The paradoxes of material implication are a group of formulae that are intuitively false but treated as true in systems of logic that interpret the conditional connective as material conditional. On the material implication interpretation, a conditional formula is true unless is true and is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis won World War Two, everybody would be happy" is true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.
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:
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 because although the above is a logically valid argument, it is not sound (not all of its premises are true).
Validity is defined in classical logic as follows:
For example a valid argument might run:
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:
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.
The classical paradox formulae are closely tied to the formula,
the principle of Simplification, which can be derived from the paradox formulae rather easily (e.g. from (1) by Importation). In addition, there are serious problems with trying to use material implication as representing the English "if ... then ...". For example, the following are valid inferences:
but mapping these back to English sentences using "if" gives paradoxes. The first might be read "If John is in London then he is in England, and if he is in Paris then he is in France. Therefore, it is either true that (a) if John is in London then he is in France, or (b) that if he is in Paris then he is in England." Using material implication, if John really is in London, then (since he is not in Paris) (b) is true; whereas if he is in Paris, then (a) is true. Since he cannot be in both places, the conclusion that at least one of (a) or (b) is true is valid.
But this does not match how "if ... then ..." is used in natural language: the most likely scenario in which one would say "If John is in London then he is in England" is if one does not know where John is, but nonetheless knows that if he is in London, he is in England. Under this interpretation, both premises are true, but both clauses of the conclusion are false.
The second example can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that if switch A is closed, the light is on, or that if switch B is closed, the light is on." Here, the most likely natural-language interpretation of the "if ... then ..." statements would be "whenever switch A is closed, the light is on", and "whenever switch B is closed, the light is on". Again, under this interpretation both clauses of the conclusion may be false (for instance in a series circuit, with a light that only comes on when both switches are closed).