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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4455/#Item_16" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #71 to #72: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <blockquote> <p>‘Contrariwise,’ continued Tweedledee, ‘if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’</p> <p>(Lewis Carroll, <em><a href='http://sabian.org/looking_glass4.php'>Through the Looking Glass</a></em>)</p> </blockquote> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='foundations'>Foundations</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/foundation+of+mathematics'>foundations</a></strong></p> <h2 id='the_basis_of_it_all'>The basis of it all</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mathematical+logic'>mathematical logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/deductive+system'>deduction system</a>, <a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>natural deduction</a>, <a class='existingWikiWord' href='/nlab/show/diff/sequent+calculus'>sequent calculus</a>, <a class='existingWikiWord' href='/nlab/show/diff/lambda-calculus'>lambda-calculus</a>, <a class='existingWikiWord' href='/nlab/show/diff/judgment'>judgment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/simple+type+theory'>simple type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/collection'>collection</a>, <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a>, <a class='existingWikiWord' href='/nlab/show/diff/type'>type</a>, <a class='existingWikiWord' href='/nlab/show/diff/term'>term</a>, <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a>, <a class='existingWikiWord' href='/nlab/show/diff/element'>element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a>, <a class='existingWikiWord' href='/nlab/show/diff/judgmental+equality'>judgmental equality</a>, <a class='existingWikiWord' href='/nlab/show/diff/typal+equality'>typal equality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a>, <a class='existingWikiWord' href='/nlab/show/diff/size+issues'>size issues</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher-order+logic'>higher-order logic</a></p> </li> </ul> <h2 id='set_theory'> Set theory</h2> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a></strong></p> <ul> <li>fundamentals of set theory <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/propositional+logic'>propositional logic</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/predicate+logic'>first-order logic</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/typed+predicate+logic'>typed predicate logic</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/membership+relation'>membership relation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/propositional+equality'>propositional equality</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/set'>set</a>, <a class='existingWikiWord' href='/nlab/show/diff/element'>element</a>, <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a>, <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a>, <a class='existingWikiWord' href='/nlab/show/diff/small+set'>small set</a>, <a class='existingWikiWord' href='/nlab/show/diff/large+set'>large set</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/material+set+theory'>material set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/membership+relation'>membership relation</a>, <a class='existingWikiWord' href='/nlab/show/diff/propositional+equality'>propositional equality</a>, <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+extensionality'>axiom of extensionality</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/pairing+structure'>pairing structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+pairing'>axiom of pairing</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/union+structure'>union structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+union'>axiom of union</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/powerset+structure'>powerset structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/power+set'>axiom of power sets</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/natural+numbers+structure'>natural numbers structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+infinity'>axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/first-order+set+theory'>first-order set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/unsorted+set+theory'>unsorted set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simply+sorted+set+theory'>simply sorted set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/unsorted+set+theory'>one-sorted set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/two-sorted+set+theory'>two-sorted set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/three-sorted+set+theory'>three-sorted set theory</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/dependently+sorted+set+theory'>dependently sorted set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/structurally+presented+set+theory'>structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/material+set+theory'>material set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/ZFC'>ZFC</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/ZFA'>ZFA</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Mostowski+set+theory'>Mostowski set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/New+Foundations'>New Foundations</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/structural+set+theory'>structural set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/categorical+set+theory'>categorical set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/ETCS'>ETCS</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fully+formal+ETCS'>fully formal ETCS</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/ETCS+with+elements'>ETCS with elements</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+on+ETCS+I'>Trimble on ETCS I</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+on+ETCS+II'>Trimble on ETCS II</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+on+ETCS+III'>Trimble on ETCS III</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/structural+ZFC'>structural ZFC</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/allegorical+set+theory'>allegorical set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/SEAR'>SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/class+theory'>class-set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/class'>class</a>, <a class='existingWikiWord' href='/nlab/show/diff/class'>proper class</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/universal+class'>universal class</a>, <a class='existingWikiWord' href='/nlab/show/diff/universe'>universe</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/category+of+classes'>category of classes</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/category+with+class+structure'>category with class structure</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/constructive+set+theory'>constructive set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+set+theory'>algebraic set theory</a></li> </ul> <h2 id='foundational_axioms'>Foundational axioms</h2> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/foundational+axiom'>foundational</a> <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axioms</a></strong></p> <ul> <li> <p>basic constructions:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>axiom of cartesian products</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>axiom of disjoint unions</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/empty+set'>axiom of the empty set</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/subset+collection'>axiom of fullness</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/function+set'>axiom of function sets</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/power+set'>axiom of power sets</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/quotient+set'>axiom of quotient sets</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/material+set+theory'>material axioms</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+extensionality'>axiom of extensionality</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+foundation'>axiom of foundation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+foundation'>axiom of anti-foundation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Mostowski%27s+collapsing+lemma'>Mostowski's axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+pairing'>axiom of pairing</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/transitive+set'>axiom of transitive closure</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+union'>axiom of union</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/structural+set+theory'>structural axioms</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+materialization'>axiom of materialization</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theoretic axioms</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+set+truncation'>axioms of set truncation</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/uniqueness+of+identity+proofs'>uniqueness of identity proofs</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+K'>axiom K</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/boundary+separation'>boundary separation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/extensional+type+theory'>equality reflection</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+circle+type+localization'>axiom of circle type localization</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theoretic axioms</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/univalence+axiom'>univalence axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axioms of choice</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/countable+choice'>axiom of countable choice</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/dependent+choice'>axiom of dependent choice</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>axiom of excluded middle</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/choice+operator'>axiom of existence</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+multiple+choice'>axiom of multiple choice</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Markov%27s+principle'>Markov's axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/presentation+axiom'>presentation axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/small+cardinality+selection+axiom'>small cardinality selection axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/small+violations+of+choice'>axiom of small violations of choice</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/WISC'>axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/large+cardinal'>large cardinal axioms</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+infinity'>axiom of infinity</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+universe'>axiom of universes</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/regular+extension+axiom'>regular extension axiom</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/inaccessible+cardinal'>inaccessible cardinal</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/measurable+cardinal'>measurable cardinal</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/elementary+embedding'>elementary embedding</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/supercompact+cardinal'>supercompact cardinal</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Vop%C4%9Bnka%27s+principle'>Vopěnka's principle</a></li> </ul> </li> <li> <p>strong axioms</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+separation'>axiom of separation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+replacement'>axiom of replacement</a></li> </ul> </li> <li> <p>further</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/reflection+principle'>reflection principle</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tight+apartness+relation'>axiom of tight apartness</a></p> </li> </ul> <h2 id='removing_axioms'>Removing axioms</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/predicative+mathematics'>predicative mathematics</a></li> </ul> <div> <p> <a href='/nlab/edit/foundations+-+contents'>Edit this sidebar</a> </p> </div></div> <h4 id='category_theory'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(0,1)</annotation></semantics></math>-Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category+theory'>(0,1)-category theory</a></strong>: <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a>, <a class='existingWikiWord' href='/nlab/show/diff/order+theory'>order theory</a></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category'>(0,1)-category</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+preorders+and+%280%2C1%29-categories'>relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preorder'>proset</a>, <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>partially ordered set</a> (<a class='existingWikiWord' href='/nlab/show/diff/direction'>directed set</a>, <a class='existingWikiWord' href='/nlab/show/diff/total+order'>total order</a>, <a class='existingWikiWord' href='/nlab/show/diff/linear+order'>linear order</a>)</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/top'>top</a>, <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom</a>, <a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monotone+function'>monotone function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a>, <a class='existingWikiWord' href='/nlab/show/diff/interval'>interval</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lattice'>lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/semilattice'>semilattice</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>logical conjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>and</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjunction'>logical disjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjunction'>or</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+element'>compact element</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/poset+of+subobjects'>lattice of subobjects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complete+lattice'>complete lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+lattice'>algebraic lattice</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/distributive+lattice'>distributive lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/completely+distributive+lattice'>completely distributive lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/canonical+extension'>canonical extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperdoctrine'>hyperdoctrine</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/first-order+hyperdoctrine'>first-order</a>, <a class='existingWikiWord' href='/nlab/show/diff/Boolean+hyperdoctrine'>Boolean</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherent+hyperdoctrine'>coherent</a>, <a class='existingWikiWord' href='/nlab/show/diff/tripos'>tripos</a></li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heyting+algebra'>Heyting algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/regular+element'>regular element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a></li> </ul> </div> <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_39a0238135deecf9c07a36390a26276b42284593_2' 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> </div> </div> <h1 id='logic'>Logic</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#MathematicalLogic'>Mathematical logic</a><ul><li><a href='#classical_subfields'>Classical subfields</a></li><li><a href='#CategoricalLogic'>Categorical logic</a></li></ul></li><li><a href='#entries_on_logic'>Entries on logic</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#General'>General</a></li><li><a href='#ReferencesCategoricalLogic'>On categorical logic</a></li><li><a href='#logic_in_natural_languages'>Logic in natural languages</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>Traditionally, as a discipline, <strong>logic</strong> is the study of correct methods of reasoning. Logicians have principally studied <a class='existingWikiWord' href='/nlab/show/diff/deduction'>deduction</a>, the process of passing from premises to conclusion in such a way that the <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>truth</a> of the former necessitates the truth of the latter. In other words, deductive logic studies what it is for an argument to be <em>valid</em>. A second branch of logic studies <a class='existingWikiWord' href='/nlab/show/diff/inductive+reasoning'>induction</a>, reasoning about how to assess the plausibility of general propositions from observations of their instances. This has often been done in terms of <a class='existingWikiWord' href='/nlab/show/diff/probability+theory'>probability theory</a>, particularly <a class='existingWikiWord' href='/nlab/show/diff/Bayesian+reasoning'>Bayesian</a>.</p> <p>Some <a class='existingWikiWord' href='/nlab/show/diff/philosophy'>philosophers</a>, notably <a class='existingWikiWord' href='/nlab/show/diff/Charles+Sanders+Peirce'>Charles Peirce</a>, considered there to be third variety of reasoning for logic to study, namely, <a class='existingWikiWord' href='/nlab/show/diff/abductive+reasoning'>abduction</a>. This is a process whereby one reasons to the truth of an explanation from its ability to account for what is observed. It is therefore sometimes also known as <em>inference to the best explanation</em>. At least some aspects of this can also be studied using Bayesian probability.</p> <p>Deductive logic is the best developed of the branches. For centuries, treatments of the <a class='existingWikiWord' href='/nlab/show/diff/syllogism'>syllogism</a> were at the forefront of the discipline. In the nineteenth century, however, spurred largely by the needs of <a class='existingWikiWord' href='/nlab/show/diff/mathematics'>mathematics</a>, in particular the need to handle <a class='existingWikiWord' href='/nlab/show/diff/relation'>relations</a> and <a class='existingWikiWord' href='/nlab/show/diff/quantification'>quantifiers</a>, a new logic emerged, known today as <a class='existingWikiWord' href='/nlab/show/diff/predicate+logic'>predicate logic</a>.</p> <p>As we said above, logic is traditionally concerned with <em>correct</em> methods of reasoning, and philosophers (and others) have had much to say <em>prescriptively</em> about logic. However, one can also study logic <em>descriptively</em>, taking it to be the study of methods of reasoning, without attempting to determine whether these methods are correct. One may study <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive logic</a>, or a <a class='existingWikiWord' href='/nlab/show/diff/substructural+logic'>substructural logic</a>, without saying that it should be adopted. Also psychologists study how people actually reason rapidly in situations without full information, such as by the <a href='http://fastandfrugal.com/'>fast and frugal</a> approach.</p> <p>A <strong>logic</strong> is a specific method of reasoning. There are several ways to <em>formalise</em> a logic as a <a class='existingWikiWord' href='/nlab/show/diff/mathematics'>mathematical</a> object; see at <em><a href='#MathematicalLogic'>Mathematical Logic</a></em> below.</p> <h2 id='MathematicalLogic'>Mathematical logic</h2> <p><em>Mathematical logic</em> or <em>symbolic logic</em> is the study of logic and <a class='existingWikiWord' href='/nlab/show/diff/foundation+of+mathematics'>foundations</a> of mathematics as, or via, formal systems – <em><a class='existingWikiWord' href='/nlab/show/diff/theory'>theories</a></em> – such as <a class='existingWikiWord' href='/nlab/show/diff/predicate+logic'>first-order logic</a> or <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>.</p> <h3 id='classical_subfields'>Classical subfields</h3> <p>The classical subfields of mathematical logic are</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+theory'>model theory</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/computability'>recursion theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proof+theory'>proof theory</a></p> </li> </ul> <h3 id='CategoricalLogic'>Categorical logic</h3> <p>By a convergence and unification of concepts that has been named <em><a class='existingWikiWord' href='/nlab/show/diff/computational+trilogy'>computational trinitarianism</a></em>, mathematical logic is equivalently incarnated in</p> <ol> <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/category+theory'>category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/computer+science'>programming theory</a></p> </li> </ol> <p>The logical theory that is specified by and specifies a given <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> – called its <em><a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a></em>, see there for more details and also see <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal language</a>, <a class='existingWikiWord' href='/nlab/show/diff/syntactic+category'>syntactic category</a>. – is the one</p> <ul> <li> <p>whose <a class='existingWikiWord' href='/nlab/show/diff/type'>types</a> are the <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>whose <a class='existingWikiWord' href='/nlab/show/diff/context'>contexts</a> are the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice categories</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_{/A}</annotation></semantics></math>;</p> </li> <li> <p>whose <a class='existingWikiWord' href='/nlab/show/diff/proposition'>propositions</a> in context are the <a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>(-1)-truncated objects</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy='false'>/</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_{/A}</annotation></semantics></math>;</p> </li> <li> <p>whose <a class='existingWikiWord' href='/nlab/show/diff/proof'>proofs</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊢</mo><mi>PhiIsTrue</mi><mo>:</mo><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>A \vdash PhiIsTrue : \phi </annotation></semantics></math> are the <a class='existingWikiWord' href='/nlab/show/diff/generalized+element'>generalized elements</a> of <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math>.</p> </li> </ul> <p>Hence pure mathematical logic in the sense of the study of <a class='existingWikiWord' href='/nlab/show/diff/proposition'>propositions</a> is identified with <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category+theory'>(0,1)-category theory</a>: where one concentrates only on <a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>(-1)-truncated objects</a>. Genuine <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>, which is about <a class='existingWikiWord' href='/nlab/show/diff/truncated+object'>0-truncated objects</a>, is the home for logic and <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a>, or rather <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>, the 0-truncated objects being the <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>/<a class='existingWikiWord' href='/nlab/show/diff/type'>types</a>/<a class='existingWikiWord' href='/nlab/show/diff/h-set'>h-sets</a>.</p> <p>For instance,</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> and <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a>, <a class='existingWikiWord' href='/nlab/show/diff/exponential+map'>exponentials</a>, and <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifiers</a> belong to the <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>;</p> </li> <li> <p>while their (-1)-truncation, in this order: <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersections</a>/(<a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>and</a>), <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a>(<a class='existingWikiWord' href='/nlab/show/diff/disjunction'>or</a>), <a class='existingWikiWord' href='/nlab/show/diff/implication'>implications</a>, and <a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifiers</a>, belong to the logic.</p> </li> </ul> <p>Generally, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, which is about untruncated objects, is the home for logic and types with a <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive</a> notion of <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a>, the <a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity types</a> in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>.</p> <p>See also at <em><a href='model%20theory#CategoricalModelTheory'>categorical model theory</a></em>.</p> <h2 id='entries_on_logic'>Entries on logic</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+model+for+modal+logics'>Boolean algebra with operators</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/boolean+domain'>boolean domain</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/boolean+function'>boolean function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/boolean-valued+function'>boolean-valued function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+logic'>classical logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/context'>context</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dependence+logic'>dependence logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+theory'>geometric logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heyting+algebra'>Heyting algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher-order+logic'>higher-order logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intuitionistic+logic'>intuitionistic logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/logicality+and+invariance'>logicality and invariance</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/minimal+logic'>minimal logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modal+logic'>modal logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+logic'>adjoint logic</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+model+for+modal+logics'>algebraic models for modal logics</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/closure+algebra'>closure algebra</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/monadic+algebra'>monadic algebra</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/temporal+algebra'>temporal algebra</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/arrow+structure'>arrow logic</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/epistemic+modal+logic'>epistemic logic</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/K+modal+logic'>the logic K(m)</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/the+logic+T%28m%29'>the logic T(m)</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/S4+modal+logic'>the logic S4(m)</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/S5%28m%29+modal+logic'>the logic S5(m)</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/Kripke+frame'>frame (modal logic)</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+model+for+modal+logics'>geometric models for modal logics</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/modal+type+theory'>modal type theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/temporal+logic'>temporal logic</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peirce%27s+law'>Peirce's law</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proposition'>predicate</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/predicate+logic'>predicate logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/predicative+mathematics'>predicative mathematics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/propositional+logic'>propositional logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/team+logic'>team logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth value</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type'>type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a></p> </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mathematical+logic'>mathematical logic</a></p> </li> <li> <p><span class='newWikiWord'>logic symbols<a href='/nlab/new/logic+symbols'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a>, <strong>logic</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-type+theory'>2-type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-logic'>2-logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>(∞,1)-type theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-logic'>(∞,1)-logic</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/objective+and+subjective+logic'>objective and subjective logic</a></p> </li> </ul> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math>symbol<math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math>in <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∈</mo></mrow><annotation encoding='application/x-tex'>\in</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/element'>element</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /><mo>:</mo></mrow><annotation encoding='application/x-tex'>\,:</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/type'>typing</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊢</mo></mrow><annotation encoding='application/x-tex'>\vdash</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/implication'>entailment</a> / <a class='existingWikiWord' href='/nlab/show/diff/sequent'>sequent</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊤</mo></mrow><annotation encoding='application/x-tex'>\top</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a> / <a class='existingWikiWord' href='/nlab/show/diff/top'>top</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊥</mo></mrow><annotation encoding='application/x-tex'>\bot</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a> / <a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇒</mo></mrow><annotation encoding='application/x-tex'>\Rightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇔</mo></mrow><annotation encoding='application/x-tex'>\Leftrightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+equivalence'>logical equivalence</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo></mrow><annotation encoding='application/x-tex'>\not</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≠</mo></mrow><annotation encoding='application/x-tex'>\neq</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a> of <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a> / <a class='existingWikiWord' href='/nlab/show/diff/apartness+relation'>apartness</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∉</mo></mrow><annotation encoding='application/x-tex'>\notin</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a> of <a class='existingWikiWord' href='/nlab/show/diff/element'>element</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> <math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding='application/x-tex'>\not \not</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/double+negation'>negation of negation</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo></mrow><annotation encoding='application/x-tex'>\exists</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantification</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∀</mo></mrow><annotation encoding='application/x-tex'>\forall</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∧</mo></mrow><annotation encoding='application/x-tex'>\wedge</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>logical conjunction</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∨</mo></mrow><annotation encoding='application/x-tex'>\vee</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/disjunction'>logical disjunction</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><strong>symbol</strong></td><td style='text-align: left;'><strong>in <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> (<a class='existingWikiWord' href='/nlab/show/diff/propositions+as+types'>propositions as types</a>)</strong></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a> (<a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>×</mo></mrow><annotation encoding='application/x-tex'>\times</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/product+type'>product type</a> (<a class='existingWikiWord' href='/nlab/show/diff/conjunction'>conjunction</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo></mrow><annotation encoding='application/x-tex'>+</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/sum+type'>sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/disjunction'>disjunction</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/empty+type'>empty type</a> (<a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/unit+type'>unit type</a> (<a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity type</a> (<a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+types'>equivalence of types</a> (<a class='existingWikiWord' href='/nlab/show/diff/logical+equivalence'>logical equivalence</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo></mrow><annotation encoding='application/x-tex'>\sum</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/dependent+sum+type'>dependent sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantifier</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo></mrow><annotation encoding='application/x-tex'>\prod</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/dependent+product+type'>dependent product type</a> (<a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantifier</a>)</td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><strong>symbol</strong></td><td style='text-align: left;'><strong>in <a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a></strong></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊸</mo></mrow><annotation encoding='application/x-tex'>\multimap</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/linear+implication'>linear implication</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+conjunction'>multiplicative conjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊕</mo></mrow><annotation encoding='application/x-tex'>\oplus</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/additive+disjunction'>additive disjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>&</mi></mrow><annotation encoding='application/x-tex'>\&</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>additive conjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⅋</mo></mrow><annotation encoding='application/x-tex'>\invamp</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+disjunction'>multiplicative disjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mo>!</mo></mrow><annotation encoding='application/x-tex'>\;!</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%21-modality'>exponential conjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mo>?</mo></mrow><annotation encoding='application/x-tex'>\;?</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%3F-modality'>exponential disjunction</a><math class='maruku-mathml' display='inline' id='mathml_39a0238135deecf9c07a36390a26276b42284593_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <h2 id='references'>References</h2> <h3 id='General'>General</h3> <p>For centuries, logic was <a class='existingWikiWord' href='/nlab/show/diff/syllogism'>Aristotle's logic</a> of <a class='existingWikiWord' href='/nlab/show/diff/deduction'>deduction</a> by <a class='existingWikiWord' href='/nlab/show/diff/syllogism'>syllogism</a>. In the 19th century the idea of <a class='existingWikiWord' href='/nlab/show/diff/objective+and+subjective+logic'>objective logic</a> as <a class='existingWikiWord' href='/nlab/show/diff/metaphysics'>metaphysics</a> was influential</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Georg+Hegel'>Hegel</a>, <em>Wissenschaft der Logik</em> ( <em><a class='existingWikiWord' href='/nlab/show/diff/Science+of+Logic'>Science of Logic</a></em> )</li> </ul> <p>This “old logic” was famously criticized</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+Russell'>Bertrand Russell</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Logic+as+the+Essence+of+Philosophy'>Logic as the Essence of Philosophy</a></em>, 1914</li> </ul> <p>as opposed to the “new logic” of <a class='existingWikiWord' href='/nlab/show/diff/Giuseppe+Peano'>Peano</a> and <a class='existingWikiWord' href='/nlab/show/diff/Gottlob+Frege'>Frege</a>, contemporary <a class='existingWikiWord' href='/nlab/show/diff/predicate+logic'>predicate logic</a>.</p> <p>Textbooks on mathematical logic:</p> <ul> <li id='HilbertAckermann59'> <p><a class='existingWikiWord' href='/nlab/show/diff/David+Hilbert'>David Hilbert</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilhelm+Ackermann'>Wilhelm Ackermann</a>, <em>Grundzüge der Theoretischen Logik</em>, 4th ed. Springer Heidelberg 1959 [1928]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Clarence+I.+Lewis'>Clarence I. Lewis</a>, <a class='existingWikiWord' href='/nlab/show/diff/Cooper+H.+Langford'>Cooper H. Langford</a>, <em>Symbolic Logic</em> (1932), Dover (2000) [App. II pdf]</p> <blockquote> <p>(giving the first modern formalizations of <a class='existingWikiWord' href='/nlab/show/diff/modal+logic'>modal logic</a>)</p> </blockquote> </li> <li id='BellMach77'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Lane+Bell'>John Lane Bell</a>, M. Machover, <em>A Course in Mathematical Logic</em>, North-Holland Amsterdam 1977. (ch. 10,§5) (<a href='https://www.elsevier.com/books/a-course-in-mathematical-logic/bell/978-0-7204-2844-5'>ISBN:9780720428445</a>)</p> </li> <li id='Girard89'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean-Yves+Girard'>Jean-Yves Girard</a> (translated and with appendiced by <a class='existingWikiWord' href='/nlab/show/diff/Paul+Taylor'>Paul Taylor</a> and <a class='existingWikiWord' href='/nlab/show/diff/Yves+Lafont'>Yves Lafont</a>), <em>Proofs and Types</em>, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), <a href='http://www.paultaylor.eu/stable/Proofs+Types.html'>webpage</a>, <a href='https://www.paultaylor.eu/stable/prot.pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Open+Logic+Project'>Open Logic Project</a>, <em>The open logic text</em> (<a href='http://people.ucalgary.ca/~rzach/static/open-logic/open-logic-complete.pdf'>pdf</a>)</p> </li> </ul> <h3 id='ReferencesCategoricalLogic'>On categorical logic</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/William+Lawvere'>William Lawvere</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Adjointness+in+Foundations'>Adjointness in Foundations</a></em>, Dialectica 23 (1969), 281-296</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/William+Lawvere'>William Lawvere</a> <em>Equality in hyperdoctrines and comprehension schema as an adjoint functor</em>. In A. Heller, ed., <em>Proc. New York Symp. on Applications of Categorical Algebra</em>, pp. 1–14. AMS, 1970. (<a class='existingWikiWord' href='/nlab/files/LawvereComprehension.pdf' title='pdf'>pdf</a>)</p> </li> <li id='MakkaiReyes77'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Makkai'>Michael Makkai</a> and <a class='existingWikiWord' href='/nlab/show/diff/Gonzalo+E.+Reyes'>Gonzalo Reyes</a>, <em>First Order Categorical Logic: Model-theoretical methods in the theory of topoi and related categories</em>, Lecture Notes in Math. <strong>611</strong>, Springer-Verlag, 1977. re-written by Francisco Marmolejo in 2018 (<a href='https://reyes-reyes.com/2018/04/02/first-order-categorical-logic-2/'>web announcement</a>, <a href='https://marieetgonzalo.files.wordpress.com/2018/04/makkai-reyes-book.pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Cartier'>Pierre Cartier</a>, <em>Logique, catégories et faisceaux</em>, Séminaire Bourbaki, 20 (1977-1978), Exp. No. 513, 24 p. (<a href='http://www.numdam.org/item?id=SB_1977-1978__20__123_0'>numdam</a>)</p> </li> <li id='LambekScott86'> <p><a class='existingWikiWord' href='/nlab/show/diff/Joachim+Lambek'>Joachim Lambek</a>, <a class='existingWikiWord' href='/nlab/show/diff/Philip+Scott'>Philip J. Scott</a>, <em>Introduction to higher order categorical logic</em>, Cambridge Studies in Advanced Mathematics <strong>7</strong> (1986) [[ISBN: 0-521-24665-2](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534), <a href='https://raw.githubusercontent.com/Mzk-Levi/texts/master/Lambek%20J.%2C%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Joachim+Lambek'>Jim Lambek</a>, <a class='existingWikiWord' href='/nlab/show/diff/Philip+Scott'>Phil Scott</a>, <em>Reflections on the categorical foundations of mathematics</em> (<a href='https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ieke+Moerdijk'>Ieke Moerdijk</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sheaves+in+Geometry+and+Logic'>Sheaves in Geometry and Logic</a></em></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bart+Jacobs'>Bart Jacobs</a>, <em>Categorical Logic and Type Theory</em>, (1999) Elsevier</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Bell'>John Bell</a>, <em>The development of categorical logic</em> (<a href='http://publish.uwo.ca/~jbell/catlogprime.pdf'>pdf</a>)</p> </li> <li id='MarquisReyes09'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean-Pierre+Marquis'>Jean-Pierre Marquis</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gonzalo+E.+Reyes'>Gonzalo Reyes</a>, (2009) <em>The History of Categorical Logic 1963-1977</em> (<a href='https://www.webdepot.umontreal.ca/Usagers/marquisj/MonDepotPublic/HistofCatLog.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/MarquisReyes_CategoricalLogic.pdf' title='pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean-Yves+Girard'>Jean-Yves Girard</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/The+Blind+Spot'>Lectures on Logic</a></em>, European Mathematical Society 2011</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em>Categorical Logic</em>, Lecture notes 2018 (<a href='https://www.math.ias.edu/~lurie/278x.html'>web</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Steve+Awodey'>Steve Awodey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Andrej+Bauer'>Andrej Bauer</a>: <em>Introduction to categorical logic</em>, lecture notes for a course at Carnegie Mellon University (January-April 2024) [[web](https://awodey.github.io/catlog/notes/), 0:<a href='https://awodey.github.io/catlog/notes/catlog0.pdf'>pdf</a>, 1:<a href='https://awodey.github.io/catlog/notes/catlog1.pdf'>pdf</a>, 2:<a href='https://awodey.github.io/catlog/notes/catlog2.pdf'>pdf</a>, 3:<a href='https://awodey.github.io/catlog/notes/catlog3.pdf'>pdf</a>, 4:<a href='https://awodey.github.io/catlog/notes/catlog4.pdf'>pdf</a>]</p> </li> </ul> <h3 id='logic_in_natural_languages'>Logic in natural languages</h3> <ul> <li>Pieter A. M. Seuren, <em>The logic of language</em>, vol. II of Language from within; (vol. I: <em>Language in cognition</em>) Oxford University Press 2010</li> </ul><ins class='diffins'> </ins><ins class='diffins'><p><div class='property'> category: <a class='category_link' href='/nlab/list/logic'>logic</a></div></p></ins> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 30, 2024 at 17:30:55. 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