Destructive dilemma

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In logic, a 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

$P \rightarrow Q$

$R \rightarrow S$

$\neg Q \lor \neg S$

$\vdash \neg P \lor \neg R$

where $\vdash$ 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 Kucinich runs for President, he will be President.

Either Howard Dean will not be President or Dennis Kucinich will not be President.

Therefore, either Howard Dean doesn't run for President or Dennis Kucinich doesn't 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.

$\lnot(p\land\lnot p)$