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"?

Duality in Boolean Logic or Propositional Calculus is "built in" through the semantics of the connecetives: see truth table. The t-t for $\lnot (A \lor B)$ has the same computed output of that for $(\lnot A \land \lnot B)$. — Mauro ALLEGRANZA, Sep 01 '17 at 09:29
You can see Boolean algebra : Duality principle and the related post: is-the-duality-principle-of-boolean-algebra-always-true ? — Mauro ALLEGRANZA, Sep 01 '17 at 09:33
@MauroALLEGRANZA Thank you, As I can understand You mean the Demorgan's Laws are actualy a defintion and not a result? — Logan Luther, Sep 01 '17 at 09:38
NO; they are a result of the basic "semantical def" via truth table. We satrt with the t-t for $\land, \lor$ and $\lnot$ and we compute the t-t for $\lnot (A \lor B)$ and $(\lnot A \land \lnot B)$ and we verify that they have the same output. — Mauro ALLEGRANZA, Sep 01 '17 at 09:45

score 1 · Answer 1 · answered Sep 01 '17 at 13:28

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

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

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

The structure and intuition behind the logical duality principle

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