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Topic: Theory of Types

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Intuitionistic type theory

Denotational semantics

Categorical logic

Formal semantics

Category (mathematics)

Typologist

Topos theory

Category theory

Sheaf theory

Domain theory

Mathematical logic

Hypercomputation

Algorithmic information theory

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Lambda calculus

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	PlanetMath: Russell's theory of types
		Type theory is based on the idea that impredicative definitions are the root of all evil.
		A formula for which there is an assignment of types (degrees) to the variables and constants so that it accords to the restrictions of type theory is said to be stratified.
		This is version 7 of Russell's theory of types, born on 2003-08-06, modified 2004-02-16.
	www.planetmath.org /encyclopedia/AxiomOfReducibility.html (958 words)

	Russell's Paradox [Internet Encyclopedia of Philosophy]
		This is to insist that properties fall into different types, and that the type of a property is never the same as the entities to which it applies.
		To be philosophically adequate, the adoption of the theory of types for properties requires developing an account of the nature of properties such that one would be able to explain why they cannot apply to themselves.
		The Theory of Types for Classes: It was mentioned earlier that Russell advocated a more comprehensive theory of types than Frege's distinction of levels, one that divided not only properties or concepts into various types, but classes as well.
	www.utm.edu /research/iep/p/par-russ.htm (2714 words)

	NationMaster - Encyclopedia: Type theory
		Types in this sense are related to the metaphysical notion of type.
		The main attributes of this type are: a sanguine complexion, synergistic merging of unbridled narcissism and aggression, hyperactivity, non-perfectionism, a tendency toward extreme self-adornment, exhibitionism in the limelight, a "flashy" extroverted smile, a tendency toward hypersexuality, and the capacity to exhibit the narcissistic, aggressive-vindictive or combined narcissistic-aggressive rages.
		The main attributes of this type are: a sanguine complexion, a loud voice, dynamism with a tendency to be overbearing, bombastic garrulity, intense eye contact, a strong sense of duty, a bent toward conventional values, unpretentious self-adornment, an outgoing smile of moderate intensity, and the capacity to exhibit the narcissistic, aggressive, or explosive narcissistic-aggressive rages.
	www.nationmaster.com /encyclopedia/Type-theory (789 words)

	Type Theory (Stanford Encyclopedia of Philosophy)
		The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory (Russell, 1903).
		Type theory can be used as a foundation for mathematics, and indeed, it was presented as such by Russell in his 1908 paper, which appeared the same year as Zermelo's paper, presenting set theory as a foundation for mathematics.
		We limit ourselves to presenting two applications of type theory to category theory: the constructions of the free cartesian closed category and of the free topos (see the entry on category theory for an explanation of "cartesian closed" and "topos").
	plato.stanford.edu /entries/type-theory (6514 words)

	NPA personality theory at AllExperts
		The main attributes of this type are: a sanguine complexion, synergistic merging of unbridled narcissism and aggression, hyperactivity, non-perfectionism, a tendency toward extreme self-adornment, exhibitionism in the limelight, a "flashy" extroverted smile, a tendency toward hypersexuality, and the capacity to exhibit the narcissistic, aggressive-vindictive or combined narcissistic-aggressive rages.
		The main qualities of this type are: a tendency toward a sanguine complexion, industriousness, orderliness, an intense sense of duty, unaggressiveness, stubbornness, negativism, a tendency to ruminate, perfectionistic rather than unbridled self-adornment, an uncommonly seen gingival smile of recognition, and the capacity to exhibit the florid narcissistic rage.
		The main attributes of this type are: a sanguine complexion, a loud voice, dynamism with a tendency to be overbearing, bombastic garrulity, intense eye contact, a strong sense of duty, a bent toward conventional values, unpretentious self-adornment, an outgoing smile of moderate intensity, and the capacity to exhibit the narcissistic, aggressive, or explosive narcissistic-aggressive rages.
	en.allexperts.com /e/n/np/npa_personality_theory.htm (4163 words)

	The Many Types of String Theory
		In the two heterotic theories, clockwise vibrations resemble those of a Type II string (clockwise vibrations in IIA and IIB are the same), but counterclockwise vibrations are those of the original bosonic theory.
		The Type I theory is similar to Type IIB except that it includes open strings, or strings whose ends are loose and unconnected, as well as the usual closed strings.
		String theory includes an equation to find the constant, but it is currently only an approximate version solvable only to yield the answer that the constant times zero is zero (thus implying that it could be any number according to the approximate equations).
	library.advanced.org /27930/stringtheory6.htm (2156 words)

	Type Theory, Set Theory and Domain Theory
		Whenever an apparent variable occurs in a proposition, the range of values of the apparent variable is a type, the type being fixed by the function of which "all values" are concerned.
		Zermelo set theory, Z. The constructive version of second order type theory allows a fully impredicative notion of proposition, which at first sight seems to violate the notion of construction in that the propositions cannot be built up systematically from simpler ones.
		Type theory opened the door to another fundamental idea in logic - the notion that propositions are types which, if true, are inhabited by proofs.
	www.cs.cornell.edu /Info/Projects/NuPrl/Intro/TypeSetDomain/typesetd.html (1988 words)

	Type theory - Biocrawler (Site not responding. Last check: 2007-11-05)
		At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into sets called types.
		Modern type theory was invented partly in response to Russell's paradox, and features prominently in Russell and Whitehead's Principia Mathematica.
		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.
	www.biocrawler.com /encyclopedia/Theory_of_types (487 words)

	Using Theory
		The primary use of theory by a practitioner is to rationalize or enhance the outcomes of his undertakings.
		A theory is a formal theory if, to put it technically, its variables are not independently definable, that is, if some variables are constituents of others.
		The distinction between formal and causal theories is important to the practitioner and policy-maker because policies are usually formulated on the basis of some causal theory, "Under conditions C, do X (so as to effect I)".
	www.newfoundations.com /EGR/UseTheory.html (2396 words)

	Typed Lambda Logic's
		The simple theory of types is arrived at from the lambda calculus by adding a type system, and then some logical constants, and then beefing up the inference rules to reflect the intended meaning of the constants.
		A term is either a constant name decorated with a type, or a variable name decorated with a type, or the application of one term to another (shown by juxtaposition with the function on the left), or a lambda abstraction in which the bound variable name is decorated with a type.
		They determine the type of a function formed by abstraction, ensuring that the type of the bound variable in the abstraction is the domain of the function type of the abstraction and also that the type of the body of the lambda expression is the type of the co-domain of the resulting function.
	www.rbjones.com /rbjpub/logic/cl/cl013.htm (378 words)

	MATHS: Introduction to Types (Site not responding. Last check: 2007-11-05)
		Symbols for types have two meanings, one of these is in a normal part of an expression where it represents the set of objects or an identity map from objects to objects depending on the context.
		But the collection of all types is not a type so we can not define a set of all sets and so should avoid the paradoxes discovered at the beginning of the 20th century.
		If there is a map from one type to another then there is an easy way to generate a new type - by grouping the elements of the second type according to their values under the mapping - this is called the quotient type with respect to the mapping.
	www.csci.csusb.edu /dick/maths/types.html (2822 words)

	Type Theory
		The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory (Russell, 1903).
		Type theory can be used as a foundation for mathematics, and indeed, it was presented as such by Russell in his 1908 paper, which appeared the same year as Zermelo's paper, presenting set theory as a foundation for mathematics.
		Given (1) and (2) we should have a type of propositions (as in simple type theory), and given (3) this should also be the type of all types.
	www.seop.leeds.ac.uk /archives/sum2006/entries/type-theory (6470 words)

	IDtheory
		Theories provide patterns for the interpretation of data, linking one study with another, supplying frameworks within which concepts and variables acquire special significance and allow the researcher to interpret the larger meaning of his/her findings.
		Theory construction is not a random process dependent on the interests of the theorist only, but also a specialised type of decision-making and problem solving- process.
		Types of theories that relate to theory development in IT are taxonomies, conceptual frameworks and theoretical systems (Seels, 1997, p.14).
	hagar.up.ac.za /catts/IDtheory.htm (3988 words)

	Simple Theory of Types
		Theory developed by Bertrand Russell (1872-1970) to deal with paradoxes like his paradox of classes: is the class of all classes that are not members of themselves a member of itself?
		The class of all type n classes is of type n+1, but there is no class of all classes (or property which applies to all properties).
		This is the simple theory, which Russell developed only as part of the ramified theory (see ramified theory of types).
	www.philosophyprofessor.com /philosophies/types-simple-theory.php (219 words)

	THEORY OF LOGICAL TYPES (Site not responding. Last check: 2007-11-05)
		A theory proposed by B. Russell that rules out self-reference in order to prevent the emergence of antinomies and paradoxes in logic.
		The theory has been influential in linguistics by recognizing the importance of logical as well as grammatical restrictions on the combinations of words (see language).
		However, by exorcising self-reference, the theory of logical types bas retarded the development of theory, largely cognitive theory, in areas where self-reference is prevalent.
	pespmc1.vub.ac.be /ASC/THEORY_TYPES.html (245 words)

	Bertrand Russell (Stanford Encyclopedia of Philosophy)
		Russell's contributions to logic and the foundations of mathematics include his discovery of Russell's paradox, his defense of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his development of the theory of types, and his refining of the first-order predicate calculus.
		Although first introduced in 1903, the theory of types was further developed by Russell in his 1908 article "Mathematical Logic as Based on the Theory of Types" and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913).
		Urquhart, Alasdair (1988) "Russell's Zig-Zag Path to the Ramified Theory of Types," Russell, 8, 82-91.
	plato.stanford.edu /entries/russell (3965 words)

	[No title]
		Martin-Löf type theory (MLTT) is a functional programming language with a highly expressive type system.
		Whereas in traditional programming languages types are typically used to reduce run-time erros, in MLTT types can be used not only for that, but...
		To be philosophically adequate, the adoption of the theory of types for properties requires developing an account of the nature of properties such that one would be able to explain why they cannot apply to themselves.
	www.lycos.com /info/type-theory.html (407 words)

	The Many Types of String Theory
		In the two heterotic theories, clockwise vibrations resemble those of a Type II string (clockwise vibrations in IIA and IIB are the same), but counterclockwise vibrations are those of the original bosonic theory.
		The Type I theory is similar to Type IIB except that it includes open strings, or strings whose ends are loose and unconnected, as well as the usual closed strings.
		String theory includes an equation to find the constant, but it is currently only an approximate version solvable only to yield the answer that the constant times zero is zero (thus implying that it could be any number according to the approximate equations).
	library.thinkquest.org /27930/stringtheory6.htm (2156 words)

	AmosWEB is Economics: Encyclonomic WEBpedia (Site not responding. Last check: 2007-11-05)*
		A theory is both the "starting point" for doing science and the "end product" of the scientific method.
		A theory that explains ALL types of consumer behavior is better than one which ONLY explains how consumers behave when on roller skates.
		The basic belief axioms of an economic theory inevitably include world views about the political system, individual freedoms and responsibilities, the nature of humanity, and the role of government.
	www.amosweb.com /cgi-bin/awb_nav.pl?s=wpd&c=dsp&k=theory (763 words)

	Jung's Theory of Temperaments
		Abstract: Jung's theory of psychological types is sketched as a prelude to developing a naturalistic ethics.
		No one lives completely as one type or the other; your type might be innate, at least your type begins very early in life.
		Jung says that there is a marked tendency for either type to marry its opposite (each secretly hopes that the other will take care of the side of the life each lacks).
	philosophy.lander.edu /ethics/jung.html (900 words)

	Career Development Theory
		This theory describes 6 general types that can be applied to people’s personalities and to work environments (see diagram page 46).
		Ginzberg’s theory addressed 3 diffeent stages: fantasy which involves role playing and imagination (up to age 12), tentative which involves recognition of one’s interests abilities and values (12 to 17), and realistic which involves identifying an occupational choice (over 17.
		This theory assumes there are 8 occupational groups: service, business contact, organization, technology, outdoor, science, general cultural, and arts and entertainment.
	www.educ.drake.edu /nri/syllabi/reha220/class4notes.html (741 words)

	Computer-Assisted Reasoning based on Type Theory (TYPES) (Site not responding. Last check: 2007-11-05)
		The current TYPES project is administered by Chalmers (Sweden).
		The aim of our research activities is to develop the technology of formal reasoning based on Type Theory by improving the languages and tools of reasoning and by applying the technology in several domains such as programming languages, certified software, and formalisation of mathematics.
		The theory and the associated systems developed by sites in the consortium define the state of the art in type theory and its applications.
	www.dur.ac.uk /TYPES (230 words)

	Jung's Theory of Types
		The most likely personality type of a mystic, according to Jung, is the extroverted intuitive type.
		The feeling type has intutition as it most undervalued function and is noted for emotional outbursts.
		Jung's Theory of Temperaments: The still influential theory of Carl Jung's personality types is studied as one kind of naturalistic ethics.
	philosophy.lander.edu /ethics/jung_quiz.html (195 words)

	Lecture 21 -- The intentional theory of types: Syntax
		This material is from Chapter 5 of Gamut, and is mainly relevant if we want to interpret statements of natural language directly into a model, rather than first translating into a statement of intensional propositional logic (i.e., interpreting rather than compiling).
		The first thing new is that we introdcue a new basic type, s, to go along with e and t (for truth values).
		That is, we can talk about functions from worlds to other types, but we can't talk directly about the worlds.
	www.cs.pomona.edu /classes/cs66/Lectures/Lecture21/Lecture21_3.html (345 words)

	Set theory
		Cantor's early work was in number theory and he published a number of articles on this topic between 1867 and 1871.
		By this stage, however, set theory was beginning to have a major impact on other areas of mathematics.
		However the method of avoiding the paradoxes by introducing a 'theory of types' made it impossible to say that a class was or was not a member of itself.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Beginnings_of_set_theory.html (2182 words)

	Amazon.ca: Axiomatic Set Theory: Books: Patrick Suppes (Site not responding. Last check: 2007-11-05)
		Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning.
		Chapter 5 then goes into the theory of ordinal numbers, wherein it is emphasized that no special axioms are needed for the development of this theory.
		The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.
	www.amazon.ca /Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304 (1773 words)

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