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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/logic">logic</a>. For the notion of the same name in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> see at <em><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</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> <ul> <li><a href='#SyntacticView'>Syntactic view</a></li> <li><a href='#SemanticView'>Semantic view</a></li> <li><a href='#CategoricalView'>Categorical view</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#classes_of_theories'>Classes of theories</a></li> <ul> <li><a href='#hierarchy_of_theories_cartesian_regular_coherent_geometric'>Hierarchy of theories: cartesian, regular, coherent, geometric</a></li> </ul> <li><a href='#abelian_theories'>Abelian theories</a></li> <li><a href='#SpecificExamples'>Specific examples</a></li> </ul> <li><a href='#models_for_a_theory'>Models for a theory</a></li> <ul> <li><a href='#settheoretic_models_for_a_firstorder_theory_in_syntactic_approach'>Set-theoretic models for a first-order theory in syntactic approach</a></li> <ul> <li><a href='#categorical_point_of_view_and_models_in_topoi'>Categorical point of view and models in topoi</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/mathematical+logic">mathematical logic</a>, a <em>theory</em> is a formal <a class="existingWikiWord" href="/nlab/show/language">language</a> used to precisely <a class="existingWikiWord" href="/nlab/show/axiom">axiomatize</a> a certain class of <a class="existingWikiWord" href="/nlab/show/models">models</a>.</p> <p>In principle also all other notions of <em>theory</em>, such as in the sense of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> should be special cases of this, but in practice of course there are many systems called “theories” which are not (yet) as fully formalized as in mathematical logic.</p> <h2 id="definition">Definition</h2> <p>There are several different viewpoint on theories:</p> <ul> <li> <p>The</p> <p><a href="#SyntacticView">syntactic view</a></p> <p>is that the theory itself consists of a set of <a class="existingWikiWord" href="/nlab/show/formulas">formulas</a> in the first order language <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lang</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lang(\Sigma)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/signature+%28in+logic%29">signature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. Classically, these formulas are assumed to have no <a class="existingWikiWord" href="/nlab/show/free+variables">free variables</a> (i.e. to be “sentences”), but in weaker logics that lack <a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a> it is better to take them to be formulas-in-<a class="existingWikiWord" href="/nlab/show/context">context</a>. One also sometimes considers the theory to include all logical consequences (aka theorems) of the axioms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, relative to (some specified) fragment of <a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a> — that is, to be “saturated” with respect to provability.</p> </li> <li> <p>The</p> <p><a href="#SemanticView">semantic view</a></p> <p>is that the theory is given by the class of its <a class="existingWikiWord" href="/nlab/show/model">model</a>s appropriate to that fragment of logic. Gödel’s <a class="existingWikiWord" href="/nlab/show/completeness+theorem">completeness theorem</a> is that a sentence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> is a theorem iff it is satisfied in every model.</p> </li> <li> <p>The</p> <p><a href="#CategoricalView">categorical view</a></p> <p>is that the logical formalism of a theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> can frequently be embodied in a <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> of terms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}}</annotation></semantics></math>, so that models of a theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> are identified with <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}} \to Set</annotation></semantics></math></div> <p>that preserve some (typically <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">property-like</a>) <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structures</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}}</annotation></semantics></math>, such as certain classes of <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s or of <a class="existingWikiWord" href="/nlab/show/limits">limits</a>, pertinent to the fragment of logic at hand. Then a completeness theorem would be the statement that the canonical map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>models</mi><mspace width="thickmathspace"></mspace><mi>F</mi><mspace width="thickmathspace"></mspace><mi>in</mi><mspace width="thickmathspace"></mspace><mi>Set</mi></mrow></munder><mi>Set</mi></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}} \to \prod_{models\ F\ in\ Set} Set</annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full faithful embedding</a> (one that preserves all relevant logical structure). For this reason, completeness theorems are also known as <em>embedding theorems</em>.</p> </li> </ul> <div class="query"> <p>Hm, is that the way it should be said?</p> </div> <p>In fact, the notion of model can be generalized away from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> to more general <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, namely those that have enough structure to “internalize” the fragment of logic at hand. From this very general point of view on model, the syntactic category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}}</annotation></semantics></math> is the generic or universal model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math>, and if we simply call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>𝒯</mi></msub></mrow><annotation encoding="application/x-tex">C_{\mathcal{T}}</annotation></semantics></math> the theory, then models and theories are placed on the same footing.</p> <p>In this article we mostly consider the categorical view on “theory”.</p> <h3 id="SyntacticView">Syntactic view</h3> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>In first-order <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, a <strong>theory</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> is presented by</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/signature+%28in+logic%29">signature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, specifying the allowed <a class="existingWikiWord" href="/nlab/show/type">type</a>s of variables and the <a class="existingWikiWord" href="/nlab/show/function">function</a>s and <a class="existingWikiWord" href="/nlab/show/relation">relation</a>s between these;</p> </li> <li> <p>a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a>s of <a class="existingWikiWord" href="/nlab/show/formula">formula</a>s in this signature, called the <strong>axioms</strong> of the theory: expressed in the first-order language <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Lang</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Lang(\Sigma)</annotation></semantics></math> with equality generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></p> </li> </ol> </div> <p>For instance (<a href="#Johnstone">Johnstone, def. D1.1.6</a>).</p> <h3 id="SemanticView">Semantic view</h3> <p>(…)</p> <h3 id="CategoricalView">Categorical view</h3> <p>(…)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></li> </ul> <p>(…)</p> <h2 id="examples">Examples</h2> <h3 id="classes_of_theories">Classes of theories</h3> <p>There are many different kinds of “theory” depending on the strength of the “logic”: a by-no-means complete list includes</p> <ul> <li> <p><span class="newWikiWord">equational logic<a href="/nlab/new/equational+logic">?</a></span>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/first-order+theory">first-order theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Horn+theory">Horn theory</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+logic">geometric logic</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+logic">regular logic</a>,</p> </li> <li> <p><span class="newWikiWord">exact logic<a href="/nlab/new/exact+logic">?</a></span>,</p> </li> <li> <p><span class="newWikiWord">extensive logic<a href="/nlab/new/extensive+logic">?</a></span>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+logic">coherent logic</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a> (aka pretopos logic),</p> </li> </ul> <p>and corresponding theories for these logics.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+algebraic+theory">generalized algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+algebraic+theory">commutative algebraic theory</a></p> </li> <li> <p>finitary algebraic theory: <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></li> </ul> </li> </ul> </li> </ul> <h4 id="hierarchy_of_theories_cartesian_regular_coherent_geometric">Hierarchy of theories: cartesian, regular, coherent, geometric</h4> <p>There is a hierarchy of theories that can be interpreted in the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of a hierarchy of types of <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. Since <a class="existingWikiWord" href="/nlab/show/predicates">predicates</a> in the internal logic are represented by <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a>, in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved. For instance:</p> <ul> <li> <p><strong>cartesian theories</strong> Since <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> 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">\top</annotation></semantics></math> are represented by <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a> and identities (top elements), these can be interpreted in any <a class="existingWikiWord" href="/nlab/show/lex+category">lex category</a>. Theories involving only these are <a class="existingWikiWord" href="/nlab/show/cartesian+theory">cartesian theories</a>.</p> </li> <li> <p><strong>regular theories</strong> Since <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> is represented by the <a class="existingWikiWord" href="/nlab/show/image">image</a> of a subobject, it can be interpreted in any <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a>. Theories involving only <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>, <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>, 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">\exists</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/regular+theory">regular theories</a>.</p> </li> <li> <p><strong>coherent theories</strong> Since <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> 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">\bot</annotation></semantics></math> are represented by <a class="existingWikiWord" href="/nlab/show/union">union</a> and <a class="existingWikiWord" href="/nlab/show/bottom+elements">bottom elements</a>, these can be interpreted in any <a class="existingWikiWord" href="/nlab/show/coherent+category">coherent category</a>. Theories which add these to regular logic are called <a class="existingWikiWord" href="/nlab/show/coherent+theory">coherent theories</a>.</p> </li> <li> <p><strong>geometric theories</strong> Finally, theories which also involve infinitary <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">\bigvee</annotation></semantics></math>, which is again represented by an infinitary union, can be interpreted in <a class="existingWikiWord" href="/nlab/show/infinitary+coherent+categories">infinitary coherent categories</a>, aka <em><a class="existingWikiWord" href="/nlab/show/geometric+categories">geometric categories</a></em>. These are <strong><a class="existingWikiWord" href="/nlab/show/geometric+theories">geometric theories</a></strong>.</p> </li> </ul> <p>Note that the axioms of one of these theories are actually of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>φ</mi><msub><mo>⊢</mo> <mover><mi>x</mi><mo stretchy="false">→</mo></mover></msub><mi>ψ</mi></mrow><annotation encoding="application/x-tex"> \varphi \vdash_{\vec{x}} \psi </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> are formulas involving only the specified connectives and quantifiers, <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> means entailment, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec{x}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/context">context</a>. Such an axiom can also be written as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>.</mo><mo stretchy="false">(</mo><mi>φ</mi><mo>⇒</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \forall \vec{x}. (\varphi \Rightarrow \psi) </annotation></semantics></math></div> <p>so that although <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> 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">\forall</annotation></semantics></math> are not strictly part of any of the above logics, they can be applied “once at top level.” In an axiom of this form for geometric logic, the formulas <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> which must be built out 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">\top</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>, <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><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo></mrow><annotation encoding="application/x-tex">\bigvee</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">\exists</annotation></semantics></math> are sometimes called <em>positive</em> formulas.</p> <h3 id="abelian_theories">Abelian theories</h3> <p>Interestingly, one form of logic which made an early appearance but is not ordinarily thought of as logic at all is the logic of <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>, which is characterized by certain exactness properties. Here a small abelian category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> can be thought of as a syntactic site for some “abelian theory”; models of the theory are exact additive functors with domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The classical models would in fact be exact additive functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">A \to Ab</annotation></semantics></math>, or exact additive functors to a category of modules. A “Freyd-Heron-Lubkin-Mitchell” embedding theorem is then a completeness theorem with respect to the classical models, and assures us that a statement in the language of abelian category theory is provable if and only if it is true when interpreted in any module category.</p> <h3 id="SpecificExamples">Specific examples</h3> <p>The simplest nontrivial theory is the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/theory+of+objects">theory of objects</a></li> </ul> <p>A still pretty simple but very nontrivial theory is</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+function+arithmetic">elementary function arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second+order+arithmetic">second order arithmetic</a></p> </li> </ul> <p>The theories</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(Cat)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/categories">categories</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>Lex</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(Lex)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/finitely+complete+categories">finitely complete categories</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>Topos</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(Topos)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/elementary+toposes">elementary toposes</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>ETCS</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(ETCS)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> (<a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a>)</p> </li> </ul> <p>are discussed at <em><a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a></em>.</p> <h2 id="models_for_a_theory">Models for a theory</h2> <h3 id="settheoretic_models_for_a_firstorder_theory_in_syntactic_approach">Set-theoretic models for a first-order theory in syntactic approach</h3> <p>The basic concept is of a structure for a first-order language <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>: a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> together with an interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. A theory is specified by a language and a set of sentences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>. An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>-structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <strong>model</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> if for every sentence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, its interpretation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>M</mi></msup></mrow><annotation encoding="application/x-tex">\phi^M</annotation></semantics></math> is true (“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> holds in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>”). We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is <strong>consistent</strong> or satisfiable (relative to the universe in which we do <a class="existingWikiWord" href="/nlab/show/model+theory">model theory</a>) if there exist at least one model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> (in our universe). Two theories, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T_2</annotation></semantics></math> are said to be <strong>equivalent</strong> if they have the same models.</p> <p>Given a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> of structures for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>, there is a theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(K)</annotation></semantics></math> consisting of all sentences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> which hold in every structure from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. Two structures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <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> are <strong>elementary equivalent</strong> (sometimes written by equality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">M=N</annotation></semantics></math>, sometimes said “elementarily equivalent”) if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Th</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(M)=Th(N)</annotation></semantics></math>, i.e. if they satisfy the same sentences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>. Any set of sentences which is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(K)</annotation></semantics></math> is called a <strong>set of axioms</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. A theory is said to be <strong>finitely axiomatizable</strong> if there exist a finite set of axioms for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>A theory is said to be <strong>complete</strong> if it is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(M)</annotation></semantics></math> for some structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> <h4 id="categorical_point_of_view_and_models_in_topoi">Categorical point of view and models in topoi</h4> <p>From the categorical point of view, for every theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> there exists a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math> – the <strong><a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub></mrow><annotation encoding="application/x-tex">C_T</annotation></semantics></math> – such that a model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">C_T \to \mathcal{T}</annotation></semantics></math> into some <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math>, satisfying certain conditions.</p> <p>For instance the syntactic categories of <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theories</a> are precisely those categories that have finite cartesian <a class="existingWikiWord" href="/nlab/show/product">product</a>s and in which every object is <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a> to a finite cartesian power <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x^n</annotation></semantics></math> of a distinguished object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. A model for a Lawvere theory is precisely a finite product preserving functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">C_T \to \mathcal{T}</annotation></semantics></math>.</p> <p>We say a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒯</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{T}_1 \to \mathcal{T}_2</annotation></semantics></math> of toposes (for instance a <a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a> or a <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>) <em>preserves a theory</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> if for every model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><msub><mi>𝒯</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">C_T \to \mathcal{T}_1</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{T}_1</annotation></semantics></math>, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>T</mi></msub><mo>→</mo><msub><mi>𝒯</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒯</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">C_T \to \mathcal{T}_1 \to \mathcal{T}_2</annotation></semantics></math> is a model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒯</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{T}_2</annotation></semantics></math>.</p> <p>For instance, every <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> preserves every <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> since, being a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, it preserves <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s, hence finite products.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+extension">conservative extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a>, <a class="existingWikiWord" href="/nlab/show/globular+theory">globular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theory+of+presheaf+type">theory of presheaf type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinal+analysis">ordinal analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-theory">2-theory</a>, <a class="existingWikiWord" href="/nlab/show/3-theory">3-theory</a></p> </li> </ul> <p><br /></p> <div> <p><strong>mathematical statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/theory">theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/judgement">judgement</a>, <a class="existingWikiWord" href="/nlab/show/assertion">assertion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypothesis">hypothesis</a>, <a class="existingWikiWord" href="/nlab/show/consequence">consequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom">axiom</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/assignment">assignment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/definition">definition</a> (<a class="existingWikiWord" href="/nlab/show/inductive+definition">inductive</a>, <a class="existingWikiWord" href="/nlab/show/coinductive+definition">coinductive</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lemma">lemma</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/theorem">theorem</a></p> </li> <li> <p><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/example">example</a>, <a class="existingWikiWord" href="/nlab/show/counterexample">counterexample</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conjecture">conjecture</a>, <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exercise">exercise</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analogy">analogy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautology">tautology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paradox">paradox</a></p> </li> </ul> </div> <p><br /></p> <h2 id="references">References</h2> <ul> <li>H. Jerome Keisler, <em>Fundamentals of model theory</em>, A2 in JOhn Barwise, Handbook of mathematical logic (pp.47-103), North Holland 1977</li> </ul> <p>A standard textbook reference for the <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> is section D of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></li> </ul> <p>A discussion of the relation between theories and their <a class="existingWikiWord" href="/nlab/show/syntactic+categories">syntactic categories</a> is at</p> <ul> <li>n-category cafe 2010: <a href="http://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html">what is a theory</a></li> </ul> <p>Other references include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mamuka+Jibladze">Mamuka Jibladze</a>, <a class="existingWikiWord" href="/nlab/show/Teimuraz+Pirashvili">Teimuraz Pirashvili</a>, <em>Cohomology of algebraic theories</em>, J. Algebra <strong>137</strong> (1991), no. 2, 253–296, <a href="http://dx.doi.org/10.1016/0021-8693(91)90093-N">doi</a></p> </li> <li> <p>wikipedia: <a href="http://en.wikipedia.org/wiki/List_of_first-order_theories">list of first-order theories</a>, <a href="http://en.wikipedia.org/wiki/Morley%27s_categoricity_theorem">Morley’s categoricity theorem</a>, <a href="http://en.wikipedia.org/wiki/Decidability_%28logic%29">Decidability (logic)</a>, <a href="http://en.wikipedia.org/wiki/Signature_%28logic%29">signature (logic)</a></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a> theories are specified with the</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/Gallina+specification+language">Gallina specification language</a></em>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 29, 2024 at 15:33:33. 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