14440:
251:
12662:
12672:
12682:
9767:, which means that any order of applying the rules will always end in the same result. However, some are not. In a normalizing type theory, the one-directional computation rules are called "reduction rules", and applying the rules "reduces" the term. If a rule is not one-directional, it is called a "conversion rule".
3982:
B" resembles a function from type "A" to type "B". For a variety of logics, the rules are similar to expressions in a programming language's types. The similarity goes farther, as applications of the rules resemble programs in the programming languages. Thus, the correspondence is often summarized as
4039:
be viewed as type theories of a certain kind; this fact alone indicates that type theory is much more closely related to category theory than it is to set theory." In brief, a category can be viewed as a type theory by regarding its objects as types (or sorts), i.e. "Roughly speaking, a category may
10065:
nature of type theory, where proving that a value exists requires a method to compute the value. The Axiom of Choice is less powerful in type theory than most set theories, because type theory's functions must be computable and, being syntax-driven, the number of terms in a type must be countable.
9713:
Type theory has a built-in notion of computation. Thus, "1+1" and "2" are different terms in type theory, but they compute to the same value. Moreover, functions are defined computationally as lambda terms. In set theory, "1+1=2" means that "1+1" is just another way to refer the value "2". Type
9307:
2524:
3745:
It is possible to include the law of excluded middle and double negation into a type theory, by rule or assumption. However, terms may not compute down to canonical terms and it will interfere with the ability to determine if two terms are judgementally equal to each other.
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Most of the type theories proposed as foundations are constructive, and this includes most of the ones used by proof assistants. It is possible to add non-constructive features to a type theory, by rule or assumption. These include operators on continuations such as
8151:
163:
published in 1910, 1912, and 1913. This system avoided contradictions suggested in
Russell's paradox by creating a hierarchy of types and then assigning each concrete mathematical entity to a specific type. Entities of a given type were built exclusively of
8754:, where the second type depends on the value of the first term. This arises naturally in computer science where functions may return different types of outputs based on the input. For example, the Boolean type is usually defined with an eliminator function
313:. This led to proposals such as Lawvere's Elementary Theory of the Category of Sets (ETCS). Homotopy type theory continues in this line using type theory. Researchers are exploring connections between dependent types (especially the identity type) and
1792:
3039:(statements that can be proven), and terms inhabiting the type are interpreted to be proofs of that proposition. When some types are interpreted as propositions, there is a set of common types that can be used to connect them to make a
8631:
10331:
Provides a historical survey of the developments of the theory of types with a focus on the decline of the theory as a foundation of mathematics over the four decades following the publication of the second edition of 'Principia
10024:
Sometimes, a type theory will add a few axioms. An axiom is a judgment that is accepted without a derivation using the rules of inference. They are often added to ensure properties that cannot be added cleanly through the rules.
9665:
In set theory, an element is not restricted to one set. The element can appear in subsets and unions with other sets. In type theory, terms (generally) belong to only one type. Where a subset would be used, type theory can use a
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Inductive types are a general template for creating a large variety of types. In fact, all the types described above and more can be defined using the rules of inductive types. Two methods of generating inductive types are
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10017:. This is because a type theory is defined by its rules of inference. This is a source of confusion for people familiar with Set Theory, where a theory is defined by both the rules of inference for a logic (such as
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Some combinations of types are equivalent to other combinations of types. When functions are considered "exponentiation", the combinations of types can be written similarly to algebraic identities. Thus,
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134:) is that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself. Between 1902 and 1908,
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2654:, where the top-most rules need no assumptions. One example of a rule that does not require any inputs is one that states the type of a constant term. For example, to assert that there is a term
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3740:
347:. Most of these systems use a type theory as the mathematical foundation for encoding proofs, which is not surprising, given the close connection between type theory and programming languages:
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Many programs that work with type theory (e.g., interactive theorem provers) also do type inferencing. It lets them select the rules that the user intends, with fewer actions by the user.
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To generate a particular judgment in type theory, there must be a rule to generate it, as well as rules to generate all of that rule's required inputs, and so on. The applied rules form a
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Some type theories allow for types of complex terms, such as functions or lists, to depend on the types of its arguments. For example, a type theory could have the dependent type
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Language
Engineering and Rigorous Software Development: International LerNet ALFA Summer School 2008, Piriapolis, Uruguay, February 24 - March 1, 2008, Revised Tutorial Lectures
9528:
5840:, so parentheses can be dropped where appropriate. In the case of the three examples above, all parentheses could be omitted from the first two, and the third may simplified to
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of that type, thus preventing an entity from being defined using itself. This resolution of
Russell's paradox is similar to approaches taken in other formal systems, such as
4877:
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3998:, (the computational counterpart of) type inhabitation can be used to construct (all or parts of) programs from the specification given in the form of type information.
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using a horizontal line, with the required input judgments above the line and the resulting judgment below the line. For example, the following inference rule states a
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1982:
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919:". A type theory has judgments that define types and assign them to a collection of formal objects, known as terms. A term and its type are often written together as
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10585:'s type system, for example, abstract types have no instances, but can have subtype, whereas concrete types do not have subtypes but can have instances, for "
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7613:
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is a function from sets of entities to truth-values, i.e. a (indicator function of a) set of sets. This latter type is standardly taken to be the type of
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6419:
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2707:
4120:
The most basic types are called atoms, and a term whose type is an atom is known as an atomic term. Common atomic terms included in type theories are
3016:
5747:
4057:(categories with products and exponentials and one non-terminal object) correspond to the untyped λ-calculus (observed independently by Lambek and
10541:
Montague, R. (1973) "The proper treatment of quantification in ordinary
English". In K. J. J. Hintikka, J. M. E. Moravcsik, and P. Suppes (eds.),
4031:
Although the initial motivation for category theory was far removed from foundationalism, the two fields turned out to have deep connections. As
13494:
5843:
10061:
does not need to be added to type theory, because in most type theories it can be derived from the rules of inference. This is because of the
9302:{\displaystyle (\Pi \,\sigma \,\tau .{\mathsf {bool}}\to \sigma \to \tau \to (\Sigma \,x:{\textsf {bool}}.\mathrm {TF} \,x\,\sigma \,\tau ))}
4782:
142:
8431:, respectively. Dependent product and sum types commonly appear in function types and are frequently incorporated in programming languages.
5629:
1801:
1200:
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12718:
10855:
4847:, which would be a function which takes in a function of natural numbers and returns a natural number. The convention is that the arrow is
7325:
There are foundational issues that can arise from dependent types if a theory is not careful about what dependencies are allowed, such as
11013:
9718:
8963:
2519:{\displaystyle {\begin{array}{c}\Gamma \vdash t:T_{1}\qquad \Delta \vdash T_{1}=T_{2}\\\hline \Gamma ,\Delta \vdash t:T_{2}\end{array}}}
14476:
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9846:
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notion of equality. The judgments enforce that every term has a type. The type will restrict which rules can be applied to a term.
11126:
10028:
Axioms can cause problems if they introduce terms without a way to compute on those terms. That is, axioms can interfere with the
8159:
13891:
9651:, while type theories only have rules. Type theories, in general, do not have axioms and are defined by their rules of inference.
1903:, are common choices to represent some or all of the assumptions. The 4 different judgments are thus usually written as follows.
10815:
4677:
Strictly speaking, a simple type only allows for one input and one output, so a more faithful reading of the above type is that
12415:
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10417:
10324:
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10253:
10247:
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A good introduction to simple type theory for computer scientists; the system described is not exactly Church's STT though.
13904:
13227:
12291:
11148:
Heineman, George T.; Bessai, Jan; Düdder, Boris; Rehof, Jakob (2016). "A long and winding road towards modular synthesis".
8855:
741:
11659:
11272:
8785:
3847:. In type theory, existence is accomplished using the dependent product type, and its proof requires a term of that type.
14818:
14485:
12445:
11717:
11471:
10391:
9636:
202:
In the modern literature, "type theory" refers to a typed system based around lambda calculus. One influential system is
10039:"Axiom K" ensures "uniqueness of identity proofs". That is, that every term of an identity type is equal to reflexivity.
5278:
4708:
3686:
298:
mathematics to serve as a new foundation for mathematics. There is ongoing research into mathematical foundations using
13909:
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12842:
11950:
11359:
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8146:{\displaystyle \mathrm {match} :(\Pi \,\rho .(\sigma \to \rho )\to (\tau \to \rho )\to (\sigma \sqcup \tau )\to \rho )}
6110:
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1156:
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922:
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Some theories also permit types to be dependent on terms instead of types. For example, a theory could have the type
4064:
13387:
11582:
11203:
9949:
9890:
4090:
attempts to combine type theory and category theory. It focuses on equalities, especially equalities between types.
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12685:
11779:
11492:
10029:
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may actually consist of complex terms and types that contain many function applications, not just single symbols.
14905:
13447:
13140:
12073:
10545:(Synthese Library, 49), Dordrecht: Reidel, 221–242; reprinted in Portner and Partee (eds.) 2002, pp. 17–35. See:
9628:
8376:
3959:
2214:
418:
310:
169:
85:
12881:
10042:"Univalence Axiom" holds that equivalence of types is equality of types. The research into this property led to
622:
nothing except the basic types, and what can be constructed from them by means of the previous clause are types.
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14900:
14559:
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14403:
14105:
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13863:
13688:
13109:
12793:
12364:
12326:
11983:
11691:
11522:
9667:
5956:-reduction. They generalize the notion of function application to lambda terms. Symbolically, they are written
3914:
exists. Constructive mathematics does not allow the last step of removing the double negation to conclude that
2088:
809:
469:
457:
7763:
1594:
14678:
14499:
14398:
14181:
14098:
13811:
13742:
13619:
12861:
12506:
12483:
12213:
12203:
11374:
Bove, Ana; Dybjer, Peter (2009), Bove, Ana; Barbosa, Luís Soares; Pardo, Alberto; Pinto, Jorge Sousa (eds.),
10067:
9774:
9639:
may also act as a foundation of mathematics. There are a number of differences between these two approaches.
9328:
7691:
2386:
866:
438:
370:
131:
11664:
11655:
9736:
In type theory, proofs are types whereas in set theory, proofs are part of the underlying first-order logic.
2651:
14323:
14149:
13835:
13469:
13068:
12587:
12175:
12083:
11988:
11764:
11749:
10746:
10615:, for example, a function with no name, but with two parameters in some tuple (x,y) can be denoted by say,
10612:
10582:
10084:
9745:
7220:
4554:
3874:
does not exist and refuting it by contradiction. The conclusion from that step is "it is not the case that
2925:
of a type system with Curry–Howard correspondence. To be sound, such a system must have uninhabited types.
211:
184:
10171:
5275:, which is seen by applying the function application inference rule twice. Thus, the lambda term has type
5212:
3639:
14564:
14201:
14196:
13806:
13545:
13474:
12803:
12704:
12675:
12410:
11908:
11628:
9581:
8713:
8656:
8474:
7900:
7435:
7073:
6272:. The second reduction makes explicit the relationship between lambda expressions and function types: if
4040:
be thought of as a type theory shorn of its syntax." A number of significant results follow in this way:
3542:
3442:
3040:
823:
705:
629:
590:
446:
402:
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4266:
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13114:
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10817:
Elements of Formal
Semantics: An Introduction to the Mathematical Theory of Meaning in Natural Language
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5592:
5562:
5532:
4354:
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1991:
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449:, a programming language which uses UTT (Luo's Unified Theory of dependent Types) for its type system.
414:
291:
207:
192:
70:
54:
10484:
Ferreirós, José; Domínguez, José Ferreirós (2007). "X. Logic and Type Theory in the
Interwar Period".
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Proponents of type theory will also point out its connection to constructive mathematics through the
9600:
9447:
7924:
7283:
7108:
6895:
5248:
5182:
4161:
4127:
4079:, has been a subject of active research since then; see the monograph of Jacobs (1999) for instance.
4044:
3587:
2677:
1919:
1637:
1564:
1020:
814:
453:
383:
344:
219:
188:
93:
12876:
8528:, and returns the list with the element at the end. The type annotation of such a function would be
4891:, and are called lambda terms. These terms are also defined inductively: a lambda term has the form
4854:
4754:
4680:
4566:
4068:
3939:
3062:
Under this intuitionistic interpretation, there are common types that act as the logical operators:
2989:"term introduction" rules define the canonical terms and constructor functions, like "pair" and "S".
2405:
say what judgments can be made, based on the existence of other judgments. Rules are expressed as a
1787:{\displaystyle x:{\mathsf {bool}},y:{\mathsf {nat}}\vdash ({\textrm {if}}\,x\,y\,y):{\mathsf {nat}}}
15111:
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14261:
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12768:
12516:
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11935:
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because it lacks a name. The concept of anonymous functions appears in many programming languages.
4894:
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3435:
1079:
805:
336:
10309:
A History of the Theory of Types: Developments After the Second
Edition of 'Principia Mathematica'
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is the observed similarity between logics and programming languages. The implication in logic, "A
3487:
1054:
262:
191:
that afflicted the original untyped lambda calculus. Church demonstrated that it could serve as a
20:
15106:
14647:
14093:
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13858:
13504:
13424:
13268:
13212:
12825:
12577:
12511:
12402:
12218:
11878:
11649:
11625:— moderated e-mail forum focusing on type theory in computer science, operating since 1987.
11588:
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10375:
9710:. Where a union would be used, type theory uses the sum type, which contains new canonical terms.
827:
434:
11609:
10852:
9760:
Terms usually belong to a single type. However, there are set theories that define "subtyping".
6769:
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14773:
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11150:
Leveraging
Applications of Formal Methods, Verification and Validation: Foundational Techniques
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input. If the type theory allows for dependent types, then it is possible to define a function
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3052:
990:
783:
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105:
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which are interpreted as elements of the set of functions from entities to truth-values, i.e.
15031:
14925:
14890:
14778:
14753:
14597:
14514:
14313:
14166:
13958:
13676:
13412:
13318:
13177:
13162:
13043:
13018:
12612:
12582:
12572:
12468:
12382:
12258:
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12165:
12155:
12038:
12003:
11993:
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11774:
11769:
11734:
11570:
11339:
11152:. ISoLA 2016. Lecture Notes in Computer Science. Vol. 9952. Springer. pp. 303–317.
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7383:
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2897:
2817:
2556:
2536:
2368:
2342:
2190:
2064:
1967:
1886:
1865:
1670:
997:, where a term is applied to another term. Constant symbols could include the natural number
893:
366:
159:
146:
61:
11297:
7167:
6502:
6397:
6025:
5919:
5482:
4888:
3253:
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15167:
15016:
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14248:
14125:
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12013:
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11783:
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10208:
Aarts, C.; Backhouse, R.; Hoogendijk, P.; Voermans, E.; van der Woude, J. (December 1992).
10165:
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9659:
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9149:
8626:{\displaystyle \mathrm {append} :(\Pi \,a.{\mathsf {list}}\,a\to a\to {\mathsf {list}}\,a)}
7501:
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6559:
6522:
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5939:
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3015:, or constructive, logic. Formally, type theory is often cited as an implementation of the
3012:
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2623:
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2414:
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2136:
1437:
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994:
982:
299:
235:
123:
11622:
10673:
8394:
8322:
8226:
3965:
3781:
2995:"computation" rules specify how computation is performed with the type-specific functions.
290:, used type theory to encode mathematics on a computer. Martin-Löf specifically developed
8:
15101:
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13873:
13666:
13661:
13609:
13377:
13366:
13038:
12938:
12866:
12857:
12853:
12788:
12783:
12637:
12562:
12478:
12463:
12228:
12008:
11965:
11960:
11857:
11847:
11819:
11087:
10452:
9577:
9573:
9533:
9383:
9357:
9324:
7838:
7326:
7318:: functions with vector length restrictions or length matching requirements, such as the
5833:
5072:. The following lambda term represents a function which doubles an input natural number.
4563:
Some terms may be declared directly as having a simple type, such as the following term,
3405:
2963:
916:
10265:
Covers type theory in depth, including polymorphic and dependent type extensions. Gives
2918:
15177:
15146:
15071:
15041:
15006:
14986:
14915:
14895:
14833:
14828:
14738:
14728:
14713:
14657:
14444:
14213:
14176:
14161:
14154:
14137:
13941:
13923:
13789:
13715:
13698:
13651:
13464:
13373:
13207:
13192:
13152:
13104:
13089:
13077:
13033:
13008:
12778:
12727:
12602:
12501:
12377:
12334:
12243:
12185:
12170:
12160:
11945:
11744:
11308:
11264:
11236:
11118:
10728:
10720:
10184:
10088:
9741:
9693:
9673:
9478:
9337:
8906:
8833:
8693:
8636:
8511:
8368:
8352:
8040:
7856:
7816:
7741:
7521:
7481:
7263:
7193:
7114:
7006:
6795:
6749:
6651:
6605:
6374:
6354:
6334:
6314:
6255:
6235:
6215:
6195:
6116:
6092:
5462:
5442:
5402:
5356:
5329:
5162:
5055:
5015:
4949:
4929:
3917:
3897:
3877:
3857:
3830:
3810:
3761:
2959:
2955:
2857:
2657:
2576:
2322:
2044:
2024:
1947:
1665:
1544:
1491:
1464:
1327:
1304:
1278:
1104:
1049:
1000:
858:
735:
685:
665:
570:
550:
522:
502:
481:
362:
314:
196:
14454:
13397:
7603:{\displaystyle \mathrm {first} :(\Pi \,\sigma \,\tau .\sigma \times \tau \to \sigma )}
4668:{\displaystyle \mathrm {add} :{\mathsf {nat}}\to ({\mathsf {nat}}\to {\mathsf {nat}})}
15126:
15081:
15061:
15021:
14960:
14930:
14910:
14703:
14632:
14439:
14379:
14186:
13996:
13986:
13878:
13759:
13594:
13570:
13351:
13335:
13240:
13217:
13094:
13063:
13028:
12923:
12758:
12622:
12552:
12531:
12493:
12301:
12268:
12248:
11940:
11852:
11726:
11391:
11268:
11254:
11193:
11161:
11060:
10959:
10821:
10783:
10521:
10489:
10470:
10440:
10407:
10351:
10340:
10320:
10282:
10243:
10222:
10209:
10136:
10103:
10018:
9763:
Computation takes place by repeated application of rules. Many types of theories are
9644:
9616:
8380:
7678:{\displaystyle \mathrm {second} :(\Pi \,\sigma \,\tau .\sigma \times \tau \to \tau )}
6599:
5346:
4076:
3995:
2763:
2757:
2402:
862:
351:
10732:
9334:
An identity type requires two terms of the same type and is written with the symbol
6157:
The first reduction describes how to evaluate a lambda term: if a lambda expression
2992:"term elimination" rules define the other functions like "first", "second", and "R".
234:. Type theory is an active area of research, one direction being the development of
203:
74:
15131:
15056:
14945:
14723:
14393:
14388:
14281:
14238:
14060:
14021:
14016:
14001:
13827:
13784:
13681:
13479:
13429:
13003:
12965:
12455:
12339:
12306:
12101:
12023:
11912:
11898:
11893:
11842:
11829:
11754:
11707:
11439:
11431:
11383:
11382:, Lecture Notes in Computer Science, Berlin, Heidelberg: Springer, pp. 57–99,
11318:
11250:
11246:
11153:
10951:
10900:
10712:
10513:
10432:
10312:
10146:
6966:
6486:{\displaystyle (\lambda x.\mathrm {add} \,x\,x)\,2\rightarrow \mathrm {add} \,2\,2}
5837:
5620:
5150:{\displaystyle (\lambda x.\mathrm {add} \,x\,x):{\mathsf {nat}}\to {\mathsf {nat}}}
2406:
986:
831:
795:
485:
477:
473:
154:
135:
115:
31:
10504:
10401:
10274:
10237:
9658:. When a type theory encodes the concepts of "and" and "or" as types, it leads to
14935:
14838:
14733:
14698:
14374:
14364:
14318:
14301:
14256:
14218:
14120:
14040:
13847:
13774:
13747:
13735:
13641:
13555:
13529:
13484:
13452:
13253:
13055:
12998:
12948:
12913:
12871:
12526:
12420:
12392:
12286:
12238:
12223:
12208:
12063:
12058:
11998:
11888:
11862:
11814:
11759:
11157:
11054:
10859:
10777:
10680:
10659:
10652:
10459:
10386:
10382:
10058:
10050:
9749:
9722:
9632:
9620:
9588:
7337:
as a framework for studying various restrictions and levels of dependent typing.
7330:
4705:
is a function which takes in a natural number and returns a function of the form
4591:, which takes in two natural numbers in sequence and returns one natural number.
4026:
3056:
3044:
3020:
2946:
1796:
If there are no assumptions, there will be nothing to the left of the turnstile.
842:
340:
330:
306:
231:
215:
180:
89:
81:
11387:
11323:
10486:
Labyrinth of thought: a history of set theory and its role in modern mathematics
10242:. Studies in Logic and the Foundations of Mathematics. Vol. 141. Elsevier.
5916:
Type theories that allow for lambda terms also include inference rules known as
5345:
The power of type theories is in specifying how terms may be combined by way of
15121:
15116:
15036:
14920:
14798:
14693:
14534:
14359:
14338:
14296:
14276:
14171:
14026:
13624:
13614:
13604:
13599:
13533:
13407:
13283:
13172:
13167:
13145:
12746:
12632:
12536:
12435:
12281:
12253:
11671:
11576:
11228:
10602:
with his simple theory of types, and explained his method in 1956, pages 47-68.
10127:
9592:
8751:
8384:
8364:
4121:
4032:
4007:
387:
223:
11463:
11435:
11375:
9563:
is a notable area of research that mainly deals with equality in type theory.
9559:
The complexities of equality in type theory make it an active research topic;
7853:
The sum type depends on two types, and it is commonly written with the symbol
845:
introduced a theory of logical types into the social sciences; his notions of
15161:
14808:
14783:
14617:
14333:
14011:
13518:
13303:
13293:
13263:
13248:
12918:
12521:
11809:
10963:
10700:
10219:
An
Introduction to Mathematical Logic and Type Theory: To Truth Through Proof
9444:
is a possible type. Canonical terms are created with a reflexivity function,
9320:
7345:
The product type depends on two types, and its terms are commonly written as
5822:{\displaystyle ((\mathrm {add} \,1)((\mathrm {add} \,2)\,0)):{\textsf {nat}}}
4155:
3947:
2410:
176:
127:
65:
57:. Two influential type theories that have been proposed as foundations are:
39:
11558:
11417:
11298:"Cubical Type Theory: A constructive interpretation of the univalence axiom"
11174:
10904:
8367:, dependent product and dependent sum types, allow for the theory to encode
6824:
The
Boolean type has exactly 2 canonical terms. The type is usually written
1488:
Judgments may follow from assumptions. For example, one might say "assuming
15076:
14996:
14863:
14743:
14627:
14607:
14233:
14080:
13981:
13973:
13853:
13801:
13710:
13646:
13629:
13560:
13419:
13278:
12980:
12763:
12617:
12276:
11614:
11305:
21st International Conference on Types for Proofs and Programs (TYPES 2015)
10923:
Handbook of the Philosophy of Science. Volume 14: Philosophy of Linguistics
9730:
7894:
7842:
7346:
7311:
7005:
for zero. Canonical values larger than zero use iterated applications of a
6746:. The unit type is also used in proofs of type inhabitation. If for a type
3754:
Per Martin-Löf proposed his intuitionistic type theory as a foundation for
3636:
Because the law of excluded middle does not hold, there is no term of type
3036:
2999:
For examples of rules, an interested reader may follow Appendix A.2 of the
2531:
978:
422:
119:
14991:
14848:
14612:
14539:
14223:
13402:
13392:
13339:
13023:
12943:
12928:
12753:
12607:
12233:
12145:
11182:
11052:
10370:
10142:
7334:
7319:
7315:
5901:{\displaystyle \mathrm {add} \,1\,(\mathrm {add} \,2\,0):{\textsf {nat}}}
3307:
3246:
2922:
870:
846:
799:
540:
43:
27:
250:
15141:
14788:
14554:
14509:
14504:
13273:
13128:
13099:
12905:
12627:
12557:
12150:
11883:
11739:
11597:
book, which proposed homotopy type theory as a mathematical foundation.
11594:
11296:
Cohen, Cyril; Coquand, Thierry; Huber, Simon; Mörtberg, Anders (2016).
11233:
2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
10985:
10955:
10724:
10316:
8388:
8348:
6553:
4840:{\displaystyle ({\mathsf {nat}}\to {\mathsf {nat}})\to {\mathsf {nat}}}
4260:, whose type may vary. For example, the following may be atomic terms.
4058:
3894:
does not exist". The last step is, by double negation, concluding that
3683:. Likewise, double negation does not hold, so there is no term of type
3371:
2425:
50:
11444:
10940:"Truth of a proposition, evidence of a judgement, validity of a proof"
5677:{\displaystyle (\mathrm {add} \,1):{\textsf {nat}}\to {\textsf {nat}}}
4435:
In addition to atomic terms, most modern type theories also allow for
3942:. However, these operators tend to break desirable properties such as
2762:
Generally, the desired conclusion of a proof in type theory is one of
1848:{\displaystyle \vdash \mathrm {S} :{\mathsf {nat}}\to {\mathsf {nat}}}
1256:{\displaystyle (\mathrm {if} \,\mathrm {true} \,0\,(\mathrm {S} \,0))}
309:
already had difficulty working with the widely accepted foundation of
14940:
14758:
14683:
14662:
14592:
14544:
14524:
14425:
14328:
13381:
13298:
13258:
13222:
13158:
12970:
12960:
12933:
12696:
12132:
12093:
10939:
10891:
Cooper, Robin (2005). "Records and Record Types in Semantic Theory".
10757:
10517:
6681:
5349:. Type theories which have functions also have the inference rule of
4510:
2527:
543:, respectively, and defines the set of types recursively as follows:
165:
10872:
10716:
10350:. Nato Science Series II. Vol. 62. Springer. pp. 213–259.
9714:
theory's computation does require a complicated concept of equality.
9599:, are based on the calculus for inductive constructions, which is a
2982:
Also, for each "by rule" type, there are 4 different kinds of rules
417:. Isabelle also supports foundations besides type theories, such as
14853:
14642:
14410:
14208:
13656:
13361:
12955:
12193:
11676:
11313:
11241:
10122:
10098:
9015:{\displaystyle \mathrm {TF} \colon {\mathsf {bool}}\to U\to U\to U}
442:
318:
287:
10049:"Law of Excluded Middle" is often added to satisfy users who want
4102:
was proposed, which is a homotopy type theory with normalization.
3758:. Constructive mathematics requires when proving "there exists an
3019:
of intuitionistic logic. Additionally, connections can be made to
2742:{\displaystyle {\begin{array}{c}\hline \vdash 0:nat\\\end{array}}}
14006:
12798:
10983:
10467:
A modern perspective on type theory: from its origins until today
9880:{\displaystyle {\mathbb {1} }+{\mathbb {1} }\cong {\mathbb {2} }}
111:
19:"Theory of types" redirects here. For an architectural term, see
10207:
7974:{\displaystyle \mathrm {left} :\sigma \to (\sigma \sqcup \tau )}
7538:. The product type is usually defined with eliminator functions
6499:
for types and terms, there are corresponding inference rules of
10465:
Kamareddine, Fairouz D.; Laan, Twan; Nederpelt, Rob P. (2004).
9139:{\displaystyle \mathrm {TF} \,\mathrm {false} \,\sigma \,\tau }
8032:{\displaystyle \mathrm {right} :\tau \to (\sigma \sqcup \tau )}
7055:{\displaystyle \mathrm {S} :{\mathsf {nat}}\to {\mathsf {nat}}}
4048:
3986:
The opposition of terms and types can also be viewed as one of
3035:
When used as a foundation, certain types are interpreted to be
425:
is an example of a proof system that only supports set theory.
391:
11053:
Henk Barendregt; Wil Dekkers; Richard Statman (20 June 2013).
9066:{\displaystyle \mathrm {TF} \,\mathrm {true} \,\sigma \,\tau }
3011:
The logical framework of a type theory bears a resemblance to
187:. Church's theory of types helped the formal system avoid the
14718:
13550:
12896:
12741:
10189:
10014:
9648:
9624:
8312:{\displaystyle \mathrm {match} \,f\,g\,(\mathrm {right} \,y)}
7837:
Besides ordered pairs, this type is used for the concepts of
5321:, which means it is a function taking a natural number as an
1862:
377:
355:
10862:." Handbook of the Philosophy of Science 14 (2012): 271-323.
9690:
is paired with a proof that the subset's property holds for
9670:
or use a dependently-typed product type, where each element
8216:{\displaystyle \mathrm {match} \,f\,g\,(\mathrm {left} \,x)}
7893:. In programming languages, sum types may be referred to as
2928:
A type theory usually has several rules, including ones to:
2766:. The decision problem of type inhabitation (abbreviated by
11192:. Handbook of the History of Logic. Vol. 6. Elsevier.
10761:
9662:, and does not necessarily have the law of excluded middle.
2972:
list all the interactions of equality, such as substitution
849:
and logical levels are based on Russell's theory of types.
335:
Much of the current research into type theory is driven by
10987:
Homotopy Type Theory: Univalent Foundations of Mathematics
5179:
and (implicit from the lambda term's type) must have type
460:) and its own type system was heavily influenced by them.
11295:
11147:
9319:
Following the notion of Curry-Howard Correspondence, the
6351:. Therefore, the lambda expression is equivalent to just
401:
UTT (Luo's Unified Theory of dependent Types) is used by
10464:
6965:
Natural numbers are usually implemented in the style of
5736:{\displaystyle ((\mathrm {add} \,2)\,0):{\textsf {nat}}}
3003:
book, or read Martin-Löf's Intuitionistic Type Theory.
2921:
shows that type inhabitation is strongly related to the
405:
which is both a programming language and proof assistant
175:
Type theory is particularly popular in conjunction with
14484:
10776:
Chatzikyriakidis, Stergios; Luo, Zhaohui (2017-02-07).
10703:(1940). "A formulation of the simple theory of types".
488:
extensively use type constructors to define the types (
183:. One notable early example of type theory is Church's
8779:, which takes three arguments and behaves as follows.
4098:
mostly by its handling of the equality type. In 2016,
2713:
445:, has a connection to type theory. A prime example is
46:. Type theory is the academic study of type systems.
11665:
Summer 2015 Types, Logic, Semantics, and Verification
10506:
The evolution of type theory in logic and mathematics
9952:
9893:
9849:
9813:
9777:
9696:
9676:
9536:
9501:
9481:
9450:
9412:
9386:
9360:
9340:
9201:
9176:
9152:
9101:
9079:
9031:
8966:
8933:
8909:
8896:{\displaystyle \mathrm {if} \,\mathrm {false} \,x\,y}
8858:
8836:
8788:
8760:
8716:
8696:
8659:
8639:
8534:
8514:
8477:
8440:
8417:
8397:
8325:
8255:
8229:
8162:
8049:
7987:
7932:
7903:
7879:
7859:
7819:
7766:
7744:
7694:
7616:
7544:
7524:
7504:
7484:
7464:
7438:
7406:
7386:
7354:
7286:
7266:
7223:
7196:
7170:
7137:
7117:
7076:
7014:
6975:
6929:
6898:
6876:
6854:
6830:
6798:
6772:
6752:
6732:
6710:
6690:
6654:
6628:
6608:
6582:
6562:
6525:
6505:
6422:
6400:
6377:
6357:
6337:
6317:
6278:
6258:
6238:
6218:
6198:
6163:
6139:
6119:
6095:
6050:
6028:
5965:
5942:
5922:
5846:
5750:
5691:
5632:
5595:
5565:
5535:
5515:
5485:
5465:
5445:
5425:
5405:
5379:
5359:
5281:
5251:
5215:
5185:
5165:
5080:
5058:
5038:
5018:
4998:
4972:
4952:
4932:
4897:
4857:
4785:
4757:
4711:
4683:
4599:
4569:
4539:
4519:
4489:
4469:
4449:
4394:
4357:
4306:
4269:
4229:
4195:
4164:
4130:
3968:
3920:
3900:
3880:
3860:
3833:
3813:
3784:
3764:
3689:
3642:
3590:
3545:
3490:
3445:
3408:
3381:
3344:
3317:
3280:
3256:
3219:
3192:
3161:
3140:
3111:
3090:
2900:
2880:
2860:
2840:
2820:
2772:
2710:
2680:
2660:
2626:
2599:
2579:
2559:
2539:
2423:
2371:
2345:
2325:
2298:
2271:
2217:
2193:
2166:
2139:
2091:
2067:
2047:
2027:
1994:
1970:
1950:
1922:
1889:
1868:
1804:
1695:
1673:
1640:
1597:
1567:
1547:
1514:
1494:
1467:
1440:
1413:
1383:
1353:
1330:
1307:
1281:
1203:
1159:
1127:
1107:
1082:
1057:
1023:
1003:
925:
901:
878:
775:{\displaystyle \langle \langle e,t\rangle ,t\rangle }
744:
708:
688:
668:
632:
593:
573:
553:
525:
505:
10046:, where the property holds without needing an axiom.
9721:. Type theory can encode numbers as functions using
8823:{\displaystyle \mathrm {if} \,\mathrm {true} \,x\,y}
7322:, can encode this requirement as part of the type.
4439:. Function types introduce an arrow symbol, and are
10775:
9635:, abbreviated ZFC. Type theories having sufficient
2969:
define substitution for application of lambda terms
499:The most common construction takes the basic types
11567:article at the Stanford Encyclopedia of Philosophy
11088:"Rules to Martin-Löf's Intuitionistic Type Theory"
10483:
10346:. In Schwichtenberg, H.; Steinbruggen, R. (eds.).
10068:Axiom of choice § In constructive mathematics
9997:
9938:
9879:
9835:
9799:
9702:
9682:
9548:
9522:
9487:
9467:
9436:
9398:
9372:
9346:
9301:
9187:
9158:
9138:
9085:
9065:
9014:
8952:
8915:
8895:
8842:
8822:
8771:
8739:
8702:
8682:
8645:
8625:
8520:
8500:
8463:
8423:
8403:
8335:
8311:
8239:
8215:
8145:
8031:
7973:
7915:
7885:
7865:
7825:
7805:
7750:
7730:
7677:
7602:
7530:
7510:
7490:
7470:
7450:
7424:
7392:
7372:
7302:
7272:
7252:
7202:
7182:
7156:
7123:
7099:
7054:
6997:
6949:
6915:
6884:
6862:
6840:
6816:, which is to say it must have one or more terms.
6804:
6784:
6758:
6738:
6718:
6696:
6684:has exactly 1 canonical term. The type is written
6660:
6640:
6614:
6590:
6568:
6531:
6511:
6485:
6406:
6383:
6363:
6343:
6331:is a function term because it is being applied to
6323:
6303:
6264:
6244:
6224:
6204:
6184:
6145:
6125:
6101:
6081:
6034:
6014:
5948:
5928:
5900:
5836:; however, by convention, function application is
5821:
5735:
5676:
5611:
5581:
5551:
5521:
5501:
5471:
5451:
5431:
5411:
5391:
5365:
5314:{\displaystyle {\mathsf {nat}}\to {\mathsf {nat}}}
5313:
5267:
5237:
5201:
5171:
5149:
5064:
5044:
5024:
5004:
4984:
4958:
4938:
4918:
4871:
4839:
4771:
4744:{\displaystyle {\mathsf {nat}}\to {\mathsf {nat}}}
4743:
4697:
4667:
4583:
4545:
4525:
4501:
4475:
4455:
4419:
4379:
4342:
4291:
4248:
4215:
4181:
4146:
3974:
3926:
3906:
3886:
3866:
3839:
3819:
3799:
3770:
3735:{\displaystyle \Pi A.((A\to \bot )\to \bot )\to A}
3734:
3675:
3620:
3575:
3520:
3475:
3420:
3393:
3356:
3329:
3292:
3265:
3231:
3204:
3167:
3146:
3117:
3096:
2906:
2886:
2866:
2846:
2826:
2802:
2741:
2696:
2666:
2639:
2612:
2585:
2565:
2545:
2518:
2377:
2351:
2331:
2311:
2284:
2255:
2199:
2179:
2152:
2123:
2073:
2053:
2033:
2012:
1976:
1956:
1934:
1895:
1874:
1847:
1786:
1679:
1656:
1626:
1583:
1553:
1533:
1500:
1473:
1453:
1426:
1396:
1366:
1336:
1313:
1287:
1255:
1189:
1145:
1113:
1093:
1065:
1040:
1009:
961:
907:
884:
774:
726:
694:
674:
650:
611:
579:
559:
531:
511:
456:was developed for manipulating type theories (see
11577:Calculus of Constructions / Typed Lambda Calculus
11344:: CS1 maint: DOI inactive as of September 2024 (
11227:Sterling, Jonathan; Angiuli, Carlo (2021-06-29).
10779:Modern Perspectives in Type-Theoretical Semantics
2986:"type formation" rules say how to create the type
1190:{\displaystyle (\mathrm {S} \,(\mathrm {S} \,0))}
15159:
14951:Segmented discourse representation theory (SDRT)
8927:The return type of this function depends on its
6766:, it is consistent to derive a function of type
6622:, it is consistent to derive a function of type
6495:In type theories that also establish notions of
4343:{\displaystyle \mathrm {true} :{\mathsf {bool}}}
2803:{\displaystyle \exists t.\Gamma \vdash t:\tau ?}
1861:of the judgment. Capital greek letters, such as
1664:". Such judgments are formally written with the
962:{\displaystyle \mathrm {term} :{\mathsf {type}}}
11607:
11464:"Programming with Math (Exploring Type Theory)"
11226:
9530:returns the canonical term inhabiting the type
5529:. For example, if one knows the type notations
9998:{\displaystyle A^{B\times C}\cong (A^{B})^{C}}
9939:{\displaystyle A^{B+C}\cong A^{B}\times A^{C}}
9755:
9606:
8358:
3953:
437:, such as the type checking algorithms in the
14470:
12712:
11716:Note: This template roughly follows the 2012
11692:
11059:. Cambridge University Press. pp. 1–66.
10592:
10279:The Computer Science and Engineering Handbook
10132:UTT (Luo's Unified Theory of dependent Types)
9836:{\displaystyle {\mathbb {1} }\times A\cong A}
6082:{\displaystyle (\lambda v.t\,v)\rightarrow t}
6015:{\displaystyle (\lambda v.t)\,s\rightarrow t}
3006:
11190:Sets and Extensions in the Twentieth Century
10873:Generalized quantifiers and natural language
10174:is an implementation of homotopy type theory
4887:New function terms may be constructed using
869:. Most logics have judgments asserting "The
769:
760:
748:
745:
721:
709:
645:
633:
606:
594:
138:proposed various solutions to this problem.
49:Some type theories serve as alternatives to
10512:(PhD). Eindhoven University of Technology.
10106:is often used to define other type theories
5619:, then the following type notations can be
3749:
2256:{\displaystyle \Gamma \vdash t_{1}=t_{2}:T}
1857:The list of assumptions on the left is the
738:of sets of entities. An expression of type
286:The first computer proof assistant, called
281:
14477:
14463:
12904:
12719:
12705:
11699:
11685:
11650:Oregon Programming Languages Summer School
11418:"Introduction to generalized type systems"
11373:
10984:The Univalent Foundations Program (2013).
10937:
10632:
10575:
10216:
9580:. A method that only uses lambda terms is
8391:, they are often written with the symbols
7314:. This allows for greater specificity and
6598:. One use for the empty type is proofs of
6556:has no terms. The type is usually written
3850:An example of a non-constructive proof is
2939:add an assumption to the context (context
11443:
11415:
11322:
11312:
11240:
10674:(rev. Mon Oct 12, 2020) Russell’s Paradox
10645:
10587:documentation, optimization, and dispatch
10369:Intended as a type theory counterpart of
10335:
9872:
9862:
9852:
9816:
9780:
9729:, and the construction closely resembles
9516:
9289:
9285:
9281:
9265:
9256:
9212:
9208:
9132:
9128:
9110:
9059:
9055:
9040:
8889:
8885:
8867:
8816:
8812:
8797:
8733:
8676:
8616:
8587:
8564:
8494:
8347:The sum type is used for the concepts of
8329:
8302:
8281:
8277:
8273:
8233:
8206:
8188:
8184:
8180:
8076:
7787:
7712:
7650:
7646:
7575:
7571:
7246:
7111:of terms, where each term must have type
7093:
6878:
6856:
6833:
6726:and the single canonical term is written
6712:
6584:
6542:
6479:
6475:
6457:
6450:
6446:
6371:, as both take in one argument and apply
6294:
6066:
5984:
5893:
5881:
5877:
5862:
5858:
5814:
5799:
5792:
5768:
5728:
5716:
5709:
5669:
5659:
5647:
5604:
5574:
5544:
5492:
5328:A lambda term is often referred to as an
5231:
5227:
5108:
5104:
4851:, so the parentheses may be dropped from
3017:Brouwer–Heyting–Kolmogorov interpretation
2124:{\displaystyle \Gamma \vdash T_{1}=T_{2}}
1761:
1757:
1753:
1617:
1613:
1609:
1243:
1234:
1230:
1215:
1177:
1168:
1136:
852:
210:, which was proposed as a foundation for
11552:
11352:
10747:Introduction To Mathematical Logic Vol 1
10695:
10693:
10691:
10689:
10399:
10272:
10073:
9654:Classical set theory and logic have the
8373:universal and existential quantification
7806:{\displaystyle \mathrm {second} \,(s,t)}
4509:is the type of a function which takes a
3043:out of types. However, the logic is not
1627:{\displaystyle (\mathrm {if} \,x\,y\,y)}
861:, which is to say it is a collection of
428:
11589:Programming in Martin-Löf's Type Theory
11229:"Normalization for Cubical Type Theory"
10431:
10381:
10306:
9800:{\displaystyle {\mathbb {0} }+A\cong A}
7731:{\displaystyle \mathrm {first} \,(s,t)}
6394:For example, the following term may be
6311:is a lambda term, then it must be that
5340:
4082:
3051:, which is to say it does not have the
3030:
2365:Some textbooks use a triple equal sign
15160:
12726:
12416:Knowledge representation and reasoning
10890:
10813:
10699:
10640:Getting Started With Julia Programming
10549:, Stanford Encyclopedia of Philosophy.
10403:Type Theory and Functional Programming
10235:
10035:Some commonly encountered axioms are:
9230:
9227:
9224:
9221:
8989:
8986:
8983:
8980:
8945:
8942:
8939:
8936:
8728:
8725:
8722:
8719:
8671:
8668:
8665:
8662:
8611:
8608:
8605:
8602:
8582:
8579:
8576:
8573:
8489:
8486:
8483:
8480:
7295:
7292:
7289:
7241:
7238:
7235:
7232:
7229:
7226:
7149:
7146:
7143:
7140:
7088:
7085:
7082:
7079:
7047:
7044:
7041:
7031:
7028:
7025:
6990:
6987:
6984:
5306:
5303:
5300:
5290:
5287:
5284:
5260:
5257:
5254:
5194:
5191:
5188:
5142:
5139:
5136:
5126:
5123:
5120:
4832:
4829:
4826:
4813:
4810:
4807:
4797:
4794:
4791:
4736:
4733:
4730:
4720:
4717:
4714:
4657:
4654:
4651:
4641:
4638:
4635:
4622:
4619:
4616:
4412:
4409:
4406:
4403:
4372:
4369:
4366:
4335:
4332:
4329:
4326:
4284:
4281:
4278:
4241:
4238:
4235:
4232:
4139:
4136:
4133:
2951:use an assumption to create a variable
2689:
2686:
2683:
1840:
1837:
1834:
1824:
1821:
1818:
1779:
1776:
1773:
1735:
1732:
1729:
1713:
1710:
1707:
1704:
1649:
1646:
1643:
1576:
1573:
1570:
1526:
1523:
1520:
1517:
1481:
954:
951:
948:
945:
358:, often to define other type theories;
14906:Discourse representation theory (DRT)
14458:
12700:
12441:Philosophy of artificial intelligence
11680:
11579:textbook style paper by Helmut Brandl
11503:from the original on 22 December 2021
11457:
11455:
11369:
11367:
11291:
11289:
11082:
11080:
11078:
11076:
11048:
11046:
11044:
11042:
11040:
11038:
11036:
11034:
10979:
10977:
10975:
10973:
10933:
10931:
10686:
10664:
10563:
8633:, which can be read as "for any type
7253:{\displaystyle {\mathsf {vector}}\,n}
2854:, decide whether there exists a term
2396:
1271:Most type theories have 4 judgments:
376:computational type theory is used by
222:, which is used as the foundation by
11760:Energy consumption (Green computing)
11706:
11601:
11561:, which has articles on many topics.
11461:
11235:. Rome, Italy: IEEE. pp. 1–15.
11180:
10739:
10605:
10570:
10502:
10212:. Technische Universiteit Eindhoven.
5238:{\displaystyle \mathrm {add} \,x\,x}
4047:correspond to the typed λ-calculus (
3676:{\displaystyle \Pi a.A+(A\to \bot )}
2975:define a hierarchy of type universes
2751:
2526:The rules are syntactic and work by
2210:
480:and its descendants. In particular,
409:Many type theories are supported by
361:many type theories which fall under
245:
114:in a mathematical equation based on
14819:Quantificational variability effect
14486:Formal semantics (natural language)
12446:Distributed artificial intelligence
11718:ACM Computing Classification System
11129:from the original on 13 August 2023
10871:Barwise, Jon; Cooper, Robin (1981)
10671:Stanford Encyclopedia of Philosophy
10392:Stanford Encyclopedia of Philosophy
9331:that type theory already provides.
8740:{\displaystyle {\mathsf {list}}\,a}
8683:{\displaystyle {\mathsf {list}}\,a}
8501:{\displaystyle {\mathsf {list}}\,a}
7916:{\displaystyle \sigma \sqcup \tau }
7451:{\displaystyle \sigma \times \tau }
7100:{\displaystyle {\mathsf {list}}\,a}
7065:
6672:, which is to say it has no terms.
6212:, one replaces every occurrence of
4966:is a term, and its type is notated
4065:locally cartesian closed categories
3807:", one must construct a particular
3576:{\displaystyle \exists a\in A,P(a)}
3476:{\displaystyle \forall a\in A,P(a)}
727:{\displaystyle \langle e,t\rangle }
651:{\displaystyle \langle a,b\rangle }
612:{\displaystyle \langle a,b\rangle }
468:Type theory is also widely used in
324:
110:Type theory was created to avoid a
94:Calculus of Inductive Constructions
21:Form (architecture) § Theories
13:
11951:Integrated development environment
11452:
11364:
11286:
11073:
11031:
10970:
10928:
10210:"A Relational Theory of Datatypes"
10200:
10159:
10053:, instead of intuitionistic logic.
9566:
9523:{\displaystyle \mathrm {refl} \,t}
9512:
9509:
9506:
9503:
9461:
9458:
9455:
9452:
9329:judgmental (syntactic) equivalence
9277:
9274:
9253:
9205:
9181:
9178:
9124:
9121:
9118:
9115:
9112:
9106:
9103:
9051:
9048:
9045:
9042:
9036:
9033:
8971:
8968:
8881:
8878:
8875:
8872:
8869:
8863:
8860:
8808:
8805:
8802:
8799:
8793:
8790:
8765:
8762:
8561:
8551:
8548:
8545:
8542:
8539:
8536:
8457:
8454:
8451:
8448:
8445:
8442:
8418:
8398:
8298:
8295:
8292:
8289:
8286:
8269:
8266:
8263:
8260:
8257:
8202:
8199:
8196:
8193:
8176:
8173:
8170:
8167:
8164:
8073:
8063:
8060:
8057:
8054:
8051:
8001:
7998:
7995:
7992:
7989:
7943:
7940:
7937:
7934:
7783:
7780:
7777:
7774:
7771:
7768:
7708:
7705:
7702:
7699:
7696:
7643:
7633:
7630:
7627:
7624:
7621:
7618:
7568:
7558:
7555:
7552:
7549:
7546:
7016:
6960:
6943:
6940:
6937:
6934:
6931:
6909:
6906:
6903:
6900:
6892:. The canonical terms are usually
6773:
6691:
6635:
6563:
6471:
6468:
6465:
6442:
6439:
6436:
5873:
5870:
5867:
5854:
5851:
5848:
5788:
5785:
5782:
5764:
5761:
5758:
5705:
5702:
5699:
5643:
5640:
5637:
5335:
5223:
5220:
5217:
5100:
5097:
5094:
4865:
4862:
4859:
4765:
4762:
4759:
4691:
4688:
4685:
4607:
4604:
4601:
4577:
4574:
4571:
4553:. Types of this form are known as
4420:{\displaystyle y:{\mathsf {bool}}}
4317:
4314:
4311:
4308:
4292:{\displaystyle 42:{\mathsf {nat}}}
4209:
4206:
4203:
4200:
4197:
4175:
4172:
4169:
4166:
4110:
4020:
3854:. The first step is assuming that
3720:
3711:
3690:
3667:
3643:
3591:
3546:
3491:
3446:
3287:
3257:
3162:
3141:
3112:
3091:
2901:
2821:
2782:
2773:
2712:
2560:
2540:
2490:
2484:
2451:
2428:
2346:
2218:
2194:
2092:
2068:
1995:
1971:
1923:
1890:
1869:
1809:
1605:
1602:
1321:
1295:
1239:
1226:
1223:
1220:
1217:
1211:
1208:
1173:
1164:
1132:
1087:
1084:
1059:
1034:
1031:
1028:
1025:
936:
933:
930:
927:
837:
82:computerized proof-writing systems
14:
15189:
12426:Automated planning and scheduling
11956:Software configuration management
11547:
11423:Journal of Functional Programming
11360:(2015) Getting Started with Julia
11000:
10919:Type theory and semantics in flux
10853:Type theory and semantics in flux
10683:3. Early Responses to the Paradox
10341:"Naïve Computational Type Theory"
10239:Categorical Logic and Type Theory
9744:, its connection to logic by the
8464:{\displaystyle \mathrm {append} }
8434:For example, consider a function
6998:{\displaystyle 0:{\mathsf {nat}}}
5612:{\displaystyle 2:{\textsf {nat}}}
5582:{\displaystyle 1:{\textsf {nat}}}
5552:{\displaystyle 0:{\textsf {nat}}}
4430:
4380:{\displaystyle x:{\mathsf {nat}}}
4015:
4001:
3827:and a proof that it has property
2978:assert the existence of new types
2339:and are equal (under assumptions
2013:{\displaystyle \Gamma \vdash t:T}
1146:{\displaystyle (\mathrm {S} \,0)}
14438:
12680:
12670:
12661:
12660:
11660:Robert Harper's talks on YouTube
11652:, many lectures and some notes.
11631:: "Introduction to Type Theory."
11334:from the original on 2022-10-09.
10893:Journal of Logic and Computation
10537:from the original on 2022-10-09.
10367:from the original on 2022-10-09.
9314:
8953:{\displaystyle {\mathsf {bool}}}
7157:{\displaystyle {\mathsf {list}}}
6950:{\displaystyle \mathrm {false} }
6841:{\displaystyle {\textsf {bool}}}
6304:{\displaystyle (\lambda v.t\,v)}
5392:{\displaystyle \sigma \to \tau }
5325:and returning a natural number.
4985:{\displaystyle \sigma \to \tau }
4502:{\displaystyle \sigma \to \tau }
4249:{\displaystyle {\mathsf {bool}}}
4216:{\displaystyle \mathrm {false} }
4035:writes: "In fact categories can
2704:, one would write the following.
1987:
1915:
1534:{\displaystyle {\mathsf {bool}}}
812:representation framework, using
386:and its derivatives are used by
249:
12671:
12074:Computational complexity theory
11533:from the original on 2022-01-19
11515:
11485:
11474:from the original on 2022-01-22
11409:
11275:from the original on 2023-08-13
11220:
11209:from the original on 2018-04-17
11141:
11111:
11100:from the original on 2021-10-21
11019:from the original on 2022-10-09
10994:
10911:
10884:
10865:
10845:
10834:from the original on 2023-08-10
10807:
10796:from the original on 2023-08-10
10769:
10420:from the original on 2021-03-23
10311:. Lambert Academic Publishing.
10295:from the original on 2008-04-10
10281:. CRC Press. pp. 2208–36.
10256:from the original on 2023-08-10
10013:Most type theories do not have
9468:{\displaystyle \mathrm {refl} }
9323:is a type introduced to mirror
7340:
7303:{\displaystyle {\mathsf {nat}}}
6916:{\displaystyle \mathrm {true} }
6819:
5268:{\displaystyle {\mathsf {nat}}}
5202:{\displaystyle {\mathsf {nat}}}
4882:
4751:. The parentheses clarify that
4182:{\displaystyle \mathrm {true} }
4147:{\displaystyle {\mathsf {nat}}}
4115:
3621:{\displaystyle \Sigma a:A.P(a)}
2697:{\displaystyle {\mathsf {nat}}}
2450:
1935:{\displaystyle \Gamma \vdash T}
1657:{\displaystyle {\mathsf {nat}}}
1584:{\displaystyle {\mathsf {nat}}}
1374:
1041:{\displaystyle \mathrm {true} }
241:
14901:Combinatory categorial grammar
11858:Network performance evaluation
11416:Barendegt, Henk (April 1991).
11251:10.1109/LICS52264.2021.9470719
11188:. In Kanamory, Akihiro (ed.).
10938:Martin-Löf, Per (1987-12-01).
10820:. Edinburgh University Press.
10751:
10543:Approaches to Natural Language
10439:. Cambridge University Press.
9986:
9972:
9296:
9293:
9250:
9247:
9241:
9235:
9202:
9006:
9000:
8994:
8620:
8597:
8591:
8558:
8306:
8282:
8210:
8189:
8140:
8134:
8131:
8119:
8116:
8113:
8107:
8101:
8098:
8095:
8089:
8083:
8070:
8026:
8014:
8011:
7968:
7956:
7953:
7800:
7788:
7725:
7713:
7672:
7666:
7640:
7597:
7591:
7565:
7419:
7407:
7367:
7355:
7174:
7036:
6776:
6632:
6461:
6454:
6423:
6298:
6279:
6179:
6164:
6073:
6070:
6051:
6009:
5994:
5988:
5981:
5966:
5885:
5863:
5806:
5803:
5796:
5778:
5775:
5772:
5754:
5751:
5720:
5713:
5695:
5692:
5664:
5651:
5633:
5496:
5486:
5383:
5295:
5131:
5112:
5081:
4976:
4913:
4898:
4872:{\displaystyle \mathrm {add} }
4821:
4818:
4802:
4786:
4772:{\displaystyle \mathrm {add} }
4725:
4698:{\displaystyle \mathrm {add} }
4662:
4646:
4630:
4627:
4584:{\displaystyle \mathrm {add} }
4493:
4124:, often notated with the type
4105:
3969:
3940:call with current continuation
3794:
3788:
3726:
3723:
3717:
3714:
3708:
3702:
3699:
3670:
3664:
3658:
3615:
3609:
3570:
3564:
3515:
3509:
3470:
3464:
3284:
3223:
3196:
2932:create a judgment (known as a
2874:that can be assigned the type
1829:
1765:
1743:
1621:
1598:
1250:
1247:
1235:
1204:
1184:
1181:
1169:
1160:
1140:
1128:
463:
141:By 1908, Russell arrived at a
1:
14679:Antecedent-contained deletion
14399:History of mathematical logic
12229:Multimedia information system
12214:Geographic information system
12204:Enterprise information system
11793:Computer systems organization
10705:The Journal of Symbolic Logic
10626:
10277:. In Tucker, Allen B. (ed.).
9188:{\displaystyle \mathrm {if} }
8772:{\displaystyle \mathrm {if} }
8043:, and an eliminator function
7214:of all types in the theory.
7107:, which should correspond to
6547:
6185:{\displaystyle (\lambda v.t)}
5911:
4919:{\displaystyle (\lambda v.t)}
4483:are types, then the notation
2417:rule for judgmental equality.
2084:
1964:is a type (under assumptions
1909:Formal notation for judgments
1094:{\displaystyle \mathrm {if} }
132:The Foundations of Arithmetic
16:Concept in mathematical logic
14324:Primitive recursive function
12588:Computational social science
12176:Theoretical computer science
11989:Software development process
11765:Electronic design automation
11750:Very Large Scale Integration
11637:of summer schools 2005–2008
11183:"Types, Sets and Categories"
11158:10.1007/978-3-319-47166-2_21
10348:Proof and System-Reliability
10085:Simply typed lambda calculus
8371:by acting as equivalents to
6969:. There is a canonical term
6885:{\displaystyle \mathbb {2} }
6863:{\displaystyle \mathbb {B} }
6719:{\displaystyle \mathbb {1} }
6675:
6591:{\displaystyle \mathbb {0} }
3521:{\displaystyle \Pi a:A.P(a)}
1266:
1066:{\displaystyle \mathrm {S} }
1048:, and functions such as the
784:natural language quantifiers
470:formal theories of semantics
195:and it was referred to as a
185:simply typed lambda calculus
149:, both of which appeared in
32:theoretical computer science
7:
12411:Natural language processing
12199:Information storage systems
11635:Types Project lecture notes
11610:"Computational type theory"
11608:Robert L. Constable (ed.).
11448:– via Cambridge Core.
11388:10.1007/978-3-642-03153-3_2
11324:10.4230/LIPIcs.CVIT.2016.23
10814:Winter, Yoad (2016-04-08).
10619:, as an anonymous function.
10307:Collins, Jordan E. (2012).
10178:
9756:Properties of type theories
9629:Zermelo–Fraenkel set theory
9611:The most commonly accepted
9607:Differences from set theory
8379:. As they also connect to
8377:Curry–Howard Correspondence
8359:Dependent products and sums
7848:
7310:encoding the length of the
5623:from function application.
4533:and returns a term of type
4045:cartesian closed categories
3960:Curry–Howard correspondence
3954:Curry-Howard correspondence
1101:. Thus some terms could be
824:natural language processing
311:Zermelo–Fraenkel set theory
170:Zermelo-Fraenkel set theory
10:
15194:
14560:Syntax–semantics interface
13388:Schröder–Bernstein theorem
13115:Monadic predicate calculus
12774:Foundations of mathematics
12327:Human–computer interaction
12297:Intrusion detection system
12209:Social information systems
12194:Database management system
11583:Intuitionistic Type Theory
11565:Intuitionistic Type Theory
11056:Lambda Calculus with Types
10876:Linguistics and Philosophy
10679:December 18, 2021, at the
10488:(2nd ed.). Springer.
10217:Andrews B., Peter (2002).
10168:explores equality of types
10094:intuitionistic type theory
9613:foundation for mathematics
6785:{\displaystyle \top \to a}
6641:{\displaystyle a\to \bot }
5439:, then the application of
4096:intuitionistic type theory
4005:
3293:{\displaystyle A\to \bot }
3007:Connections to foundations
2755:
702:. Thus one has types like
328:
305:Mathematicians working in
292:intuitionistic type theory
208:intuitionistic type theory
103:
99:
71:Intuitionistic type theory
18:
15094:
15052:Question under discussion
15002:Conversational scoreboard
14979:
14883:
14876:
14779:Intersective modification
14764:Homogeneity (linguistics)
14671:
14580:
14573:
14492:
14434:
14421:Philosophy of mathematics
14370:Automated theorem proving
14352:
14247:
14079:
13972:
13824:
13541:
13517:
13495:Von Neumann–Bernays–Gödel
13440:
13334:
13238:
13136:
13127:
13054:
12989:
12895:
12817:
12734:
12656:
12593:Computational engineering
12568:Computational mathematics
12545:
12492:
12454:
12401:
12363:
12325:
12267:
12184:
12130:
12092:
12037:
11974:
11907:
11871:
11828:
11792:
11725:
11714:
11644:has introductory lectures
11571:Lambda Calculi with Types
11436:10.1017/S0956796800020025
11376:"Dependent Types at Work"
10110:calculus of constructions
10021:) and axioms about sets.
10008:
9748:, and its connections to
9601:calculus of constructions
9325:propositional equivalence
5832:Parentheses indicate the
4946:is a formal variable and
4223:), notated with the type
3357:{\displaystyle A\times B}
2947:rearrange the assumptions
452:The programming language
384:calculus of constructions
345:automated theorem provers
220:calculus of constructions
193:foundation of mathematics
55:foundation of mathematics
15112:Distributional semantics
12603:Computational healthcare
12598:Differentiable computing
12517:Graphics processing unit
11936:Domain-specific language
11805:Computational complexity
11493:"Axioms and Computation"
11119:"proof by contradiction"
10598:Church demonstrated his
10556:
10437:Basic Simple Type Theory
10400:Thompson, Simon (1991).
10221:(2nd ed.). Kluwer.
10116:
10078:
9746:Curry–Howard isomorphism
8375:; this is formalized by
8369:BHK intuitionistic logic
7923:is usually defined with
4075:The interplay, known as
4069:Martin-Löf type theories
3756:constructive mathematics
3750:Constructive mathematics
3330:{\displaystyle A\land B}
2894:in the type environment
972:
908:{\displaystyle \varphi }
885:{\displaystyle \varphi }
806:Type theory with records
282:Mathematical foundations
212:constructive mathematics
143:ramified theory of types
15173:Systems of formal logic
15107:Computational semantics
14844:Subsective modification
14648:Propositional attitudes
14071:Self-verifying theories
13892:Tarski's axiomatization
12843:Tarski's undefinability
12838:incompleteness theorems
12578:Computational chemistry
12512:Photograph manipulation
12403:Artificial intelligence
12219:Decision support system
11585:notes by Per Martin-Löf
11573:book by Henk Barendregt
11497:Theorem Proving in Lean
11327:(inactive 2024-09-12).
11003:"Types of proof system"
10990:. Homotopy Type Theory.
10273:Cardelli, Luca (1996).
9725:, or more naturally as
9719:encodes numbers as sets
9437:{\displaystyle x+1=1+x}
9195:may then be written as
9086:{\displaystyle \sigma }
8424:{\displaystyle \Sigma }
7886:{\displaystyle \sqcup }
7471:{\displaystyle \sigma }
7393:{\displaystyle \times }
5432:{\displaystyle \sigma }
5005:{\displaystyle \sigma }
4779:does not have the type
4526:{\displaystyle \sigma }
4456:{\displaystyle \sigma }
3394:{\displaystyle A\lor B}
3302:Function to Empty Type
2966:for judgmental equality
2907:{\displaystyle \Gamma }
2827:{\displaystyle \Gamma }
2566:{\displaystyle \Delta }
2546:{\displaystyle \Gamma }
2385:to stress that this is
2378:{\displaystyle \equiv }
2352:{\displaystyle \Gamma }
2200:{\displaystyle \Gamma }
2074:{\displaystyle \Gamma }
1977:{\displaystyle \Gamma }
1896:{\displaystyle \Delta }
1875:{\displaystyle \Gamma }
1680:{\displaystyle \vdash }
828:computational semantics
435:static program analysis
15137:Philosophy of language
14774:Inalienable possession
14754:Free choice inferences
14749:Faultless disagreement
14520:Generalized quantifier
14445:Mathematics portal
14056:Proof of impossibility
13704:propositional variable
13014:Propositional calculus
12643:Educational technology
12474:Reinforcement learning
12224:Process control system
12122:Computational geometry
12112:Algorithmic efficiency
12107:Analysis of algorithms
11755:Systems on Chip (SoCs)
11181:Bell, John L. (2012).
10917:Cooper, Robin (2010).
10642:ISBN 978-1-78328-479-5
10571:§ Terms and types
10195:Type–token distinction
10152:others under the name
10141:others defined in the
9999:
9940:
9881:
9837:
9801:
9704:
9684:
9656:law of excluded middle
9603:with inductive types.
9550:
9524:
9489:
9469:
9438:
9400:
9374:
9348:
9303:
9189:
9160:
9140:
9087:
9067:
9016:
8954:
8917:
8897:
8844:
8824:
8773:
8750:Sum types are seen in
8741:
8704:
8684:
8647:
8627:
8522:
8502:
8465:
8425:
8405:
8337:
8313:
8241:
8217:
8147:
8033:
7975:
7917:
7887:
7867:
7827:
7807:
7752:
7732:
7679:
7604:
7532:
7512:
7492:
7472:
7452:
7426:
7394:
7374:
7304:
7274:
7254:
7204:
7184:
7183:{\displaystyle U\to U}
7158:
7125:
7101:
7056:
6999:
6951:
6917:
6886:
6864:
6842:
6806:
6786:
6760:
6740:
6720:
6698:
6662:
6642:
6616:
6592:
6570:
6543:Common terms and types
6533:
6513:
6512:{\displaystyle \beta }
6487:
6408:
6407:{\displaystyle \beta }
6385:
6365:
6345:
6325:
6305:
6266:
6246:
6226:
6206:
6186:
6147:
6127:
6103:
6083:
6036:
6035:{\displaystyle \beta }
6016:
5950:
5930:
5929:{\displaystyle \beta }
5902:
5823:
5737:
5678:
5613:
5583:
5553:
5523:
5503:
5502:{\displaystyle (t\,s)}
5473:
5453:
5433:
5413:
5393:
5367:
5315:
5269:
5239:
5203:
5173:
5151:
5066:
5046:
5026:
5006:
4986:
4960:
4940:
4920:
4873:
4841:
4773:
4745:
4699:
4669:
4585:
4547:
4527:
4503:
4477:
4457:
4421:
4381:
4344:
4293:
4250:
4217:
4183:
4148:
3983:"proofs as programs".
3976:
3928:
3908:
3888:
3868:
3852:proof by contradiction
3841:
3821:
3801:
3772:
3736:
3677:
3622:
3577:
3522:
3477:
3422:
3395:
3358:
3331:
3294:
3267:
3266:{\displaystyle \neg A}
3233:
3232:{\displaystyle A\to B}
3206:
3205:{\displaystyle A\to B}
3169:
3148:
3119:
3098:
3053:law of excluded middle
2908:
2888:
2868:
2848:
2828:
2804:
2743:
2698:
2668:
2641:
2614:
2587:
2567:
2547:
2520:
2379:
2353:
2333:
2313:
2286:
2257:
2201:
2181:
2154:
2125:
2075:
2055:
2035:
2014:
1978:
1958:
1936:
1897:
1876:
1849:
1788:
1681:
1658:
1628:
1585:
1555:
1535:
1502:
1475:
1455:
1428:
1398:
1368:
1338:
1315:
1289:
1257:
1191:
1147:
1115:
1095:
1067:
1042:
1011:
963:
909:
886:
853:Type theory as a logic
822:. It has been used in
776:
728:
696:
676:
662:from entities of type
652:
613:
587:are types, then so is
581:
561:
533:
513:
106:History of type theory
84:use a type theory for
15032:Plural quantification
14926:Inquisitive semantics
14891:Alternative semantics
14314:Kolmogorov complexity
14267:Computably enumerable
14167:Model complete theory
13959:Principia Mathematica
13019:Propositional formula
12848:Banach–Tarski paradox
12613:Electronic publishing
12583:Computational biology
12573:Computational physics
12469:Unsupervised learning
12383:Distributed computing
12259:Information retrieval
12166:Mathematical analysis
12156:Mathematical software
12039:Theory of computation
12004:Software construction
11994:Requirements analysis
11872:Software organization
11800:Computer architecture
11770:Hardware acceleration
11735:Printed circuit board
11553:Introductory material
10905:10.1093/logcom/exi004
10745:Alonzo Church (1956)
10638:Balbaert, Ivo (2015)
10503:Laan, T.D.L. (1997).
10267:categorical semantics
10236:Jacobs, Bart (1999).
10154:typed lambda calculus
10074:List of type theories
10000:
9941:
9882:
9838:
9802:
9705:
9685:
9551:
9525:
9490:
9470:
9439:
9401:
9375:
9349:
9304:
9190:
9161:
9159:{\displaystyle \tau }
9141:
9088:
9068:
9017:
8955:
8918:
8898:
8845:
8825:
8774:
8742:
8705:
8685:
8648:
8628:
8523:
8503:
8466:
8426:
8406:
8338:
8314:
8242:
8218:
8148:
8034:
7976:
7918:
7888:
7868:
7828:
7808:
7753:
7733:
7680:
7605:
7533:
7513:
7511:{\displaystyle \tau }
7493:
7473:
7453:
7432:has the product type
7427:
7425:{\displaystyle (s,t)}
7395:
7375:
7373:{\displaystyle (s,t)}
7305:
7275:
7255:
7205:
7185:
7159:
7126:
7102:
7057:
7000:
6952:
6918:
6887:
6865:
6843:
6807:
6787:
6761:
6741:
6739:{\displaystyle \ast }
6721:
6699:
6697:{\displaystyle \top }
6663:
6643:
6617:
6593:
6571:
6569:{\displaystyle \bot }
6534:
6532:{\displaystyle \eta }
6514:
6488:
6409:
6386:
6366:
6346:
6326:
6306:
6267:
6247:
6227:
6207:
6192:is applied to a term
6187:
6148:
6146:{\displaystyle \eta }
6128:
6104:
6084:
6037:
6017:
5951:
5949:{\displaystyle \eta }
5931:
5903:
5824:
5738:
5679:
5614:
5584:
5554:
5524:
5522:{\displaystyle \tau }
5504:
5474:
5454:
5434:
5414:
5394:
5368:
5316:
5270:
5240:
5204:
5174:
5152:
5067:
5047:
5045:{\displaystyle \tau }
5027:
5007:
4987:
4961:
4941:
4921:
4874:
4842:
4774:
4746:
4700:
4670:
4586:
4548:
4546:{\displaystyle \tau }
4528:
4504:
4478:
4476:{\displaystyle \tau }
4458:
4422:
4382:
4345:
4294:
4251:
4218:
4184:
4149:
3977:
3929:
3909:
3889:
3869:
3842:
3822:
3802:
3773:
3737:
3678:
3623:
3578:
3523:
3478:
3423:
3396:
3359:
3332:
3295:
3268:
3234:
3207:
3170:
3168:{\displaystyle \bot }
3149:
3147:{\displaystyle \bot }
3120:
3118:{\displaystyle \top }
3099:
3097:{\displaystyle \top }
2909:
2889:
2887:{\displaystyle \tau }
2869:
2849:
2847:{\displaystyle \tau }
2829:
2805:
2744:
2699:
2669:
2642:
2640:{\displaystyle T_{2}}
2615:
2613:{\displaystyle T_{1}}
2588:
2568:
2548:
2521:
2380:
2354:
2334:
2314:
2312:{\displaystyle t_{2}}
2287:
2285:{\displaystyle t_{1}}
2258:
2202:
2182:
2180:{\displaystyle T_{2}}
2155:
2153:{\displaystyle T_{1}}
2126:
2076:
2056:
2036:
2015:
1979:
1959:
1937:
1898:
1877:
1850:
1789:
1682:
1659:
1629:
1586:
1556:
1536:
1503:
1476:
1456:
1454:{\displaystyle t_{2}}
1429:
1427:{\displaystyle t_{1}}
1399:
1397:{\displaystyle T_{2}}
1369:
1367:{\displaystyle T_{1}}
1339:
1316:
1290:
1258:
1192:
1148:
1116:
1096:
1068:
1043:
1012:
964:
910:
887:
777:
729:
697:
677:
653:
614:
582:
562:
534:
514:
429:Programming languages
367:HOL family of provers
230:, and other computer
189:Kleene–Rosser paradox
160:Principia Mathematica
147:axiom of reducibility
15017:Function application
14824:Responsive predicate
14814:Privative adjectives
14262:Church–Turing thesis
14249:Computability theory
13458:continuum hypothesis
12976:Square of opposition
12834:Gödel's completeness
12373:Concurrent computing
12345:Ubiquitous computing
12317:Application security
12312:Information security
12141:Discrete mathematics
12117:Randomized algorithm
12069:Computability theory
12047:Model of computation
12019:Software maintenance
12014:Software engineering
11976:Software development
11926:Programming language
11921:Programming paradigm
11838:Network architecture
11656:Summer 2013 lectures
11595:Homotopy Type Theory
10337:Constable, Robert L.
10166:Homotopy type theory
10032:of the type theory.
10030:normalizing property
9950:
9891:
9847:
9811:
9775:
9765:strongly normalizing
9694:
9674:
9660:intuitionistic logic
9643:Set theory has both
9561:homotopy type theory
9534:
9499:
9479:
9448:
9410:
9384:
9358:
9338:
9327:, as opposed to the
9199:
9174:
9150:
9099:
9077:
9029:
8964:
8931:
8907:
8856:
8834:
8786:
8758:
8714:
8694:
8657:
8637:
8532:
8512:
8475:
8438:
8415:
8404:{\displaystyle \Pi }
8395:
8336:{\displaystyle g\,y}
8323:
8253:
8240:{\displaystyle f\,x}
8227:
8160:
8047:
7985:
7930:
7901:
7877:
7857:
7817:
7764:
7742:
7692:
7614:
7542:
7522:
7502:
7482:
7462:
7436:
7404:
7384:
7352:
7284:
7264:
7221:
7194:
7168:
7135:
7115:
7074:
7012:
6973:
6927:
6896:
6874:
6852:
6828:
6796:
6770:
6750:
6730:
6708:
6688:
6652:
6626:
6606:
6580:
6560:
6523:
6503:
6420:
6398:
6375:
6355:
6335:
6315:
6276:
6256:
6236:
6216:
6196:
6161:
6137:
6117:
6093:
6048:
6026:
5963:
5940:
5920:
5844:
5748:
5689:
5630:
5593:
5563:
5533:
5513:
5483:
5463:
5443:
5423:
5403:
5377:
5357:
5351:function application
5341:Function application
5279:
5249:
5213:
5183:
5163:
5078:
5056:
5036:
5016:
4996:
4970:
4950:
4930:
4895:
4855:
4783:
4755:
4709:
4681:
4597:
4567:
4537:
4517:
4487:
4467:
4447:
4392:
4355:
4304:
4267:
4227:
4193:
4162:
4128:
4092:Homotopy type theory
4088:Homotopy type theory
4083:Homotopy type theory
3975:{\displaystyle \to }
3966:
3918:
3898:
3878:
3858:
3831:
3811:
3800:{\displaystyle P(x)}
3782:
3762:
3687:
3640:
3588:
3543:
3488:
3443:
3406:
3379:
3342:
3315:
3278:
3254:
3217:
3190:
3159:
3138:
3109:
3088:
3049:intuitionistic logic
3031:Intuitionistic logic
3001:Homotopy Type Theory
2898:
2878:
2858:
2838:
2818:
2770:
2737:
2708:
2678:
2658:
2624:
2597:
2577:
2557:
2537:
2421:
2369:
2343:
2323:
2296:
2269:
2215:
2191:
2164:
2137:
2089:
2065:
2045:
2025:
1992:
1968:
1948:
1920:
1887:
1866:
1802:
1693:
1671:
1638:
1595:
1565:
1545:
1512:
1492:
1465:
1438:
1411:
1381:
1351:
1328:
1305:
1279:
1201:
1157:
1125:
1105:
1080:
1075:conditional operator
1055:
1021:
1017:, the Boolean value
1001:
995:function application
923:
899:
876:
742:
706:
686:
682:to entities of type
666:
630:
591:
571:
551:
539:for individuals and
523:
503:
300:homotopy type theory
236:homotopy type theory
126:(first described in
15102:Cognitive semantics
15067:Strawson entailment
15012:Existential closure
14956:Situation semantics
14859:Temperature paradox
14829:Rising declaratives
14794:Modal subordination
14769:Hurford disjunction
14729:Discourse relations
14416:Mathematical object
14307:P versus NP problem
14272:Computable function
14066:Reverse mathematics
13992:Logical consequence
13869:primitive recursive
13864:elementary function
13637:Free/bound variable
13490:Tarski–Grothendieck
13009:Logical connectives
12939:Logical equivalence
12789:Logical consequence
12648:Document management
12638:Operations research
12563:Enterprise software
12479:Multi-task learning
12464:Supervised learning
12186:Information systems
12009:Software deployment
11966:Software repository
11820:Real-time computing
11672:Andrej Bauer's blog
11559:Type Theory at nLab
11462:Milewski, Bartosz.
10881:(2):159--219 (1981)
10651:docs.julialang.org
10172:Cubical Type Theory
10112:and its derivatives
10044:cubical type theory
9578:induction-induction
9574:induction-recursion
9549:{\displaystyle t=t}
9399:{\displaystyle 1+x}
9373:{\displaystyle x+1}
8508:and a term of type
8471:, which takes in a
8349:logical disjunction
7839:logical conjunction
7380:or with the symbol
5834:order of operations
4441:defined inductively
4100:cubical type theory
3421:{\displaystyle A+B}
2387:judgmental equality
2187:(under assumptions
2061:(under assumptions
983:recursively defined
917:well-formed formula
857:A type theory is a
736:indicator functions
482:categorial grammars
40:formal presentation
15147:Semantics of logic
15072:Strict conditional
15042:Quantifier raising
15007:Downward entailing
14987:Autonomy of syntax
14916:Generative grammar
14896:Categorial grammar
14834:Scalar implicature
14739:Epistemic modality
14714:De dicto and de re
14214:Transfer principle
14177:Semantics of logic
14162:Categorical theory
14138:Non-standard model
13652:Logical connective
12779:Information theory
12728:Mathematical logic
12431:Search methodology
12378:Parallel computing
12335:Interaction design
12244:Computing platform
12171:Numerical analysis
12161:Information theory
11946:Software framework
11909:Software notations
11848:Network components
11745:Integrated circuit
11642:2005 summer school
10956:10.1007/BF00484985
10858:2022-05-10 at the
10658:2022-03-24 at the
10547:Montague Semantics
10458:2011-06-07 at the
10406:. Addison–Wesley.
10185:Class (set theory)
10089:higher-order logic
9995:
9936:
9877:
9833:
9797:
9742:BHK interpretation
9700:
9680:
9668:predicate function
9546:
9520:
9485:
9465:
9434:
9396:
9370:
9354:. For example, if
9344:
9299:
9185:
9156:
9136:
9083:
9063:
9012:
8950:
8913:
8893:
8840:
8820:
8769:
8737:
8700:
8680:
8643:
8623:
8518:
8498:
8461:
8421:
8401:
8333:
8309:
8237:
8213:
8143:
8029:
7971:
7913:
7883:
7863:
7823:
7803:
7748:
7728:
7675:
7600:
7528:
7508:
7488:
7468:
7448:
7422:
7390:
7370:
7300:
7280:is a term of type
7270:
7250:
7200:
7180:
7154:
7121:
7097:
7052:
7007:successor function
6995:
6947:
6913:
6882:
6860:
6838:
6802:
6782:
6756:
6736:
6716:
6694:
6658:
6638:
6612:
6588:
6566:
6529:
6509:
6483:
6404:
6381:
6361:
6341:
6321:
6301:
6262:
6242:
6222:
6202:
6182:
6143:
6123:
6099:
6079:
6032:
6012:
5946:
5926:
5898:
5819:
5733:
5674:
5609:
5579:
5549:
5519:
5499:
5469:
5449:
5429:
5419:is a term of type
5409:
5389:
5373:is a term of type
5363:
5330:anonymous function
5311:
5265:
5235:
5199:
5169:
5147:
5062:
5042:
5022:
5002:
4982:
4956:
4936:
4916:
4889:lambda expressions
4869:
4837:
4769:
4741:
4695:
4665:
4581:
4543:
4523:
4499:
4473:
4453:
4417:
4377:
4340:
4289:
4246:
4213:
4179:
4144:
3972:
3924:
3904:
3884:
3864:
3837:
3817:
3797:
3768:
3732:
3673:
3618:
3573:
3530:Dependent Product
3518:
3473:
3418:
3391:
3354:
3327:
3290:
3263:
3229:
3202:
3165:
3144:
3115:
3094:
2904:
2884:
2864:
2844:
2824:
2800:
2739:
2736:
2694:
2664:
2637:
2610:
2583:
2563:
2543:
2516:
2514:
2397:Rules of Inference
2375:
2349:
2329:
2309:
2282:
2253:
2197:
2177:
2150:
2121:
2071:
2051:
2041:is a term of type
2031:
2010:
1974:
1954:
1932:
1893:
1872:
1845:
1784:
1677:
1654:
1634:is a term of type
1624:
1591:, it follows that
1581:
1561:is a term of type
1551:
1531:
1508:is a term of type
1498:
1471:
1451:
1424:
1394:
1364:
1334:
1311:
1285:
1253:
1187:
1143:
1111:
1091:
1063:
1050:successor function
1038:
1007:
959:
905:
892:is true", or "The
882:
863:rules of inference
859:mathematical logic
802:and Cooper 1981).
772:
724:
692:
672:
648:
609:
577:
557:
529:
509:
496:, etc.) of words.
363:higher-order logic
315:algebraic topology
261:. You can help by
197:higher-order logic
88:. A common one is
15155:
15154:
15127:Logic translation
15090:
15089:
15082:Universal grinder
15062:Squiggle operator
15022:Meaning postulate
14961:Supervaluationism
14931:Intensional logic
14911:Dynamic semantics
14872:
14871:
14704:Crossover effects
14653:Tense–aspect–mood
14633:Lexical semantics
14452:
14451:
14384:Abstract category
14187:Theories of truth
13997:Rule of inference
13987:Natural deduction
13968:
13967:
13513:
13512:
13218:Cartesian product
13123:
13122:
13029:Many-valued logic
13004:Boolean functions
12887:Russell's paradox
12862:diagonal argument
12759:First-order logic
12694:
12693:
12623:Electronic voting
12553:Quantum Computing
12546:Applied computing
12532:Image compression
12302:Hardware security
12292:Security services
12249:Digital marketing
12029:Open-source model
11941:Modeling language
11853:Network scheduler
11602:Advanced material
11397:978-3-642-03153-3
11260:978-1-6654-4895-6
11199:978-0-08-093066-4
11167:978-3-319-47165-5
11066:978-0-521-76614-2
10827:978-0-7486-7777-1
10789:978-3-319-50422-3
10617:(x,y) -> x^5+y
10495:978-3-7643-8349-7
10446:978-0-521-05422-5
10433:Hindley, J. Roger
10326:978-3-8473-2963-3
10249:978-0-444-50170-7
10228:978-1-4020-0763-7
10147:pure type systems
10137:combinatory logic
10019:first-order logic
9703:{\displaystyle x}
9683:{\displaystyle x}
9617:first-order logic
9488:{\displaystyle t}
9347:{\displaystyle =}
9267:
8916:{\displaystyle y}
8843:{\displaystyle x}
8703:{\displaystyle a}
8646:{\displaystyle a}
8521:{\displaystyle a}
8365:type dependencies
7866:{\displaystyle +}
7826:{\displaystyle t}
7751:{\displaystyle s}
7531:{\displaystyle t}
7491:{\displaystyle s}
7273:{\displaystyle n}
7203:{\displaystyle U}
7124:{\displaystyle a}
6835:
6805:{\displaystyle a}
6759:{\displaystyle a}
6661:{\displaystyle a}
6615:{\displaystyle a}
6600:type inhabitation
6384:{\displaystyle t}
6364:{\displaystyle t}
6344:{\displaystyle v}
6324:{\displaystyle t}
6265:{\displaystyle s}
6245:{\displaystyle t}
6225:{\displaystyle v}
6205:{\displaystyle s}
6126:{\displaystyle t}
6102:{\displaystyle v}
5895:
5816:
5730:
5671:
5661:
5606:
5576:
5546:
5472:{\displaystyle s}
5452:{\displaystyle t}
5412:{\displaystyle s}
5366:{\displaystyle t}
5172:{\displaystyle x}
5065:{\displaystyle t}
5025:{\displaystyle v}
4959:{\displaystyle t}
4939:{\displaystyle v}
4849:right associative
4077:categorical logic
3996:program synthesis
3927:{\displaystyle x}
3907:{\displaystyle x}
3887:{\displaystyle x}
3867:{\displaystyle x}
3840:{\displaystyle P}
3820:{\displaystyle x}
3771:{\displaystyle x}
3634:
3633:
3025:computer programs
2867:{\displaystyle t}
2764:type inhabitation
2758:Type inhabitation
2752:Type inhabitation
2667:{\displaystyle 0}
2586:{\displaystyle t}
2363:
2362:
2332:{\displaystyle T}
2319:are both of type
2160:is equal to type
2054:{\displaystyle T}
2034:{\displaystyle t}
1957:{\displaystyle T}
1750:
1554:{\displaystyle y}
1501:{\displaystyle x}
1474:{\displaystyle T}
1337:{\displaystyle T}
1314:{\displaystyle t}
1288:{\displaystyle T}
1114:{\displaystyle 0}
1010:{\displaystyle 0}
820:type theory types
695:{\displaystyle b}
675:{\displaystyle a}
580:{\displaystyle b}
560:{\displaystyle a}
532:{\displaystyle t}
512:{\displaystyle e}
486:pregroup grammars
474:natural languages
439:semantic analysis
279:
278:
145:together with an
124:Russell's paradox
15185:
15132:Linguistics wars
15057:Semantic parsing
14946:Montague grammar
14881:
14880:
14724:Deontic modality
14578:
14577:
14565:Truth conditions
14500:Compositionality
14493:Central concepts
14479:
14472:
14465:
14456:
14455:
14443:
14442:
14394:History of logic
14389:Category of sets
14282:Decision problem
14061:Ordinal analysis
14002:Sequent calculus
13900:Boolean algebras
13840:
13839:
13814:
13785:logical/constant
13539:
13538:
13525:
13448:Zermelo–Fraenkel
13199:Set operations:
13134:
13133:
13071:
12902:
12901:
12882:Löwenheim–Skolem
12769:Formal semantics
12721:
12714:
12707:
12698:
12697:
12684:
12683:
12674:
12673:
12664:
12663:
12484:Cross-validation
12456:Machine learning
12340:Social computing
12307:Network security
12102:Algorithm design
12024:Programming team
11984:Control variable
11961:Software library
11899:Software quality
11894:Operating system
11843:Network protocol
11708:Computer science
11701:
11694:
11687:
11678:
11677:
11619:
11542:
11541:
11539:
11538:
11519:
11513:
11512:
11510:
11508:
11489:
11483:
11482:
11480:
11479:
11459:
11450:
11449:
11447:
11413:
11407:
11406:
11405:
11404:
11371:
11362:
11356:
11350:
11349:
11343:
11335:
11333:
11326:
11316:
11302:
11293:
11284:
11283:
11281:
11280:
11244:
11224:
11218:
11217:
11215:
11214:
11208:
11187:
11178:
11172:
11171:
11145:
11139:
11138:
11136:
11134:
11115:
11109:
11108:
11106:
11105:
11099:
11092:
11084:
11071:
11070:
11050:
11029:
11028:
11026:
11024:
11018:
11010:logicmatters.net
11007:
10998:
10992:
10991:
10981:
10968:
10967:
10935:
10926:
10915:
10909:
10908:
10888:
10882:
10869:
10863:
10851:Cooper, Robin. "
10849:
10843:
10842:
10840:
10839:
10811:
10805:
10804:
10802:
10801:
10773:
10767:
10755:
10749:
10743:
10737:
10736:
10697:
10684:
10668:
10662:
10649:
10643:
10636:
10620:
10618:
10609:
10603:
10596:
10590:
10579:
10573:
10567:
10538:
10536:
10518:10.6100/IR498552
10511:
10499:
10480:
10450:
10428:
10426:
10425:
10396:
10383:Coquand, Thierry
10376:Naïve Set Theory
10368:
10366:
10345:
10330:
10303:
10301:
10300:
10264:
10262:
10261:
10232:
10213:
10004:
10002:
10001:
9996:
9994:
9993:
9984:
9983:
9968:
9967:
9945:
9943:
9942:
9937:
9935:
9934:
9922:
9921:
9909:
9908:
9886:
9884:
9883:
9878:
9876:
9875:
9866:
9865:
9856:
9855:
9842:
9840:
9839:
9834:
9820:
9819:
9806:
9804:
9803:
9798:
9784:
9783:
9709:
9707:
9706:
9701:
9689:
9687:
9686:
9681:
9589:proof assistants
9555:
9553:
9552:
9547:
9529:
9527:
9526:
9521:
9515:
9494:
9492:
9491:
9486:
9474:
9472:
9471:
9466:
9464:
9443:
9441:
9440:
9435:
9406:are terms, then
9405:
9403:
9402:
9397:
9379:
9377:
9376:
9371:
9353:
9351:
9350:
9345:
9308:
9306:
9305:
9300:
9280:
9269:
9268:
9234:
9233:
9194:
9192:
9191:
9186:
9184:
9165:
9163:
9162:
9157:
9145:
9143:
9142:
9137:
9127:
9109:
9092:
9090:
9089:
9084:
9072:
9070:
9069:
9064:
9054:
9039:
9021:
9019:
9018:
9013:
8993:
8992:
8974:
8959:
8957:
8956:
8951:
8949:
8948:
8922:
8920:
8919:
8914:
8902:
8900:
8899:
8894:
8884:
8866:
8849:
8847:
8846:
8841:
8829:
8827:
8826:
8821:
8811:
8796:
8778:
8776:
8775:
8770:
8768:
8746:
8744:
8743:
8738:
8732:
8731:
8709:
8707:
8706:
8701:
8689:
8687:
8686:
8681:
8675:
8674:
8652:
8650:
8649:
8644:
8632:
8630:
8629:
8624:
8615:
8614:
8586:
8585:
8554:
8527:
8525:
8524:
8519:
8507:
8505:
8504:
8499:
8493:
8492:
8470:
8468:
8467:
8462:
8460:
8430:
8428:
8427:
8422:
8410:
8408:
8407:
8402:
8342:
8340:
8339:
8334:
8318:
8316:
8315:
8310:
8301:
8272:
8246:
8244:
8243:
8238:
8222:
8220:
8219:
8214:
8205:
8179:
8152:
8150:
8149:
8144:
8066:
8038:
8036:
8035:
8030:
8004:
7980:
7978:
7977:
7972:
7946:
7922:
7920:
7919:
7914:
7892:
7890:
7889:
7884:
7872:
7870:
7869:
7864:
7832:
7830:
7829:
7824:
7812:
7810:
7809:
7804:
7786:
7757:
7755:
7754:
7749:
7737:
7735:
7734:
7729:
7711:
7684:
7682:
7681:
7676:
7636:
7609:
7607:
7606:
7601:
7561:
7537:
7535:
7534:
7529:
7517:
7515:
7514:
7509:
7497:
7495:
7494:
7489:
7477:
7475:
7474:
7469:
7457:
7455:
7454:
7449:
7431:
7429:
7428:
7423:
7399:
7397:
7396:
7391:
7379:
7377:
7376:
7371:
7327:Girard's Paradox
7309:
7307:
7306:
7301:
7299:
7298:
7279:
7277:
7276:
7271:
7259:
7257:
7256:
7251:
7245:
7244:
7209:
7207:
7206:
7201:
7189:
7187:
7186:
7181:
7163:
7161:
7160:
7155:
7153:
7152:
7131:. In this case,
7130:
7128:
7127:
7122:
7106:
7104:
7103:
7098:
7092:
7091:
7066:Dependent typing
7061:
7059:
7058:
7053:
7051:
7050:
7035:
7034:
7019:
7004:
7002:
7001:
6996:
6994:
6993:
6967:Peano Arithmetic
6956:
6954:
6953:
6948:
6946:
6922:
6920:
6919:
6914:
6912:
6891:
6889:
6888:
6883:
6881:
6869:
6867:
6866:
6861:
6859:
6847:
6845:
6844:
6839:
6837:
6836:
6811:
6809:
6808:
6803:
6791:
6789:
6788:
6783:
6765:
6763:
6762:
6757:
6745:
6743:
6742:
6737:
6725:
6723:
6722:
6717:
6715:
6703:
6701:
6700:
6695:
6667:
6665:
6664:
6659:
6647:
6645:
6644:
6639:
6621:
6619:
6618:
6613:
6602:. If for a type
6597:
6595:
6594:
6589:
6587:
6575:
6573:
6572:
6567:
6538:
6536:
6535:
6530:
6518:
6516:
6515:
6510:
6492:
6490:
6489:
6484:
6474:
6445:
6413:
6411:
6410:
6405:
6390:
6388:
6387:
6382:
6370:
6368:
6367:
6362:
6350:
6348:
6347:
6342:
6330:
6328:
6327:
6322:
6310:
6308:
6307:
6302:
6271:
6269:
6268:
6263:
6251:
6249:
6248:
6243:
6231:
6229:
6228:
6223:
6211:
6209:
6208:
6203:
6191:
6189:
6188:
6183:
6152:
6150:
6149:
6144:
6132:
6130:
6129:
6124:
6108:
6106:
6105:
6100:
6088:
6086:
6085:
6080:
6041:
6039:
6038:
6033:
6021:
6019:
6018:
6013:
5955:
5953:
5952:
5947:
5935:
5933:
5932:
5927:
5907:
5905:
5904:
5899:
5897:
5896:
5876:
5857:
5838:left associative
5828:
5826:
5825:
5820:
5818:
5817:
5791:
5767:
5742:
5740:
5739:
5734:
5732:
5731:
5708:
5683:
5681:
5680:
5675:
5673:
5672:
5663:
5662:
5646:
5618:
5616:
5615:
5610:
5608:
5607:
5588:
5586:
5585:
5580:
5578:
5577:
5558:
5556:
5555:
5550:
5548:
5547:
5528:
5526:
5525:
5520:
5508:
5506:
5505:
5500:
5479:, often written
5478:
5476:
5475:
5470:
5458:
5456:
5455:
5450:
5438:
5436:
5435:
5430:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5372:
5370:
5369:
5364:
5320:
5318:
5317:
5312:
5310:
5309:
5294:
5293:
5274:
5272:
5271:
5266:
5264:
5263:
5244:
5242:
5241:
5236:
5226:
5208:
5206:
5205:
5200:
5198:
5197:
5178:
5176:
5175:
5170:
5159:The variable is
5156:
5154:
5153:
5148:
5146:
5145:
5130:
5129:
5103:
5071:
5069:
5068:
5063:
5051:
5049:
5048:
5043:
5031:
5029:
5028:
5023:
5011:
5009:
5008:
5003:
4991:
4989:
4988:
4983:
4965:
4963:
4962:
4957:
4945:
4943:
4942:
4937:
4925:
4923:
4922:
4917:
4878:
4876:
4875:
4870:
4868:
4846:
4844:
4843:
4838:
4836:
4835:
4817:
4816:
4801:
4800:
4778:
4776:
4775:
4770:
4768:
4750:
4748:
4747:
4742:
4740:
4739:
4724:
4723:
4704:
4702:
4701:
4696:
4694:
4674:
4672:
4671:
4666:
4661:
4660:
4645:
4644:
4626:
4625:
4610:
4590:
4588:
4587:
4582:
4580:
4552:
4550:
4549:
4544:
4532:
4530:
4529:
4524:
4508:
4506:
4505:
4500:
4482:
4480:
4479:
4474:
4462:
4460:
4459:
4454:
4426:
4424:
4423:
4418:
4416:
4415:
4386:
4384:
4383:
4378:
4376:
4375:
4349:
4347:
4346:
4341:
4339:
4338:
4320:
4298:
4296:
4295:
4290:
4288:
4287:
4258:formal variables
4255:
4253:
4252:
4247:
4245:
4244:
4222:
4220:
4219:
4214:
4212:
4188:
4186:
4185:
4180:
4178:
4153:
4151:
4150:
4145:
4143:
4142:
3981:
3979:
3978:
3973:
3933:
3931:
3930:
3925:
3913:
3911:
3910:
3905:
3893:
3891:
3890:
3885:
3873:
3871:
3870:
3865:
3846:
3844:
3843:
3838:
3826:
3824:
3823:
3818:
3806:
3804:
3803:
3798:
3777:
3775:
3774:
3769:
3741:
3739:
3738:
3733:
3682:
3680:
3679:
3674:
3627:
3625:
3624:
3619:
3582:
3580:
3579:
3574:
3527:
3525:
3524:
3519:
3482:
3480:
3479:
3474:
3427:
3425:
3424:
3419:
3400:
3398:
3397:
3392:
3363:
3361:
3360:
3355:
3336:
3334:
3333:
3328:
3299:
3297:
3296:
3291:
3272:
3270:
3269:
3264:
3238:
3236:
3235:
3230:
3211:
3209:
3208:
3203:
3174:
3172:
3171:
3166:
3153:
3151:
3150:
3145:
3124:
3122:
3121:
3116:
3103:
3101:
3100:
3095:
3065:
3064:
2919:Girard's paradox
2913:
2911:
2910:
2905:
2893:
2891:
2890:
2885:
2873:
2871:
2870:
2865:
2853:
2851:
2850:
2845:
2833:
2831:
2830:
2825:
2814:Given a context
2809:
2807:
2806:
2801:
2748:
2746:
2745:
2740:
2738:
2703:
2701:
2700:
2695:
2693:
2692:
2673:
2671:
2670:
2665:
2646:
2644:
2643:
2638:
2636:
2635:
2619:
2617:
2616:
2611:
2609:
2608:
2592:
2590:
2589:
2584:
2572:
2570:
2569:
2564:
2552:
2550:
2549:
2544:
2525:
2523:
2522:
2517:
2515:
2511:
2510:
2479:
2478:
2466:
2465:
2449:
2448:
2401:A type theory's
2384:
2382:
2381:
2376:
2358:
2356:
2355:
2350:
2338:
2336:
2335:
2330:
2318:
2316:
2315:
2310:
2308:
2307:
2291:
2289:
2288:
2283:
2281:
2280:
2262:
2260:
2259:
2254:
2246:
2245:
2233:
2232:
2206:
2204:
2203:
2198:
2186:
2184:
2183:
2178:
2176:
2175:
2159:
2157:
2156:
2151:
2149:
2148:
2130:
2128:
2127:
2122:
2120:
2119:
2107:
2106:
2080:
2078:
2077:
2072:
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2019:
2017:
2016:
2011:
1983:
1981:
1980:
1975:
1963:
1961:
1960:
1955:
1941:
1939:
1938:
1933:
1906:
1905:
1902:
1900:
1899:
1894:
1881:
1879:
1878:
1873:
1854:
1852:
1851:
1846:
1844:
1843:
1828:
1827:
1812:
1793:
1791:
1790:
1785:
1783:
1782:
1752:
1751:
1748:
1739:
1738:
1717:
1716:
1686:
1684:
1683:
1678:
1666:turnstile symbol
1663:
1661:
1660:
1655:
1653:
1652:
1633:
1631:
1630:
1625:
1608:
1590:
1588:
1587:
1582:
1580:
1579:
1560:
1558:
1557:
1552:
1540:
1538:
1537:
1532:
1530:
1529:
1507:
1505:
1504:
1499:
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1450:
1449:
1433:
1431:
1430:
1425:
1423:
1422:
1403:
1401:
1400:
1395:
1393:
1392:
1373:
1371:
1370:
1365:
1363:
1362:
1343:
1341:
1340:
1335:
1320:
1318:
1317:
1312:
1294:
1292:
1291:
1286:
1262:
1260:
1259:
1254:
1242:
1229:
1214:
1196:
1194:
1193:
1188:
1176:
1167:
1152:
1150:
1149:
1144:
1135:
1120:
1118:
1117:
1112:
1100:
1098:
1097:
1092:
1090:
1072:
1070:
1069:
1064:
1062:
1047:
1045:
1044:
1039:
1037:
1016:
1014:
1013:
1008:
968:
966:
965:
960:
958:
957:
939:
914:
912:
911:
906:
891:
889:
888:
883:
832:dialogue systems
810:formal semantics
781:
779:
778:
773:
733:
731:
730:
725:
701:
699:
698:
693:
681:
679:
678:
673:
657:
655:
654:
649:
618:
616:
615:
610:
586:
584:
583:
578:
566:
564:
563:
558:
538:
536:
535:
530:
518:
516:
515:
510:
478:Montague grammar
365:are used by the
341:proof assistants
325:Proof assistants
274:
271:
253:
246:
232:proof assistants
136:Bertrand Russell
116:naive set theory
86:their foundation
62:Typed λ-calculus
15193:
15192:
15188:
15187:
15186:
15184:
15183:
15182:
15158:
15157:
15156:
15151:
15086:
14975:
14936:Lambda calculus
14868:
14839:Sloppy identity
14799:Opaque contexts
14734:Donkey anaphora
14699:Counterfactuals
14667:
14569:
14488:
14483:
14453:
14448:
14437:
14430:
14375:Category theory
14365:Algebraic logic
14348:
14319:Lambda calculus
14257:Church encoding
14243:
14219:Truth predicate
14075:
14041:Complete theory
13964:
13833:
13829:
13825:
13820:
13812:
13532: and
13528:
13523:
13509:
13485:New Foundations
13453:axiom of choice
13436:
13398:Gödel numbering
13338: and
13330:
13234:
13119:
13069:
13050:
12999:Boolean algebra
12985:
12949:Equiconsistency
12914:Classical logic
12891:
12872:Halting problem
12860: and
12836: and
12824: and
12823:
12818:Theorems (
12813:
12730:
12725:
12695:
12690:
12681:
12652:
12633:Word processing
12541:
12527:Virtual reality
12488:
12450:
12421:Computer vision
12397:
12393:Multiprocessing
12359:
12321:
12287:Security hacker
12263:
12239:Digital library
12180:
12131:Mathematics of
12126:
12088:
12064:Automata theory
12059:Formal language
12033:
11999:Software design
11970:
11903:
11889:Virtual machine
11867:
11863:Network service
11824:
11815:Embedded system
11788:
11721:
11710:
11705:
11623:The TYPES Forum
11604:
11555:
11550:
11545:
11536:
11534:
11521:
11520:
11516:
11506:
11504:
11491:
11490:
11486:
11477:
11475:
11460:
11453:
11414:
11410:
11402:
11400:
11398:
11372:
11365:
11357:
11353:
11337:
11336:
11331:
11300:
11294:
11287:
11278:
11276:
11261:
11225:
11221:
11212:
11210:
11206:
11200:
11185:
11179:
11175:
11168:
11146:
11142:
11132:
11130:
11117:
11116:
11112:
11103:
11101:
11097:
11090:
11086:
11085:
11074:
11067:
11051:
11032:
11022:
11020:
11016:
11005:
10999:
10995:
10982:
10971:
10936:
10929:
10916:
10912:
10889:
10885:
10870:
10866:
10860:Wayback Machine
10850:
10846:
10837:
10835:
10828:
10812:
10808:
10799:
10797:
10790:
10774:
10770:
10756:
10752:
10744:
10740:
10717:10.2307/2266170
10698:
10687:
10681:Wayback Machine
10669:
10665:
10660:Wayback Machine
10650:
10646:
10637:
10633:
10629:
10624:
10623:
10616:
10610:
10606:
10600:logistic method
10597:
10593:
10580:
10576:
10568:
10564:
10559:
10554:
10534:
10528:
10509:
10496:
10477:
10460:Wayback Machine
10447:
10423:
10421:
10414:
10364:
10358:
10343:
10327:
10298:
10296:
10289:
10259:
10257:
10250:
10229:
10203:
10201:Further reading
10181:
10162:
10160:Active research
10145:(also known as
10119:
10081:
10076:
10059:Axiom of Choice
10051:classical logic
10011:
9989:
9985:
9979:
9975:
9957:
9953:
9951:
9948:
9947:
9930:
9926:
9917:
9913:
9898:
9894:
9892:
9889:
9888:
9871:
9870:
9861:
9860:
9851:
9850:
9848:
9845:
9844:
9815:
9814:
9812:
9809:
9808:
9779:
9778:
9776:
9773:
9772:
9758:
9750:Category theory
9727:inductive types
9723:Church encoding
9695:
9692:
9691:
9675:
9672:
9671:
9633:axiom of choice
9609:
9569:
9567:Inductive types
9535:
9532:
9531:
9502:
9500:
9497:
9496:
9480:
9477:
9476:
9451:
9449:
9446:
9445:
9411:
9408:
9407:
9385:
9382:
9381:
9359:
9356:
9355:
9339:
9336:
9335:
9317:
9273:
9264:
9263:
9220:
9219:
9200:
9197:
9196:
9177:
9175:
9172:
9171:
9151:
9148:
9147:
9111:
9102:
9100:
9097:
9096:
9078:
9075:
9074:
9041:
9032:
9030:
9027:
9026:
8979:
8978:
8967:
8965:
8962:
8961:
8935:
8934:
8932:
8929:
8928:
8908:
8905:
8904:
8868:
8859:
8857:
8854:
8853:
8835:
8832:
8831:
8798:
8789:
8787:
8784:
8783:
8761:
8759:
8756:
8755:
8752:dependent pairs
8718:
8717:
8715:
8712:
8711:
8710:, and return a
8695:
8692:
8691:
8661:
8660:
8658:
8655:
8654:
8638:
8635:
8634:
8601:
8600:
8572:
8571:
8535:
8533:
8530:
8529:
8513:
8510:
8509:
8479:
8478:
8476:
8473:
8472:
8441:
8439:
8436:
8435:
8416:
8413:
8412:
8396:
8393:
8392:
8361:
8324:
8321:
8320:
8285:
8256:
8254:
8251:
8250:
8228:
8225:
8224:
8192:
8163:
8161:
8158:
8157:
8050:
8048:
8045:
8044:
7988:
7986:
7983:
7982:
7933:
7931:
7928:
7927:
7902:
7899:
7898:
7878:
7875:
7874:
7858:
7855:
7854:
7851:
7818:
7815:
7814:
7767:
7765:
7762:
7761:
7743:
7740:
7739:
7695:
7693:
7690:
7689:
7617:
7615:
7612:
7611:
7545:
7543:
7540:
7539:
7523:
7520:
7519:
7518:is the type of
7503:
7500:
7499:
7483:
7480:
7479:
7478:is the type of
7463:
7460:
7459:
7437:
7434:
7433:
7405:
7402:
7401:
7385:
7382:
7381:
7353:
7350:
7349:
7343:
7333:introduced the
7329:. The logician
7288:
7287:
7285:
7282:
7281:
7265:
7262:
7261:
7225:
7224:
7222:
7219:
7218:
7195:
7192:
7191:
7169:
7166:
7165:
7139:
7138:
7136:
7133:
7132:
7116:
7113:
7112:
7078:
7077:
7075:
7072:
7071:
7068:
7040:
7039:
7024:
7023:
7015:
7013:
7010:
7009:
6983:
6982:
6974:
6971:
6970:
6963:
6961:Natural numbers
6930:
6928:
6925:
6924:
6899:
6897:
6894:
6893:
6877:
6875:
6872:
6871:
6855:
6853:
6850:
6849:
6832:
6831:
6829:
6826:
6825:
6822:
6797:
6794:
6793:
6771:
6768:
6767:
6751:
6748:
6747:
6731:
6728:
6727:
6711:
6709:
6706:
6705:
6689:
6686:
6685:
6678:
6653:
6650:
6649:
6627:
6624:
6623:
6607:
6604:
6603:
6583:
6581:
6578:
6577:
6561:
6558:
6557:
6550:
6545:
6524:
6521:
6520:
6504:
6501:
6500:
6464:
6435:
6421:
6418:
6417:
6399:
6396:
6395:
6376:
6373:
6372:
6356:
6353:
6352:
6336:
6333:
6332:
6316:
6313:
6312:
6277:
6274:
6273:
6257:
6254:
6253:
6237:
6234:
6233:
6217:
6214:
6213:
6197:
6194:
6193:
6162:
6159:
6158:
6138:
6135:
6134:
6118:
6115:
6114:
6094:
6091:
6090:
6049:
6046:
6045:
6027:
6024:
6023:
5964:
5961:
5960:
5941:
5938:
5937:
5936:-reduction and
5921:
5918:
5917:
5914:
5892:
5891:
5866:
5847:
5845:
5842:
5841:
5813:
5812:
5781:
5757:
5749:
5746:
5745:
5727:
5726:
5698:
5690:
5687:
5686:
5668:
5667:
5658:
5657:
5636:
5631:
5628:
5627:
5603:
5602:
5594:
5591:
5590:
5573:
5572:
5564:
5561:
5560:
5543:
5542:
5534:
5531:
5530:
5514:
5511:
5510:
5484:
5481:
5480:
5464:
5461:
5460:
5444:
5441:
5440:
5424:
5421:
5420:
5404:
5401:
5400:
5378:
5375:
5374:
5358:
5355:
5354:
5347:inference rules
5343:
5338:
5336:Inference Rules
5299:
5298:
5283:
5282:
5280:
5277:
5276:
5253:
5252:
5250:
5247:
5246:
5216:
5214:
5211:
5210:
5187:
5186:
5184:
5181:
5180:
5164:
5161:
5160:
5135:
5134:
5119:
5118:
5093:
5079:
5076:
5075:
5057:
5054:
5053:
5052:is the type of
5037:
5034:
5033:
5017:
5014:
5013:
5012:is the type of
4997:
4994:
4993:
4971:
4968:
4967:
4951:
4948:
4947:
4931:
4928:
4927:
4896:
4893:
4892:
4885:
4858:
4856:
4853:
4852:
4825:
4824:
4806:
4805:
4790:
4789:
4784:
4781:
4780:
4758:
4756:
4753:
4752:
4729:
4728:
4713:
4712:
4710:
4707:
4706:
4684:
4682:
4679:
4678:
4650:
4649:
4634:
4633:
4615:
4614:
4600:
4598:
4595:
4594:
4570:
4568:
4565:
4564:
4538:
4535:
4534:
4518:
4515:
4514:
4488:
4485:
4484:
4468:
4465:
4464:
4448:
4445:
4444:
4433:
4402:
4401:
4393:
4390:
4389:
4365:
4364:
4356:
4353:
4352:
4325:
4324:
4307:
4305:
4302:
4301:
4277:
4276:
4268:
4265:
4264:
4231:
4230:
4228:
4225:
4224:
4196:
4194:
4191:
4190:
4165:
4163:
4160:
4159:
4132:
4131:
4129:
4126:
4125:
4122:natural numbers
4118:
4113:
4111:Terms and types
4108:
4085:
4027:Category theory
4023:
4021:Category theory
4018:
4010:
4004:
3967:
3964:
3963:
3956:
3919:
3916:
3915:
3899:
3896:
3895:
3879:
3876:
3875:
3859:
3856:
3855:
3832:
3829:
3828:
3812:
3809:
3808:
3783:
3780:
3779:
3763:
3760:
3759:
3752:
3688:
3685:
3684:
3641:
3638:
3637:
3589:
3586:
3585:
3544:
3541:
3540:
3489:
3486:
3485:
3444:
3441:
3440:
3407:
3404:
3403:
3380:
3377:
3376:
3343:
3340:
3339:
3316:
3313:
3312:
3279:
3276:
3275:
3255:
3252:
3251:
3218:
3215:
3214:
3191:
3188:
3187:
3160:
3157:
3156:
3139:
3136:
3135:
3110:
3107:
3106:
3089:
3086:
3085:
3057:double negation
3045:classical logic
3041:Boolean algebra
3033:
3021:category theory
3009:
2899:
2896:
2895:
2879:
2876:
2875:
2859:
2856:
2855:
2839:
2836:
2835:
2819:
2816:
2815:
2771:
2768:
2767:
2760:
2754:
2735:
2734:
2711:
2709:
2706:
2705:
2682:
2681:
2679:
2676:
2675:
2659:
2656:
2655:
2631:
2627:
2625:
2622:
2621:
2604:
2600:
2598:
2595:
2594:
2578:
2575:
2574:
2558:
2555:
2554:
2538:
2535:
2534:
2513:
2512:
2506:
2502:
2481:
2480:
2474:
2470:
2461:
2457:
2444:
2440:
2424:
2422:
2419:
2418:
2403:inference rules
2399:
2370:
2367:
2366:
2344:
2341:
2340:
2324:
2321:
2320:
2303:
2299:
2297:
2294:
2293:
2276:
2272:
2270:
2267:
2266:
2241:
2237:
2228:
2224:
2216:
2213:
2212:
2192:
2189:
2188:
2171:
2167:
2165:
2162:
2161:
2144:
2140:
2138:
2135:
2134:
2115:
2111:
2102:
2098:
2090:
2087:
2086:
2066:
2063:
2062:
2046:
2043:
2042:
2026:
2023:
2022:
1993:
1990:
1989:
1969:
1966:
1965:
1949:
1946:
1945:
1921:
1918:
1917:
1888:
1885:
1884:
1867:
1864:
1863:
1833:
1832:
1817:
1816:
1808:
1803:
1800:
1799:
1772:
1771:
1747:
1746:
1728:
1727:
1703:
1702:
1694:
1691:
1690:
1672:
1669:
1668:
1642:
1641:
1639:
1636:
1635:
1601:
1596:
1593:
1592:
1569:
1568:
1566:
1563:
1562:
1546:
1543:
1542:
1516:
1515:
1513:
1510:
1509:
1493:
1490:
1489:
1466:
1463:
1462:
1445:
1441:
1439:
1436:
1435:
1418:
1414:
1412:
1409:
1408:
1388:
1384:
1382:
1379:
1378:
1358:
1354:
1352:
1349:
1348:
1329:
1326:
1325:
1306:
1303:
1302:
1280:
1277:
1276:
1269:
1238:
1216:
1207:
1202:
1199:
1198:
1172:
1163:
1158:
1155:
1154:
1131:
1126:
1123:
1122:
1106:
1103:
1102:
1083:
1081:
1078:
1077:
1058:
1056:
1053:
1052:
1024:
1022:
1019:
1018:
1002:
999:
998:
987:constant symbol
975:
944:
943:
926:
924:
921:
920:
900:
897:
896:
877:
874:
873:
865:that result in
855:
843:Gregory Bateson
840:
838:Social sciences
743:
740:
739:
707:
704:
703:
687:
684:
683:
667:
664:
663:
658:is the type of
631:
628:
627:
626:A complex type
592:
589:
588:
572:
569:
568:
552:
549:
548:
524:
521:
520:
504:
501:
500:
466:
431:
333:
331:Proof assistant
327:
307:category theory
284:
275:
269:
266:
259:needs expansion
244:
216:Thierry Coquand
181:lambda calculus
108:
102:
90:Thierry Coquand
24:
17:
12:
11:
5:
15191:
15181:
15180:
15175:
15170:
15153:
15152:
15150:
15149:
15144:
15139:
15134:
15129:
15124:
15122:Inferentialism
15119:
15117:Formal grammar
15114:
15109:
15104:
15098:
15096:
15092:
15091:
15088:
15087:
15085:
15084:
15079:
15074:
15069:
15064:
15059:
15054:
15049:
15044:
15039:
15037:Possible world
15034:
15029:
15024:
15019:
15014:
15009:
15004:
14999:
14994:
14989:
14983:
14981:
14977:
14976:
14974:
14973:
14968:
14963:
14958:
14953:
14948:
14943:
14938:
14933:
14928:
14923:
14921:Glue semantics
14918:
14913:
14908:
14903:
14898:
14893:
14887:
14885:
14884:Formal systems
14878:
14874:
14873:
14870:
14869:
14867:
14866:
14861:
14856:
14851:
14846:
14841:
14836:
14831:
14826:
14821:
14816:
14811:
14809:Polarity items
14806:
14801:
14796:
14791:
14786:
14781:
14776:
14771:
14766:
14761:
14756:
14751:
14746:
14741:
14736:
14731:
14726:
14721:
14716:
14711:
14706:
14701:
14696:
14694:Conservativity
14691:
14686:
14681:
14675:
14673:
14669:
14668:
14666:
14665:
14660:
14658:Quantification
14655:
14650:
14645:
14640:
14635:
14630:
14625:
14620:
14615:
14610:
14605:
14600:
14595:
14590:
14584:
14582:
14575:
14571:
14570:
14568:
14567:
14562:
14557:
14552:
14547:
14542:
14537:
14535:Presupposition
14532:
14527:
14522:
14517:
14512:
14507:
14502:
14496:
14494:
14490:
14489:
14482:
14481:
14474:
14467:
14459:
14450:
14449:
14435:
14432:
14431:
14429:
14428:
14423:
14418:
14413:
14408:
14407:
14406:
14396:
14391:
14386:
14377:
14372:
14367:
14362:
14360:Abstract logic
14356:
14354:
14350:
14349:
14347:
14346:
14341:
14339:Turing machine
14336:
14331:
14326:
14321:
14316:
14311:
14310:
14309:
14304:
14299:
14294:
14289:
14279:
14277:Computable set
14274:
14269:
14264:
14259:
14253:
14251:
14245:
14244:
14242:
14241:
14236:
14231:
14226:
14221:
14216:
14211:
14206:
14205:
14204:
14199:
14194:
14184:
14179:
14174:
14172:Satisfiability
14169:
14164:
14159:
14158:
14157:
14147:
14146:
14145:
14135:
14134:
14133:
14128:
14123:
14118:
14113:
14103:
14102:
14101:
14096:
14089:Interpretation
14085:
14083:
14077:
14076:
14074:
14073:
14068:
14063:
14058:
14053:
14043:
14038:
14037:
14036:
14035:
14034:
14024:
14019:
14009:
14004:
13999:
13994:
13989:
13984:
13978:
13976:
13970:
13969:
13966:
13965:
13963:
13962:
13954:
13953:
13952:
13951:
13946:
13945:
13944:
13939:
13934:
13914:
13913:
13912:
13910:minimal axioms
13907:
13896:
13895:
13894:
13883:
13882:
13881:
13876:
13871:
13866:
13861:
13856:
13843:
13841:
13822:
13821:
13819:
13818:
13817:
13816:
13804:
13799:
13798:
13797:
13792:
13787:
13782:
13772:
13767:
13762:
13757:
13756:
13755:
13750:
13740:
13739:
13738:
13733:
13728:
13723:
13713:
13708:
13707:
13706:
13701:
13696:
13686:
13685:
13684:
13679:
13674:
13669:
13664:
13659:
13649:
13644:
13639:
13634:
13633:
13632:
13627:
13622:
13617:
13607:
13602:
13600:Formation rule
13597:
13592:
13591:
13590:
13585:
13575:
13574:
13573:
13563:
13558:
13553:
13548:
13542:
13536:
13519:Formal systems
13515:
13514:
13511:
13510:
13508:
13507:
13502:
13497:
13492:
13487:
13482:
13477:
13472:
13467:
13462:
13461:
13460:
13455:
13444:
13442:
13438:
13437:
13435:
13434:
13433:
13432:
13422:
13417:
13416:
13415:
13408:Large cardinal
13405:
13400:
13395:
13390:
13385:
13371:
13370:
13369:
13364:
13359:
13344:
13342:
13332:
13331:
13329:
13328:
13327:
13326:
13321:
13316:
13306:
13301:
13296:
13291:
13286:
13281:
13276:
13271:
13266:
13261:
13256:
13251:
13245:
13243:
13236:
13235:
13233:
13232:
13231:
13230:
13225:
13220:
13215:
13210:
13205:
13197:
13196:
13195:
13190:
13180:
13175:
13173:Extensionality
13170:
13168:Ordinal number
13165:
13155:
13150:
13149:
13148:
13137:
13131:
13125:
13124:
13121:
13120:
13118:
13117:
13112:
13107:
13102:
13097:
13092:
13087:
13086:
13085:
13075:
13074:
13073:
13060:
13058:
13052:
13051:
13049:
13048:
13047:
13046:
13041:
13036:
13026:
13021:
13016:
13011:
13006:
13001:
12995:
12993:
12987:
12986:
12984:
12983:
12978:
12973:
12968:
12963:
12958:
12953:
12952:
12951:
12941:
12936:
12931:
12926:
12921:
12916:
12910:
12908:
12899:
12893:
12892:
12890:
12889:
12884:
12879:
12874:
12869:
12864:
12852:Cantor's
12850:
12845:
12840:
12830:
12828:
12815:
12814:
12812:
12811:
12806:
12801:
12796:
12791:
12786:
12781:
12776:
12771:
12766:
12761:
12756:
12751:
12750:
12749:
12738:
12736:
12732:
12731:
12724:
12723:
12716:
12709:
12701:
12692:
12691:
12689:
12688:
12678:
12668:
12657:
12654:
12653:
12651:
12650:
12645:
12640:
12635:
12630:
12625:
12620:
12615:
12610:
12605:
12600:
12595:
12590:
12585:
12580:
12575:
12570:
12565:
12560:
12555:
12549:
12547:
12543:
12542:
12540:
12539:
12537:Solid modeling
12534:
12529:
12524:
12519:
12514:
12509:
12504:
12498:
12496:
12490:
12489:
12487:
12486:
12481:
12476:
12471:
12466:
12460:
12458:
12452:
12451:
12449:
12448:
12443:
12438:
12436:Control method
12433:
12428:
12423:
12418:
12413:
12407:
12405:
12399:
12398:
12396:
12395:
12390:
12388:Multithreading
12385:
12380:
12375:
12369:
12367:
12361:
12360:
12358:
12357:
12352:
12347:
12342:
12337:
12331:
12329:
12323:
12322:
12320:
12319:
12314:
12309:
12304:
12299:
12294:
12289:
12284:
12282:Formal methods
12279:
12273:
12271:
12265:
12264:
12262:
12261:
12256:
12254:World Wide Web
12251:
12246:
12241:
12236:
12231:
12226:
12221:
12216:
12211:
12206:
12201:
12196:
12190:
12188:
12182:
12181:
12179:
12178:
12173:
12168:
12163:
12158:
12153:
12148:
12143:
12137:
12135:
12128:
12127:
12125:
12124:
12119:
12114:
12109:
12104:
12098:
12096:
12090:
12089:
12087:
12086:
12081:
12076:
12071:
12066:
12061:
12056:
12055:
12054:
12043:
12041:
12035:
12034:
12032:
12031:
12026:
12021:
12016:
12011:
12006:
12001:
11996:
11991:
11986:
11980:
11978:
11972:
11971:
11969:
11968:
11963:
11958:
11953:
11948:
11943:
11938:
11933:
11928:
11923:
11917:
11915:
11905:
11904:
11902:
11901:
11896:
11891:
11886:
11881:
11875:
11873:
11869:
11868:
11866:
11865:
11860:
11855:
11850:
11845:
11840:
11834:
11832:
11826:
11825:
11823:
11822:
11817:
11812:
11807:
11802:
11796:
11794:
11790:
11789:
11787:
11786:
11777:
11772:
11767:
11762:
11757:
11752:
11747:
11742:
11737:
11731:
11729:
11723:
11722:
11715:
11712:
11711:
11704:
11703:
11696:
11689:
11681:
11675:
11674:
11669:
11668:
11667:
11662:
11647:
11646:
11645:
11632:
11629:The Nuprl Book
11626:
11620:
11603:
11600:
11599:
11598:
11592:
11586:
11580:
11574:
11568:
11562:
11554:
11551:
11549:
11548:External links
11546:
11544:
11543:
11514:
11484:
11451:
11430:(2): 125–154.
11408:
11396:
11363:
11351:
11285:
11259:
11219:
11198:
11173:
11166:
11140:
11110:
11072:
11065:
11030:
11001:Smith, Peter.
10993:
10969:
10950:(3): 407–420.
10927:
10910:
10883:
10864:
10844:
10826:
10806:
10788:
10768:
10750:
10738:
10701:Church, Alonzo
10685:
10663:
10644:
10630:
10628:
10625:
10622:
10621:
10604:
10591:
10574:
10561:
10560:
10558:
10555:
10553:
10552:
10539:
10526:
10500:
10494:
10481:
10475:
10462:
10445:
10429:
10412:
10397:
10379:
10356:
10333:
10325:
10304:
10287:
10275:"Type Systems"
10270:
10248:
10233:
10227:
10214:
10204:
10202:
10199:
10198:
10197:
10192:
10187:
10180:
10177:
10176:
10175:
10169:
10161:
10158:
10157:
10156:
10150:
10139:
10135:some forms of
10133:
10130:
10128:ST type theory
10125:
10118:
10115:
10114:
10113:
10107:
10101:
10096:
10091:
10080:
10077:
10075:
10072:
10055:
10054:
10047:
10040:
10010:
10007:
9992:
9988:
9982:
9978:
9974:
9971:
9966:
9963:
9960:
9956:
9933:
9929:
9925:
9920:
9916:
9912:
9907:
9904:
9901:
9897:
9874:
9869:
9864:
9859:
9854:
9832:
9829:
9826:
9823:
9818:
9796:
9793:
9790:
9787:
9782:
9757:
9754:
9738:
9737:
9734:
9731:Peano's axioms
9715:
9711:
9699:
9679:
9663:
9652:
9637:expressibility
9608:
9605:
9582:Scott encoding
9568:
9565:
9545:
9542:
9539:
9519:
9514:
9511:
9508:
9505:
9484:
9463:
9460:
9457:
9454:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9395:
9392:
9389:
9369:
9366:
9363:
9343:
9316:
9313:
9298:
9295:
9292:
9288:
9284:
9279:
9276:
9272:
9262:
9259:
9255:
9252:
9249:
9246:
9243:
9240:
9237:
9232:
9229:
9226:
9223:
9218:
9215:
9211:
9207:
9204:
9183:
9180:
9168:
9167:
9155:
9135:
9131:
9126:
9123:
9120:
9117:
9114:
9108:
9105:
9094:
9082:
9062:
9058:
9053:
9050:
9047:
9044:
9038:
9035:
9011:
9008:
9005:
9002:
8999:
8996:
8991:
8988:
8985:
8982:
8977:
8973:
8970:
8947:
8944:
8941:
8938:
8925:
8924:
8912:
8892:
8888:
8883:
8880:
8877:
8874:
8871:
8865:
8862:
8851:
8839:
8819:
8815:
8810:
8807:
8804:
8801:
8795:
8792:
8767:
8764:
8736:
8730:
8727:
8724:
8721:
8699:
8679:
8673:
8670:
8667:
8664:
8642:
8622:
8619:
8613:
8610:
8607:
8604:
8599:
8596:
8593:
8590:
8584:
8581:
8578:
8575:
8570:
8567:
8563:
8560:
8557:
8553:
8550:
8547:
8544:
8541:
8538:
8517:
8497:
8491:
8488:
8485:
8482:
8459:
8456:
8453:
8450:
8447:
8444:
8420:
8400:
8360:
8357:
8345:
8344:
8332:
8328:
8308:
8305:
8300:
8297:
8294:
8291:
8288:
8284:
8280:
8276:
8271:
8268:
8265:
8262:
8259:
8248:
8236:
8232:
8212:
8209:
8204:
8201:
8198:
8195:
8191:
8187:
8183:
8178:
8175:
8172:
8169:
8166:
8142:
8139:
8136:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8075:
8072:
8069:
8065:
8062:
8059:
8056:
8053:
8028:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
8003:
8000:
7997:
7994:
7991:
7970:
7967:
7964:
7961:
7958:
7955:
7952:
7949:
7945:
7942:
7939:
7936:
7912:
7909:
7906:
7882:
7862:
7850:
7847:
7835:
7834:
7822:
7802:
7799:
7796:
7793:
7790:
7785:
7782:
7779:
7776:
7773:
7770:
7759:
7747:
7727:
7724:
7721:
7718:
7715:
7710:
7707:
7704:
7701:
7698:
7674:
7671:
7668:
7665:
7662:
7659:
7656:
7653:
7649:
7645:
7642:
7639:
7635:
7632:
7629:
7626:
7623:
7620:
7599:
7596:
7593:
7590:
7587:
7584:
7581:
7578:
7574:
7570:
7567:
7564:
7560:
7557:
7554:
7551:
7548:
7527:
7507:
7487:
7467:
7447:
7444:
7441:
7421:
7418:
7415:
7412:
7409:
7389:
7369:
7366:
7363:
7360:
7357:
7342:
7339:
7331:Henk Barendegt
7297:
7294:
7291:
7269:
7249:
7243:
7240:
7237:
7234:
7231:
7228:
7199:
7179:
7176:
7173:
7151:
7148:
7145:
7142:
7120:
7096:
7090:
7087:
7084:
7081:
7067:
7064:
7049:
7046:
7043:
7038:
7033:
7030:
7027:
7022:
7018:
6992:
6989:
6986:
6981:
6978:
6962:
6959:
6945:
6942:
6939:
6936:
6933:
6911:
6908:
6905:
6902:
6880:
6858:
6821:
6818:
6801:
6781:
6778:
6775:
6755:
6735:
6714:
6693:
6677:
6674:
6657:
6637:
6634:
6631:
6611:
6586:
6565:
6549:
6546:
6544:
6541:
6528:
6519:-equality and
6508:
6482:
6478:
6473:
6470:
6467:
6463:
6460:
6456:
6453:
6449:
6444:
6441:
6438:
6434:
6431:
6428:
6425:
6403:
6380:
6360:
6340:
6320:
6300:
6297:
6293:
6290:
6287:
6284:
6281:
6261:
6241:
6221:
6201:
6181:
6178:
6175:
6172:
6169:
6166:
6155:
6154:
6142:
6122:
6098:
6078:
6075:
6072:
6069:
6065:
6062:
6059:
6056:
6053:
6043:
6031:
6011:
6008:
6005:
6002:
5999:
5996:
5993:
5990:
5987:
5983:
5980:
5977:
5974:
5971:
5968:
5945:
5925:
5913:
5910:
5890:
5887:
5884:
5880:
5875:
5872:
5869:
5865:
5861:
5856:
5853:
5850:
5830:
5829:
5811:
5808:
5805:
5802:
5798:
5795:
5790:
5787:
5784:
5780:
5777:
5774:
5771:
5766:
5763:
5760:
5756:
5753:
5743:
5725:
5722:
5719:
5715:
5712:
5707:
5704:
5701:
5697:
5694:
5684:
5666:
5656:
5653:
5650:
5645:
5642:
5639:
5635:
5601:
5598:
5571:
5568:
5541:
5538:
5518:
5498:
5495:
5491:
5488:
5468:
5448:
5428:
5408:
5388:
5385:
5382:
5362:
5342:
5339:
5337:
5334:
5308:
5305:
5302:
5297:
5292:
5289:
5286:
5262:
5259:
5256:
5234:
5230:
5225:
5222:
5219:
5196:
5193:
5190:
5168:
5144:
5141:
5138:
5133:
5128:
5125:
5122:
5117:
5114:
5111:
5107:
5102:
5099:
5096:
5092:
5089:
5086:
5083:
5061:
5041:
5021:
5001:
4981:
4978:
4975:
4955:
4935:
4915:
4912:
4909:
4906:
4903:
4900:
4884:
4881:
4867:
4864:
4861:
4834:
4831:
4828:
4823:
4820:
4815:
4812:
4809:
4804:
4799:
4796:
4793:
4788:
4767:
4764:
4761:
4738:
4735:
4732:
4727:
4722:
4719:
4716:
4693:
4690:
4687:
4664:
4659:
4656:
4653:
4648:
4643:
4640:
4637:
4632:
4629:
4624:
4621:
4618:
4613:
4609:
4606:
4603:
4579:
4576:
4573:
4542:
4522:
4498:
4495:
4492:
4472:
4452:
4432:
4431:Function terms
4429:
4428:
4427:
4414:
4411:
4408:
4405:
4400:
4397:
4387:
4374:
4371:
4368:
4363:
4360:
4350:
4337:
4334:
4331:
4328:
4323:
4319:
4316:
4313:
4310:
4299:
4286:
4283:
4280:
4275:
4272:
4243:
4240:
4237:
4234:
4211:
4208:
4205:
4202:
4199:
4177:
4174:
4171:
4168:
4141:
4138:
4135:
4117:
4114:
4112:
4109:
4107:
4104:
4084:
4081:
4073:
4072:
4071:(Seely, 1984).
4067:correspond to
4062:
4052:
4033:John Lane Bell
4025:Main article:
4022:
4019:
4017:
4016:Research areas
4014:
4008:Type inference
4006:Main article:
4003:
4002:Type inference
4000:
3988:implementation
3971:
3955:
3952:
3923:
3903:
3883:
3863:
3836:
3816:
3796:
3793:
3790:
3787:
3778:with property
3767:
3751:
3748:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3632:
3631:
3630:Dependent Sum
3628:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3583:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3538:
3532:
3531:
3528:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3483:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3438:
3432:
3431:
3428:
3417:
3414:
3411:
3401:
3390:
3387:
3384:
3374:
3368:
3367:
3364:
3353:
3350:
3347:
3337:
3326:
3323:
3320:
3310:
3304:
3303:
3300:
3289:
3286:
3283:
3273:
3262:
3259:
3249:
3243:
3242:
3239:
3228:
3225:
3222:
3212:
3201:
3198:
3195:
3185:
3179:
3178:
3175:
3164:
3154:
3143:
3133:
3129:
3128:
3125:
3114:
3104:
3093:
3083:
3079:
3078:
3075:
3072:
3071:Logic Notation
3069:
3032:
3029:
3013:intuitionistic
3008:
3005:
2997:
2996:
2993:
2990:
2987:
2980:
2979:
2976:
2973:
2970:
2967:
2952:
2949:
2944:
2937:
2916:
2915:
2903:
2883:
2863:
2843:
2823:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2756:Main article:
2753:
2750:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2714:
2691:
2688:
2685:
2663:
2634:
2630:
2607:
2603:
2582:
2562:
2542:
2509:
2505:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2482:
2477:
2473:
2469:
2464:
2460:
2456:
2453:
2447:
2443:
2439:
2436:
2433:
2430:
2427:
2426:
2398:
2395:
2374:
2361:
2360:
2348:
2328:
2306:
2302:
2279:
2275:
2263:
2252:
2249:
2244:
2240:
2236:
2231:
2227:
2223:
2220:
2209:
2208:
2196:
2174:
2170:
2147:
2143:
2131:
2118:
2114:
2110:
2105:
2101:
2097:
2094:
2083:
2082:
2070:
2050:
2030:
2020:
2009:
2006:
2003:
2000:
1997:
1986:
1985:
1973:
1953:
1943:
1931:
1928:
1925:
1914:
1913:
1910:
1892:
1871:
1842:
1839:
1836:
1831:
1826:
1823:
1820:
1815:
1811:
1807:
1781:
1778:
1775:
1770:
1767:
1764:
1760:
1756:
1745:
1742:
1737:
1734:
1731:
1726:
1723:
1720:
1715:
1712:
1709:
1706:
1701:
1698:
1676:
1651:
1648:
1645:
1623:
1620:
1616:
1612:
1607:
1604:
1600:
1578:
1575:
1572:
1550:
1528:
1525:
1522:
1519:
1497:
1486:
1485:
1470:
1448:
1444:
1421:
1417:
1405:
1391:
1387:
1361:
1357:
1345:
1333:
1310:
1299:
1284:
1268:
1265:
1252:
1249:
1246:
1241:
1237:
1233:
1228:
1225:
1222:
1219:
1213:
1210:
1206:
1186:
1183:
1180:
1175:
1171:
1166:
1162:
1142:
1139:
1134:
1130:
1110:
1089:
1086:
1061:
1036:
1033:
1030:
1027:
1006:
974:
971:
956:
953:
950:
947:
942:
938:
935:
932:
929:
904:
881:
854:
851:
839:
836:
826:, principally
771:
768:
765:
762:
759:
756:
753:
750:
747:
723:
720:
717:
714:
711:
691:
671:
647:
644:
641:
638:
635:
624:
623:
620:
608:
605:
602:
599:
596:
576:
556:
528:
508:
465:
462:
430:
427:
407:
406:
399:
381:
374:
359:
339:, interactive
337:proof checkers
329:Main article:
326:
323:
317:(specifically
283:
280:
277:
276:
256:
254:
243:
240:
204:Per Martin-Löf
104:Main article:
101:
98:
78:
77:
75:Per Martin-Löf
68:
42:of a specific
15:
9:
6:
4:
3:
2:
15190:
15179:
15176:
15174:
15171:
15169:
15166:
15165:
15163:
15148:
15145:
15143:
15140:
15138:
15135:
15133:
15130:
15128:
15125:
15123:
15120:
15118:
15115:
15113:
15110:
15108:
15105:
15103:
15100:
15099:
15097:
15093:
15083:
15080:
15078:
15075:
15073:
15070:
15068:
15065:
15063:
15060:
15058:
15055:
15053:
15050:
15048:
15045:
15043:
15040:
15038:
15035:
15033:
15030:
15028:
15025:
15023:
15020:
15018:
15015:
15013:
15010:
15008:
15005:
15003:
15000:
14998:
14995:
14993:
14990:
14988:
14985:
14984:
14982:
14978:
14972:
14969:
14967:
14964:
14962:
14959:
14957:
14954:
14952:
14949:
14947:
14944:
14942:
14939:
14937:
14934:
14932:
14929:
14927:
14924:
14922:
14919:
14917:
14914:
14912:
14909:
14907:
14904:
14902:
14899:
14897:
14894:
14892:
14889:
14888:
14886:
14882:
14879:
14875:
14865:
14862:
14860:
14857:
14855:
14852:
14850:
14847:
14845:
14842:
14840:
14837:
14835:
14832:
14830:
14827:
14825:
14822:
14820:
14817:
14815:
14812:
14810:
14807:
14805:
14804:Performatives
14802:
14800:
14797:
14795:
14792:
14790:
14787:
14785:
14784:Logophoricity
14782:
14780:
14777:
14775:
14772:
14770:
14767:
14765:
14762:
14760:
14757:
14755:
14752:
14750:
14747:
14745:
14742:
14740:
14737:
14735:
14732:
14730:
14727:
14725:
14722:
14720:
14717:
14715:
14712:
14710:
14707:
14705:
14702:
14700:
14697:
14695:
14692:
14690:
14687:
14685:
14682:
14680:
14677:
14676:
14674:
14670:
14664:
14661:
14659:
14656:
14654:
14651:
14649:
14646:
14644:
14641:
14639:
14636:
14634:
14631:
14629:
14626:
14624:
14621:
14619:
14618:Evidentiality
14616:
14614:
14611:
14609:
14606:
14604:
14601:
14599:
14596:
14594:
14591:
14589:
14586:
14585:
14583:
14579:
14576:
14572:
14566:
14563:
14561:
14558:
14556:
14553:
14551:
14548:
14546:
14543:
14541:
14538:
14536:
14533:
14531:
14528:
14526:
14523:
14521:
14518:
14516:
14513:
14511:
14508:
14506:
14503:
14501:
14498:
14497:
14495:
14491:
14487:
14480:
14475:
14473:
14468:
14466:
14461:
14460:
14457:
14447:
14446:
14441:
14433:
14427:
14424:
14422:
14419:
14417:
14414:
14412:
14409:
14405:
14402:
14401:
14400:
14397:
14395:
14392:
14390:
14387:
14385:
14381:
14378:
14376:
14373:
14371:
14368:
14366:
14363:
14361:
14358:
14357:
14355:
14351:
14345:
14342:
14340:
14337:
14335:
14334:Recursive set
14332:
14330:
14327:
14325:
14322:
14320:
14317:
14315:
14312:
14308:
14305:
14303:
14300:
14298:
14295:
14293:
14290:
14288:
14285:
14284:
14283:
14280:
14278:
14275:
14273:
14270:
14268:
14265:
14263:
14260:
14258:
14255:
14254:
14252:
14250:
14246:
14240:
14237:
14235:
14232:
14230:
14227:
14225:
14222:
14220:
14217:
14215:
14212:
14210:
14207:
14203:
14200:
14198:
14195:
14193:
14190:
14189:
14188:
14185:
14183:
14180:
14178:
14175:
14173:
14170:
14168:
14165:
14163:
14160:
14156:
14153:
14152:
14151:
14148:
14144:
14143:of arithmetic
14141:
14140:
14139:
14136:
14132:
14129:
14127:
14124:
14122:
14119:
14117:
14114:
14112:
14109:
14108:
14107:
14104:
14100:
14097:
14095:
14092:
14091:
14090:
14087:
14086:
14084:
14082:
14078:
14072:
14069:
14067:
14064:
14062:
14059:
14057:
14054:
14051:
14050:from ZFC
14047:
14044:
14042:
14039:
14033:
14030:
14029:
14028:
14025:
14023:
14020:
14018:
14015:
14014:
14013:
14010:
14008:
14005:
14003:
14000:
13998:
13995:
13993:
13990:
13988:
13985:
13983:
13980:
13979:
13977:
13975:
13971:
13961:
13960:
13956:
13955:
13950:
13949:non-Euclidean
13947:
13943:
13940:
13938:
13935:
13933:
13932:
13928:
13927:
13925:
13922:
13921:
13919:
13915:
13911:
13908:
13906:
13903:
13902:
13901:
13897:
13893:
13890:
13889:
13888:
13884:
13880:
13877:
13875:
13872:
13870:
13867:
13865:
13862:
13860:
13857:
13855:
13852:
13851:
13849:
13845:
13844:
13842:
13837:
13831:
13826:Example
13823:
13815:
13810:
13809:
13808:
13805:
13803:
13800:
13796:
13793:
13791:
13788:
13786:
13783:
13781:
13778:
13777:
13776:
13773:
13771:
13768:
13766:
13763:
13761:
13758:
13754:
13751:
13749:
13746:
13745:
13744:
13741:
13737:
13734:
13732:
13729:
13727:
13724:
13722:
13719:
13718:
13717:
13714:
13712:
13709:
13705:
13702:
13700:
13697:
13695:
13692:
13691:
13690:
13687:
13683:
13680:
13678:
13675:
13673:
13670:
13668:
13665:
13663:
13660:
13658:
13655:
13654:
13653:
13650:
13648:
13645:
13643:
13640:
13638:
13635:
13631:
13628:
13626:
13623:
13621:
13618:
13616:
13613:
13612:
13611:
13608:
13606:
13603:
13601:
13598:
13596:
13593:
13589:
13586:
13584:
13583:by definition
13581:
13580:
13579:
13576:
13572:
13569:
13568:
13567:
13564:
13562:
13559:
13557:
13554:
13552:
13549:
13547:
13544:
13543:
13540:
13537:
13535:
13531:
13526:
13520:
13516:
13506:
13503:
13501:
13498:
13496:
13493:
13491:
13488:
13486:
13483:
13481:
13478:
13476:
13473:
13471:
13470:Kripke–Platek
13468:
13466:
13463:
13459:
13456:
13454:
13451:
13450:
13449:
13446:
13445:
13443:
13439:
13431:
13428:
13427:
13426:
13423:
13421:
13418:
13414:
13411:
13410:
13409:
13406:
13404:
13401:
13399:
13396:
13394:
13391:
13389:
13386:
13383:
13379:
13375:
13372:
13368:
13365:
13363:
13360:
13358:
13355:
13354:
13353:
13349:
13346:
13345:
13343:
13341:
13337:
13333:
13325:
13322:
13320:
13317:
13315:
13314:constructible
13312:
13311:
13310:
13307:
13305:
13302:
13300:
13297:
13295:
13292:
13290:
13287:
13285:
13282:
13280:
13277:
13275:
13272:
13270:
13267:
13265:
13262:
13260:
13257:
13255:
13252:
13250:
13247:
13246:
13244:
13242:
13237:
13229:
13226:
13224:
13221:
13219:
13216:
13214:
13211:
13209:
13206:
13204:
13201:
13200:
13198:
13194:
13191:
13189:
13186:
13185:
13184:
13181:
13179:
13176:
13174:
13171:
13169:
13166:
13164:
13160:
13156:
13154:
13151:
13147:
13144:
13143:
13142:
13139:
13138:
13135:
13132:
13130:
13126:
13116:
13113:
13111:
13108:
13106:
13103:
13101:
13098:
13096:
13093:
13091:
13088:
13084:
13081:
13080:
13079:
13076:
13072:
13067:
13066:
13065:
13062:
13061:
13059:
13057:
13053:
13045:
13042:
13040:
13037:
13035:
13032:
13031:
13030:
13027:
13025:
13022:
13020:
13017:
13015:
13012:
13010:
13007:
13005:
13002:
13000:
12997:
12996:
12994:
12992:
12991:Propositional
12988:
12982:
12979:
12977:
12974:
12972:
12969:
12967:
12964:
12962:
12959:
12957:
12954:
12950:
12947:
12946:
12945:
12942:
12940:
12937:
12935:
12932:
12930:
12927:
12925:
12922:
12920:
12919:Logical truth
12917:
12915:
12912:
12911:
12909:
12907:
12903:
12900:
12898:
12894:
12888:
12885:
12883:
12880:
12878:
12875:
12873:
12870:
12868:
12865:
12863:
12859:
12855:
12851:
12849:
12846:
12844:
12841:
12839:
12835:
12832:
12831:
12829:
12827:
12821:
12816:
12810:
12807:
12805:
12802:
12800:
12797:
12795:
12792:
12790:
12787:
12785:
12782:
12780:
12777:
12775:
12772:
12770:
12767:
12765:
12762:
12760:
12757:
12755:
12752:
12748:
12745:
12744:
12743:
12740:
12739:
12737:
12733:
12729:
12722:
12717:
12715:
12710:
12708:
12703:
12702:
12699:
12687:
12679:
12677:
12669:
12667:
12659:
12658:
12655:
12649:
12646:
12644:
12641:
12639:
12636:
12634:
12631:
12629:
12626:
12624:
12621:
12619:
12616:
12614:
12611:
12609:
12606:
12604:
12601:
12599:
12596:
12594:
12591:
12589:
12586:
12584:
12581:
12579:
12576:
12574:
12571:
12569:
12566:
12564:
12561:
12559:
12556:
12554:
12551:
12550:
12548:
12544:
12538:
12535:
12533:
12530:
12528:
12525:
12523:
12522:Mixed reality
12520:
12518:
12515:
12513:
12510:
12508:
12505:
12503:
12500:
12499:
12497:
12495:
12491:
12485:
12482:
12480:
12477:
12475:
12472:
12470:
12467:
12465:
12462:
12461:
12459:
12457:
12453:
12447:
12444:
12442:
12439:
12437:
12434:
12432:
12429:
12427:
12424:
12422:
12419:
12417:
12414:
12412:
12409:
12408:
12406:
12404:
12400:
12394:
12391:
12389:
12386:
12384:
12381:
12379:
12376:
12374:
12371:
12370:
12368:
12366:
12362:
12356:
12355:Accessibility
12353:
12351:
12350:Visualization
12348:
12346:
12343:
12341:
12338:
12336:
12333:
12332:
12330:
12328:
12324:
12318:
12315:
12313:
12310:
12308:
12305:
12303:
12300:
12298:
12295:
12293:
12290:
12288:
12285:
12283:
12280:
12278:
12275:
12274:
12272:
12270:
12266:
12260:
12257:
12255:
12252:
12250:
12247:
12245:
12242:
12240:
12237:
12235:
12232:
12230:
12227:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12202:
12200:
12197:
12195:
12192:
12191:
12189:
12187:
12183:
12177:
12174:
12172:
12169:
12167:
12164:
12162:
12159:
12157:
12154:
12152:
12149:
12147:
12144:
12142:
12139:
12138:
12136:
12134:
12129:
12123:
12120:
12118:
12115:
12113:
12110:
12108:
12105:
12103:
12100:
12099:
12097:
12095:
12091:
12085:
12082:
12080:
12077:
12075:
12072:
12070:
12067:
12065:
12062:
12060:
12057:
12053:
12050:
12049:
12048:
12045:
12044:
12042:
12040:
12036:
12030:
12027:
12025:
12022:
12020:
12017:
12015:
12012:
12010:
12007:
12005:
12002:
12000:
11997:
11995:
11992:
11990:
11987:
11985:
11982:
11981:
11979:
11977:
11973:
11967:
11964:
11962:
11959:
11957:
11954:
11952:
11949:
11947:
11944:
11942:
11939:
11937:
11934:
11932:
11929:
11927:
11924:
11922:
11919:
11918:
11916:
11914:
11910:
11906:
11900:
11897:
11895:
11892:
11890:
11887:
11885:
11882:
11880:
11877:
11876:
11874:
11870:
11864:
11861:
11859:
11856:
11854:
11851:
11849:
11846:
11844:
11841:
11839:
11836:
11835:
11833:
11831:
11827:
11821:
11818:
11816:
11813:
11811:
11810:Dependability
11808:
11806:
11803:
11801:
11798:
11797:
11795:
11791:
11785:
11781:
11778:
11776:
11773:
11771:
11768:
11766:
11763:
11761:
11758:
11756:
11753:
11751:
11748:
11746:
11743:
11741:
11738:
11736:
11733:
11732:
11730:
11728:
11724:
11719:
11713:
11709:
11702:
11697:
11695:
11690:
11688:
11683:
11682:
11679:
11673:
11670:
11666:
11663:
11661:
11657:
11654:
11653:
11651:
11648:
11643:
11639:
11638:
11636:
11633:
11630:
11627:
11624:
11621:
11617:
11616:
11611:
11606:
11605:
11596:
11593:
11590:
11587:
11584:
11581:
11578:
11575:
11572:
11569:
11566:
11563:
11560:
11557:
11556:
11532:
11528:
11524:
11518:
11502:
11498:
11494:
11488:
11473:
11469:
11465:
11458:
11456:
11446:
11441:
11437:
11433:
11429:
11425:
11424:
11419:
11412:
11399:
11393:
11389:
11385:
11381:
11377:
11370:
11368:
11361:
11358:Balbaert,Ivo
11355:
11347:
11341:
11330:
11325:
11320:
11315:
11310:
11306:
11299:
11292:
11290:
11274:
11270:
11266:
11262:
11256:
11252:
11248:
11243:
11238:
11234:
11230:
11223:
11205:
11201:
11195:
11191:
11184:
11177:
11169:
11163:
11159:
11155:
11151:
11144:
11128:
11124:
11120:
11114:
11096:
11089:
11083:
11081:
11079:
11077:
11068:
11062:
11058:
11057:
11049:
11047:
11045:
11043:
11041:
11039:
11037:
11035:
11015:
11011:
11004:
10997:
10989:
10988:
10980:
10978:
10976:
10974:
10965:
10961:
10957:
10953:
10949:
10945:
10941:
10934:
10932:
10924:
10920:
10914:
10906:
10902:
10899:(2): 99–112.
10898:
10894:
10887:
10880:
10877:
10874:
10868:
10861:
10857:
10854:
10848:
10833:
10829:
10823:
10819:
10818:
10810:
10795:
10791:
10785:
10781:
10780:
10772:
10766:
10764:
10759:
10754:
10748:
10742:
10734:
10730:
10726:
10722:
10718:
10714:
10710:
10706:
10702:
10696:
10694:
10692:
10690:
10682:
10678:
10675:
10672:
10667:
10661:
10657:
10654:
10648:
10641:
10635:
10631:
10614:
10608:
10601:
10595:
10588:
10584:
10578:
10572:
10566:
10562:
10551:
10548:
10544:
10540:
10533:
10529:
10527:90-386-0531-5
10523:
10519:
10515:
10508:
10507:
10501:
10497:
10491:
10487:
10482:
10478:
10476:1-4020-2334-0
10472:
10468:
10463:
10461:
10457:
10454:
10448:
10442:
10438:
10434:
10430:
10419:
10415:
10413:0-201-41667-0
10409:
10405:
10404:
10398:
10394:
10393:
10388:
10387:"Type Theory"
10384:
10380:
10378:
10377:
10372:
10363:
10359:
10357:9789401004138
10353:
10349:
10342:
10338:
10334:
10332:Mathematica'.
10328:
10322:
10318:
10314:
10310:
10305:
10294:
10290:
10288:9780849329098
10284:
10280:
10276:
10271:
10268:
10255:
10251:
10245:
10241:
10240:
10234:
10230:
10224:
10220:
10215:
10211:
10206:
10205:
10196:
10193:
10191:
10188:
10186:
10183:
10182:
10173:
10170:
10167:
10164:
10163:
10155:
10151:
10148:
10144:
10140:
10138:
10134:
10131:
10129:
10126:
10124:
10121:
10120:
10111:
10108:
10105:
10102:
10100:
10097:
10095:
10092:
10090:
10086:
10083:
10082:
10071:
10069:
10064:
10060:
10052:
10048:
10045:
10041:
10038:
10037:
10036:
10033:
10031:
10026:
10022:
10020:
10016:
10006:
9990:
9980:
9976:
9969:
9964:
9961:
9958:
9954:
9931:
9927:
9923:
9918:
9914:
9910:
9905:
9902:
9899:
9895:
9867:
9857:
9830:
9827:
9824:
9821:
9794:
9791:
9788:
9785:
9768:
9766:
9761:
9753:
9751:
9747:
9743:
9735:
9732:
9728:
9724:
9720:
9716:
9712:
9697:
9677:
9669:
9664:
9661:
9657:
9653:
9650:
9646:
9642:
9641:
9640:
9638:
9634:
9630:
9626:
9622:
9618:
9614:
9604:
9602:
9598:
9594:
9590:
9585:
9583:
9579:
9575:
9564:
9562:
9557:
9543:
9540:
9537:
9517:
9482:
9475:. For a term
9431:
9428:
9425:
9422:
9419:
9416:
9413:
9393:
9390:
9387:
9367:
9364:
9361:
9341:
9332:
9330:
9326:
9322:
9321:identity type
9315:Identity type
9312:
9310:
9290:
9286:
9282:
9270:
9260:
9257:
9244:
9238:
9216:
9213:
9209:
9153:
9133:
9129:
9095:
9080:
9060:
9056:
9025:
9024:
9023:
9009:
9003:
8997:
8975:
8910:
8890:
8886:
8852:
8837:
8817:
8813:
8782:
8781:
8780:
8753:
8748:
8734:
8697:
8677:
8640:
8617:
8594:
8588:
8568:
8565:
8555:
8515:
8495:
8432:
8390:
8386:
8382:
8378:
8374:
8370:
8366:
8356:
8354:
8350:
8330:
8326:
8303:
8278:
8274:
8249:
8234:
8230:
8207:
8185:
8181:
8156:
8155:
8154:
8137:
8128:
8125:
8122:
8110:
8104:
8092:
8086:
8080:
8077:
8067:
8042:
8023:
8020:
8017:
8008:
8005:
7965:
7962:
7959:
7950:
7947:
7926:
7910:
7907:
7904:
7896:
7895:tagged unions
7880:
7860:
7846:
7844:
7840:
7820:
7797:
7794:
7791:
7760:
7745:
7722:
7719:
7716:
7688:
7687:
7686:
7669:
7663:
7660:
7657:
7654:
7651:
7647:
7637:
7594:
7588:
7585:
7582:
7579:
7576:
7572:
7562:
7525:
7505:
7485:
7465:
7445:
7442:
7439:
7416:
7413:
7410:
7387:
7364:
7361:
7358:
7348:
7347:ordered pairs
7338:
7336:
7332:
7328:
7323:
7321:
7317:
7313:
7267:
7247:
7215:
7213:
7197:
7177:
7171:
7164:has the type
7118:
7110:
7094:
7063:
7020:
7008:
6979:
6976:
6968:
6958:
6817:
6815:
6799:
6779:
6753:
6733:
6683:
6673:
6671:
6655:
6629:
6609:
6601:
6555:
6540:
6526:
6506:
6498:
6493:
6480:
6476:
6458:
6451:
6447:
6432:
6429:
6426:
6415:
6401:
6392:
6378:
6358:
6338:
6318:
6295:
6291:
6288:
6285:
6282:
6259:
6239:
6219:
6199:
6176:
6173:
6170:
6167:
6140:
6120:
6112:
6111:free variable
6096:
6076:
6067:
6063:
6060:
6057:
6054:
6044:
6029:
6006:
6003:
6000:
5997:
5991:
5985:
5978:
5975:
5972:
5969:
5959:
5958:
5957:
5943:
5923:
5909:
5888:
5882:
5878:
5859:
5839:
5835:
5809:
5800:
5793:
5769:
5744:
5723:
5717:
5710:
5685:
5654:
5648:
5626:
5625:
5624:
5622:
5599:
5596:
5569:
5566:
5539:
5536:
5516:
5493:
5489:
5466:
5446:
5426:
5406:
5386:
5380:
5360:
5352:
5348:
5333:
5331:
5326:
5324:
5232:
5228:
5166:
5157:
5115:
5109:
5105:
5090:
5087:
5084:
5073:
5059:
5039:
5019:
4999:
4979:
4973:
4953:
4933:
4910:
4907:
4904:
4901:
4890:
4880:
4850:
4675:
4611:
4592:
4561:
4559:
4557:
4540:
4520:
4512:
4496:
4490:
4470:
4450:
4442:
4438:
4398:
4395:
4388:
4361:
4358:
4351:
4321:
4300:
4273:
4270:
4263:
4262:
4261:
4259:
4157:
4156:Boolean logic
4123:
4103:
4101:
4097:
4094:differs from
4093:
4089:
4080:
4078:
4070:
4066:
4063:
4061:around 1980);
4060:
4056:
4053:
4050:
4046:
4043:
4042:
4041:
4038:
4034:
4029:
4028:
4013:
4009:
3999:
3997:
3993:
3992:specification
3989:
3984:
3961:
3951:
3949:
3948:parametricity
3945:
3941:
3935:
3921:
3901:
3881:
3861:
3853:
3848:
3834:
3814:
3791:
3785:
3765:
3757:
3747:
3743:
3729:
3705:
3696:
3693:
3661:
3655:
3652:
3649:
3646:
3629:
3612:
3606:
3603:
3600:
3597:
3594:
3584:
3567:
3561:
3558:
3555:
3552:
3549:
3539:
3537:
3534:
3533:
3529:
3512:
3506:
3503:
3500:
3497:
3494:
3484:
3467:
3461:
3458:
3455:
3452:
3449:
3439:
3437:
3434:
3433:
3429:
3415:
3412:
3409:
3402:
3388:
3385:
3382:
3375:
3373:
3370:
3369:
3366:Product Type
3365:
3351:
3348:
3345:
3338:
3324:
3321:
3318:
3311:
3309:
3306:
3305:
3301:
3281:
3274:
3260:
3250:
3248:
3245:
3244:
3240:
3226:
3220:
3213:
3199:
3193:
3186:
3184:
3181:
3180:
3176:
3155:
3134:
3131:
3130:
3126:
3105:
3084:
3081:
3080:
3076:
3074:Type Notation
3073:
3070:
3067:
3066:
3063:
3060:
3058:
3054:
3050:
3046:
3042:
3038:
3028:
3026:
3022:
3018:
3014:
3004:
3002:
2994:
2991:
2988:
2985:
2984:
2983:
2977:
2974:
2971:
2968:
2965:
2961:
2957:
2953:
2950:
2948:
2945:
2942:
2938:
2936:in this case)
2935:
2931:
2930:
2929:
2926:
2924:
2920:
2881:
2861:
2841:
2813:
2812:
2811:
2797:
2794:
2791:
2788:
2785:
2779:
2776:
2765:
2759:
2749:
2731:
2728:
2725:
2722:
2719:
2716:
2661:
2653:
2648:
2632:
2628:
2605:
2601:
2580:
2533:
2532:metavariables
2529:
2507:
2503:
2499:
2496:
2493:
2487:
2475:
2471:
2467:
2462:
2458:
2454:
2445:
2441:
2437:
2434:
2431:
2416:
2412:
2408:
2404:
2394:
2392:
2388:
2372:
2326:
2304:
2300:
2277:
2273:
2264:
2250:
2247:
2242:
2238:
2234:
2229:
2225:
2221:
2211:
2172:
2168:
2145:
2141:
2132:
2116:
2112:
2108:
2103:
2099:
2095:
2085:
2048:
2028:
2021:
2007:
2004:
2001:
1998:
1988:
1951:
1944:
1929:
1926:
1916:
1911:
1908:
1907:
1904:
1882:
1860:
1855:
1813:
1805:
1797:
1794:
1768:
1762:
1758:
1754:
1740:
1724:
1721:
1718:
1699:
1696:
1688:
1674:
1667:
1618:
1614:
1610:
1548:
1495:
1483:
1468:
1461:both of type
1446:
1442:
1419:
1415:
1406:
1389:
1385:
1376:
1359:
1355:
1346:
1331:
1323:
1308:
1300:
1297:
1282:
1274:
1273:
1272:
1264:
1244:
1231:
1178:
1137:
1108:
1076:
1051:
1004:
996:
992:
988:
984:
980:
979:term in logic
970:
940:
918:
902:
895:
879:
872:
868:
864:
860:
850:
848:
844:
835:
833:
829:
825:
821:
817:
816:
811:
807:
803:
801:
797:
793:
789:
785:
766:
763:
757:
754:
751:
737:
718:
715:
712:
689:
669:
661:
642:
639:
636:
621:
603:
600:
597:
574:
554:
546:
545:
544:
542:
526:
506:
497:
495:
491:
487:
483:
479:
476:, especially
475:
471:
461:
459:
455:
450:
448:
444:
440:
436:
426:
424:
420:
416:
412:
404:
400:
397:
393:
389:
385:
382:
379:
375:
372:
368:
364:
360:
357:
353:
350:
349:
348:
346:
342:
338:
332:
322:
320:
316:
312:
308:
303:
301:
297:
293:
289:
273:
264:
260:
257:This section
255:
252:
248:
247:
239:
237:
233:
229:
225:
221:
217:
214:. Another is
213:
209:
205:
200:
198:
194:
190:
186:
182:
178:
177:Alonzo Church
173:
171:
167:
162:
161:
156:
152:
148:
144:
139:
137:
133:
129:
128:Gottlob Frege
125:
121:
117:
113:
107:
97:
95:
91:
87:
83:
76:
72:
69:
67:
66:Alonzo Church
63:
60:
59:
58:
56:
52:
47:
45:
41:
37:
33:
29:
22:
15077:Type shifter
15047:Quantization
14997:Continuation
14965:
14864:Veridicality
14744:Exhaustivity
14709:Cumulativity
14628:Indexicality
14608:Definiteness
14603:Conditionals
14530:Logical form
14436:
14343:
14234:Ultraproduct
14081:Model theory
14046:Independence
13982:Formal proof
13974:Proof theory
13957:
13930:
13887:real numbers
13859:second-order
13770:Substitution
13647:Metalanguage
13588:conservative
13561:Axiom schema
13505:Constructive
13475:Morse–Kelley
13441:Set theories
13420:Aleph number
13413:inaccessible
13319:Grothendieck
13203:intersection
13090:Higher-order
13078:Second-order
13024:Truth tables
12981:Venn diagram
12808:
12764:Formal proof
12618:Cyberwarfare
12277:Cryptography
11615:Scholarpedia
11613:
11535:. Retrieved
11526:
11517:
11505:. Retrieved
11496:
11487:
11476:. Retrieved
11467:
11427:
11421:
11411:
11401:, retrieved
11379:
11354:
11340:cite journal
11304:
11277:. Retrieved
11232:
11222:
11211:. Retrieved
11189:
11176:
11149:
11143:
11131:. Retrieved
11122:
11113:
11102:. Retrieved
11055:
11021:. Retrieved
11009:
10996:
10986:
10947:
10943:
10922:
10918:
10913:
10896:
10892:
10886:
10878:
10875:
10867:
10847:
10836:. Retrieved
10816:
10809:
10798:. Retrieved
10782:. Springer.
10778:
10771:
10762:
10753:
10741:
10711:(2): 56–68.
10708:
10704:
10670:
10666:
10647:
10639:
10634:
10607:
10599:
10594:
10577:
10565:
10550:
10542:
10505:
10485:
10469:. Springer.
10466:
10436:
10422:. Retrieved
10402:
10390:
10374:
10347:
10308:
10297:. Retrieved
10278:
10258:. Retrieved
10238:
10218:
10063:constructive
10056:
10034:
10027:
10023:
10012:
9769:
9762:
9759:
9739:
9610:
9586:
9570:
9558:
9333:
9318:
9311:
9170:The type of
9169:
8926:
8749:
8653:, pass in a
8433:
8362:
8346:
8039:, which are
7925:constructors
7852:
7843:intersection
7836:
7344:
7341:Product type
7324:
7216:
7210:denotes the
7069:
6964:
6823:
6820:Boolean type
6813:
6679:
6669:
6551:
6494:
6416:
6393:
6156:
6153:-reduction).
6042:-reduction).
5915:
5831:
5344:
5327:
5158:
5074:
4886:
4883:Lambda terms
4676:
4593:
4562:
4555:
4434:
4119:
4116:Atomic terms
4086:
4074:
4036:
4030:
4024:
4011:
3991:
3987:
3985:
3957:
3936:
3849:
3753:
3744:
3635:
3061:
3037:propositions
3034:
3010:
3000:
2998:
2981:
2964:transitivity
2940:
2933:
2927:
2917:
2761:
2649:
2415:substitution
2400:
2389:and thus an
2364:
1912:Description
1858:
1856:
1798:
1795:
1689:
1487:
1270:
976:
856:
841:
819:
813:
804:
791:
787:
625:
541:truth-values
498:
493:
489:
467:
451:
432:
408:
334:
304:
295:
285:
267:
263:adding to it
258:
242:Applications
201:
174:
158:
140:
120:formal logic
109:
79:
48:
35:
25:
15168:Type theory
14992:Context set
14966:Type theory
14849:Subtrigging
14613:Disjunction
14540:Proposition
14344:Type theory
14292:undecidable
14224:Truth value
14111:equivalence
13790:non-logical
13403:Enumeration
13393:Isomorphism
13340:cardinality
13324:Von Neumann
13289:Ultrafilter
13254:Uncountable
13188:equivalence
13105:Quantifiers
13095:Fixed-point
13064:First-order
12944:Consistency
12929:Proposition
12906:Traditional
12877:Lindström's
12867:Compactness
12809:Type theory
12754:Cardinality
12628:Video games
12608:Digital art
12365:Concurrency
12234:Data mining
12146:Probability
11879:Interpreter
11133:29 December
11023:29 December
10925:. Elsevier.
10453:Book review
10371:Paul Halmos
10317:11375/12315
10143:lambda cube
10087:which is a
9717:Set theory
9495:, the call
8363:Two common
7897:. The type
7400:. The pair
7335:lambda cube
7320:dot product
7316:type safety
6670:uninhabited
6539:-equality.
5509:, has type
5209:. The term
4106:Definitions
3183:Implication
3177:Empty Type
2956:reflexivity
2923:consistency
2834:and a type
1375:is equal to
871:proposition
847:double bind
818:to express
464:Linguistics
354:is used by
44:type system
36:type theory
28:mathematics
15162:Categories
15142:Pragmatics
14789:Mirativity
14555:Speech act
14510:Entailment
14505:Denotation
14155:elementary
13848:arithmetic
13716:Quantifier
13694:functional
13566:Expression
13284:Transitive
13228:identities
13213:complement
13146:hereditary
13129:Set theory
12686:Glossaries
12558:E-commerce
12151:Statistics
12094:Algorithms
12052:Stochastic
11884:Middleware
11740:Peripheral
11658:including
11537:2022-01-21
11507:21 January
11478:2022-01-22
11445:2066/17240
11403:2024-01-18
11314:1611.02108
11279:2022-06-21
11242:2101.11479
11213:2012-11-03
11104:2022-01-22
10838:2022-07-29
10800:2022-07-29
10627:References
10424:2006-04-03
10373:'s (1960)
10299:2004-06-26
10260:2020-07-19
9591:, such as
9022:such that
8389:set theory
8153:such that
6554:empty type
6548:Empty type
6414:-reduced.
5912:Reductions
4059:Dana Scott
4037:themselves
3944:canonicity
3127:Unit Type
3077:Type Name
3068:Logic Name
2652:proof tree
294:to encode
51:set theory
15178:Hierarchy
14941:Mereology
14877:Formalism
14759:Givenness
14684:Cataphora
14672:Phenomena
14663:Vagueness
14593:Ambiguity
14545:Reference
14525:Intension
14515:Extension
14426:Supertask
14329:Recursion
14287:decidable
14121:saturated
14099:of models
14022:deductive
14017:axiomatic
13937:Hilbert's
13924:Euclidean
13905:canonical
13828:axiomatic
13760:Signature
13689:Predicate
13578:Extension
13500:Ackermann
13425:Operation
13304:Universal
13294:Recursive
13269:Singleton
13264:Inhabited
13249:Countable
13239:Types of
13223:power set
13193:partition
13110:Predicate
13056:Predicate
12971:Syllogism
12961:Soundness
12934:Inference
12924:Tautology
12826:paradoxes
12507:Rendering
12502:Animation
12133:computing
12084:Semantics
11775:Processor
11523:"Axiom K"
11269:231719089
10964:1573-0964
10653:v.1 Types
10435:(2008) .
10385:(2018) .
10339:(2012) .
9970:≅
9962:×
9924:×
9911:≅
9868:≅
9828:≅
9822:×
9792:≅
9631:with the
9619:with the
9291:τ
9287:σ
9254:Σ
9248:→
9245:τ
9242:→
9239:σ
9236:→
9214:τ
9210:σ
9206:Π
9154:τ
9134:τ
9130:σ
9081:σ
9061:τ
9057:σ
9007:→
9001:→
8995:→
8976::
8598:→
8592:→
8562:Π
8419:Σ
8399:Π
8138:ρ
8135:→
8129:τ
8126:⊔
8123:σ
8117:→
8111:ρ
8108:→
8105:τ
8099:→
8093:ρ
8090:→
8087:σ
8078:ρ
8074:Π
8041:injective
8024:τ
8021:⊔
8018:σ
8012:→
8009:τ
7966:τ
7963:⊔
7960:σ
7954:→
7951:σ
7911:τ
7908:⊔
7905:σ
7881:⊔
7670:τ
7667:→
7664:τ
7661:×
7658:σ
7652:τ
7648:σ
7644:Π
7595:σ
7592:→
7589:τ
7586:×
7583:σ
7577:τ
7573:σ
7569:Π
7506:τ
7466:σ
7446:τ
7443:×
7440:σ
7388:×
7175:→
7037:→
6814:inhabited
6777:→
6774:⊤
6734:∗
6692:⊤
6682:unit type
6676:Unit type
6636:⊥
6633:→
6564:⊥
6527:η
6507:β
6462:→
6427:λ
6402:β
6283:λ
6168:λ
6141:η
6109:is not a
6074:→
6055:λ
6030:β
6001::
5989:→
5970:λ
5944:η
5924:β
5665:→
5517:τ
5427:σ
5387:τ
5384:→
5381:σ
5296:→
5245:has type
5132:→
5085:λ
5040:τ
5000:σ
4980:τ
4977:→
4974:σ
4902:λ
4879:'s type.
4822:→
4803:→
4726:→
4647:→
4628:→
4541:τ
4521:σ
4511:parameter
4497:τ
4494:→
4491:σ
4471:τ
4451:σ
4437:functions
4055:C-monoids
3970:→
3727:→
3721:⊥
3718:→
3712:⊥
3709:→
3691:Π
3668:⊥
3665:→
3644:Π
3592:Σ
3553:∈
3547:∃
3492:Π
3453:∈
3447:∀
3430:Sum Type
3386:∨
3349:×
3322:∧
3288:⊥
3285:→
3258:¬
3241:Function
3224:→
3197:→
3163:⊥
3142:⊥
3113:⊤
3092:⊤
2941:weakening
2902:Γ
2882:τ
2842:τ
2822:Γ
2795:τ
2786:⊢
2783:Γ
2774:∃
2717:⊢
2561:Δ
2541:Γ
2528:rewriting
2494:⊢
2491:Δ
2485:Γ
2455:⊢
2452:Δ
2432:⊢
2429:Γ
2411:deduction
2391:extrinsic
2373:≡
2347:Γ
2222:⊢
2219:Γ
2195:Γ
2096:⊢
2093:Γ
2069:Γ
1999:⊢
1996:Γ
1972:Γ
1927:⊢
1924:Γ
1891:Δ
1870:Γ
1830:→
1806:⊢
1741:⊢
1675:⊢
1482:are equal
1322:is a term
1296:is a type
1267:Judgments
903:φ
880:φ
867:judgments
788:everybody
770:⟩
761:⟩
749:⟨
746:⟨
722:⟩
710:⟨
660:functions
646:⟩
634:⟨
607:⟩
595:⟨
441:phase of
151:Whitehead
15095:See also
14980:Concepts
14854:Telicity
14689:Coercion
14643:Negation
14638:Modality
14588:Anaphora
14411:Logicism
14404:timeline
14380:Concrete
14239:Validity
14209:T-schema
14202:Kripke's
14197:Tarski's
14192:semantic
14182:Strength
14131:submodel
14126:spectrum
14094:function
13942:Tarski's
13931:Elements
13918:geometry
13874:Robinson
13795:variable
13780:function
13753:spectrum
13743:Sentence
13699:variable
13642:Language
13595:Relation
13556:Automata
13546:Alphabet
13530:language
13384:-jection
13362:codomain
13348:Function
13309:Universe
13279:Infinite
13183:Relation
12966:Validity
12956:Argument
12854:theorem,
12666:Category
12494:Graphics
12269:Security
11931:Compiler
11830:Networks
11727:Hardware
11531:Archived
11501:Archived
11472:Archived
11329:Archived
11273:Archived
11204:Archived
11127:Archived
11095:Archived
11014:Archived
10944:Synthese
10856:Archived
10832:Archived
10794:Archived
10733:15889861
10677:Archived
10656:Archived
10532:Archived
10456:Archived
10418:Archived
10362:Archived
10293:Archived
10254:Archived
10179:See also
10123:Automath
10099:system F
9621:language
9146:returns
9073:returns
8903:returns
8830:returns
8381:products
8319:returns
8223:returns
7849:Sum type
7813:returns
7738:returns
7458:, where
7260:, where
7212:universe
7190:, where
6497:equality
5323:argument
4992:, where
4926:, where
4513:of type
4158:values (
4051:, 1970);
3934:exists.
2960:symmetry
2674:of type
1324:of type
991:variable
796:Montague
443:compiler
415:Isabelle
319:homotopy
288:Automath
270:May 2008
166:subtypes
14598:Binding
14353:Related
14150:Diagram
14048: (
14027:Hilbert
14012:Systems
14007:Theorem
13885:of the
13830:systems
13610:Formula
13605:Grammar
13521: (
13465:General
13178:Forcing
13163:Element
13083:Monadic
12858:paradox
12799:Theorem
12735:General
12676:Outline
11468:YouTube
10760:at the
10725:2266170
8690:and an
6792:, then
6648:, then
6391:to it.
5621:deduced
3436:For All
2954:define
2934:context
2810:) is:
2409:-style
2407:Gentzen
1859:context
1407:"Terms
993:, or a
894:formula
815:records
800:Barwise
786:, like
155:Russell
112:paradox
100:History
38:is the
15027:Monads
14574:Topics
14116:finite
13879:Skolem
13832:
13807:Theory
13775:Symbol
13765:String
13748:atomic
13625:ground
13620:closed
13615:atomic
13571:ground
13534:syntax
13430:binary
13357:domain
13274:Finite
13039:finite
12897:Logics
12856:
12804:Theory
11394:
11267:
11257:
11196:
11164:
11063:
10962:
10824:
10786:
10731:
10723:
10524:
10492:
10473:
10443:
10410:
10354:
10323:
10285:
10246:
10225:
10015:axioms
10009:Axioms
9649:axioms
9625:axioms
7312:vector
5589:, and
5399:, and
5032:, and
4556:simple
4256:, and
4049:Lambek
3536:Exists
2620:, and
2530:. The
2265:Terms
1347:"Type
1197:, and
798:1973,
792:nobody
394:, and
392:Matita
343:, and
14719:De se
14623:Focus
14581:Areas
14550:Scope
14106:Model
13854:Peano
13711:Proof
13551:Arity
13480:Naive
13367:image
13299:Fuzzy
13259:Empty
13208:union
13153:Class
12794:Model
12784:Lemma
12742:Axiom
12079:Logic
11913:tools
11332:(PDF)
11309:arXiv
11301:(PDF)
11265:S2CID
11237:arXiv
11207:(PDF)
11186:(PDF)
11098:(PDF)
11091:(PDF)
11017:(PDF)
11006:(PDF)
10729:S2CID
10721:JSTOR
10613:Julia
10583:Julia
10557:Notes
10535:(PDF)
10510:(PDF)
10365:(PDF)
10344:(PDF)
10190:Logic
10117:Minor
10079:Major
10066:(See
9645:rules
9587:Some
9093:, and
8850:, and
8353:union
8247:, and
7758:, and
7109:lists
6252:with
6089:, if
5353:: if
4558:types
4443:: If
3994:. By
3132:False
2133:Type
1377:type
985:as a
973:Terms
915:is a
808:is a
423:Mizar
378:NuPRL
356:Twelf
80:Most
53:as a
14229:Type
14032:list
13836:list
13813:list
13802:Term
13736:rank
13630:open
13524:list
13336:Maps
13241:sets
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