"If P, then Q. Q. Therefore, P."

I have taught this as a fallacy for the last two semesters in a propositional logic section of my course. However, given the prevalence of this fallacy, as well as the observed resistance of students to assimilating it, I would like to propose that this really isn't a fallacy but a disconnect between what we often naturally mean by "p implies q" and what it technically means in propositional logic.

I argue that what we tend to understand by the statement "p implies q" is that p is the set of things that result in q. This can be tested by the fact that if Q is known to have two causes P1 and P2, and someone says that P1 implies Q, it would be rather natural and logical for another person to "correct" them by adding that P2 also causes Q. But the word correct is too strong ... it is rather an augmentation.

I think that, like many statements in ordinary usage, "P implies Q" is a fundamentally Bayesian statement; a person means: "in my experience, P is the set of things that cause Q". The 'in my experience' means that it is possible (but not observed) that Q might not happen after P, and also that other things besides P might cause Q, but again this hasn't been observed or isn't recalled at the moment. A person is normally willing to be flexible on both points in the face of new evidence, but is taught in logic class to be absolute for a moment while considering the truth or falsity of the statement.

So finally, when an untrained person is told that "p implies q" is a true logical statement, in their efforts to be "absolute" they may think that both implied meanings are true: p always causes q AND only p causes q. It takes some training (deprogramming of what it has seemed to mean their whole natural lives) to understand that only the first is intended (p always causes q) when the statement is "True" in formal logic.