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</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> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> </div> </div> <h1 id="implication">Implication</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#relations_between_the_definitions'>Relations between the definitions</a></li> </ul> <li><a href='#in_various_formalizations'>In various formalizations</a></li> <ul> <li><a href='#in_heyting_algebras'>In Heyting algebras</a></li> <li><a href='#in_natural_deduction'>In natural deduction</a></li> <li><a href='#in_type_theory'>In type theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>An <em>implication</em> may be either an <em>entailment</em> or a <em>conditional</em> statement; these are closely related but not quite the same thing.</p> <ol> <li> <p><strong>Entailment</strong> is a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> on <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> within a given <a class="existingWikiWord" href="/nlab/show/context">context</a> in a given <a class="existingWikiWord" href="/nlab/show/logic">logic</a>.</p> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> entails <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <em><a class="existingWikiWord" href="/nlab/show/syntax">syntactically</a></em>, written as a <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊢</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vdash q</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> can be <a class="existingWikiWord" href="/nlab/show/proof">proved</a> from the assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> entails <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> <em><a class="existingWikiWord" href="/nlab/show/semantics">semantically</a></em>, written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vDash q</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> holds in every <a class="existingWikiWord" href="/nlab/show/model">model</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> holds.</p> <p>(These relations are often equivalent, by various <span class="newWikiWord">soundness<a href="/nlab/new/soundness+theorem">?</a></span> and <a class="existingWikiWord" href="/nlab/show/completeness+theorem">completeness</a> theorems.)</p> </li> <li> <p>A <strong>conditional</strong> statement is the result of a <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> on <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a> within a given <a class="existingWikiWord" href="/nlab/show/context">context</a> in a given <a class="existingWikiWord" href="/nlab/show/logic">logic</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> are propositions in some context, then so is the conditional statement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \to q</annotation></semantics></math>, at least if the logic has a notion of conditional.</p> </li> </ol> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \to q</annotation></semantics></math> are all statements in the <em>object</em> language (the language that we are talking about), whereas the <a class="existingWikiWord" href="/nlab/show/hypothetical+judgements">hypothetical judgements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊢</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vdash q</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vDash q</annotation></semantics></math> are statements in the <em><a class="existingWikiWord" href="/nlab/show/metalanguage">metalanguage</a></em> (the language that we are using to talk about the <a class="existingWikiWord" href="/nlab/show/object+language">object language</a>).</p> <h3 id="relations_between_the_definitions">Relations between the definitions</h3> <p>Depending on what logic one is using, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \to q</annotation></semantics></math> might be anything, but it's probably not fair to consider it a conditional statement unless it is related to entailment as follows:</p> <div class="standout"> <p>If, in some context, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> entails <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> (either syntactically or semantically), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \to q</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/theorem">theorem</a> (syntactically) or a <a class="existingWikiWord" href="/nlab/show/tautology">tautology</a> (semantically) in that context, and conversely.</p> </div> <p>In particular, this holds for <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> and <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a>.</p> <p>You can think of entailment as being an <a class="existingWikiWord" href="/nlab/show/external+hom">external hom</a> (taking values in the poset of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>) and the conditional as being an <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> (taking values in the poset of <a class="existingWikiWord" href="/nlab/show/propositions">propositions</a>). In particular, we expect these to be related as in a <a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>→</mo><mi>r</mi><mo>⊢</mo><mo stretchy="false">(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>p</mi><mo>→</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> q \to r \vdash (p \to q) \to (p \to r) </annotation></semantics></math>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≡</mo><mo>⊤</mo><mo>→</mo><mi>p</mi></mrow><annotation encoding="application/x-tex"> p \equiv \top \to p </annotation></semantics></math>,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>⊢</mo><mi>p</mi><mo>→</mo><mi>p</mi></mrow><annotation encoding="application/x-tex"> \top \vdash p \to p </annotation></semantics></math>,</li> </ul> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math> is an appropriate constant statement (often satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊢</mo><mo>⊤</mo></mrow><annotation encoding="application/x-tex">p \vdash \top</annotation></semantics></math>, although not always, as in <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math>).</p> <p>Most kinds of logic used in practice have a notion of entailment from a <a class="existingWikiWord" href="/nlab/show/list">list</a> of multiple premises; then we expect entailment and the conditional to be related as in a closed <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>.</p> <p>Just as we may identify the internal and external hom in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, so we may identify the entailment and conditional of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>. In the <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>Lab, we tend to write this as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math>, a symbol that is variously used by other authors in place of <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>, <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>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math>.</p> <h2 id="in_various_formalizations">In various formalizations</h2> <h3 id="in_heyting_algebras">In Heyting algebras</h3> <p>Although <a class="existingWikiWord" href="/nlab/show/Heyting+algebras">Heyting algebras</a> were first developed as a way to discuss <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a>, they appear in other contexts; but their characterstic feature is that they have an operation analogous to the conditional operation in logic, usually called <strong>Heyting implication</strong> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math>. If you use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> and replace <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> above with the Heyting algebra's partial order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\leq</annotation></semantics></math>, then everything above applies.</p> <h3 id="in_natural_deduction">In natural deduction</h3> <p>In <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> the <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for implication are given as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \to Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}{\Gamma \vdash P \to Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \to Q \; \mathrm{true}}{\Gamma, P \; \mathrm{true} \vdash Q \; \mathrm{true}}</annotation></semantics></math></div> <h3 id="in_type_theory">In type theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> <ul> <li> <p>a <em>conditional statement</em> is, under <a class="existingWikiWord" href="/nlab/show/propositions-as-types">propositions-as-types</a> a <a class="existingWikiWord" href="/nlab/show/function+type">function type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>→</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \to q</annotation></semantics></math> (or the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> thereof).</p> </li> <li> <p>an <em>entailment</em> is a <a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a> or <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/function+type">function type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothetical+judgement">hypothetical judgement</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contrapositive">contrapositive</a></p> </li> </ul> <div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>symbol<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>in <a class="existingWikiWord" href="/nlab/show/logic">logic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mo>:</mo></mrow><annotation encoding="application/x-tex">\,:</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/type">typing</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><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><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/entailment">entailment</a> / <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/true">true</a> / <a class="existingWikiWord" href="/nlab/show/top">top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/false">false</a> / <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/equality">equality</a> / <a class="existingWikiWord" href="/nlab/show/apartness">apartness</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not \not</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/double+negation">negation of negation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (<a class="existingWikiWord" href="/nlab/show/implication">implication</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/product+type">product type</a> (<a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a> (<a class="existingWikiWord" href="/nlab/show/false">false</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a> (<a class="existingWikiWord" href="/nlab/show/true">true</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> (<a class="existingWikiWord" href="/nlab/show/equality">equality</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> (<a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></mrow><annotation encoding="application/x-tex">\prod</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (<a class="existingWikiWord" href="/nlab/show/universal+quantifier">universal quantifier</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+disjunction">additive disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+conjunction">additive conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\invamp</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+disjunction">multiplicative disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>!</mo></mrow><annotation encoding="application/x-tex">\;!</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+conjunction">exponential conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>?</mo></mrow><annotation encoding="application/x-tex">\;?</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+disjunction">exponential disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on November 13, 2023 at 14:51:35. 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