Quantification Totally Explained

Everything about Quantifier totally explained

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. Quantification in this sense is fundamental to the scientific method.
In logic, quantification refers to an operator that binds a variable ranging over a domain of discourse. The variable thereby becomes bound. Academic discussion of quantification refers more often to this meaning of the term than the preceding one.

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. Its proponents argue that it provides a much more natural formal rendering of natural language than the traditional treatments of Frege, Russell and Quine.

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. 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, for example, 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've quantified over the predicate or function expression whose free variable is bound by the quantifier. Quantification is used in both natural languages and formal languages. 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. Quantification is an example of a variable-binding operation.
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.
The traditional symbol for the universal quantifier "all" is "∀", an inverted letter "A", and for the existential quantifier "exists" is "∃", a rotated letter "E". These quantifiers have been generalized beginning with the work of Mostowski and Lindström. See generalized quantifier and Lindström quantifier for further details.

Mathematics

We will begin by discussing quantification in informal mathematical discourse. Consider the following statement ť 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's a procedure to generate all the conjuncts. However, if we wanted to assert something about every irrational number, we'd have no way enumerating all the conjuncts since irrationals can't be enumerated. A succinct formulation which avoids these problems uses universal quantification: ť For any natural number n, n·2 = n + n.

A similar analysis applies to the disjunction, ť 1 is prime, or 2 is prime, or 3 is prime, etc.

which can be rephrased using existential quantification: ť For some natural number n, n is prime.

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. Three approaches have been devised to date:

Relation algebra, invented by DeMorgan, and developed by Ernst Schroder, Tarski, and Tarski's students. Relation algebra can't 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.

Notation

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, ť

exists 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.
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.

History

Term logic treats quantification in a manner that's 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. 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 didn't 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), Charles Sanders Peirce and his student O. H. Mitchell independently invented universal and existential qunatifiers, and bound variables. 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. Most notably, it's 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 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. 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. In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ didn't become canonical until the 1960s.
   Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. 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.

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're quantitative facts and the basis of science. It seems to be held as universally true that the foundation of quantification is measurement. There is little doubt that quantification provided a basis for the objectivity of science. 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. Any reasonable comparison between Aristotle and Galileo shows clearly that there can be no unique lawfulness discovered without detailed quantification. Even today, universities use imperfect instruments called 'exams' to indirectly quantify something they call knowledge. This meaning of quantification comes under the heading of pragmatics.

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.

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