A dilemma is a reasoning either of form

(1) if A then B

(2) if C then D

(3) A or C

(4) So, B or D

or form

(1) if A then B

(2) if C then D

(3) not B or not D

(4) So not A or not C

Are there situations in mathematics where a conclusion of the form (X or Y) or the form (not X or not Y) would be interesting? Are there situations in mathematics in which the form of the dilemma argument would be useful? Are there classic examples of mathematical evidence using a dilemma?