Paradox Compilation

Paradox of Entailment

As the best known of the paradoxes, and most formally simple, the paradox of entailment makes the best introduction.

In natural language, an instance of the paradox of entailment arises:

It is raining
And

It is not raining
Therefore

Water exists.
This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical, as it suggests that the above is a valid argument.

Understanding the paradox

Validity defined in classical logic as follows:

An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false.
For example a valid argument might run:

If it is raining, water exists (1st premise)
It is raining (2nd premise)
Water exists (Conclusion)
In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid.

But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid argument since there would be no possible situation in which all the premises are true and therefore no possible situation in which all the premises are true and the conclusion is false.

For example an argument with inconsistent premises might run:

Matter has mass (1st premise; true)
Matter does not have mass (2nd premise; false)
All numbers are equal to 12 (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 the argument is valid whatever the conclusion is; inconsistent premises imply all conclusions.