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Paradox of entailment

From Academic Kids

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:

  • 2+2=4
  • 2+2=5

therefore

Obviously, this is a bad argument.

Valid arguments

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.

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 at least one of the premises is false, the conclusion can also be, while the argument is still valid. The point is that there is no counterexample: no case with true premises and a false conclusion.

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.

Example:

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

therefore

  • 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.

(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.)

Assessment

The paradox of entailment is perfectly true; it is what Quine has called a "veridical" paradox. Its seeming strangeness 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. Suggested improvements to the notion of logical validity include strict implication and relevant implication.

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