Logical hexagon

<h2 id="summary-of-relationships">Summary of relationships</h2>
<p>The traditional square of opposition demonstrates two sets of contradictories A and O, and E and I (i.e. they cannot both be true and cannot both be false), two contraries A and E (i.e. they can both be false, but cannot both be true), and two subcontraries I and O (i.e. they can both be true, but cannot both be false) according to Aristotle’s definitions. However, the logical hexagon provides that U and Y are also contradictory.
</p>
<h2 id="interpretations">Interpretations</h2>
<p>The logical hexagon may be interpreted in various ways, including as a model of <a href="/facts/Traditional_logic/Ke2jxPC0">traditional logic</a>, <a href="/facts/Quantifier_(logic)/bgmvwxoo">quantifications</a>, <a href="/facts/Modal_logic/I7uXvYYT">modal logic</a>, <a href="/facts/Order_theory/gn6fM3oJ">order theory</a>, or <a href="/facts/Paraconsistent_logic/6HLVEvrr">paraconsistent logic</a>.
</p><p>For instance, the statement A may be interpreted as "Whatever x may be, if x is a man, then x is white."
</p>
   (x)(M(x) → W(x))

<p>The statement E may be interpreted as "Whatever x may be, if x is a man, then x is non-white."
</p>
   (x)(M(x) → ~W(x))

<p>The statement I may be interpreted as "There exists at least one x that is both a man and white."
</p>
   (∃x)(M(x) & W(x))

<p>The statement O may be interpreted as "There exists at least one x that is both a man and non-white."
</p>
   (∃x)(M(x) & ~W(x))

<p>The statement Y may be interpreted as "There exists at least one x that is both a man and white and there exists at least one x that is both a man and non-white."
</p>
   (∃x)(M(x) & W(x)) & (∃x)(M(x) & ~W(x))

<p>The statement U may be interpreted as "One of two things, either whatever x may be, if x is a man, then x is white or whatever x may be, if x is a man, then x is non-white."
</p>
   (x)(M(x) → W(x)) w (x)(M(x) → ~W(x))

<h3>Modal logic</h3>
<p>The logical hexagon may be interpreted as a model of modal logic such that
</p>
<ul><li>A is interpreted as <a href="/facts/Necessary_and_sufficient_condition/G45aDCY9">necessity</a></li>
<li>E is interpreted as impossibility</li>
<li>I is interpreted as <a href="/facts/Logical_possibility/jGJ6gH7d">possibility</a></li>
<li>O is interpreted as non-necessity</li>
<li>U is interpreted as non-contingency</li>
<li>Y is interpreted as <a href="/facts/Contingency_(philosophy)/cCyDDBmw">contingency</a></li></ul>
<h2 id="further-extension">Further extension</h2>
<p>It has been proven that both the square and the hexagon, followed by a “<a href="/facts/Logical_cube/SAxwxhfs">logical cube</a>”, belong to a regular series of n-dimensional objects called “logical bi-simplexes of dimension n.” The pattern also goes even beyond this.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lambda_cube/tSR2laCs">Lambda cube</a></li>
<li><a href="/facts/Logical_cube/SAxwxhfs">Logical cube</a></li>
<li><a href="/facts/Square_of_opposition/z9S71UVr">Square of opposition</a></li>
<li><a href="/facts/Triangle_of_opposition/2F9pWjEG">Triangle of opposition</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="http://www.unine.ch/unilog/jyb/sep.pdf">Jean-Yves Béziau, New Light on the Square of Oppositions and its Nameless Corner</a></li>
<li><a href="/facts/Jean-Yves_Beziau/7h5dFLwI">Jean-Yves Beziau</a> (2012), "The power of the hexagon", <i><a href="/facts/Logica_Universalis/N6xglbGD">Logica Universalis</a></i> 6, 2012, 1-43. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs11787-012-0046-9">10.1007/s11787-012-0046-9</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>N-opposition theory logical hexagon <a href="http://alessiomoretti.perso.sfr.fr/NOTLogicalHexagon.html" target="_blank">http://alessiomoretti.perso.sfr.fr/NOTLogicalHexagon.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Moretti, Alessio. "The oppositional cube (or logical cube)". N-Opposition Theory: Oppositional Geometry—Homepage. Archived from the original on 2014-08-08. <a href="https://web.archive.org/web/20140808034620/http://alessiomoretti.perso.sfr.fr/NOTLogicalCube.html" target="_blank">https://web.archive.org/web/20140808034620/http://alessiomoretti.perso.sfr.fr/NOTLogicalCube.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Logical hexagon open-in-new

Logical hexagon