Logical duality

A curious symmetry sits at the very heart of logic

I should like to call attention to a certain symmetry to be found in the logical relation between “and” and “or.” One begins to see it already in the familiar De Morgan laws:

\( p ∧ q\ ≡ \ ¬(¬p ∨ ¬q)\qquad\qquad p ∨ q\ ≡ \ ¬(¬p ∧ ¬q). \)

In plain language, these laws express respectively that two statements p and q are both true exactly when neither is false, and at least one is true exactly when not both are false.

The curious symmetry I have in mind with these identities is their common algebraic form. In each case on the right-hand side we perform the same kind of calculation—we first negate the propositional atoms, then apply one of the connectives, and finally negate the result. The surprise is that by doing so with either operator we achieve the same final result we would have gotten with a direct application of the other operator. With this strange inside-out doubly negated process we thus transform “and” into “or” and vice versa.

This relationship is precisely what it means to say that conjunction and disjunction are dual logical operators.

And we can undertake the same process with any logical connective—for any binary connective ○* we define the dual connective ○* by: