A logical Destructive Dilemma is a formal logical argument that takes the form:

1a) P → Q.

b) R → S.

2) Either not-Q or not-S is true.

Therefore, either not-P or not-R is true.

In logical operator notation

where represents the logical assertion.

In sum, this argument states that if a first premise implies one conclusion, and a second premise implies a second, separate conclusion, and if one of the conclusions must be false, one of the premises must be false.

If Howard Dean runs for president, he will be President.

If Dennis Kuchinich runs for president, he will be President.

Either Howard Dean or Dennis Kuchinich is not president.

Therefore, either Howard Dean or Dennis Kucinich did not run for president.

The Dilemma derives its name in part because it is the contrapositive of the Constructive Dilemma . Because either Q or S is false, either P or R must be false. In this example, the statement provides limitations because Dean or Kucinich is not president, the possibility that both of them ran for President would create a contradiction.