Paradox of entailment

The paradox of entailment is an apparent paradox derived from the observation that, in classical logic, inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical, as it suggests that the following is a good argument:

Inconsistent premises

A result of this definition is that inconsistent premises - i.e. premises that cannot all be true in any one situation - always satisfy this definition, regardless of the conclusion. If there is no situation where the premises are all true, then obviously there is no situation in which the premises are all true AND the conclusion is false. Thus there is (by definition) no counterexample: the argument is valid.

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Example:

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• It is raining (1st premise)
• It is not raining (2nd premise)

therefore

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• water exists (conclusion)

As there is no possible situation where both premises could be true, then there is certainly no possible situation in which the premises could be true while the conclusion was false. So whatever the conclusion, the argument is valid; inconsistent premises imply anything at all.

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(Note the argument would not be sound, in the sense that its premises and conclusion are all true and the conclusion follows from the premises, but it would be valid.)

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