The structure and intuition behind the logical duality principle

Question

It is my first time exposure to duality, and it bothers me that the book i'm using takes this theorem for granted and only says that it is a clear fact from truth tables.

I want to know Concrete and also abstract examples of duality and also how it relates to the concepts of "And" and "Or"?

Answer 1

You can nicely see the duality between And and Or in the long list of pairs of equivalence principles involving them: every time some equivalence holds involving And and Or, there is always a dual principle that also holds when systematically swapping them:

Commutation

$P \land Q \Leftrightarrow Q \land P$

$P \lor Q \Leftrightarrow Q \lor P$

Association

$P \land (Q \land R) \Leftrightarrow (P \land Q) \land R$

$P \lor (Q \lor R) \Leftrightarrow (P \lor Q) \lor R$

DeMorgan

$\neg(P \land Q) \Leftrightarrow \neg P \lor \neg Q$

$\neg(P \lor Q) \Leftrightarrow \neg P \land \neg Q$

Distribution

$P \land (Q \lor R) \Leftrightarrow (P \land Q) \lor (P \land R)$

$P \lor (Q \land R) \Leftrightarrow (P \lor Q) \land (P \lor R)$

Absorption

$P \land (P \lor Q) \Leftrightarrow P$

$P \lor (P \land Q) \Leftrightarrow P$

Reduction

$P \land (\neg P \lor Q) \Leftrightarrow P \land Q$

$P \lor (\neg P \land Q) \Leftrightarrow P \lor Q$

Adjacency

$P \Leftrightarrow (P \lor Q) \land (P \lor \neg Q)$

$P \Leftrightarrow (P \land Q) \lor (P \land \neg Q)$

Consensus

$(P \lor Q) \land (\neg Q \lor R) \land (P \lor R) \Leftrightarrow (P \lor Q) \land (\neg Q \lor R)$

$(P \land Q) \lor (\neg Q \land R) \lor (P \land R) \Leftrightarrow (P \land Q) \lor (\neg Q \land R)$