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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #10 to #11: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='type_theory'>Type theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>natural deduction</a></strong> <a class='existingWikiWord' href='/nlab/show/diff/metalanguage'>metalanguage</a>, <a class='existingWikiWord' href='/nlab/show/diff/practical+foundation+of+mathematics'>practical foundations</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/judgment'>judgement</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/judgment'>hypothetical judgement</a>, <a class='existingWikiWord' href='/nlab/show/diff/sequent'>sequent</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/antecedent'>antecedents</a> <math class='maruku-mathml' display='inline' id='mathml_a16efa5254d4794124cb3138766b1b5f0e1e9f61_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊢</mo></mrow><annotation encoding='application/x-tex'>\vdash</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/consequent'>consequent</a>, <a class='existingWikiWord' href='/nlab/show/diff/succedent'>succedents</a></li> </ul> </li> </ul> <ol> <li><a class='existingWikiWord' href='/nlab/show/diff/type+formation'>type formation rule</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>term introduction rule</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/term+elimination'>term elimination rule</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/conversion+rule'>computation rule</a></li> </ol> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent</a>, <a class='existingWikiWord' href='/nlab/show/diff/intensional+type+theory'>intensional</a>, <a class='existingWikiWord' href='/nlab/show/diff/observational+type+theory'>observational type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>)</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/calculus+of+constructions'>calculus of constructions</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/syntax'>syntax</a></strong> <a class='existingWikiWord' href='/nlab/show/diff/metalanguage'>object language</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/theory'>theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axiom</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proposition'>proposition</a>/<a class='existingWikiWord' href='/nlab/show/diff/type'>type</a> (<a class='existingWikiWord' href='/nlab/show/diff/propositions+as+types'>propositions as types</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/definition'>definition</a>/<a class='existingWikiWord' href='/nlab/show/diff/proof'>proof</a>/<a class='existingWikiWord' href='/nlab/show/diff/program'>program</a> (<a class='existingWikiWord' href='/nlab/show/diff/proofs+as+programs'>proofs as programs</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/theorem'>theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/computational+trilogy'>computational trinitarianism</a></strong> = \linebreak <strong><a class='existingWikiWord' href='/nlab/show/diff/propositions+as+types'>propositions as types</a></strong> +<strong><a class='existingWikiWord' href='/nlab/show/diff/proofs+as+programs'>programs as proofs</a></strong> +<strong><a class='existingWikiWord' href='/nlab/show/diff/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/diff/logic'>logic</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a> (<a class='existingWikiWord' href='/nlab/show/diff/internal+logic+of+set+theory'>internal logic</a> of)</th><th><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a></th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/proposition'>proposition</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/object'>object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type'>type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/proposition'>predicate</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/family+of+sets'>family of sets</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/display+map'>display morphism</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dependent+type'>dependent type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/proof'>proof</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/element'>element</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/generalized+element'>generalized element</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/term'>term</a>/<a class='existingWikiWord' href='/nlab/show/diff/program'>program</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cut+rule'>cut rule</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> of <a class='existingWikiWord' href='/nlab/show/diff/classifying+morphism'>classifying morphisms</a> / <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> of <a class='existingWikiWord' href='/nlab/show/diff/display+map'>display maps</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/substitution'>substitution</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>introduction rule</a> for <a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/unit+of+an+adjunction'>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/diff/term+elimination'>elimination rule</a> for <a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/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/diff/cut+rule'>cut elimination</a> for <a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td><td style='text-align: left;' /><td style='text-align: left;'>one of the <a class='existingWikiWord' href='/nlab/show/diff/triangle+identities'>zigzag identities</a> for hom-tensor adjunction</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/beta-reduction'>beta reduction</a></td></tr> <tr><td style='text-align: left;'>identity elimination for <a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td><td style='text-align: left;' /><td style='text-align: left;'>the other <a class='existingWikiWord' href='/nlab/show/diff/triangle+identities'>zigzag identity</a> for hom-tensor adjunction</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/eta-conversion'>eta conversion</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>/<a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>(-2)-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/contractible+type'>h-level 0</a>-<a class='existingWikiWord' href='/nlab/show/diff/type'>type</a>/<a class='existingWikiWord' href='/nlab/show/diff/unit+type'>unit type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/empty+type'>empty type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/proposition'>proposition</a>, <a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth value</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/subsingleton'>subsingleton</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/subterminal+object'>subterminal object</a>/<a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>(-1)-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>h-proposition</a>, <a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>mere proposition</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>logical conjunction</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/product+type'>product type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/disjunction'>disjunction</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> (<a class='existingWikiWord' href='/nlab/show/diff/support'>support</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> (<a class='existingWikiWord' href='/nlab/show/diff/truncation'>(-1)-truncation</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/sum+type'>sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/bracket+type'>bracket type</a> of)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/function+set'>function set</a> (into <a class='existingWikiWord' href='/nlab/show/diff/subsingleton'>subsingleton</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> (into <a class='existingWikiWord' href='/nlab/show/diff/subterminal+object'>subterminal object</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a> (into <a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>h-proposition</a>)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/function+set'>function set</a> into <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> into <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a> into <a class='existingWikiWord' href='/nlab/show/diff/empty+type'>empty type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a></td><td style='text-align: left;'>indexed <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a> (of family of <a class='existingWikiWord' href='/nlab/show/diff/subsingleton'>subsingletons</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dependent+product'>dependent product</a> (of family of <a class='existingWikiWord' href='/nlab/show/diff/subterminal+object'>subterminal objects</a>)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dependent+product+type'>dependent product type</a> (of family of <a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>h-propositions</a>)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantification</a></td><td style='text-align: left;'>indexed <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> (<a class='existingWikiWord' href='/nlab/show/diff/support'>support</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dependent+sum'>dependent sum</a> (<a class='existingWikiWord' href='/nlab/show/diff/truncation'>(-1)-truncation</a> of)</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/dependent+sum+type'>dependent sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/bracket+type'>bracket type</a> of)</td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/logical+equivalence'>logical equivalence</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/bijection+set'>bijection set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/object+of+isomorphisms'>object of isomorphisms</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/equivalence+type'>equivalence type</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/support+of+a+set'>support set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/support+object'>support object</a>/<a class='existingWikiWord' href='/nlab/show/diff/truncation'>(-1)-truncation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/bracket+type'>propositional truncation</a>/<a class='existingWikiWord' href='/nlab/show/diff/bracket+type'>bracket type</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/n-image'>n-image</a> of <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> into <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>/<a class='existingWikiWord' href='/nlab/show/diff/truncation'>n-truncation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/n-truncation+modality'>n-truncation modality</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/diagonal+function'>diagonal function</a>/<a class='existingWikiWord' href='/nlab/show/diff/diagonal+subset'>diagonal subset</a>/<a class='existingWikiWord' href='/nlab/show/diff/equality'>diagonal relation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path space object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity type</a>/<a class='existingWikiWord' href='/nlab/show/diff/path+type'>path type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/presentation+axiom'>completely presented set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete object</a>/<a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>0-truncated object</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/h-set'>h-level 2</a>-<a class='existingWikiWord' href='/nlab/show/diff/type'>type</a>/<a class='existingWikiWord' href='/nlab/show/diff/set'>set</a>/<a class='existingWikiWord' href='/nlab/show/diff/h-set'>h-set</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> with <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>internal 0-groupoid</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Bishop+set'>Bishop set</a>/<a class='existingWikiWord' href='/nlab/show/diff/setoid'>setoid</a> with its <a class='existingWikiWord' href='/nlab/show/diff/pseudo-equivalence+relation'>pseudo-equivalence relation</a> an actual <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence class</a>/<a class='existingWikiWord' href='/nlab/show/diff/quotient+set'>quotient set</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/quotient+object'>quotient</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/quotient+type'>quotient type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/induction'>induction</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/inductive+type'>inductive type</a>, <a class='existingWikiWord' href='/nlab/show/diff/W-type'>W-type</a>, <a class='existingWikiWord' href='/nlab/show/diff/M-type'>M-type</a></td></tr> <tr><td style='text-align: left;'>higher <a class='existingWikiWord' href='/nlab/show/diff/induction'>induction</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>higher colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/higher+inductive+type'>higher inductive type</a></td></tr> <tr><td style='text-align: left;'>-</td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/truncation'>0-truncated</a> <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>higher colimit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/quotient+inductive+type'>quotient inductive type</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/coinduction'>coinduction</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/coinductive+type'>coinductive type</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/preset'>preset</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type'>type</a> without <a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity types</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of <a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth values</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type+of+propositions'>type of propositions</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type'>domain of discourse</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/type+universe'>type universe</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/modality'>modality</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/closure+operator'>closure operator</a>, (<a class='existingWikiWord' href='/nlab/show/diff/idempotent+monad'>idempotent</a>) <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/modal+type+theory'>modal type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/monad+%28in+computer+science%29'>monad (in computer science)</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a></td><td style='text-align: left;' /><td style='text-align: left;'>(<a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed</a>) <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/linear+type+theory'>linear type theory</a>/<a class='existingWikiWord' href='/nlab/show/diff/quantum+computation'>quantum computation</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/proof+net'>proof net</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/string+diagram'>string diagram</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/quantum+circuit+diagram'>quantum circuit</a></td></tr> <tr><td style='text-align: left;'>(absence of) <a class='existingWikiWord' href='/nlab/show/diff/contraction+rule'>contraction rule</a></td><td style='text-align: left;' /><td style='text-align: left;'>(absence of) <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/no-cloning+theorem'>no-cloning theorem</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/synthetic+mathematics'>synthetic mathematics</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/domain+specific+embedded+programming+language'>domain specific embedded programming language</a></td></tr> </tbody></table> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy+level'>homotopy levels</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-type+theory'>2-type theory</a>, <a class='existingWikiWord' href='/michaelshulman/show/diff/2-categorical+logic' title='michaelshulman'>2-categorical logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory+-+contents'>homotopy type theory - contents</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/function+extensionality'>function extensionality</a>, <a class='existingWikiWord' href='/nlab/show/diff/internal+logic+of+an+%28infinity%2C1%29-topos'>internal logic of an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+type+theory'>cohesive homotopy type theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+type+theory'>directed homotopy type theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/HoTT+methods+for+homotopy+theorists'>HoTT methods for homotopy theorists</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/semantics'>semantics</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a>, <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/display+map'>display map</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Mitchell-B%C3%A9nabou+language'>internal logic of a topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Mitchell-B%C3%A9nabou+language'>Mitchell-Benabou language</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kripke-Joyal+semantics'>Kripke-Joyal semantics</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/internal+logic+of+an+%28infinity%2C1%29-topos'>internal logic of an (∞,1)-topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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> <h4 id='linguistics'>Linguistics</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/linguistics'>Linguistics</a></strong></p> <h2 id='general'>General</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/structuralism'>structuralism</a></li> </ul> <h3 id='philosophy_of_language'>Philosophy of Language</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/transcendental+grammar'>transcendental grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/language+game'>language game</a></p> </li> </ul> <h3 id='<a class='><a class='existingWikiWord' href='/nlab/show/diff/mathematical+linguistics'>Mathematical linguistics</a></h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/language'>language</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/transcendental+syntax'>transcendental syntax</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/syntax-semantics+duality'>syntax-semantics duality</a></p> </li> </ul> <h4 id='<a class='><a class='existingWikiWord' href='/nlab/show/diff/syntax'>Syntax</a></h4> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+grammar'>formal grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/regular+language'>regular language</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/context-free+grammar'>context-free grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pregroup+grammar'>pregroup grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorial+grammar'>categorial grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/combinatory+categorial+grammar'>combinatory categorial grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mildly+context-sensitive+grammar'>mildly context-sensitive grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/context-sensitive+grammar'>context-sensitive grammar</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dependency+grammar'>dependency grammar</a></p> </li> </ul> <h4 id='<a class='><a class='existingWikiWord' href='/nlab/show/diff/semantics'>Semantics</a></h4> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Montague+semantics'>Montague semantics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theoretic+methods+in+natural+language+semantics'>dependent type theoretic methods in natural language semantics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+compositional+distributional+semantics'>categorical compositional distributional semantics</a></p> </li> </ul> <h4 id='<span class='><span class='newWikiWord'>Pragmatics<a href='/nlab/new/pragmatics'>?</a></span></h4> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/monad+%28in+linguistics%29'>monad (in linguistics)</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/language+game'>language game</a></li> </ul> <h3 id='concepts'>Concepts</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compositionality'>compositionality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+quantifier'>generalized quantifier</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tree'>tree</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chomsky+hierarchy'>Chomsky hierarchy</a></p> </li> </ul> <h3 id='people'>People</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Noam+Chomsky'>Noam Chomsky</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/G%C3%BCnter+Hotz'>Günter Hotz</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Marcus+Kracht'>Marcus Kracht</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Joachim+Lambek'>Joachim Lambek</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Matilde+Marcolli'>Matilde Marcolli</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Richard+Montague'>Richard Montague</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Aarne+Ranta'>Aarne Ranta</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Mehrnoosh+Sadrzadeh'>Mehrnoosh Sadrzadeh</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stephan+Wannowski'>Stephan Wannowski</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/linguistics+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a></li><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/diff/formal+logic'>formal logic</a>, by <em>syntax</em> one means formal specification of the (<a class='existingWikiWord' href='/nlab/show/diff/language'>language</a> of) a <a class='existingWikiWord' href='/nlab/show/diff/theory'>theory</a> – this in contrast to <em><a class='existingWikiWord' href='/nlab/show/diff/semantics'>semantics</a></em>.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/linguistics'>linguistics</a> the meaning of syntax is similar, but a little different and narrower (does not include phonological, morphological and lexical levels of descriptions), see <a class='existingWikiWord' href='/nlab/show/diff/syntax+%28linguistics%29'>syntax (linguistics)</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/abstract+syntax+tree'>abstract syntax tree</a></p> </li></ins><ins class='diffins'> </ins><li> <p><a class='existingWikiWord' href='/nlab/show/diff/syntax-semantics+duality'>syntax - semantics duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/transcendental+syntax'>transcendental syntax</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/name+binding'>name binding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/language'>language</a>, <a class='existingWikiWord' href='/nlab/show/diff/theory'>theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/word'>word</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+%28computer+science%29'>string</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/linguistics'>linguistics</a>, <a class='existingWikiWord' href='/nlab/show/diff/syntax+%28linguistics%29'>syntax (linguistics)</a>.</p> </li> </ul> <h2 id='references'>References</h2> <p>See also</p> <ul> <li> <p>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Syntax_(logic)'>Syntax (logic)</a></em></p> </li> <li> <p>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Syntax'>Syntax</a></em></p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 30, 2022 at 07:28:14. See the <a href="/nlab/history/syntax" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/syntax" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/15738/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/syntax/10" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/syntax" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/syntax" accesskey="S" class="navlink" id="history" rel="nofollow">History (10 revisions)</a> <a href="/nlab/show/syntax/cite" style="color: black">Cite</a> <a href="/nlab/print/syntax" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/syntax" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>