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disjunction in nLab
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content="application/xhtml+xml;charset=utf-8" /><title>Logical disjunction</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> </div> </div> <h1 id="logical_disjunction">Logical disjunction</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <li><a href='#remarks'>Remarks</a></li> <li><a href='#in_dependent_type_theory'>In dependent type theory</a></li> <li><a href='#classical_vs_constructive'>Classical vs constructive</a></li> <li><a href='#RoI'>Rules of inference</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>In <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, logical disjunction is the <a class="existingWikiWord" href="/nlab/show/join">join</a> in the <a class="existingWikiWord" href="/nlab/show/poset">poset</a> of <a class="existingWikiWord" href="/nlab/show/truth+values">truth values</a>.</p> <p>Assuming that (as in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a>) the only truth values are true (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>) and false (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>), then the disjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vee q</annotation></semantics></math> of the truth values <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> may be defined by a truth table:</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></th><th></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vee q</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></td></tr> </tbody></table> <p>That is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vee q</annotation></semantics></math> is true if and only if at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> is true. Disjunction also exists in nearly every non-classical logic.</p> <p>More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> are any two <a class="existingWikiWord" href="/nlab/show/relations">relations</a> on the same domain, then we define their disjunction pointwise, thinking of a relation as a <a class="existingWikiWord" href="/nlab/show/function">function</a> to truth values. If instead we think of a relation as a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of its domain, then disjunction becomes <a class="existingWikiWord" href="/nlab/show/union">union</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> the <a class="existingWikiWord" href="/nlab/show/inference+rules">inference rules</a> for disjunction are given as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mo>∨</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>P</mi><mo>∨</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow></mfrac><mspace width="2em"></mspace><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>P</mi><mo>∨</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma \vdash P \vee Q \; \mathrm{prop}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, P \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}} \qquad \frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop}}{\Gamma, Q \; \mathrm{true} \vdash P \vee Q \; \mathrm{true}}</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>⊢</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>P</mi><mo>∨</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">prop</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>P</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mspace width="1em"></mspace><mi>Γ</mi><mo>,</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>P</mi><mo>∨</mo><mi>Q</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi><mo>⊢</mo><mi>R</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">true</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \vdash P \; \mathrm{prop} \quad \Gamma \vdash Q \; \mathrm{prop} \quad \Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{prop} \quad \Gamma, P \; \mathrm{true} \vdash R \; \mathrm{true} \quad \Gamma, Q \; \mathrm{true} \vdash R \; \mathrm{true}}{\Gamma, P \vee Q \; \mathrm{true} \vdash R \; \mathrm{true}}</annotation></semantics></math></div> <h2 id="remarks">Remarks</h2> <p>Disjunction as defined above is sometimes called <strong>inclusive disjunction</strong> to distinguish it from <a class="existingWikiWord" href="/nlab/show/exclusive+disjunction">exclusive disjunction</a>, where <em>exactly</em> one of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> must be true.</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/substructural+logics">substructural logics</a> such as <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, we often have both <em><a class="existingWikiWord" href="/nlab/show/additive+disjunction">additive disjunction</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math> and <em><a class="existingWikiWord" href="/nlab/show/multiplicative+disjunction">multiplicative disjunction</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\parr</annotation></semantics></math>; see the <a href="#RoI">Rules of Inference</a> below for the distinction. In <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, additive disjunction is the <a class="existingWikiWord" href="/nlab/show/join">join</a> under the <a class="existingWikiWord" href="/nlab/show/entailment">entailment</a> relation, just like disjunction in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> (and <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a>), while multiplicative disjunction is something different.</p> <p>Disjunction is <a class="existingWikiWord" href="/nlab/show/de+Morgan+dual">de Morgan dual</a> to <a class="existingWikiWord" href="/nlab/show/logical+conjunction">conjunction</a>.</p> <p>Like any join, disjunction is an associative operation, so we can take the disjunction of any finite positive whole number of truth values; the disjunction is true if and only if at least one of the various truth values is true. Disjunction also has an <a class="existingWikiWord" href="/nlab/show/identity+element">identity element</a>, which is the <a class="existingWikiWord" href="/nlab/show/falsity">false</a> truth value. Some logics allow a notion of infinitary disjunction. Indexed disjunction is <a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a>.</p> <h2 id="in_dependent_type_theory">In dependent type theory</h2> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, the disjunction of two <a class="existingWikiWord" href="/nlab/show/mere+propositions">mere propositions</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/bracket+type">bracket type</a> of their <a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>P</mi><mo>+</mo><mi>Q</mi><mo stretchy="false">‖</mo></mrow><annotation encoding="application/x-tex">\| P + Q \|</annotation></semantics></math>. Disjunction types in general could also be regarded as a particular sort of <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a>. In <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a> syntax:</p> <pre><code>Inductive disjunction (P Q:Type) : Type := | inl : P -> disjunction P Q | inr : Q -> disjunction P Q | contr0 : forall (p q : disjunction P Q) p == q</code></pre> <p>If the <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> has a <a class="existingWikiWord" href="/nlab/show/type+of+propositions">type of propositions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">Prop</mi></mrow><annotation encoding="application/x-tex">\mathrm{Prop}</annotation></semantics></math>, such as the one derived from a type universe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> - <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>A</mi><mo>:</mo><mi>U</mi></mrow></msub><mi mathvariant="normal">isProp</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sum_{A:U} \mathrm{isProp}(A)</annotation></semantics></math>, then the disjunction of two types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is defined as the <a class="existingWikiWord" href="/nlab/show/dependent+function+type">dependent function type</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∨</mo><mi>B</mi><mo>≡</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">A \vee B \equiv \prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P</annotation></semantics></math></div> <p>By <a class="existingWikiWord" href="/nlab/show/weak+function+extensionality">weak function extensionality</a>, the disjunction of two types is a proposition.</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>The two definitions above are equivalent.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/propositional+truncation">propositional truncation</a> of a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is equivalent to the following dependent function type</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo stretchy="false">‖</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\| A \| \simeq \prod_{P:\mathrm{Prop}} (A \to P) \to P</annotation></semantics></math></div> <p>Substituting the sum type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A + B</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">‖</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\| A + B \| \simeq \prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P</annotation></semantics></math></div> <p>Given any type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, there is an equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((A + B) \to C) \simeq (A \to C) \times (B \to C)</annotation></semantics></math></div> <p>and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≃</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \simeq B</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A \to C) \simeq (B \to C)</annotation></semantics></math>. In addition, for all type families <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash B(x)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi><mo>⊢</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x:A \vdash C(x)</annotation></semantics></math>, if there is a family of equivalences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e:\prod_{x:A} B(x) \simeq C(x)</annotation></semantics></math>, then there is an equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(\prod_{x:A} B(x)\right) \simeq \left(\prod_{x:A} C(x)\right)</annotation></semantics></math>. All this taken together means that there are equivalences</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">‖</mo><mo>≃</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>P</mi><mo>:</mo><mi mathvariant="normal">Prop</mi></mrow></munder><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>P</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\| A + B \| \simeq \left(\prod_{P:\mathrm{Prop}} ((A + B) \to P) \to P\right) \simeq \left(\prod_{P:\mathrm{Prop}} ((A \to P) \times (B \to P)) \to P\right)</annotation></semantics></math></div> <p></p> </div> </p> <p>If one has the <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a> and the <a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>, then the disjunction of two types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is given by the following type:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∨</mo><mi>B</mi><mo>≔</mo><mo>∃</mo><mi>b</mi><mo>:</mo><mi mathvariant="normal">bool</mi><mo>.</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi mathvariant="normal">true</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mi mathvariant="normal">false</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \vee B \coloneqq \exists b:\mathrm{bool}.((b = \mathrm{true}) \to A) \times ((b = \mathrm{false}) \to B)</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∨</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \vee Q</annotation></semantics></math> of two <a class="existingWikiWord" href="/nlab/show/mere+propositions">mere propositions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is also the <a class="existingWikiWord" href="/nlab/show/join+type">join type</a> of the two types <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>*</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P * Q</annotation></semantics></math>. This is because every mere proposition is a <a class="existingWikiWord" href="/nlab/show/subtype">subtype</a> of the <a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a>, and the disjunction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> as two subtypes of the unit types, and the union of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> as subtypes of the unit type is defined to be the <a class="existingWikiWord" href="/nlab/show/join+type">join type</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/pushout+type">pushout type</a> of the two product projection functions from the <a class="existingWikiWord" href="/nlab/show/product+type">product type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>×</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \times Q</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> respectively.</p> <h2 id="classical_vs_constructive">Classical vs constructive</h2> <p>There are a variety of connectives that are distinct in <a class="existingWikiWord" href="/nlab/show/intuitionistic+logic">intuitionistic logic</a> but are all equivalent to disjunction in <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a>. Here is a Hasse diagram of some of them, with the strongest statement at the bottom and the weakest at the top (so that each statement entails those above it):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>∧</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>⇗</mi></mtd> <mtd></mtd> <mtd><mi>⇖</mi></mtd></mtr> <mtr><mtd><mo>¬</mo><mi>P</mi><mo>→</mo><mi>Q</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>P</mi><mo>←</mo><mo>¬</mo><mi>Q</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>⇖</mi></mtd> <mtd></mtd> <mtd><mi>⇗</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>P</mi><mo>←</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>P</mi><mo>∨</mo><mi>Q</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { & & \neg(\neg{P} \wedge \neg{Q}) \\ & &#x21d7; & & &#x21d6; \\ \neg{P} \rightarrow Q & & & & P \leftarrow \neg{Q} \\ & &#x21d6; & & &#x21d7; \\ & & (\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q}) \\ & & \Uparrow \\ & & P \vee Q } </annotation></semantics></math></div> <p>(A single arrow is implication in the object language; a double arrow is entailment in the metalanguage.) Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mi>P</mi><mo>∧</mo><mo>¬</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">\neg{P} \wedge \neg{Q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/negation">negation</a> of every item in this diagram.</p> <p>In the <span class="newWikiWord">double-negation interpretation<a href="/nlab/new/double-negation+interpretation">?</a></span> of classical logic in intuitionistic logic, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>∧</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\neg(\neg{P} \wedge \neg{Q})</annotation></semantics></math> is the interpretation in intuitionistic logic of disjunction in classical logic. For this reason, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>∧</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\neg(\neg{P} \wedge \neg{Q})</annotation></semantics></math> is sometimes called <em>classical disjunction</em>. But this doesn't mean that it should always be used when turning classical mathematics into <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>. Indeed, a stronger statement is almost always preferable, if one is valid; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>∧</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\neg(\neg{P} \wedge \neg{Q})</annotation></semantics></math> is merely the fallback position when nothing better can be found. (And as can be seen in the example in the paragraph after next, sometimes even this is not valid.)</p> <p>In the <a class="existingWikiWord" href="/nlab/show/antithesis+interpretation">antithesis interpretation</a> of <a class="existingWikiWord" href="/nlab/show/affine+logic">affine logic</a> in intuitionistic logic, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>¬</mo><mi>P</mi><mo>→</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>P</mi><mo>←</mo><mo>¬</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\neg{P} \rightarrow Q) \wedge (P \leftarrow \neg{Q})</annotation></semantics></math> is the interpretation of the <a class="existingWikiWord" href="/nlab/show/multiplicative+disjunction">multiplicative disjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⅋</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \parr Q</annotation></semantics></math> for affirmative propositions. More generally, a statement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> in affine logic is interpreted as a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>P</mi> <mo>−</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P^+,P^-)</annotation></semantics></math> of mutually contradictory statements in intuitionistic logic; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">P^-</annotation></semantics></math> is simply the negation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">P^+</annotation></semantics></math> for affirmative propositions, but in general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">P^-</annotation></semantics></math> only entails <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mrow><msup><mi>P</mi> <mo>+</mo></msup></mrow></mrow><annotation encoding="application/x-tex">\neg{P^+}</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⅋</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \parr Q</annotation></semantics></math> is interpreted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>−</mo></msup><mo>→</mo><msup><mi>Q</mi> <mo>+</mo></msup><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>+</mo></msup><mo>←</mo><msup><mi>Q</mi> <mo>−</mo></msup><mo stretchy="false">)</mo><mo>,</mo><msup><mi>P</mi> <mo>−</mo></msup><mo>∧</mo><msup><mi>Q</mi> <mo>−</mo></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big((P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-), P^- \wedge Q^-\big)</annotation></semantics></math>; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>⅋</mo><mi>Q</mi><msup><mo stretchy="false">)</mo> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">(P \parr Q)^+</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>−</mo></msup><mo>→</mo><msup><mi>Q</mi> <mo>+</mo></msup><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>+</mo></msup><mo>←</mo><msup><mi>Q</mi> <mo>−</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P^- \rightarrow Q^+) \wedge (P^+ \leftarrow Q^-)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>⅋</mo><mi>Q</mi><msup><mo stretchy="false">)</mo> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">(P \parr Q)^-</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>−</mo></msup><mo>∧</mo><msup><mi>Q</mi> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">P^- \wedge Q^-</annotation></semantics></math>. (In contrast, the additive disjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊕</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \oplus Q</annotation></semantics></math> is interpreted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>P</mi> <mo>+</mo></msup><mo>∨</mo><msup><mi>Q</mi> <mo>+</mo></msup><mo>,</mo><msup><mi>P</mi> <mo>−</mo></msup><mo>∧</mo><msup><mi>Q</mi> <mo>−</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P^+ \vee Q^+, P^- \wedge Q^-)</annotation></semantics></math>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊕</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \oplus Q</annotation></semantics></math> entails <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⅋</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">P \parr Q</annotation></semantics></math> in affine logic, even though they are independent in linear logic.)</p> <p>For a non-affirmative example, in the arithmetic of (located) <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, it is not constructively valid to derive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a = 0) \vee (b = 0)</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a b = 0</annotation></semantics></math>, and it's not even valid to derive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo>¬</mo><mo stretchy="false">(</mo><mi>a</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo>¬</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\neg\big(\neg(a = 0) \wedge \neg(b = 0)\big)</annotation></semantics></math> without <a class="existingWikiWord" href="/nlab/show/Markov%27s+principle">Markov's principle</a> (or at least some weak version of it), but it <em>is</em> valid to derive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>#</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>b</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a \# 0) \rightarrow (b = 0)</annotation></semantics></math> (and conversely), where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>#</mo></mrow><annotation encoding="application/x-tex">\#</annotation></semantics></math> is the usual <a class="existingWikiWord" href="/nlab/show/apartness+relation">apartness relation</a> between real numbers. (Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">P^+</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">P^-</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>#</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a \# 0</annotation></semantics></math>, and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>.) Of course, it's also valid to derive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><mi>a</mi><mo>#</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>b</mi><mo>#</mo><mn>0</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\neg\big((a \# 0) \wedge (b \# 0)\big)</annotation></semantics></math> (which is actually equivalent).</p> <h2 id="RoI">Rules of inference</h2> <p>The <a class="existingWikiWord" href="/nlab/show/rule+of+inference">rules of inference</a> for disjunction in <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a> are dual to those for <a class="existingWikiWord" href="/nlab/show/logical+conjunction">conjunction</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" rowspacing="1.0ex"><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>Σ</mi><mo>;</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>Σ</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>,</mo><mi>Δ</mi><mo>⊢</mo><mi>Σ</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>left additive</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>,</mo><mi>Σ</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>right additive 0</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>,</mo><mi>Σ</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>right additive 1</mtext></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin {gathered} \frac { \Gamma , p , \Delta \vdash \Sigma ; \; \Gamma , q , \Delta \vdash \Sigma } { \Gamma , p \vee q , \Delta \vdash \Sigma } \; \text {left additive} \\ \frac { \Gamma \vdash \Delta , p , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 0} \\ \frac { \Gamma \vdash \Delta , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \; \text {right additive 1} \\ \end {gathered} </annotation></semantics></math></div> <p>Equivalently, we can use the following rules with weakened contexts:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" rowspacing="1.0ex"><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>⊢</mo><mi>Δ</mi><mo>;</mo><mspace width="thickmathspace"></mspace><mi>q</mi><mo>,</mo><mi>Σ</mi><mo>⊢</mo><mi>Π</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>,</mo><mi>Σ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>Π</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>left multiplicative</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>Σ</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>,</mo><mi>Σ</mi></mrow></mfrac><mtext>right multiplicative</mtext></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin {gathered} \frac { \Gamma , p \vdash \Delta ; \; q , \Sigma \vdash \Pi } { \Gamma , p \vee q , \Sigma \vdash \Delta , \Pi } \; \text {left multiplicative} \\ \frac { \Gamma \vdash \Delta , p , q , \Sigma } { \Gamma \vdash \Delta , p \vee q , \Sigma } \text {right multiplicative} \\ \end {gathered} </annotation></semantics></math></div> <p>The rules above are written so as to remain valid in logics without the <a class="existingWikiWord" href="/nlab/show/exchange+rule">exchange rule</a>. In <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, the first batch of sequent rules apply to additive disjunction (interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vee q</annotation></semantics></math> in these rules as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⊕</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \oplus q</annotation></semantics></math>), while the second batch of rules apply to multiplicative disjunction (interpret <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∨</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \vee q</annotation></semantics></math> in those rules as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>⅋</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \parr q</annotation></semantics></math>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a> rules for disjunction are a little more complicated than those for conjunction:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" rowspacing="1.0ex"><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>⊢</mo><mi>r</mi><mo>;</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mo>,</mo><mi>q</mi><mo>⊢</mo><mi>r</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>p</mi><mo>∨</mo><mi>q</mi><mo>⊢</mo><mi>r</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>elimination</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>∨</mo><mi>q</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>introduction 0</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>q</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>∨</mo><mi>q</mi></mrow></mfrac><mspace width="thickmathspace"></mspace><mtext>introduction 1</mtext></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin {gathered} \frac { \Gamma , p \vdash r ; \; \Gamma , q \vdash r } { \Gamma , p \vee q \vdash r } \; \text {elimination} \\ \frac { \Gamma \vdash p } { \Gamma \vdash p \vee q } \; \text {introduction 0} \\ \frac { \Gamma \vdash q } { \Gamma \vdash p \vee q } \; \text {introduction 1} \\ \end {gathered} </annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>symbol<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math>in <a class="existingWikiWord" href="/nlab/show/logic">logic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo lspace="verythinmathspace" rspace="0em">−</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{-}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mo>:</mo></mrow><annotation encoding="application/x-tex">\,:</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/type">typing</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equality">equality</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊢</mo></mrow><annotation encoding="application/x-tex">\vdash</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/entailment">entailment</a> / <a class="existingWikiWord" href="/nlab/show/sequent">sequent</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo></mrow><annotation encoding="application/x-tex">\top</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/true">true</a> / <a class="existingWikiWord" href="/nlab/show/top">top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo></mrow><annotation encoding="application/x-tex">\bot</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/false">false</a> / <a class="existingWikiWord" href="/nlab/show/bottom">bottom</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/implication">implication</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇔</mo></mrow><annotation encoding="application/x-tex">\Leftrightarrow</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo></mrow><annotation encoding="application/x-tex">\neq</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/equality">equality</a> / <a class="existingWikiWord" href="/nlab/show/apartness">apartness</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∉</mo></mrow><annotation encoding="application/x-tex">\notin</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/negation">negation</a> of <a class="existingWikiWord" href="/nlab/show/element">element</a> <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding="application/x-tex">\not \not</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/double+negation">negation of negation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∃</mo></mrow><annotation encoding="application/x-tex">\exists</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/existential+quantification">existential quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo></mrow><annotation encoding="application/x-tex">\forall</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a> (<a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>)</strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/function+type">function type</a> (<a class="existingWikiWord" href="/nlab/show/implication">implication</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/product+type">product type</a> (<a class="existingWikiWord" href="/nlab/show/conjunction">conjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/sum+type">sum type</a> (<a class="existingWikiWord" href="/nlab/show/disjunction">disjunction</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/empty+type">empty type</a> (<a class="existingWikiWord" href="/nlab/show/false">false</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unit+type">unit type</a> (<a class="existingWikiWord" href="/nlab/show/true">true</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> (<a class="existingWikiWord" href="/nlab/show/equality">equality</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> (<a class="existingWikiWord" href="/nlab/show/logical+equivalence">logical equivalence</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo></mrow><annotation encoding="application/x-tex">\sum</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+sum+type">dependent sum type</a> (<a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo></mrow><annotation encoding="application/x-tex">\prod</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dependent+product+type">dependent product type</a> (<a class="existingWikiWord" href="/nlab/show/universal+quantifier">universal quantifier</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong>symbol</strong></td><td style="text-align: left;"><strong>in <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></strong></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊸</mo></mrow><annotation encoding="application/x-tex">\multimap</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊕</mo></mrow><annotation encoding="application/x-tex">\oplus</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+disjunction">additive disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/additive+conjunction">additive conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⅋</mo></mrow><annotation encoding="application/x-tex">\invamp</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/multiplicative+disjunction">multiplicative disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>!</mo></mrow><annotation encoding="application/x-tex">\;!</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+conjunction">exponential conjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>?</mo></mrow><annotation encoding="application/x-tex">\;?</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/exponential+disjunction">exponential disjunction</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <ul> <li>Landon D. C. Elkind, Richard Zach, <em>The Genealogy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math></em> (<a href="https://arxiv.org/abs/2012.06072">arXiv:2012.06072</a>)</li> </ul> <p>The definition of the disjunction of two types in dependent type theory as the propositional truncation of the sum type is found in:</p> <ul> <li id="UFP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §3.7 in: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</li> </ul> <p>The definition of the disjunction of two mere propositions in dependent type theory as the <a class="existingWikiWord" href="/nlab/show/join+type">join type</a> of propositions is found in:</p> <ul> <li id="Shulman13"><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Cohomology</em>, Homotopy type theory blog (<a href="http://homotopytypetheory.org/2013/07/24/cohomology/">web</a>)</li> </ul> <p>And the disjunction of two types defined from the type of propositions and dependent product types can be found in:</p> <ul> <li>Madeleine Birchfield, <em>Constructing coproduct types and boolean types from universes</em>, MathOverflow (<a href="https://mathoverflow.net/questions/457904/constructing-coproduct-types-and-boolean-types-from-universes">web</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 13, 2024 at 21:08:08. 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