Self-dual propositions

Problems

"When does s* = s, where s is a compound proposition"

The dual of a compound proposition that contains only the logical operators AND, OR, and NOT is the compound proposition obtained by replacing each AND by OR, each OR by AND, each T by F, and each F by T. The dual of s is denoted by s*

Solution

Let s = P(a₁, a₂, a₃,…, a_n) where a_i is a single proposition, and P is the relationship among a_i.

So s* = (¬P)(a₁, a₂, a₃,…, a_n) where (¬P) is the inverse relationship of P

s* = ¬(¬(¬P)(a₁, a₂, a₃,…, a_n) ) double negating

s* = ¬( P(¬a₁,¬ a₂,¬ a₃,…,¬ a_n) )

If s*= s

Then ¬P(a₁, a₂, a₃,…, a_n) = P(¬a₁,¬ a₂,¬ a₃,…,¬ a_n)

It means when we inverse every single proposition in s, if the truth value of s is also inversed, proposition s is self-dual (or s*=s)