Whenever we teach about fallacies involving arguments with conditional propositions, we usually tell students about two: denying the antecedent and affirming the consequent. I would suggest that it would make at least a little bit more sense to identify three, the third being the following:
If p, q
If p, r
Therefore, If q, r.
Why add this one, which (as far as I know) has no name? The reason is that it brings out more clearly an important and often overlooked parallel, namely, that every argument using conditional propositions has a corresponding categorical syllogism using A propositions. This has been something recognized off and on throughout the history of logic. But it was Jevons in the nineteenth century, as far as I am aware, who first recognized that whenever the conditional version of the argument committed the fallacy of affirming the consequent, the corresponding categorical version committed the fallacy of the undistributed middle. Similarly, whenever the conditional argument committed the fallacy of denying the antecedent, the corresponding categorical version committed the fallacy of the illicit process of the major term.
And this is where the third fallacy, above, comes in. For there is a third fallacy having to do with the distribution of terms in a categorical syllogism, namely, illicit process of the minor term. And, as you may have guessed, the above argument is the fallacious conditional argument corresponding to a categorical syllogism that commits that fallacy. As I said, I don't know of any name that goes with it; but when you think about it, it does occur quite a bit in real life. We could call it "false chaining" or something like that.
