Example of mathematical reasoning that is an instance of dilemma? A dilemma is a reasoning either of the form (1) if A then B (2) if C then D (3) A or C (4) therefore, B or D or of the form (1) if A then B (2) if C then D (3) not B or not D (4) therefore not A or not C Are there situations in mathematics where a conclusion of the form (X or Y) or of the forme ( not X or not Y) would be interesting? Are there situations in mathematics in which the dilemma argument form would be usefull? Are there classical examples of mathematical proofs using dilemma?

p: There exists two irrational $a,b$ such that $a^b$ is rational.

q: $\sqrt{2}^\sqrt{2}$ is rational

r: $\left (\sqrt{2}^\sqrt{2} \right)^\sqrt{2}$ is rational

q $\Rightarrow$ p

($\lnot$ q) $\Rightarrow$ (r $\Rightarrow$ p)

q or ($\lnot$q) therefore p.