Quantification

Quantification has two distinct meanings.

In mathematics and empirical science, it refers to human acts, known as counting and measuring that map human sense observations and experiences into members of some set of numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Philosophy, empiricism is a theory of Knowledge which asserts that knowledge arises from Experience. Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data Experience as a general concept comprises Knowledge of or skill in or Observation of some thing or some event gained through involvement in or In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. Quantification in this sense is fundamental to the scientific method. Scientific method refers to bodies of Techniques for investigating phenomena

In logic, quantification refers to an operator that binds a variable ranging over a domain of discourse. Logic is the study of the principles of valid demonstration and Inference. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive The variable thereby becomes bound. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a Academic discussion of quantification refers more often to this meaning of the term than the preceding one.

1 Natural language
2 Logic
3 Syntax
4 History
5 Science
6 Development of quantitification both across species and within humans
7 See also
8 Notes
9 References
10 External links

Natural language

All known human languages make use of quantification (Wiese 2004). For example, in English:

Every glass in my recent order was chipped.
Some of the people standing across the river have white armbands.
Most of the people I talked to didn't have a clue who the candidates were.
Everyone in the waiting room had at least one complaint against Dr. Ballyhoo.
There was somebody in his class that was able to correctly answer every one of the questions I submitted.
A lot of people are smart.

The words in italics are called quantifiers.

There exists no simple way of reformulating any one of these expressions as a conjunction or disjunction of sentences, each a simple predicate of an individual such as That wine glass was chipped. These examples also suggest that the construction of quantified expressions in natural language can be syntactically very complicated. Fortunately, for mathematical assertions, the quantification process is syntactically more straightforward.

The study of quantification in natural languages is much more difficult than the corresponding problem for formal languages. This comes in part from the fact that the grammatical structure of natural language sentences may conceal the logical structure. Moreover, mathematical conventions strictly specify the range of validity for formal language quantifiers; for natural language, specifying the range of validity requires dealing with non-trivial semantic problems.

Montague grammar gives a novel formal semantics of natural languages. Montague grammar is an approach to Natural language Semantics, named after American Logician Richard Montague. Its proponents argue that it provides a much more natural formal rendering of natural language than the traditional treatments of Frege, Russell and Quine. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van"

Logic

More specifically, in language and logic, quantification is a construct that specifies the quantity of individuals of the domain of discourse that apply to (or satisfy) an open formula. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them Logic is the study of the principles of valid demonstration and Inference. The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive For example, in arithmetic, it allows the expression of the statement that every natural number has a successor, and in logic, that something (at least one thing) in the domain of discourse has a certain property, i. e. , there exist things with that property in the domain. A language element which generates a quantification is called a quantifier. The resulting expression is a quantified expression, and we say we have quantified over the predicate or function expression whose free variable is bound by the quantifier. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a Quantification is used in both natural languages and formal languages. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related A formal language is a set of words, ie finite strings of letters, or symbols. Examples of quantifiers in a natural language are: for all, for some, many, few, a lot, and no. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted as an extent of validity. Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from Quantification is an example of a variable-binding operation.

The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. Sometimes it is inconvenient or impossible to describe a set by listing all of its elements In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain These concepts are covered in detail in their individual articles; here we discuss features of quantification that apply in both cases. Other kinds of quantification include uniqueness quantification. In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists

The traditional symbol for the universal quantifier "all" is "∀", an inverted letter "A", and for the existential quantifier "exists" is "∃", a rotated letter "E". The letter A is the first letter in the Latin alphabet. Its name in English is a (eɪ plural E is the fifth letter in the Latin alphabet. Its name in English is spelled e (iː plural es or ees (also written E's E These quantifiers have been generalized beginning with the work of Mostowski and Lindström. See generalized quantifier and Lindström quantifier for further details. In linguistic Semantics, a generalized quantifier is an expression that denotes a property of a property also called a Higher-order property In Mathematical logic, a Lindström quantifier is a generalized polyadic quantifier.

Mathematics

We will begin by discussing quantification in informal mathematical discourse. Consider the following statement

1·2 = 1 + 1, and 2·2 = 2 + 2, and 3 · 2 = 3 + 3, . . . . , and n · 2 = n + n, etc.

This has the appearance of an infinite conjunction of propositions. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of From the point of view of formal languages this is immediately a problem, since we expect syntax rules to generate finite objects. A formal language is a set of words, ie finite strings of letters, or symbols. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Putting aside this objection, also note that in this example we were lucky in that there is a procedure to generate all the conjuncts. However, if we wanted to assert something about every irrational number, we would have no way enumerating all the conjuncts since irrationals cannot be enumerated. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction A succinct formulation which avoids these problems uses universal quantification:

For any natural number n, n·2 = n + n. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

A similar analysis applies to the disjunction,

1 is prime, or 2 is prime, or 3 is prime, etc. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

which can be rephrased using existential quantification:

For some natural number n, n is prime. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

It is possible to devise abstract algebras whose models include formal languages with quantification, but progress has been slow and interest in such algebra has been limited. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models A formal language is a set of words, ie finite strings of letters, or symbols. Three approaches have been devised to date:

Relation algebra, invented by DeMorgan, and developed by Ernst Schroder, Tarski, and Tarski's students. Relation algebra is different from Relational algebra, a framework developed by Edgar Codd in 1970 for Relational databases. Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. For the actor see Ernst Schröder (actor. Ernst Schröder ( 25 November, 1841 Mannheim Germany – Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Relation algebra cannot represent any formula with quantifiers nested more than three deep. Surprisingly, the models of relation algebra include the axiomatic set theory ZFC and Peano arithmetic;
Cylindric algebra, devised by Tarski, Henkin, and others;
The polyadic algebra of Paul Halmos. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of First-order logic. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California Henkin is a Jewish last name and may refer to the following people Leon Henkin, Logician and Mathematician Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician

Notation

The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The letter A is the first letter in the Latin alphabet. Its name in English is a (eɪ plural The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". E is the fifth letter in the Latin alphabet. Its name in English is spelled e (iː plural es or ees (also written E's E Correspondingly, quantified expressions are constructed as follows,

$\exists{x}\, P \quad \forall{x}\, P$

where "P" denotes a formula. Many variant notations are used, such as

$\exists{x}\, P \quad (\exists{x}) P \quad (\exists x \ . \ P) \quad (\exists x : P) \quad \exists{x}(P) \quad \exists_{x}\, P \quad \exists{x}{,}\, P \quad \exists{x}{\in}\mathbb{N}\, P \quad \exists\, x{:}\mathbb{N}\, P$

All of these variations also apply to universal quantification. Other variations for the universal quantifier are

$(x) \, P \quad \bigwedge_{x} P$

Note that some versions of the notation explicitly mention the range of quantification. The range of quantification must always be specified, but 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. In Computer science, a type system defines how a Programming language classifies values and expressions into '''types''', how it can
Mention explicitly the range of quantification, perhaps using a symbol for the set of all objects in that domain or the type of the objects in that domain. In Mathematics, Logic and Computer science, type theory is any of several Formal systems that can serve as alternatives to Naive set theory

Also note that one can use any variable as a quantified variable in place of any other, under certain restrictions, that is in which variable capture does not occur. Even if the notation uses typed variables, one can still use any variable of that type. The issue of variable capture is exceedingly important, and we discuss that in the formal semantics below.

Informally, the "∀x" or "∃x" might well appear after P(x), or even in the middle if P(x) is a long phrase. Formally, however, the phrase that introduces the dummy variable is standardly placed in front. See also above.

Note that mathematical formulas mix symbolic expressions for quantifiers, with natural language quantifiers such as

For any 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, . In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists . . .

There is one and only one x such that . . . .

One might even avoid variable names such as x using a pronoun. In Linguistics and Grammar, a pronoun is a Pro-form that substitutes for a (including a noun phrase consisting of a single Noun) with or For example,

For any natural number, its product with 2 equals to its sum with itself

Some natural number is prime.

Nesting

Consider the following statement:

For any natural number n, there is a natural number s such that s = n × n.

This is clearly true; it just asserts that every natural number has a square.

The meaning of the assertion in which the quantifiers are turned around is quite different:

There is a natural number s such that for any natural number n, s = n × n.

This is clearly false; it asserts that there is a single natural number s that is at once the square of every natural number.

This illustrates a fundamentally important point when quantifiers are nested: The order of alternation of quantifiers is of absolute importance.

A less trivial example is the important concept of uniform continuity from analysis, which differs from the more familiar concept of pointwise continuity only by an exchange in the positions of two quantifiers. In Mathematical analysis, a function f ( x) is called uniformly continuous if roughly speaking small changes in the input x effect Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output To illustrate this, let f be a real-valued function on R.

A: Pointwise continuity of f on R:

$\underbrace{\forall x \in \mathbb{R}, \ \forall \epsilon >0}, \exists \delta > 0, \forall h \in \mathbb{R}, \quad |h| < \delta \implies |f(x) - f(x+h)| < \epsilon$

interchanging the universal quantifiers over the braces, this is the same as

A': Pointwise continuity of f on R:

$\forall \epsilon >0, \ \underbrace{\forall x \in \mathbb{R}, \exists \delta > 0}, \ \forall h \in \mathbb{R}, \quad |h| < \delta \implies |f(x) - f(x+h)| < \epsilon$

This differs from

B: Uniform continuity of f on R:

$\forall \epsilon >0, \underbrace{\exists \delta > 0, \forall x \in \mathbb{R}}, \forall h \in \mathbb{R}, \quad |h| < \delta\implies |f(x) - f(x+h)| < \epsilon$

by interchanging the existential and universal quantifiers over the braces in A'.

Ambiguity is avoided by putting the quantifiers (in symbols or words) in front:

$\exists$ A: $\forall$ B: C - unambiguous
there is an A such that $\forall$ B: C - unambiguous
there is an A such that for all B, C - unambiguous, provided that the separation between B and C is clear
there is an A such that C for all B - it is often clear that what is meant is

there is an A such that (C for all B)

but it could be interpreted as

(there is an A such that C) for all B

there is an A such that C $\forall$ B - suggests more strongly that the first is meant; this may be reinforced by the layout, for example by putting "C $\forall$ B" on a new line.

Range 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, asserting that a predicate holds for some natural number or for some real number. In Mathematics, the real numbers may be described informally in several different ways 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 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 number n, n is prime. In Mathematics, the parity of an object states whether it is even or odd

In some mathematical theories one assumes a single domain of discourse fixed in advance. For example, in Zermelo Fraenkel set theory, variables range over all sets. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above to express

For any natural number n, n·2 = n + n

in Zermelo-Fraenkel set theory, one can say

For any 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—that is, mathematically specified—language. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A formal language is a set of words, ie finite strings of letters, or symbols. 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. In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the In this article, we only address the issue of how quantifier elements are interpreted.

In this section we only consider first-order logic with function symbols. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science We refer the reader to the article on model theory for more information on the interpretation of formulas within this logical framework. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models The syntax of a formula can be given by a syntax tree. Quantifiers have scope and a variable x is free if it is not within the scope of a quantification for that variable. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a Thus in

$\forall x (\exists y B(x,y)) \vee C(y,x)$

the occurrence of both x and y in C(y,x) is free.

Syntactic tree illustrating scope and variable capture

An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x₁, . . . , x_n is interpreted as a boolean-valued function F(v₁, . . . , v_n) of n arguments, where each argument ranges over the domain X. Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood) . The interpretation of the formula

$\forall x_n A(x_1, \ldots , x_n)$

is the function G of n-1 arguments such that G(v₁, . . . ,v_n-1) = T if and only if F(v₁, . . . , v_n-1, w) = T for every w in X. If F(v₁, . . . , v_n-1, w) = F for at least one value of w, then G(v₁, . . . ,v_n-1) = F. Similarly the interpretation of the formula

$\exists x_n A(x_1, \ldots , x_n)$

is the function H of n-1 arguments such that H(v₁, . . . ,v_n-1) = T if and only if F(v₁, . . . ,v_n-1, w) = T for at least one w and H(v₁, . . . , v_n-1) = F otherwise.

The semantics for uniqueness quantification requires first-order predicate calculus with equality. In Mathematics and Logic, the phrase "there is one and only one " is used to indicate that exactly one object with a certain property exists 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

$\exists ! x_n A(x_1, \ldots , x_n)$

then is the function of n-1 arguments, which is the logical and of the interpretations of

$\exists x_n A(x_1, \ldots , x_n)$

$\forall y,z \left\{ A(x_1, \ldots ,x_{n-1}, y) \wedge A(x_1, \ldots ,x_{n-1}, z) \implies y = z \right\}$

Paucal, multal and other degree quantifiers

So far we have only considered universal, existential and uniqueness quantification as used in mathematics. None of this applies to a quantification such as

There were many dancers out on the dance floor this evening.

Although this article will not treat the semantics of natural language, we will attempt to provide a semantics for assertions in a formal language of the type

There are many integers n < 100, such that n is divisible by 2 or 3 or 5.

One possible interpretation mechanism can 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. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. If A is a formula with free variables x₁,. . . ,x_n whose interpretation is the function F of variables v₁,. . . ,v_n then the interpretation of

$\exists^{\mathrm{many}} x_n A(x_1, \ldots, x_{n-1}, x_n)$

is the function of v₁,. . . ,v_n-1 which is T if and only if

$\operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T} \} \geq b$

and F otherwise. Similarly, the interpretation of

$\exists^{\mathrm{few}} x_n A(x_1, \ldots, x_{n-1}, x_n)$

is the function of v₁,. . . ,v_n-1 which is F if and only if

$0< \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T}\} \leq a$

and T otherwise. We have completely avoided discussion of technical issues regarding measurability of the interpretation functions; some of these are technical questions that require Fubini's theorem. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematical analysis, Fubini's theorem, named after Guido Fubini, states that if \int_{A\times B} |f(xy|\d(xy

We caution the reader that the logic corresponding to such semantics is exceedingly complicated.

Syntax

Quantification in formal and natural languages falls under syntax and semantics. A formal language is a set of words, ie finite strings of letters, or symbols. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related In Linguistics, syntax (from Ancient Greek grc συν- syn-, "together" and grc τάξις táxis, "arrangement" is the Semantics is the study of meaning in communication The word derives from Greek σημαντικός ( semantikos) "significant" from

History

Term logic treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. In Philosophy, term logic, also known as traditional logic, is a loose name for the way of doing logic that began with Aristotle, and that was dominant Aristotelian logic treated All', Some and No in the 1st century BC, in an account also touching on the alethic modalities. The Organon is the name given by Aristotle 's followers the Peripatetics to the standard collection of his six works on Logic. The 1st century BC started the first day of 100 BC and ended the last day of 1 BC. Some languages distinguish between alethic moods and non-alethic moods

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. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Begriffsschrift is the title of a short book on Logic by Gottlob Frege, published in 1879, and is also the name of the Formal system The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive 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. For contraposition in the field of traditional logic see Contraposition (traditional logic. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics. Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian

In work that culminated in Peirce (1885), Charles Sanders Peirce and his student O. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher H. Mitchell independently invented universal and existential qunatifiers, and bound variables. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a Peirce and Mitchell wrote Π_x and Σ_x where we now write ∀x and ∃x. Peirce's notation can be found in the writings of Ernst Schroder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. For the actor see Ernst Schröder (actor. Ernst Schröder ( 25 November, 1841 Mannheim Germany – Leopold Löwenheim (1878 Krefeld Germany - 1957 Berlin) was a German Mathematician, known for his work in Mathematical logic. Thoralf Albert Skolem ( May 23, 1887 – March 23, 1963) (ˈtɔɾɑlf ˈskuləm was a Norwegian Mathematician known Most notably, it is the notation of Kurt Goedel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Gödel's completeness theorem is a fundamental theorem in Mathematical logic that establishes a correspondence between semantic truth and syntactic provability in First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural

Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ∃x for the existential quantification of x. William Ernest Johnson ( June 23, 1858 – January 14, 1931) was a British Logician mainly remembered for his Logic (1921–1924 Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional Hence for decades, the canonical notation in philosophy and mathematical logic was (x)P to express "all individuals in the domain of discourse have the property P," and "(∃x)P" for "there exists at least one individual in the domain of discourse having the property P. " Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. Gerhard Karl Erich Gentzen ( November 24, 1909, Greifswald, Germany &ndash August 4, 1945, Prague, Czechoslovakia ∀ did not become canonical until the 1960s.

Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. An existential graph is a type of Diagrammatic or visual notation for logical expressions proposed by Charles Sanders Peirce, who wrote his first paper on graphical Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations. ) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference. A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for

Science

Some measure of the undisputed general importance of quantification in the natural sciences can be gleaned from the following comments: these are mere facts, but they are quantitative facts and the basis of science. ^[1] It seems to be held as universally true that the foundation of quantification is measurement. ^[2] There is little doubt that quantification provided a basis for the objectivity of science. ^[3] In ancient times, musicians and artists. . . rejected quantification, but merchants, by definition, quantified their affairs, in order to survive, made them visible on parchment and paper. ^[4] Any reasonable comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification. ^[5] Even today, universities use imperfect instruments called 'exams' to indirectly quantify something they call knowledge. This article is about a type of examination for other uses see Final examination (disambiguation A final examination (or final ^[6] This meaning of quantification comes under the heading of pragmatics. Pragmatics is the study of the ability of Natural language speakers to communicate more than that which is explicitly stated

Development of quantitification both across species and within humans

In Quantitative analysis of behavior, Evolutionary Psychology and Cognitive Developmental Psychology, quantification is studied as behavior. The Society was founded in 1978 by Michael Lamport Commons and John Anthony Nevin. Evolutionary psychology ( EP) attempts to explain mental and psychological traits ��such as Memory, Perception,

Notes

^ Cattell, J. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of First-order logic. The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive In linguistic Semantics, a generalized quantifier is an expression that denotes a property of a property also called a Higher-order property An indefinite pronoun is a Pronoun that refers to one or more unspecified beings objects or places In Mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Montague grammar is an approach to Natural language Semantics, named after American Logician Richard Montague. Relation algebra is different from Relational algebra, a framework developed by Edgar Codd in 1970 for Relational databases. In Mathematics, and in other disciplines involving Formal languages including Mathematical logic and Computer science, a free variable is a M. , & Farrand, L. (1896) "Physical and mental measurements of the students of Columbia University," The Psychological Review 3: 618-648, p. 648 quoted in James McKeen Cattell (1860-1944) Psychologist, Publisher, and Editor.
^ Wilks, S. S. (1961) "Some Aspects of Quantification in Science," Isis 52: 135-142, p. 135.
^ Hong, "History of Science: Building Circuits of Trust," Science, 10 September 2004: 1569-1570.
^ Crosby (1996: 201).
^ Langs, Robert (1987) "Psychoanalysis as an Aristotelian Science—Pathways to Copernicus and a Modern-Day Approach," Contemporary Psychoanalysis 23: 555-576.
^ Lynch, Aaron 1999) "Misleading Mix of Religion and Science," Journal of Memetics - Evolutionary Models of Information Transmission 3.

References

Jon Barwise and John Etchemendy, 2000. Kenneth Jon Barwise ( June 29, 1942 - March 5, 2000) was a US John W Etchemendy (b 1952 in Reno Nevada is Stanford University 's twelfth and current Provost. 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. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science
Crosby, Alfred W. (1996) The Measure of Reality: Quantification and Western Society, 1250-1600. Cambridge University Press.
Gottlob Frege, 1879. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Begriffsschrift. Begriffsschrift is the title of a short book on Logic by Gottlob Frege, published in 1879, and is also the name of the Formal system Translated in Jean van Heijenoort, 1967. Jean Louis Maxime Van Heijenoort (pronounced highenort) ( July 23 1912, Creil France - March 29 1986, Mexico City From Frege to Godel: A Source Book on Mathematical Logic, 1879-1931. Harvard Univ. Press. The first appearance of quantification.
David Hilbert and Wilhelm Ackermann, 1950 (1928). David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Wilhelm Friedrich Ackermann ( March 29, 1896, Herscheid municipality Germany &ndash December 24, 1962 Lüdenscheid Principles of Theoretical Logic. Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert 's and Wilhelm Ackermann 's classic text Chelsea. Translation of Grundzüge der theoretischen Logik. Springer-Verlag. The 1928 first edition is the first time quantification was consciously employed in the now-standar manner, namely as binding variables ranging over some fixed domain of discourse. This is the defining aspect of first-order logic. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science
Charles Peirce, 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, American Journal of Mathematics 7: 180-202. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Reprinted in Kloesel, N. et al, eds. , 1993. Writings of C. S. Peirce, Vol. 5. Indiana Univ. Press. The first appearance of quantification in anything like its present form.
Hans Reichenbach, 1975 (1947). Hans Reichenbach ( September 26, 1891 in Hamburg – April 9, 1953 in Los Angeles Elements of Symbolic Logic, Dover Publications. The quantifiers are discussed in chapters §18 "Binding of variables" through §30 "Derivations from Synthetic Premises".
Wiese, 2003. Numbers, language, and the human mind. Cambridge University Press. ISBN 0-521-83182-2.
Westerstahl, Dag, 2001, "Quantifiers," in Goble, Lou, ed. , The Blackwell Guide to Philosophical Logic. Blackwell.

External links

Stanford Encyclopedia of Philosophy:
- "Classical Logic -- by Stewart Shapiro. The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University. Covers syntax, model theory, and metatheory for first order logic in the natural deduction style.
- "Generalized quantifiers" -- by Dag Westerstahl.
Peter, Stanley, and Dag Westerståhl (2002) "Quantifiers."

Dictionary

-noun

the act of quantifying
(economics) the expression of an economic activity in monetary units
(logic) a limitation that is imposed on the variables of a proposition

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