A valid form of logical inference in propositional logic, which infers from two conditional and a negative disjunct statement a new negative disjunct statement.
Formally, the destructive dilemma has three premises, it looks as follows:
Premise 1:A → B– (if A, then B)
Premise 2:C → D– (if C, then D)
Premise 3:⌐B ∨ ⌐D– (not B or not D)
Conclusion:⌐A ∨ ⌐C– (not A or not C)
The destructive dilemma can be seen as a combination of two Modus Tollens, which are connected by a disjunct statement.
The term “dilemma” in this context should be understood as a “decision” between two conditionals.
An example for a destructive dilemma could be:
If the sun shines tomorrow, [then] we will go to the beach.
If it rains tomorrow, [then] we will go to the museum.
Tomorrow we will either not got to the museum or not go to the beach [or both].
Therefore it will either not rain or the sun will not shine [or both].