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 [or neither]
Conclusion:⌐A ∨ ⌐C–not A or not C [or neither]
A practical example could be the following:
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 neither].
Therefore it will either not rain or the sun will not shine [or neither].
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.
The relationships between the various statements in a constructive dilemma can best be explained by showing them as a diagram: