What is the name of the fallacy characterized by "All A are B; therefore all B are A"?

What is the name of this fallacy? I can't seem to find any reference to it on the 'net. It may seem too simple to be a common fallacy, but I don't find that to be the case.

For instance, Hume (it can be argued) says that the only things that are right are things we prefer. A possible objection to this might be, "So then you're claiming that everything someone wants or prefers is right?" An answer to that, of course, is that just because everything that is right is something we prefer, that does not imply that everything we prefer is right.

I'm not interested in getting into a discussion on Hume; I'm just trying to lay out a case where this fallacy might come into play.

• You might need to be careful in your interpretation here. "Only things that are right are things we prefer" and "Everything we prefer is right" are logically equivalent. "Only X is Y" is equivalent to "All Y is X". Commented Dec 18, 2015 at 16:50
• Its intriguing how close that "logic" gets to induction. That might be why it's so hard for people. Commented Dec 18, 2015 at 21:47
• "Only things that are right are things we prefer" is not the same thing as "the only things that are right are things we prefer." The second one says 'everything that is right is something we prefer'; the first says 'everything we prefer is right.' That's just precisely the distinction I want to make here. Commented Dec 18, 2015 at 22:05
• @KevinHolmes is right, but his comment is irrelevant and apparently also confusing here. It simply doesn't have anything to do with the question.
– user2953
Commented Dec 18, 2015 at 22:17

Your example would be an example of an invalid conversion of a categorical proposition in A-form (All A are B). Such conversions are valid in E and I forms (E-form: No A are B; I-form: Some A are B), but not in A-form. I don't know about a particular term to label this fallacy, except converse error (or a categorical converse fallacy), but all the terms suggested in Keelan's answer sounds suitable.

To me, a set formulation of this is most clear, where A ⊆ B ↛ B ⊆ A.

This is called affirming the consequent:

Affirming the consequent, sometimes called converse error, fallacy of the converse or confusion of necessity and sufficiency, is a formal fallacy of inferring the converse from the original statement. The corresponding argument has the general form:

1. If P, then Q.
2. Q.
3. Therefore, P.
• This is kind of a pet peeve of mine, but should universal affirmative propositions really be glossed as material conditionals like that? Commented Dec 18, 2015 at 16:54
• @KevinHolmes this is a quote. I can't help what is written there. Anyway, there's an obvious relationship between a predicate P(x) = x ∈ P and the set P = { x | P(x) }. Here, P and Q are simply instances of predicates P(x) and Q(x) for particular x, which shows that the fallacy described in the citation also applies in the case of the OP.
– user2953
Commented Dec 18, 2015 at 17:17
• As KH implies, this does not really match the question. For instance, the "therefore" does not occur in the same logical place here as there. Commented Dec 18, 2015 at 18:35
• @JeffY yes, it does. The set A in the question is equivalent with the membership predicate P(x) of which P is a particular instance, and similarly with B and Q. I explained this already above. If you think something is incorrect, you should point out how it is incorrect instead of simply stating that it is incorrect - that isn't really constructive.
– user2953
Commented Dec 18, 2015 at 18:36
• Just do the correspondence between notations: "All A are B" <==> if (x∈A), then (x∈B); "All B are A" <==> if (x∈B), then (x∈A). There is only one "if ... then" construct in this answer, therefore it does not correspond precisely to the question. (The added "gloss" to which KH refers is going (silently) from "2.Q;3.Therefore, P." to "Therefore if Q, then P.") Commented Dec 18, 2015 at 18:43