Topics Logic

Question If the sentence "If p then q" is true, must the sentence "q because p" also be true? For example, "if it is raining, then the streets are wet" and the sentence "The streets are wet because it is raining." Are there any counter-examples where "If p then q" could be true while "q because p" could be false?

Accepted:
December 13, 2012

Comments

The answer to your first question is: No. Let's take your example: Suppose that it is true that it is raining. And suppose that it is true that the streets are wet. Then, by the truth table for the material conditional (which is the default interpretation of the English "if…then…" locution), the sentence "If it is raining, then the streets are wet" is true, because both antecedent and consequent are true. But it might have been the case that the reason that the streets are wet is that someone was cleaning the street with water before the rain began, so that it is false that the streets are wet because it is raining. And there's your counterexample.

The one possible piece of wiggle room would be for someone to claim that the material conditional is not the correct interpretation of "if…then…" in this case.

Even if the conditional isn't material, it's clear that this kind of inference has to fail. Suppose my roof leaks whenever it rains. Then it seems true to say: If my roof is leaking, then the streets are wet. But the streets aren't wet because my roof is leaking. Rather, there is a third cause of both these events. Even if there has to be a "link" between them for the conditional to be true, then, the link needn't be directly causal.