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href='/nlab/show/diff/%280%2C1%29-category+theory'>(0,1)-category theory</a></strong>: <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a>, <a class='existingWikiWord' href='/nlab/show/diff/order+theory'>order theory</a></p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category'>(0,1)-category</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+preorders+and+%280%2C1%29-categories'>relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/preorder'>proset</a>, <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>partially ordered set</a> (<a class='existingWikiWord' href='/nlab/show/diff/direction'>directed set</a>, <a class='existingWikiWord' href='/nlab/show/diff/total+order'>total order</a>, <a class='existingWikiWord' href='/nlab/show/diff/linear+order'>linear order</a>)</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/top'>top</a>, <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom</a>, <a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monotone+function'>monotone function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a>, <a class='existingWikiWord' href='/nlab/show/diff/interval'>interval</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/lattice'>lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/semilattice'>semilattice</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a>, <a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>logical conjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>and</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/join'>join</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjunction'>logical disjunction</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjunction'>or</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+element'>compact element</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/poset+of+subobjects'>lattice of subobjects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complete+lattice'>complete lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+lattice'>algebraic lattice</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/distributive+lattice'>distributive lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/completely+distributive+lattice'>completely distributive lattice</a>, <a class='existingWikiWord' href='/nlab/show/diff/canonical+extension'>canonical extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hyperdoctrine'>hyperdoctrine</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/first-order+hyperdoctrine'>first-order</a>, <a class='existingWikiWord' href='/nlab/show/diff/Boolean+hyperdoctrine'>Boolean</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherent+hyperdoctrine'>coherent</a>, <a class='existingWikiWord' href='/nlab/show/diff/tripos'>tripos</a></li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-topos</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heyting+algebra'>Heyting algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/regular+element'>regular element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a></li> </ul> </div> </div> </div> <h1 id='logical_conjunction'>Logical conjunction</h1> <div class='maruku_toc'><ul><li><a href='#definitions'>Definitions</a></li><li><a href='#remarks'>Remarks</a></li><li><a href='#rules_of_inference'>Rules of inference</a></li><li><a href='#AsALogicGate'>As a logic gate</a></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='definitions'>Definitions</h2> <p>In <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a>, logical conjunction is the <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a> in the <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> of <a class='existingWikiWord' href='/nlab/show/diff/truth+value'>truth values</a>.</p> <p>Assuming that (as in <a class='existingWikiWord' href='/nlab/show/diff/classical+logic'>classical logic</a>) the only truth values are true (<math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>) and false (<math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>), then the conjunction <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math> of the truth values <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_6' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math></th><th /><th><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></td></tr> </tbody></table> <p>That is, <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math> is true if and only if <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> are both true. Conjunction also exists in nearly every non-classical logic.</p> <p>More generally, if <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_26' xmlns='http://www.w3.org/1998/Math/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/diff/relation'>relations</a> on the same domain, then we define their conjunction pointwise, thinking of a relation as a <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> to truth values. If instead we think of a relation as a <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> of its domain, then conjunction becomes <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a>.</p> <h2 id='remarks'>Remarks</h2> <p>Conjunction is <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+duality'>de Morgan dual</a> to <a class='existingWikiWord' href='/nlab/show/diff/disjunction'>disjunction</a>.</p> <p>Like any meet, conjunction is an associative operation, so we can take the conjunction of any finite positive whole number of truth values; the conjunction is true if and only if all of the individual truth values are true. Conjunction also has an <a class='existingWikiWord' href='/nlab/show/diff/identity+element'>identity element</a>, which is the <a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a> truth value. Some logics allow a notion of infinitary conjunction. Indexed conjunction is <a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a>.</p> <p>As truth values form a poset, which is a degenerate kind of <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, so truth values under conjunction form a <a class='existingWikiWord' href='/nlab/show/diff/semilattice'>meet-semilattice</a>, which is a degenerate kind of <a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a>. Self-referentially, a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> is (up to <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence</a>) simply a <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>category enriched</a> over the cartesian monoidal category of truth values. With <a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a> as <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a>, truth values form a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>closed cartesian category</a>.</p> <p>In the context of <a class='existingWikiWord' href='/nlab/show/diff/substructural+logic'>substructural logics</a> such as <a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a>, the conjunction defined above is also called <strong>additive conjunction</strong> to disambiguate it from the <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+conjunction'>multiplicative conjunction</a>.</p> <h2 id='rules_of_inference'>Rules of inference</h2> <p>That conjunction is a <a class='existingWikiWord' href='/nlab/show/diff/meet'>meet</a> means that <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math> may be proved in a <a class='existingWikiWord' href='/nlab/show/diff/context'>context</a> <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math> if and only if both <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> may be proved in <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>. This directly yields the <a class='existingWikiWord' href='/nlab/show/diff/deductive+system'>introduction and elimination rules</a> for conjunction in <a class='existingWikiWord' href='/nlab/show/diff/natural+deduction'>natural deduction</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable displaystyle='true' rowspacing='1.0ex'><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>;</mo><mspace width='thickmathspace' /><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' /><mtext>introduction</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi></mrow></mfrac><mspace width='thickmathspace' /><mtext>elimination 0</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><mi>q</mi></mrow></mfrac><mspace width='thickmathspace' /><mtext>elimination 1</mtext></mtd></mtr> <mtr><mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin {gathered} \frac { \Gamma \vdash p ; \; \Gamma \vdash q } { \Gamma \vdash p \wedge q } \; \text {introduction} \\ \frac { \Gamma \vdash p \wedge q } { \Gamma \vdash p } \; \text {elimination 0} \\ \frac { \Gamma \vdash p \wedge q } { \Gamma \vdash q } \; \text {elimination 1} \\ \end {gathered} </annotation></semantics></math></div> <p>Alternatively, we may use these slightly more complicated (but fewer) inductive forms:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable displaystyle='true' rowspacing='1.0ex'><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>p</mi><mo>;</mo><mspace width='thickmathspace' /><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' /><mtext>introduction</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>p</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><mo>⊢</mo><mi>r</mi></mrow></mfrac><mspace width='thickmathspace' /><mtext>elimination</mtext></mtd></mtr> <mtr><mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin {gathered} \frac { \Gamma \vdash p ; \; \Gamma \vdash q } { \Gamma \vdash p \wedge q } \; \text {introduction} \\ \frac { \Gamma , p , q \vdash r } { \Gamma , p \wedge q , \vdash r } \; \text {elimination} \\ \end {gathered} </annotation></semantics></math></div> <p>In <a class='existingWikiWord' href='/nlab/show/diff/sequent+calculus'>sequent calculus</a>, the same ideas become these rules:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable displaystyle='true' rowspacing='1.0ex'><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>Σ</mi><mo>;</mo><mspace width='thickmathspace' /><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' /><mtext>right additive</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>,</mo><mi>p</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' /><mtext>left additive 0</mtext></mtd></mtr> <mtr><mtd><mfrac><mrow><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' /><mtext>left additive 1</mtext></mtd></mtr> <mtr><mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin {gathered} \frac { \Gamma \vdash \Delta , p , \Sigma ; \; \Gamma \vdash \Delta , q , \Sigma } { \Gamma \vdash \Delta , p \wedge q , \Sigma } \; \text {right additive} \\ \frac { \Gamma , p , \Delta \vdash \Sigma } { \Gamma , p \wedge q , \Delta \vdash \Sigma } \; \text {left additive 0} \\ \frac { \Gamma , q , \Delta \vdash \Sigma } { \Gamma , p \wedge q , \Delta \vdash \Sigma } \; \text {left 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 class='maruku-mathml' display='block' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable displaystyle='true' rowspacing='1.0ex'><mtr><mtd><mfrac><mrow><mi>Γ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>;</mo><mspace width='thickmathspace' /><mi>Σ</mi><mo>⊢</mo><mi>q</mi><mo>,</mo><mi>Π</mi></mrow><mrow><mi>Γ</mi><mo>,</mo><mi>Σ</mi><mo>⊢</mo><mi>Δ</mi><mo>,</mo><mi>p</mi><mo>∧</mo><mi>q</mi><mo>,</mo><mi>Π</mi></mrow></mfrac><mspace width='thickmathspace' /><mtext>right multiplicative</mtext></mtd></mtr> <mtr><mtd><mfrac><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><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><mtext>left multiplicative</mtext></mtd></mtr> <mtr><mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin {gathered} \frac { \Gamma \vdash \Delta , p ; \; \Sigma \vdash q , \Pi } { \Gamma , \Sigma \vdash \Delta , p \wedge q , \Pi } \; \text {right multiplicative} \\ \frac { \Gamma , p , q , \Delta \vdash \Sigma } { \Gamma , p \wedge q , \Delta \vdash \Sigma } \text {left 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/diff/exchange+rule'>exchange rule</a>. In <a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a>, the first batch of sequent rules apply to additive conjunction (interpret <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math> in these rules as <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>&</mi><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \& q</annotation></semantics></math>), while the second batch of rules apply to multiplicative conjunction (interpret <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>∧</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \wedge q</annotation></semantics></math> in those rules as <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>⊗</mo><mi>q</mi></mrow><annotation encoding='application/x-tex'>p \otimes q</annotation></semantics></math>).</p> <h2 id='AsALogicGate'>As a logic gate</h2> <p>Logical conjunction as</p> <ol> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/logic+gate'>logic gate</a>,</p> </li> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/reversible+computation'>reversible</a> logic gate and</p> </li> <li> <p>a (reversible) <a class='existingWikiWord' href='/nlab/show/diff/quantum+logic+gate'>quantum logic gate</a>:</p> </li> </ol> <p>\begin{imagefromfile} “file_name”: “ANDGates-221026b.jpg”, “width”: “770”, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile}</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li>in <a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a>: <em><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+conjunction'>multiplicative conjunction</a></em></li> </ul> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math>symbol<math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math>in <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{-}</annotation></semantics></math></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∈</mo></mrow><annotation encoding='application/x-tex'>\in</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/element'>element</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /><mo>:</mo></mrow><annotation encoding='application/x-tex'>\,:</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/type'>typing</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊢</mo></mrow><annotation encoding='application/x-tex'>\vdash</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/implication'>entailment</a> / <a class='existingWikiWord' href='/nlab/show/diff/sequent'>sequent</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊤</mo></mrow><annotation encoding='application/x-tex'>\top</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a> / <a class='existingWikiWord' href='/nlab/show/diff/top'>top</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊥</mo></mrow><annotation encoding='application/x-tex'>\bot</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a> / <a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇒</mo></mrow><annotation encoding='application/x-tex'>\Rightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇔</mo></mrow><annotation encoding='application/x-tex'>\Leftrightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+equivalence'>logical equivalence</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo></mrow><annotation encoding='application/x-tex'>\not</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≠</mo></mrow><annotation encoding='application/x-tex'>\neq</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a> of <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a> / <a class='existingWikiWord' href='/nlab/show/diff/apartness+relation'>apartness</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∉</mo></mrow><annotation encoding='application/x-tex'>\notin</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a> of <a class='existingWikiWord' href='/nlab/show/diff/element'>element</a> <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> <math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo><mo>¬</mo></mrow><annotation encoding='application/x-tex'>\not \not</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/double+negation'>negation of negation</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo></mrow><annotation encoding='application/x-tex'>\exists</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantification</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∀</mo></mrow><annotation encoding='application/x-tex'>\forall</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∧</mo></mrow><annotation encoding='application/x-tex'>\wedge</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>logical conjunction</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∨</mo></mrow><annotation encoding='application/x-tex'>\vee</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/disjunction'>logical disjunction</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><strong>symbol</strong></td><td style='text-align: left;'><strong>in <a class='existingWikiWord' href='/nlab/show/diff/type+theory'>type theory</a> (<a class='existingWikiWord' href='/nlab/show/diff/propositions+as+types'>propositions as types</a>)</strong></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a> (<a class='existingWikiWord' href='/nlab/show/diff/implication'>implication</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>×</mo></mrow><annotation encoding='application/x-tex'>\times</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/product+type'>product type</a> (<a class='existingWikiWord' href='/nlab/show/diff/conjunction'>conjunction</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo></mrow><annotation encoding='application/x-tex'>+</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/sum+type'>sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/disjunction'>disjunction</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/empty+type'>empty type</a> (<a class='existingWikiWord' href='/nlab/show/diff/falsehood'>false</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/unit+type'>unit type</a> (<a class='existingWikiWord' href='/nlab/show/diff/true+proposition'>true</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity type</a> (<a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+types'>equivalence of types</a> (<a class='existingWikiWord' href='/nlab/show/diff/logical+equivalence'>logical equivalence</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo></mrow><annotation encoding='application/x-tex'>\sum</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/dependent+sum+type'>dependent sum type</a> (<a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantifier</a>)</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo></mrow><annotation encoding='application/x-tex'>\prod</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/dependent+product+type'>dependent product type</a> (<a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantifier</a>)</td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><strong>symbol</strong></td><td style='text-align: left;'><strong>in <a class='existingWikiWord' href='/nlab/show/diff/linear+logic'>linear logic</a></strong></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊸</mo></mrow><annotation encoding='application/x-tex'>\multimap</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/linear+implication'>linear implication</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊗</mo></mrow><annotation encoding='application/x-tex'>\otimes</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+conjunction'>multiplicative conjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊕</mo></mrow><annotation encoding='application/x-tex'>\oplus</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/additive+disjunction'>additive disjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>&</mi></mrow><annotation encoding='application/x-tex'>\&</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/logical+conjunction'>additive conjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⅋</mo></mrow><annotation encoding='application/x-tex'>\invamp</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+disjunction'>multiplicative disjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mo>!</mo></mrow><annotation encoding='application/x-tex'>\;!</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%21-modality'>exponential conjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /><mo>?</mo></mrow><annotation encoding='application/x-tex'>\;?</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/%3F-modality'>exponential disjunction</a><math class='maruku-mathml' display='inline' id='mathml_ad1c02bafeae7121d06b1f182244df3768378b0e_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <p> </p><del class='diffdel'> </del><del class='diffdel'><p> </p></del> </div> <div class="revisedby"> <p> Last revised on March 29, 2023 at 10:24:04. See the <a href="/nlab/history/logical+conjunction" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/logical+conjunction" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/15340/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/logical+conjunction/14" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/logical+conjunction" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/logical+conjunction" accesskey="S" class="navlink" id="history" rel="nofollow">History (14 revisions)</a> <a href="/nlab/show/logical+conjunction/cite" style="color: black">Cite</a> <a href="/nlab/print/logical+conjunction" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/logical+conjunction" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>