Duality principle

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 $ A $ be a formula in the language of propositional or predicate logic not containing the implication symbol $ \supset $; a formula $ A ^ {*} $ is said to be dual to a formula $ A $ if it may be obtained from $ A $ by replacing in $ A $ each occurrence (cf. Imbedded word) of the symbols $ \& $, $ \lor $, $ \forall $, $ \exists $ by their dual operations, i.e. by the symbols $ \lor $, $ \& $, $ \exists $, and $ \forall $, respectively. The duality principle states that if $ A \supset B $ is true, then $ B ^ {*} \supset A ^ {*} $ is true as well. In particular, if two formulas $ A $ and $ B $ are equivalent, their dual formulas $ A ^ {*} $ and $ B ^ {*} $ 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 The duality principle is no longer valid if the formulas are understood in their constructive sense.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 $ \neg A \& \neg B \supset \neg ( A \lor B ) $ is constructively true, and is even deducible in a Heyting formal system, but the converse implication of the dual formula $ \neg ( A \& B ) \supset \neg A \lor \neg B $ is constructively untrue (it is not Kleene-realizable).

The following theorem is closely connected with the duality principle: If $ F ^ { * } ( A _ {1} \dots A _ {n} ) $ is a formula dual to a propositional or predicate formula $ F ( A _ {1} \dots A _ {n} ) $ constructed without making use of implications from the elementary propositions $ A _ {1} \dots A _ {n} $, then the formula $ \neg F ( A _ {1} \dots A _ {n} ) $ is equivalent to the formula $ F ^ { * } ( \neg A _ {1} \dots \neg A _ {n} ) $ in the classical propositional or predicate calculus, respectively.

References

[1]	P.S. Novikov, "Elements of mathematical logic" , Addison-Wesley (1964) (Translated from Russian)
[2]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)

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:

<tbody> </tbody> point straight line point, incident with a straight line straight line, incident with a point algebraic curve of order $ n $ algebraic bundle of straight lines of class $ n $ straight tangent line to a curve characteristic point of a bundle

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 and the 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:

<tbody> </tbody> Two triangles are congruent if the three sides of one triangle are respectively equal to three sides of the other Two triangles are congruent if the three angles of one triangle are respectively equal to three angles of the other

References

[1]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)

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. [a3].

References

[a1]	E. Artin, "Geometric algebra" , Interscience (1957)
[a2]	D. Pedoe, "A course of geometry" , Cambridge Univ. Press (1970)
[a3]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)

The duality principle in projective geometry means that to each theorem about subspaces $ S _ {a} , S _ {b} \dots $ of a projective space $ \Pi _ {n} $, their intersections and sums there corresponds a theorem concerning the subspaces $ S _ {n-} a- 1 , S _ {n-} b- 1 \dots $ 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 $ \Pi _ {n} ( X) $ over a skew-field $ K $ the duality principle is valid if and only if $ K $ permits an anti-automorphism. In the general case there is duality between the projective spaces $ \Pi _ {n} ( K) $ and $ \Pi _ {n} ( K ^ {*} ) $ whose skew-fields $ K $ and $ K ^ {*} $ are anti-isomorphic; examples are the projective spaces $ P _ {n} ^ {1} ( K) $ and $ P _ {n} ^ {r} ( K) $ over $ K $( cf. Projective algebra; Correlation), and the correspondence between them, i.e. the correspondence between $ S _ {k} $ and $ S _ {n-} k- 1 $, is determined by the choice of a pair of coordinate systems in $ \Pi _ {n} ( K) $ and $ \Pi _ {n} ( K ^ {*} ) $. The duality principle can also be based on a dual mapping of the linear spaces $ V _ {n+} 1 ( K) $ over a skew-field, which is used in the interpretation of projective spaces.

M.I. Voitsekhovskii

References

[a1]	O. Veblen, J.W. Young, "Projective geometry" , 1 , Ginn (1938)

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 $ \leq $ are replaced by $ \geq $ 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.)