{n-1}) = F. Similarly the interpretation of the formula
is the function H of n-1 arguments such that H(v_{1}, ..., v_{n-1}) = T if and only if F(v_{1}, ..., v_{n-1}, w) = T for at least one w and H(v_{1}, ..., v_{n-1}) = F otherwise.
The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X. The interpretation of
then is the function of n-1 arguments, which is the logical and of the interpretations of
Each kind of quantification defines a corresponding
Paucal, multal and other degree quantifiers
None of the quantifiers previously discussed apply to a quantification such as
- There are many integers n < 100, such that n is divisible by 2 or 3 or 5.
One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain X, we have given a probability measure P defined on X and cutoff numbers 0 < a ≤ b ≤ 1. If A is a formula with free variables x_{1},...,x_{n} whose interpretation is the function F of variables v_{1},...,v_{n} then the interpretation of
is the function of v_{1},...,v_{n-1} which is T if and only if
and F otherwise. Similarly, the interpretation of
is the function of v_{1},...,v_{n-1} which is F if and only if
and T otherwise.
Other quantifiers
A few other quantifiers have been proposed over time. In particular, the solution quantifier,^{[11]}^{: 28 } noted § (section sign) and read "those". For example,
is read "those n in N such that n^{2} ≤ 4 are in {0,1,2}." The same construct is expressible in set-builder notation as
Contrary to the other quantifiers, § yields a set rather than a formula.^{[12]}
Some other quantifiers sometimes used in mathematics include:
- There are infinitely many elements such that...
- For all but finitely many elements... (sometimes expressed as "for almost all elements...").
- There are uncountably many elements such that...
- For all but countably many elements...
- For all elements in a set of positive measure...
- For all elements except those in a set of measure zero...
History
In 1827, George Bentham published his Outline of a New System of Logic: With a Critical Examination of Dr. Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.^{[13]}
William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. Augustus De Morgan confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both all and some-not-all as quantifiers".^{[14]}
Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀x~, or contraposition. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics.
In work that culminated in Peirce (1885),
Peirce's approach to quantification also influenced
Around 1895, Peirce began developing his
See also
- Absolute generality
- Almost all
- Branching quantifier
- Conditional quantifier
- Counting quantification
- Eventually (mathematics)
- noun phrases
- Lindström quantifier — a generalized polyadic quantifier
- Quantifier shift
References
- ^ Kashef, Arman. (2023), In Quest of Univeral Logic: A brief overview of formal logic's evolution,
- ^ "Predicates and Quantifiers". www.csm.ornl.gov. Retrieved 2020-09-04.
- ^ "1.2 Quantifiers". www.whitman.edu. Retrieved 2020-09-04.
- ISBN 0-444-88074-7. Here: p.497
- .
- . Here: p.p.344
- . Here: Def. II.1.5
- .
- ^ Brown 2002.
- ^ in general, for a quantifer Q, closure makes sense only if the order of Q quantification does not matter, i.e. if Qx Qy p(x,y) is equivalent to Qy Qx p(x,y). This is satisfied for Q ∈ {∀,∃}, cf. #Order of quantifiers (nesting) above.
- Hehner, Eric C. R., 2004, Practical Theory of Programming, 2nd edition, p. 28
- ^ Hehner (2004) uses the term "quantifier" in a very general sense, also including e.g. summation.
- ISBN 1-85506-029-9
- .
Bibliography
- Barwise, Jon; and Etchemendy, John, 2000. Language Proof and Logic. CSLI (University of Chicago Press) and New York: Seven Bridges Press. A gentle introduction to first-order logic by two first-rate logicians.
- Brown, Christopher W. (July 31, 2002). "What is Quantifier Elimination". Retrieved Aug 30, 2018.
- Frege, Gottlob, 1879. Begriffsschrift. Translated in Jean van Heijenoort, 1967. From Frege to Gödel: A Source Book on Mathematical Logic, 1879-1931. Harvard University Press. The first appearance of quantification.
- Hilbert, David; and Ackermann, Wilhelm, 1950 (1928). Principles of Mathematical Logic. Chelsea. Translation of Grundzüge der theoretischen Logik. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standard manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of first-order logic.
- Peirce, C. S., 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, American Journal of Mathematics, Vol. 7, pp. 180–202. Reprinted in Kloesel, N. et al., eds., 1993. Writings of C. S. Peirce, Vol. 5. Indiana University Press. The first appearance of quantification in anything like its present form.
- Reichenbach, Hans, 1975 (1947). Elements of Symbolic Logic, Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".
- Westerståhl, Dag, 2001, "Quantifiers," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
- Wiese, Heike, 2003. Numbers, language, and the human mind. Cambridge University Press. ISBN 0-521-83182-2.
External links
- "Quantifier", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- ""For all" and "there exists" topical phrases, sentences and expressions". Archived from the original on March 1, 2000.. From College of Natural Sciences, University of Hawaii at Manoa.
- Stanford Encyclopedia of Philosophy:
- Shapiro, Stewart (2000). "Classical Logic" (Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.)
- Westerståhl, Dag (2005). "Generalized quantifiers"
- Peters, Stanley; Westerståhl, Dag (2002). "Quantifiers" Archived 2012-07-16 at the Wayback Machine