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At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type. Objects of a given type are built up from objects of the preceding type. Types in this sense are related to the metaphysical notion of type. Bertrand Russell invented type theory in response to his discovery that naive set theory was afflicted with Russell's paradox. Type theory features prominently in Whitehead and Russell's Principia Mathematica. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Logic, from Classical Greek λόγος logos (the word), is the study of the principles and criteria of valid inference and demonstration. ...
A type is a category of being. ...
Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ...
In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
(For E. W. Russells Paradox, see Religious and militarist attitudes and Paradox supported. ...
Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ...
Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ...
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. ...
The property of being a perfect square can be significantly predicated of both cardinals and ratios of cardinals; but the property of being odd (or even) is not defined for the ratios. We are thus unable to answer the question whether 2/3 is odd or even. (Nagel 1951). In programming language theory, a branch of computer science, type theory provides the formal basis for the design, analysis and study of type systems. Indeed, many computer scientists use the term type theory to refer to the formal study of type systems for programming languages, although some limit it to the study of more abstract formalisms such as typed λ-calculi. The lowercase lambda Programming language theory (commonly known as PLT) is a branch of computer science which deals with the design, implementation, analysis, characterization, and classification of programming languages and programming language features. ...
Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
In computer science, a type system defines how a programming language classifies values and expressions into types, how it can manipulate those types and how they interact. ...
Other listings of programming languages are: Categorical list of programming languages Generational list of programming languages Chronological list of programming languages Note: Esoteric programming languages have been moved to the separate List of esoteric programming languages. ...
Typed versions of the lambda calculus extend the standard lambda calculus with types. ...
Simple theory of types The following system is Mendelson's (1997, 289–293) ST. The domain of quantification is partitioned into an ascending hierarchy of types, with all individuals assigned a type. Quantified variables range over only one type; hence the underlying logic is first order logic. ST is "simple" (relative to the type theory of Principia Mathematica) primarily because all members of the domain and codomain of any relation must be of the same type. The domain of discourse, sometimes called the universe of discourse, is an analytic tool used in deductive logic, especially predicate logic. ...
As commonly used, individual refers to a person or to any specific object in a collection. ...
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. ...
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. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
There is a lowest type, whose individuals have no members and are members of the second lowest type. Individuals of the lowest type correspond to the urelements of certain set theories. Each type has a next higher type, analogous to the notion of successor in Peano arithmetic. While ST is silent as to whether there is a maximal type, a transfinite number of types poses no difficulty. These facts, reminiscent of the Peano axioms, make it convenient and conventional to assign a natural number to each type, starting with 0 for the lowest type. But type theory does not require a prior definition of the naturals. Definition In set theory a ur-element is something which is not a set, but may itself be an element of a set. ...
A successor function is the label in the literature for what is actually an operation. ...
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). ...
Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
The symbols peculiar to ST are primed variables and infix ∈. In any given formula, unprimed variables all have the same type, while primed variables (x’) range over the next higher type. The atomic formulas of ST are of two forms, x=y (identity) and y∈x ’. The infix symbol ∈ suggests the intended interpretation, set membership. In mathematics, the term identity has several important uses: identity can refer to an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...
This article or section does not cite its references or sources. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
All variables appearing in the definition of identity and in the axioms Extensionality and Comprehension, range over individuals of one of two consecutive types. Only unprimed (primed) variables, ranging over the "lower" ("higher") type, can appear to the left (right) of '∈'. The first order formulation of ST rules out quantifying over types. Hence each pair of consecutive types requires its own axiom of Extensionality and of Comprehension, which is possible if Extensionality and Comprehension below are taken as axiom schemata "ranging over" types. In mathematical logic, an axiom schema generalizes the notion of axiom. ...
- Identity, defined by x=y ↔ ∀z’ [x∈z’ ↔ y∈z’].
- Let Φ(x) denote any first order formula containing the free variable x.
- Remark. Any collection of elements of the same type may form an object of the next higher type. Comprehension is schematic with respect to Φ(x) as well as to types.
- Remark. Infinity is the only true axiom of ST and is entirely mathematical in nature. It asserts that R is a strict total order, with a domain identical to its codomain. If 0 is assigned to the lowest type, the type of R is 3. Infinity can be satisfied only if the (co)domain of R is infinite, thus forcing the existence of an infinite set. If relations are defined in terms of ordered pairs, this axiom requires a prior definition of ordered pair; the Kuratowski definition, adapted to ST, will do. The literature does not explain why the usual axiom of infinity (there exists an inductive set) of ZFC of other set theories could not be married to ST.
ST reveals how type theory can be made very similar to axiomatic set theory. Moreover, the more elaborate ontology of ST, grounded in what is now called the "iterative conception of set," makes for axiom (schemas) that are far simpler than those of conventional set theories, such as ZFC, with simpler ontologies. Set theories whose point of departure is type theory, but whose axioms, ontology, and terminology differ from the above, include New Foundations and Scott-Potter set theory. In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ...
In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...
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, 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...
In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ...
In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...
The Ouroboros something reflexive refers to itself. ...
In grammar, a verb is transitive if it takes an object. ...
In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. ...
In the context of the axiom of infinity, an inductive set is a set with the property that, for every , the successor of is also an element of An example of an inductive set is the set of natural numbers See also Natural number Peano axioms External link Mathworld: Inductive...
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. ...
This article or section is in need of attention from an expert on the subject. ...
In philosophy, ontology (from the Greek , genitive : of being (part. ...
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 philosophy, ontology (from the Greek , genitive : of being (part. ...
In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ...
An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott. ...
History of type theory
Practical impact of type theory The most obvious application of type theory is in constructing type checking algorithms in semantic analysis phase of compilers for programming languages. This article is about the computing term. ...
Type theory is also widely in use in theories of semantics of natural language. The most common construction takes the basic types e and t for individuals and truth-values, respectively, and defines the set of types recursively as follows: Semantics (Greek semantikos, giving signs, significant, symptomatic, from sema, sign) refers to the aspects of meaning that are expressed in a language, code, or other form of representation. ...
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. ...
- if a and b are types, then so is
 . - Nothing except the basic types, and what can be constructed from them by means of the previous clause are types.
A complex type is the type of functions from entities of type a to entities of type b. Thus one has types like  which are interpreted as elements of the set of functions from entities to truth-values, i.e. characteristic functions of sets if entities. An expression of type  is a function from sets of entities to truth-values, i.e. a (characteristic function of a) set of sets. This latter type is standardly taken to be the type of natural language quantifiers, like every boy or nobody (Montague 1973, Barwise and Cooper 1981). Look up Function in Wiktionary, the free dictionary. ...
In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ...
Some mathematicians use the phrase characteristic function synonymously with indicator function. ...
Connections to constructive logic -
Main article: Curry–Howard The Curry – Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry – Howard isomorphism, or the formulae-as-types correspondence. ...
Relation to other topics
Type system -
Definitions of type system vary, but the following one due to Benjamin C. Pierce roughly corresponds to the current consensus in the programming language theory community: In computer science, a type system defines how a programming language classifies values and expressions into types, how it can manipulate those types and how they interact. ...
Benjamin C. Pierce is a professor of computer science at the University of Pennsylvania. ...
[A type system is a] tractable syntactic method for proving the absence of certain program behaviors by classifying phrases according to the kinds of values they compute. (Pierce 2002). In other words, a type system divides program values into sets called types — this is called a type assignment — and makes certain program behaviors illegal on the basis of the types that are thus assigned. For example, a type system may classify the value "hello" as a string and the value 5 as a number, and prohibit the programmer from adding "hello" to 5 based on that type assignment. In this type system, the program It has been suggested that this article or section be merged with value (computer science). ...
In computer programming and some branches of mathematics, strings are sequences of various simple objects. ...
A number is an abstract entity that represents a count or measurement. ...
"hello" + 5
would be illegal. Hence, any program permitted by the type system would be provably free from the erroneous behavior of adding strings and numbers.
The design and implementation of type systems is a topic nearly as broad as the topic of programming languages itself. In fact, type theory proponents commonly proclaim that the design of type systems is the very essence of programming language design: "Design the type system correctly, and the language will design itself."
See also Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ...
In computer science, a type system defines how a programming language classifies values and expressions into types, how it can manipulate those types and how they interact. ...
In computer science, duck typing is a form of dynamic typing in which a variables value itself implicitly determines what the variable can do. ...
A data type is a constraint placed upon the interpretation of data in a type system in computer programming. ...
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
Typed versions of the lambda calculus extend the standard lambda calculus with types. ...
Bertrand Arthur William Russell, 3rd Earl Russell OM FRS (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician and advocate for social reform. ...
References - Nagel, Ernest (1951), "Causal Character of Modern Physical Theory", pp. 244–268 in Freedom and Reason, Salo W. Baron, Ernest Nagel, and Koppel B. Pinson (eds.), The Free Press. Cited on p. 759 of Jefferson Hane Weaver, The World of Physics, ISBN 0-671-49931-9.
- Pierce, Benjamin C. (2002), Types and Programming Languages, MIT Press, Cambridge, MA. ISBN 0-262-16209-1.
Ernest Nagel (November 16, 1901, Prague, Austro-Hungarian Empire -- September 22, 1985, New York City) was among the most important philosophers of science of his time. ...
Benjamin C. Pierce is a professor of computer science at the University of Pennsylvania. ...
Further reading - Barwise, Jon and Robin Cooper, 1981. Generalized quantifiers in English. Linguistics and Philosophy 4:159-219.
- Andrews, Peter B., 2002. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd ed. Kluwer Academic Publishers.
- Cardelli, Luca, 1997, "Type Systems," in Allen B. Tucker, ed., The Computer Science and Engineering Handbook. CRC Press: 2208-2236.
- Carl A. Gunter, "Semantics of Programming Languages: Structures and Techniques", MIT Press 1992.
- Mendelson, Elliot, 1997. Introduction to Mathematical Logic, 4th ed. Chapman & Hall.
- Montague, Richard,1973. The proper treatment of quantification in English. In Hintikka, K. et al., editor, Approaches to Natural Language, pages 221--242.
- Thompson, Simon, 1991. Type Theory and Functional Programming. Addison-Wesley. ISBN 0-201-41667-0.
- Winskel, Glynn, 1993. The Formal Semantics of Programming Languages, An Introduction. MIT Press. ISBN 0-262-23169-7.
- Wittgenstein, Ludwig, 1922. Tractatus Logico-Philosophicus. New York, NY: Routledge, 2005. ISBN 0-415-25562-7
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