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Encyclopedia > Quantification

In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. A language element which generates a quantification is called a quantifier. The resulting statement is a quantified statement, and we say we have quantified over the predicate. Quantification is used in both natural languages and formal languages. In natural language, examples of quantifiers are for all, for some; many, few, a lot are also quantifiers. 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. Quantification is an example of a variable-binding operation. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ... In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ... In the main, semantics (from the Greek and in greek letters σημαντικός or in latin letters semantikós, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...


The two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. 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 linguistics and logic, a predicate is an expression that can be true of something. ... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ... In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

Contents


Quantification in natural language

All known human languages make use of quantification, even languages without a fully fledged number system (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.

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. Its proponents argue that it provides a much more natural formal rendering of natural language than the traditional treatments of Frege, Russell and Quine. Montague grammar is an approach to natural language semantics, based on formal logic, especially lambda calculus and set theory. ...


Need for quantifiers in mathematical assertions

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. From the point of view of formal languages this is immediately a problem, since we expect syntax rules to generate finite objects. 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. A succinct formulation which avoids these problems uses universal quantification: In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ... For other uses, see Syntax (disambiguation). ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Look up Procedure in Wiktionary, the free dictionary. ... In mathematics, an irrational number is any real number that is not a rational number, i. ...

For any natural number n, n·2 = n + n.

A similar analysis applies to the disjunction, In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...

1 is prime, or 2 is prime, or 3 is prime, etc.

which can be rephrased using existential quantification: In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...

For some natural number n, n is prime.

In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

Nesting of quantifiers

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 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 affect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...


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. 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. In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...


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 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. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above to express In mathematics, any integer (whole number) is either even or odd. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

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.


Notation for quantifiers

The traditional symbol for the universal quantifier is "∀", an inverted letter "A", which stands for the word "all". The corresponding symbol for the existential quantifier is "∃", a rotated letter "E", which stands for the word "exists". Correspondingly, quantified expressions are constructed as follows, Look up A, a in Wiktionary, the free dictionary. ... The letter E is the fifth letter in the Latin alphabet. ...

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

Early 20th century documents do not use the ∀ symbol. The typical notation was (x)P to express "for all x, P", and "(∃x)P" for "there exists x such that P". The ∃ symbol was coined by Giuseppe Peano around 1890. Later, around 1930, Gerhard Gentzen introduced the ∀ symbol to represent universal quantification. Frege's Begriffsschrift used an entirely different notation, which did not include an existential quantifier at all; ∃x P was always represented instead with the Begriffsschrift equivalent of ¬∀x ¬P. Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... 1930 (MCMXXX) is a common year starting on Wednesday. ... Gerhard Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... 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 set out in that book. ...


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 strongly-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 type of the objects in that domain.

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. Computer programming (often simply programming or coding) is the craft of writing a set of commands or instructions that can later be compiled and/or interpreted and then inherently transformed to an executable that an electronic machine can execute or run. Programming requires mainly logic, but has elements of science... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into sets called types. ...


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.


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: In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

For exactly one natural number x, ....
There is one and only one x such that ....

One might even avoid variable names such as x using a pronoun. For example, In linguistics and grammar, a pronoun is a pro-form that substitutes for a noun phrase. ...

For any natural number, its product with 2 equals to its sum with itself
Some natural number is prime.

Formal semantics

Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal—that is, mathematically specified—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. In this article, we only address the issue of how quantifier elements are interpreted. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In mathematics, logic, and computer science, a formal language is a set of finite-length words (i. ... For other uses, see Syntax (disambiguation). ...


In this section we only consider first-order logic with function symbols. We refer the reader to the article on model theory for more information on the interpretation of formulas within this logical framework. 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. Thus in It has been suggested that Predicate calculus be merged into this article or section. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...

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
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, ..., xn is interpreted as a boolean-valued function F(v1, ..., vn) 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 tree jpg File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... tree jpg File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The adjective Boolean (sometimes boolean), coined in honor of George Boole, is used in many contexts: An evaluation that results in either TRUE or FALSE. A boolean value is a truth value, either true or false, often coded 1 and 0, respectively. ...

forall x_n A(x_1, ldots , x_n)

is the function G of n-1 arguments such that G(v1, ...,vn-1) = T if and only if F(v1, ..., vn-1, w) = T for every w in X. If F(v1, ..., vn-1, w) = F for at least one value of w, then G(v1, ...,vn-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, ...,vn-1) = T if and only if F(v1, ...,vn-1, w) = T for at least one w and H(v1, ..., vn-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 In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

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.

Though we will not consider semantics of natural language in this article, 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 < ab ≤ 1. If A is a formula with free variables x1,...,xn whose interpretation is the function F of variables v1,...,vn then the interpretation of In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...

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

is the function of v1,...,vn-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,...,vn-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, a measure is a function that assigns a number, e. ... It has been suggested that A counterexample related to Fubinis theorem be merged into this article or section. ...


We also caution the reader that the corresponding logic for such a semantics is exceedingly complicated.


History of formalization

Term logic treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Aristotelian logic treated All', Some and No in the 1st century BC, in an account also touching on the alethic modalities. Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ... Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ... (2nd millennium BC - 1st millennium BC - 1st millennium) The 1st century BC started on January 1, 100 BC and ended on December 31, 1 BC. An alternative name for this century is the last century BC. The AD/BC notation does not use a year zero. ... ...


The first variable-based treatment of quantification was Gottlob Frege's 1879 Begriffsschrift. To universally quantify a variable, Frege would make a dimple in an otherwise straight line appearing in his diagrammatic formulas, then write the quantified variable over the dimple. Frege did not have a specific notation for existential quantification, instead using the equivalent of simforall x:simldots. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... 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 set out in that book. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, and mathematician, working mostly in the 20th century. ...


Meanwhile, Charles Sanders Peirce and his student O. H. Mitchell independently invented the existential as well as the universal quantifier, in work culminating in Peirce (1885). Peirce and Mitchell wrote Πx and Σx where we now write ∀x and ∃x. This notation can be found in the writings of Ernst Schroder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. 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. Peirce's later existential graphs can be seen as featuring tacit variables whose quantification is determined by the shallowest instance. Peirce's approach to quantification influenced Ernst Schroder, William E. Johnson, and all of Europe via Giuseppe Peano. Pierce's logic has attracted fair attention in recent decades by those interested in heterogeneous reasoning and diagrammatic inference. Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... Ernst Schröder (25 November 1841 - 16 June 1902) was the most significant representative of the algebraic logic school in Germany in the second half of the nineteenth century. ... Leopold Löwenheim (born 1878 in Krefeld, died 1957 in Berlin) was a German mathematician, known for his work in mathematical logic. ... Albert Thoralf Skolem (May 23, 1887 - March 23, 1963) was a Norwegian mathematician. ... Kurt Gödel Kurt Gödel [ kurt gøːdl ], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics, whose biography lists quite a few nations, although he is usually associated with Austria. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... An existential graph is a type of diagrammatic or visual notation for logical expressions, invented by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... Ernst Schröder (25 November 1841 - 16 June 1902) was the most significant representative of the algebraic logic school in Germany in the second half of the nineteenth century. ... Willliam E. Johnson (1862-1950), better known as “Pussyfoot Johnson,” was a leader of the Anti-Saloon League. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ...


Peano notated the universal quantifier as (x). Hence "(x)φ" indicated that the formula φ was true for all values of x. He was the first to employ, in 1897, the notation (∃x) for existential quantification.The Principia Mathematica of Whitehead and Russell employed Peano's notation, as did Quine and Alonzo Church throughout their careers. Gentzen introduced the ∀ symbol 1935 by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1950s. Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher. ... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England– December 30, 1947 Cambridge, Massachusetts, USA) was an English mathematician who became an American philosopher. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, and mathematician, working mostly in the 20th century. ... In computing, a quine is a program (a form of metaprogram) that produces its complete source code as its only output. ... Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... Gerhard Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. ...


References

  • Jon Barwise and John Etchemendy, 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.
  • Gottlob Frege, 1879. Begriffsschrift. Translated in Jean van Heijenoort, 1967. 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). Principles of Theoretical 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-predominant manner, namely as the defining aspect of first order logic.
  • Charles Peirce, 1885, "On the Algebra of Logic: A Contribution to the Philosophy of Notation, American Journal of Mathematics 7: 180-202. 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). 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.

Kenneth Jon Barwise (June 29, 1942 - March 5, 2000) was a US mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. ... John W. Etchemendy is Stanfords twelfth and current Provost. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... 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 set out in that book. ... Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ... David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Wilhelm Ackermann (March 29, 1896, Herscheid municipality, Germany – December 24, 1962 Lüdenscheid, Germany ) was a German mathematician best known for the Ackermann function, an important example in the theory of computation. ... Principles of Theoretical Logic is the title of the 1950 American translation of the 1938 second edition of David Hilberts and Wilhelm Ackermanns classic text Grundzüge der theoretischen Logik, on elementary mathematical logic. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... Charles Sanders Peirce (pronounced purse), (September 10, 1839 – April 19, 1914) was an American polymath, born in Cambridge, Massachusetts. ... Hans Reichenbach (September 26, 1891, Hamburg, – April 9, 1953, Los Angeles) was a leading philosopher of science, educator and proponent of logical positivism. ...

External links

  • http://www.stanford.edu/group/nasslli/courses/peters-wes/PWbookdraft2-3.pdf

  Results from FactBites:
 
Quantification - Wikipedia, the free encyclopedia (2287 words)
In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things.
Quantification is used in both natural languages and formal languages.
Quantification is an example of a variable-binding operation.
Existential quantification - Wikipedia, the free encyclopedia (560 words)
In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain.
Quantification in general is covered in the article Quantification, while this article discusses existential quantification specifically.
Several variations in the notation for quantification (which apply to all forms) can be found in the quantification article.
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