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The duality principle in mathematical logic is a theorem on the acceptability of mutual substitution (in a certain sense) of logical operations in the formulas of formal logical and logical-objective languages. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341301.png" /> be a formula in the language of propositional or predicate logic not containing the implication symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341302.png" />; a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341303.png" /> is said to be dual to a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341304.png" /> if it may be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341305.png" /> by replacing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341306.png" /> each occurrence (cf. [[Imbedded word|Imbedded word]]) of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d0341309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413010.png" /> by their dual operations, i.e. by the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413014.png" />, respectively. The duality principle states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413015.png" /> is true, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413016.png" /> is true as well. In particular, if two formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413018.png" /> are equivalent, their dual formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413020.png" /> are equivalent too. The duality principle is valid for classical systems, and the equivalence and the truth of the formulas involved in its formulation may be understood both in terms of interpretations and in the sense of being deducible in the corresponding classical calculus. The duality principle is no longer valid if the formulas are understood in their constructive sense. For instance, in the language of propositional logic the implication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413021.png" /> is constructively true, and is even deducible in a [[Heyting formal system|Heyting formal system]], but the converse implication of the dual formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413022.png" /> is constructively untrue (it is not Kleene-realizable).

The following theorem is closely connected with the duality principle: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413023.png" /> is a formula dual to a propositional or predicate formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413024.png" /> constructed without making use of implications from the elementary propositions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413025.png" />, then the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413026.png" /> is equivalent to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413027.png" /> in the classical propositional or predicate calculus, respectively.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Novikov,   "Elements of mathematical logic" , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>

''F.A. Kabakov''

The duality principle in geometry is a principle formulated in certain fields of geometry, according to which the replacement in any true proposition of all concepts occurring in it by concepts dual to them results in another true proposition, dual to the first.

The validity of the duality principle in projective geometry follows from the fact that to each axiom of projective geometry corresponds a dual proposition which is either an axiom or a theorem.

The following concepts of projective geometry in the plane are dual:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">point</td> <td colname="2" style="background-color:white;" colspan="1">straight line</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">point, incident with a straight line</td> <td colname="2" style="background-color:white;" colspan="1">straight line, incident with a point</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">algebraic curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413028.png" /></td> <td colname="2" style="background-color:white;" colspan="1">algebraic bundle of straight lines of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413029.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">straight tangent line to a curve</td> <td colname="2" style="background-color:white;" colspan="1">characteristic point of a bundle</td> </tr> </tbody> </table>

</td></tr> </table>

If the fact that a point forms part of a second-order curve is considered to be an incidence relation between the point and the curve, while the fact that a straight line is tangent to a second-order curve is considered to be an incidence relation between the straight line and the curve, then second-order curve is a concept dual to that of curve of the second class. The [[Brianchon theorem|Brianchon theorem]] and the [[Pascal theorem|Pascal theorem]] are an example of a pair of dual propositions. The concepts of a point and a plane in three-dimensional space are dual in projective geometry; the concept of a straight line is dual to itself.

The duality principle is also valid in elliptic geometry, in which the concept of a segment and an angle are dual in addition to those in projective geometry. For instance, the following two statements are dual in elliptic geometry:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Two triangles are congruent if the three sides of one triangle are respectively equal to three sides of the other</td> <td colname="2" style="background-color:white;" colspan="1">Two triangles are congruent if the three angles of one triangle are respectively equal to three angles of the other</td> </tr> </tbody> </table>

</td></tr> </table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Efimov,   "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR></table>

''A.S. Parkhomenko''

====Comments====
Both statements in the main article above concerning triangles in elliptic geometry are false as they stand. They become true if one adds that the triangles must be of the same topological type, cf. [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Artin,   "Geometric algebra" , Interscience  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pedoe,   "A course of geometry" , Cambridge Univ. Press  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''1–2''' , Springer  (1987)  (Translated from French)</TD></TR></table>

The duality principle in projective geometry means that to each theorem about subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413030.png" /> of a [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413031.png" />, their intersections and sums there corresponds a theorem concerning the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413032.png" /> their sums and intersections. The duality principle is determined by the dual nature of the axioms of projective geometry and the theorems which follow from them. For a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413033.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413034.png" /> the duality principle is valid if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413035.png" /> permits an anti-automorphism. In the general case there is duality between the projective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413037.png" /> whose skew-fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413039.png" /> are anti-isomorphic; examples are the projective spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413042.png" /> (cf. [[Projective algebra|Projective algebra]]; [[Correlation|Correlation]]), and the correspondence between them, i.e. the correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413044.png" />, is determined by the choice of a pair of coordinate systems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413046.png" />. The duality principle can also be based on a dual mapping of the linear spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413047.png" /> over a skew-field, which is used in the interpretation of projective spaces.

''M.I. Voitsekhovskii''

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Veblen,   J.W. Young,   "Projective geometry" , '''1''' , Ginn  (1938)</TD></TR></table>

The duality principle in partially ordered sets means that if some theorem on partially ordered sets, couched in general logical terms and in terms of the order, is true, then its dual theorem is also true. In order to obtain a theorem dual to a given theorem, all propositions and concepts concerned with the order are replaced by their dual ones (i.e. all order signs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413048.png" /> are replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034130/d03413049.png" /> and vice versa), while the general logical terms remain unchanged. The truth of a given proposition concerning a given partially ordered set (or a given class of ordered sets) does not necessarily entail the truth of the dual proposition for this set (class). Thus, a partially ordered set may have a smallest, but not a largest element; it may satisfy the minimality, but not the maximality condition. The truth of the duality principle stems from the fact that the relation inverse to a partial order is itself a partial order; the duality principle itself is sometimes understood to mean this very proposition.

''T.S. Fofanova''

====Comments====
A classical example is given by the family of subsets of a set, partially ordered by inclusion.

There is yet another duality principle, generalizing that for partially ordered sets:

5) The duality principle in category theory means that if some general assertion about categories and functors is true, then so is the dual assertion obtained by reversing the direction of the arrows within each category involved (though not the direction of the functors). For example, the theorem that a left adjoint functor is faithful if and only if the unit of the adjunction is monic, is dual to the theorem that a right adjoint is faithful if and only if the co-unit is epic. (See also [[Category|Category]].)