Paradox of entailment

The paradox of entailment is an apparent paradox derived 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 following is a good argument:

It is raining
It is not raining

Therefore:

George Washington was a zombie.

Contents:
1. Understanding the paradox
2. Explaining the paradox
3. References
4. See also

1. Understanding the paradox

Validity is 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 an 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 42 (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.

2. Explaining the paradox

The strangeness of the paradox of entailment comes from the fact that the definition of validity in classical logic does not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. Other suggested improvements to the notion of logical validity include strict implication and relevant implication.

3. References

4. See also

* Correlation does not imply causation
* False dilemma