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#ExclusiveDisjunction

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Jon Awbrey<p>Logic Syllabus • Discussion 1<br />• <a href="https://inquiryintoinquiry.com/2023/06/02/logic-syllabus-discussion-1/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2023/06</span><span class="invisible">/02/logic-syllabus-discussion-1/</span></a></p><p>Re: Logic Syllabus ( <a href="https://inquiryintoinquiry.com/logic-syllabus/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/logic-s</span><span class="invisible">yllabus/</span></a> )<br />Re: Laws of Form ( <a href="https://groups.io/g/lawsofform/topic/logic_syllabus/99218507" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">groups.io/g/lawsofform/topic/l</span><span class="invisible">ogic_syllabus/99218507</span></a> )<br />Re: John Mingers ( <a href="https://groups.io/g/lawsofform/message/2326" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">groups.io/g/lawsofform/message</span><span class="invisible">/2326</span></a> )</p><p>JM: ❝In a previous post you mentioned the minimal negation operator. Is there also the converse of this, i.e. an operator which is true when exactly one of its arguments is true? Or is this just XOR?❞</p><p>Yes, the “just one true” operator is a very handy tool. We discussed it earlier under the headings of “genus and species relations” or “radio button logic”. Viewed in the form of a venn diagram it describes a partition of the universe of discourse into mutually exclusive and exhaustive regions.</p><p>Reading \(\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_m \texttt{)}\) to mean just one of \(x_1, \ldots, x_m\) is false, the form \(\texttt{((} x_1 \texttt{),} \ldots \texttt{,(} x_m \texttt{))}\) means just one of \(x_1, \ldots, x_m\) is true.</p><p>For two logical variables, though, the cases “condense” or “degenerate” and saying “just one true” is the same thing as saying “just one false”.</p><p>\[\texttt{((} x_1 \texttt{),(} x_2 \texttt{))} = \texttt{(} x_1 \texttt{,} x_2 \texttt{)} = x_1 + x_2 = \mathrm{xor} (x_1, x_2).\]</p><p>There&#39;s more information on the following pages.</p><p>Minimal Negation Operators<br />• <a href="https://oeis.org/wiki/Minimal_negation_operator" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Minimal_negation</span><span class="invisible">_operator</span></a></p><p>Related Truth Tables<br />• <a href="https://oeis.org/wiki/Minimal_negation_operator#Truth_tables" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Minimal_negation</span><span class="invisible">_operator#Truth_tables</span></a></p><p>Genus, Species, Pie Charts, Radio Buttons<br />• <a href="https://inquiryintoinquiry.com/2021/11/10/genus-species-pie-charts-radio-buttons-1/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2021/11</span><span class="invisible">/10/genus-species-pie-charts-radio-buttons-1/</span></a></p><p>Related Discussions<br />• <a href="https://inquiryintoinquiry.com/?s=Radio+Buttons" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/?s=Radi</span><span class="invisible">o+Buttons</span></a></p><p><a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="tag">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/LogicSyllabus" class="mention hashtag" rel="tag">#<span>LogicSyllabus</span></a> <a href="https://mathstodon.xyz/tags/BooleanDomain" class="mention hashtag" rel="tag">#<span>BooleanDomain</span></a> <a href="https://mathstodon.xyz/tags/BooleanFunction" class="mention hashtag" rel="tag">#<span>BooleanFunction</span></a> <a href="https://mathstodon.xyz/tags/BooleanValuedFunction" class="mention hashtag" rel="tag">#<span>BooleanValuedFunction</span></a><br /><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="tag">#<span>Peirce</span></a> <a href="https://mathstodon.xyz/tags/LogicalGraph" class="mention hashtag" rel="tag">#<span>LogicalGraph</span></a> <a href="https://mathstodon.xyz/tags/MinimalNegationOperator" class="mention hashtag" rel="tag">#<span>MinimalNegationOperator</span></a> <a href="https://mathstodon.xyz/tags/ExclusiveDisjunction" class="mention hashtag" rel="tag">#<span>ExclusiveDisjunction</span></a> <a href="https://mathstodon.xyz/tags/XOR" class="mention hashtag" rel="tag">#<span>XOR</span></a><br /><a href="https://mathstodon.xyz/tags/CactusLanguage" class="mention hashtag" rel="tag">#<span>CactusLanguage</span></a> <a href="https://mathstodon.xyz/tags/PropositionalCalculus" class="mention hashtag" rel="tag">#<span>PropositionalCalculus</span></a> <a href="https://mathstodon.xyz/tags/RadioButtonLogic" class="mention hashtag" rel="tag">#<span>RadioButtonLogic</span></a> <a href="https://mathstodon.xyz/tags/TruthTable" class="mention hashtag" rel="tag">#<span>TruthTable</span></a></p>
Jon Awbrey<p>Logic Syllabus • 2<br />• <a href="https://inquiryintoinquiry.com/logic-syllabus/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/logic-s</span><span class="invisible">yllabus/</span></a></p><p>Logical Operators<br />• <a href="https://oeis.org/wiki/Logic_Syllabus#Logical_operators" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Logic_Syllabus#L</span><span class="invisible">ogical_operators</span></a></p><p>Logical Negation • <a href="https://oeis.org/wiki/Logical_negation" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">oeis.org/wiki/Logical_negation</span><span class="invisible"></span></a><br />Logical NAND • <a href="https://oeis.org/wiki/Logical_NAND" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">oeis.org/wiki/Logical_NAND</span><span class="invisible"></span></a><br />Logical NNOR • <a href="https://oeis.org/wiki/Logical_NNOR" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">oeis.org/wiki/Logical_NNOR</span><span class="invisible"></span></a><br />Logical Conjunction • <a href="https://oeis.org/wiki/Logical_conjunction" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Logical_conjunct</span><span class="invisible">ion</span></a><br />Logical Disjunction • <a href="https://oeis.org/wiki/Logical_disjunction" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Logical_disjunct</span><span class="invisible">ion</span></a><br />Exclusive Disjunction • <a href="https://oeis.org/wiki/Exclusive_disjunction" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Exclusive_disjun</span><span class="invisible">ction</span></a><br />Logical Implication • <a href="https://oeis.org/wiki/Logical_implication" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Logical_implicat</span><span class="invisible">ion</span></a><br />Logical Equality • <a href="https://oeis.org/wiki/Logical_equality" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">oeis.org/wiki/Logical_equality</span><span class="invisible"></span></a></p><p><a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="tag">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/LogicSyllabus" class="mention hashtag" rel="tag">#<span>LogicSyllabus</span></a> <a href="https://mathstodon.xyz/tags/LogicalOperator" class="mention hashtag" rel="tag">#<span>LogicalOperator</span></a> <a href="https://mathstodon.xyz/tags/LogicalConnective" class="mention hashtag" rel="tag">#<span>LogicalConnective</span></a><br /><a href="https://mathstodon.xyz/tags/Negation" class="mention hashtag" rel="tag">#<span>Negation</span></a> <a href="https://mathstodon.xyz/tags/NAND" class="mention hashtag" rel="tag">#<span>NAND</span></a> <a href="https://mathstodon.xyz/tags/NNOR" class="mention hashtag" rel="tag">#<span>NNOR</span></a> <a href="https://mathstodon.xyz/tags/LogicalConjunction" class="mention hashtag" rel="tag">#<span>LogicalConjunction</span></a> <a href="https://mathstodon.xyz/tags/LogicalDisjunction" class="mention hashtag" rel="tag">#<span>LogicalDisjunction</span></a><br /><a href="https://mathstodon.xyz/tags/ExclusiveDisjunction" class="mention hashtag" rel="tag">#<span>ExclusiveDisjunction</span></a> <a href="https://mathstodon.xyz/tags/XOR" class="mention hashtag" rel="tag">#<span>XOR</span></a> <a href="https://mathstodon.xyz/tags/LogicalImplication" class="mention hashtag" rel="tag">#<span>LogicalImplication</span></a> <a href="https://mathstodon.xyz/tags/LogicalEquality" class="mention hashtag" rel="tag">#<span>LogicalEquality</span></a></p>
Jon Awbrey<p><a href="https://mathstodon.xyz/tags/LogicSyllabus" class="mention hashtag" rel="tag">#<span>LogicSyllabus</span></a><br />• <a href="https://inquiryintoinquiry.com/logic-syllabus/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/logic-s</span><span class="invisible">yllabus/</span></a></p><p>This page serves as a focal node for a collection of related resources.</p><p><a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="tag">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/LogicalOperators" class="mention hashtag" rel="tag">#<span>LogicalOperators</span></a></p><p><a href="https://mathstodon.xyz/tags/LogicalConjunction" class="mention hashtag" rel="tag">#<span>LogicalConjunction</span></a> <a href="https://mathstodon.xyz/tags/LogicalDisjunction" class="mention hashtag" rel="tag">#<span>LogicalDisjunction</span></a><br /><a href="https://mathstodon.xyz/tags/ExclusiveDisjunction" class="mention hashtag" rel="tag">#<span>ExclusiveDisjunction</span></a> <a href="https://mathstodon.xyz/tags/LogicalEquality" class="mention hashtag" rel="tag">#<span>LogicalEquality</span></a><br /><a href="https://mathstodon.xyz/tags/LogicalImplication" class="mention hashtag" rel="tag">#<span>LogicalImplication</span></a> <a href="https://mathstodon.xyz/tags/LogicalNegation" class="mention hashtag" rel="tag">#<span>LogicalNegation</span></a><br /><a href="https://mathstodon.xyz/tags/LogicalNAND" class="mention hashtag" rel="tag">#<span>LogicalNAND</span></a> <a href="https://mathstodon.xyz/tags/LogicalNNOR" class="mention hashtag" rel="tag">#<span>LogicalNNOR</span></a></p><p><a href="https://mathstodon.xyz/tags/LogicalConcepts" class="mention hashtag" rel="tag">#<span>LogicalConcepts</span></a></p><p><a href="https://mathstodon.xyz/tags/Ampheck" class="mention hashtag" rel="tag">#<span>Ampheck</span></a> <a href="https://mathstodon.xyz/tags/SoleSufficientOperator" class="mention hashtag" rel="tag">#<span>SoleSufficientOperator</span></a><br /><a href="https://mathstodon.xyz/tags/BooleanDomain" class="mention hashtag" rel="tag">#<span>BooleanDomain</span></a> <a href="https://mathstodon.xyz/tags/UniverseOfDiscourse" class="mention hashtag" rel="tag">#<span>UniverseOfDiscourse</span></a><br /><a href="https://mathstodon.xyz/tags/LogicalGraph" class="mention hashtag" rel="tag">#<span>LogicalGraph</span></a> <a href="https://mathstodon.xyz/tags/PropositionalCalculus" class="mention hashtag" rel="tag">#<span>PropositionalCalculus</span></a><br /><a href="https://mathstodon.xyz/tags/MinimalNegationOperator" class="mention hashtag" rel="tag">#<span>MinimalNegationOperator</span></a> <a href="https://mathstodon.xyz/tags/TruthTable" class="mention hashtag" rel="tag">#<span>TruthTable</span></a><br /><a href="https://mathstodon.xyz/tags/BooleanFunction" class="mention hashtag" rel="tag">#<span>BooleanFunction</span></a> <a href="https://mathstodon.xyz/tags/BooleanValuedFunction" class="mention hashtag" rel="tag">#<span>BooleanValuedFunction</span></a></p>
Jon Awbrey<p><a href="https://mathstodon.xyz/tags/DifferentialPropositionalCalculus" class="mention hashtag" rel="tag">#<span>DifferentialPropositionalCalculus</span></a> • 2.5<br />• <a href="https://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-2/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2020/02</span><span class="invisible">/22/differential-propositional-calculus-2/</span></a></p><p>A plus sign \(+\) may be used for <a href="https://mathstodon.xyz/tags/ExclusiveDisjunction" class="mention hashtag" rel="tag">#<span>ExclusiveDisjunction</span></a>, allowing the following equivalents.</p><p>\[\begin{matrix} x + y ~=~ \texttt{(} x \texttt{,} y \texttt{)} \\[6pt] x + y + z ~=~ \texttt{((} x \texttt{,} y \texttt{),} z \texttt{)} ~=~ \texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))} \end{matrix}\]</p><p>But note the last expressions are not equivalent to the triple bracket \(\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.\)</p>
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