Paradoxes of material implication

The paradoxes of material implication are a group of formulae that are truths of classical logic but are intuitively problematic.

The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication in natural language, and its formal interpretation in classical logic, dating back to George Boole's algebraic logic. In classical logic, implication describes conditional if-then statements using a truth-functional interpretation, i.e. "p implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional.

The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given formally as follows:

The paradoxes of material implication arise because of the truth-functional definition of material implication, which is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All paraconsistent logics must, by definition, reject (1) as invalid.) Also, any tautology is implied by anything whatsoever, since a tautology is always true.

To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material implication does not capture the meaning of "if... then".

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