Definition
The material conditional is classical logic's truth-functional rendering of if...then. Written A → B, it is defined to be false in exactly one case — when the antecedent A is true and the consequent B is false — and true in all three other cases.
This makes the material conditional automatically true whenever its antecedent is false, and automatically true whenever its consequent is true, regardless of any connection between the two. That feature is what makes it both mathematically convenient and philosophically troubling.
Why it matters
How it works
The paradoxes are easy to display. Since a false antecedent makes the whole conditional true, "If the moon is made of cheese, then 2 plus 2 is 5" counts as true. Since a true consequent does the same, "If it is raining, then 2 plus 2 is 4" is also true. Both clash with how if...then is normally heard.
Priest treats these results as a genuine problem rather than a curiosity. They suggest that ordinary conditionals demand a real connection between antecedent and consequent — something truth-functionality cannot supply. This is one of the motivations for relevance logic, which redesigns the conditional to require exactly that connection.