if two compound propositions are equivalent, their duals are also equivalent. This duality principle is a useful tool for proving logical equivalences and analyzing logical relationships.
The duals of two equivalent compound propositions are also equivalent because every logical equivalence for compound propositions containing only ∧, ∨, and ¬ has a dual equivalent. This means that when you apply the dual operation (swapping ∧ and ∨, F and T, and negating variables) to both sides of an equivalence, you still get an equivalence.
Here's why:
1. Dual Operators and Variables:** The dual operation preserves truth values. For example, the dual of ¬(p ∨ q) is ¬(p ∧ q), and both expressions have the same truth table.
2. Equivalences Remain True Under Dual: If two compound propositions p and q are equivalent (i.e., have the same truth table), then their duals p∗ and q∗ will also have the same truth table. This is because applying the dual operation to both sides of the equivalence p ⇔ q simply reverses the positions of the operators and variables without changing the underlying logic.
3. Equivalences Imply Dual Equivalences:If p ⇔ q is a logical equivalence, then p∗ ⇔ q∗ is also a logical equivalence. This is because any truth assignment that makes p and q equivalent will also make p∗ and q∗ equivalent.