Duality

Definition 12.19 (Dual). The dual of a statement formula is obtained by replacing every $\vee$ with $\wedge$, every $\wedge$ with $\vee$, every $\mathbb T$ with $\mathbb F$, and every $\mathbb F$ with $\mathbb T$.

Remarks.

1. The negation symbol $\neg$ is never changed while forming the dual.
2. The dual of a dual is the original statement itself (applying the swap twice undoes it).
3. $\mathbb T$ (tautology) and $\mathbb F$ (contradiction) are duals of each other.
4. Every $\mathbb T$ becomes $\mathbb F$ and vice versa.

Principle of Duality. If a compound statement $S_1$ contains only $\neg,\wedge,\vee$ (no $\to$ or $\leftrightarrow$), and $S_2$ is obtained from $S_1$ by swapping $\wedge\leftrightarrow\vee$, then $S_1$ is a tautology if and only if $S_2$ is a contradiction.

Worked patterns.

• The dual of $(p\vee q)\wedge(r\vee\mathbb F)$ is $(p\wedge q)\vee(r\wedge\mathbb T)$ -- every $\vee\leftrightarrow\wedge$ and $\mathbb F\to\mathbb T$.
• The dual of $p\wedge[\neg q\vee(p\wedge q)\vee\neg r]$ is $p\vee[\neg q\wedge(p\vee q)\wedge\neg r]$ -- note $\neg q$ and $\neg r$ are left completely untouched, only the $\wedge/\vee$ around them flip. …