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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the notion in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>. For the unrelated notion of the same name in <a class="existingWikiWord" href="/nlab/show/model+theory">model theory</a> see at <em><a class="existingWikiWord" href="/nlab/show/type+%28in+model+theory%29">type (in model theory)</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="type_theory">Type theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a></strong> <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>, <a class="existingWikiWord" href="/nlab/show/practical+foundations">practical foundations</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a>, <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/antecedents">antecedents</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/consequent">consequent</a>, <a class="existingWikiWord" href="/nlab/show/succedents">succedents</a></li> </ul> </li> </ul> <ol> <li><a class="existingWikiWord" href="/nlab/show/type+formation+rule">type formation rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+introduction+rule">term introduction rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/term+elimination+rule">term elimination rule</a></li> <li><a class="existingWikiWord" href="/nlab/show/computation+rule">computation rule</a></li> </ol> <p><strong><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent</a>, <a class="existingWikiWord" href="/nlab/show/intensional+type+theory">intensional</a>, <a class="existingWikiWord" href="/nlab/show/observational+type+theory">observational type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/calculus+of+constructions">calculus of constructions</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/syntax">syntax</a></strong> <a class="existingWikiWord" href="/nlab/show/object+language">object language</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a>, <a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>/<a class="existingWikiWord" href="/nlab/show/type">type</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a>/<a class="existingWikiWord" href="/nlab/show/proof">proof</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a> (<a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theorem">theorem</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></strong> = <br /> <strong><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/programs+as+proofs">programs as proofs</a></strong> +<strong><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation type theory/category theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/logic">logic</a></th><th><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> (<a class="existingWikiWord" href="/nlab/show/internal+logic+of+set+theory">internal logic</a> of)</th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th><th><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/predicate">predicate</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/family+of+sets">family of sets</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/display+morphism">display morphism</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof">proof</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/element">element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+element">generalized element</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/term">term</a>/<a class="existingWikiWord" href="/nlab/show/program">program</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+rule">cut rule</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/classifying+morphisms">classifying morphisms</a> / <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <a class="existingWikiWord" href="/nlab/show/display+maps">display maps</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/substitution">substitution</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/introduction+rule">introduction rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/counit">counit</a> for hom-tensor adjunction</td><td style="text-align: left;">lambda</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elimination+rule">elimination rule</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> for hom-tensor adjunction</td><td style="text-align: left;">application</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cut+elimination">cut elimination</a> for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">one of the <a class="existingWikiWord" href="/nlab/show/zigzag+identities">zigzag identities</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/beta+reduction">beta reduction</a></td></tr> <tr><td style="text-align: left;">identity elimination for <a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"></td><td style="text-align: left;">the other <a class="existingWikiWord" href="/nlab/show/zigzag+identity">zigzag identity</a> for hom-tensor adjunction</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/eta+conversion">eta conversion</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/true">true</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/singleton">singleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-2%29-truncated+object">(-2)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+0">h-level 0</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/false">false</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proposition">proposition</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncated+object">(-1)-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>, <a class="existingWikiWord" href="/nlab/show/mere+proposition">mere proposition</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product">product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/product+type">product type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> (into <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (into <a class="existingWikiWord" href="/nlab/show/subterminal+object">subterminal object</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (into <a class="existingWikiWord" href="/nlab/show/h-proposition">h-proposition</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+set">function set</a> into <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> into <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> into <a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subsingletons">subsingletons</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> (of family of <a class="existingWikiWord" href="/nlab/show/subterminal+objects">subterminal objects</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (of family of <a class="existingWikiWord" href="/nlab/show/h-propositions">h-propositions</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a></td><td style="text-align: left;">indexed <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> (<a class="existingWikiWord" href="/nlab/show/support">support</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> (<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a> of)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bijection+set">bijection set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+of+isomorphisms">object of isomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+type">equivalence type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+set">support set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/support+object">support object</a>/<a class="existingWikiWord" href="/nlab/show/%28-1%29-truncation">(-1)-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a>/<a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-image">n-image</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> into <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>/<a class="existingWikiWord" href="/nlab/show/n-truncation">n-truncation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-truncation+modality">n-truncation modality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/diagonal+function">diagonal function</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+subset">diagonal subset</a>/<a class="existingWikiWord" href="/nlab/show/diagonal+relation">diagonal relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>/<a class="existingWikiWord" href="/nlab/show/path+type">path type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completely+presented+set">completely presented set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>/<a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated object</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/h-level+2">h-level 2</a>-<a class="existingWikiWord" href="/nlab/show/type">type</a>/<a class="existingWikiWord" href="/nlab/show/set">set</a>/<a class="existingWikiWord" href="/nlab/show/h-set">h-set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> with <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28infinity%2C1%29-category">internal 0-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>/<a class="existingWikiWord" href="/nlab/show/setoid">setoid</a> with its <a class="existingWikiWord" href="/nlab/show/pseudo-equivalence+relation">pseudo-equivalence relation</a> an actual <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a>/<a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a>, <a class="existingWikiWord" href="/nlab/show/M-type">M-type</a></td></tr> <tr><td style="text-align: left;">higher <a class="existingWikiWord" href="/nlab/show/induction">induction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></td></tr> <tr><td style="text-align: left;">-</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncated</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-colimit">higher colimit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient+inductive+type">quotient inductive type</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduction">coinduction</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/limit">limit</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinductive+type">coinductive type</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/preset">preset</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type">type</a> without <a class="existingWikiWord" href="/nlab/show/identity+types">identity types</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+of+discourse">domain of discourse</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universe">universe</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modality">modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>, (<a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a>) <a class="existingWikiWord" href="/nlab/show/monad">monad</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a>, <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monad (in computer science)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></td><td style="text-align: left;"></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>, <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a>/<a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/proof+net">proof net</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a></td></tr> <tr><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/contraction+rule">contraction rule</a></td><td style="text-align: left;"></td><td style="text-align: left;">(absence of) <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/domain+specific+embedded+programming+language">domain specific embedded programming language</a></td></tr> </tbody></table> </div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+levels">homotopy levels</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-type+theory">2-type theory</a>, <a class="existingWikiWord" href="/michaelshulman/show/2-categorical+logic">2-categorical logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory+-+contents">homotopy type theory - contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/univalence">univalence</a>, <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a>, <a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HoTT+methods+for+homotopy+theorists">HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/semantics">semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/display+map">display map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+a+topos">internal logic of a topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mitchell-Benabou+language">Mitchell-Benabou language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic+of+an+%28%E2%88%9E%2C1%29-topos">internal logic of an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/type-theoretic+model+category">type-theoretic model category</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/type+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In modern <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, we understand that every <a class="existingWikiWord" href="/nlab/show/variable">variable</a> should have a <em>type</em>, or <em>domain of discourse</em> or be of some <em>sort</em>. For instance we say that if a variable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is constrained to be an <a class="existingWikiWord" href="/nlab/show/integer">integer</a> then “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is of integer type” or “of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>”. The usual formal expression from <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> for this – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math> – is then often written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo lspace="verythinmathspace">:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \colon \mathbb{Z}</annotation></semantics></math></p> <p>We speak of <em>typed logic</em> if this typing of variables is enforced by the <a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a>. In formulations of a <a class="existingWikiWord" href="/nlab/show/theory">theory</a> the types are often called <em>sorts</em>. More generally, <em><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></em> formalizes reasoning with such typed variables. See there for more</p> <p>(Untyped logic may be seen as simply a special case, in which there is only a single unique type. Thus, untyped logic has <em>one</em> type, not <em>no</em> type.)</p> <h2 id="definition">Definition</h2> <p>Reasoning with types is formalized in <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> (which in turn is formalized in a <a class="existingWikiWord" href="/nlab/show/logical+framework">logical framework</a> such as <a class="existingWikiWord" href="/nlab/show/Elf">Elf</a>).</p> <p>Behaviour of types is specified by a 4-step set of rules</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/type+formation">type formation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/term+introduction">term introduction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/term+elimination">term elimination</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/computation+rules">computation rules</a></p> </li> </ol> <h2 id="properties">Properties</h2> <p>Deep relations between <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/computer+science">computer science</a> relate types to other notions, such as <a class="existingWikiWord" href="/nlab/show/objects">objects</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a>. See at <em><a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a></em> for more on this.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+%28philosophy%29">category (philosophy)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibrant+type">fibrant type</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/type">type</a></strong>, <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>, <a class="existingWikiWord" href="/nlab/show/propositions+as+some+types">propositions as some types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inductive+type">inductive type</a>, <a class="existingWikiWord" href="/nlab/show/W-type">W-type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a>, <a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+type">pointed type</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a></strong>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, <a class="existingWikiWord" href="/nlab/show/Martin-L%C3%B6f+dependent+type+theory">Martin-Löf dependent type theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a>, <a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></strong>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a>, <a class="existingWikiWord" href="/nlab/show/heterogeneous+identity+type">heterogeneous identity type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a>, <a class="existingWikiWord" href="/nlab/show/connected+type">connected type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type">geometric homotopy type</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type">cohesive homotopy type</a></p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on April 27, 2017 at 13:01:46. See the <a href="/nlab/history/type" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/type" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/type/12" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/type" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/type" accesskey="S" class="navlink" id="history" rel="nofollow">History (12 revisions)</a> <a href="/nlab/show/type/cite" style="color: black">Cite</a> <a href="/nlab/print/type" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/type" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>