Paradox of entailment

The paradox of entailment states that inconsistent premises always make an argument valid.

This is because validity is defined as follows: An argument is valid if and only if there is no possible situation in which all the premises are true and the conclusion false (the premises entail the conclusion).

Example:

• If it is raining, water exists. (1st premise)
• It is raining. (2nd premise)

therefore

• water exists (conclusion)

There is no possible situation in which the premises could be true while the conclusion was false. But note that if one of the premises is false, the conclusion could also be, without affacting the validity of the argument.

A result of this definition is that inconsistent premises - ie premises that cannot all be true in any one situation - always satisfy this definition, and this holds even if the conclusion is false!

Example:

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

therefore

• water exists (conclusion)

As there is no possible situation where all the 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 would be valid, by the definition above.

Note the argument would not be correct, in the sense that its premises and conclusions are all true, but it would be valid.