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Is there a name for the following false syllogism?
A implies B
and
B implies A
therefore
A
For example: If unicorns exist then they have horns, by the definition of a unicorn. But in order for them to have horns they must surely exist. We must therefore conclude that unicorns do in fact exist.
It seems some variance of the Circular Reasoning.
Begging the question (Latin petitio principii, "assuming the initial point") is a type of logical fallacy in which a proposition is made that uses its own premise as proof of the proposition. In other words, it is a statement that refers to its own assertion to prove the assertion. Such arguments are essentially of the form "a is true because a is true" though rarely is such an argument stated as such. Often the premise 'a' is only one of many premises that go into proving that 'a' is true as a conclusion.
(A <=> B) => A?
Jun 26, 2012 at 17:58
Maybe it could be considered a fallacy of misplaced concreteness (reification), that is, treating an abstract belief or hypothetical construct as if it were a real, concrete event or physical entity.
http://en.wikipedia.org/wiki/Reification_(fallacy)#Fallacy_of_misplaced_concreteness
It would be hasty to call the circular reasoning a 'fallacy'. Because, one can in similar vein play the following game back-and-forth:
Alice: Why is circular reasoning a 'fallacy'?
Bob: What do you mean by the term 'fallacy'?
Alice: By 'fallacy', I meant "A statement or an argument based on a false or invalid inference" (fallacy)
Bob: But, how do you infer that circular reasoning is "based on a false or invalid inference"?
...
And so on...
As it turns out, non-well founded set theory, and in particular Aczel's anti-foundation axiom can be used to model circular reasoning. For an excellent intro to the subject and situation semantics, I refer to the following reference:
Jon Barwise and John Etchemendy (1987) The Liar. Oxford University Press.
Also, you may want to look at bisimulation.
Note: I am denying that it is a fallacy. But, we should be aware that if we label it a fallacy, then we are using a process to do so based on our current understanding of laws of universe to define it a fallacy.
Be careful how you state your example. I'll state it more carefully, in two different ways:
Your statement "they have horns" doesn't make clear whether you meant the variant 1B or 2B.
If you meant 1B: The statement "all unicorns have horns" is true by the definition of a unicorn (a horse-like creature with a horn), independent of the existence of unicorns. Unfortunately, for exactly this reason "All unicorns have horns" does not imply that unicorns exist.
If you meant 2B: A implies B and B implies A are both correct. However, the only logical conclusion we can draw is not that A is true, but only that A implies A. Which is a perfectly correct but useless statement. It shows that if unicorns exist, then unicorns exist. Well, that's called circular reasoning.
The basic problem with the argument is that the premises are irrelevant to the conclusion. Irving Copi divides informal fallacies into those of relevance and those of ambiguity. This argument is not ambiguous. It is all too clear that the conclusion doesn't follow.
Reification does not fit because in reification one is not concluding that something exists out of nowhere, such as A in this example, but one accepts that A exists as an abstract idea and the argument claims it is concrete.
The same goes for circular reasoning. Although the premises look circular what makes an argument circular reasoning is that one of the premises already claims A exists. Neither of these premises go that far.
A name for such a fallacy might be simply "non sequitur" although such a name seems very broad. Bo Bennett describes it as
(also known as: derailment, “that does not follow”, irrelevant reason, invalid inference, non-support, argument by scenario [form of], false premise [form of], questionable premise [form of], non-sequitur)
Description: When the conclusion does not follow from the premises. In more informal reasoning, it can be when what is presented as evidence or reason is irrelevant or adds very little support to the conclusion.
Bennet, B. Non Sequitur. Retrieved on June 14, 2019 from Logically Fallacious at https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/136/Non-Sequitur
Copi. I. M. Introduction to Logic. Sixth Edition. Macmillan (1982)
Have you come across an example of such an inference anywhere? I don't think it has a name, if only because it's very dumb :)
Notice that such a pattern of inference allows you to infer anything: take any A whatsoever. Now
Therefore A.
EDIT: I see by your comment below that you are interested in question-begging arguments in general.
The conclusion does not necessarily follow from the premises :
It does not make sense, it is a non-sequitur fallacy. You need a premise 3 (B or A).
A suggested Correction :
A implies B and B implies A can be written as a biconditional :
As for your example about Unicorns :
If unicorns exist then they have horns, by the definition of a unicorn. But in order for them to have horns they must surely exist. We must therefore conclude that unicorns do in fact exist.
There is an informal fallacy in that argument, what is in bold is begging the question.
This would be our correction version :
Unicorns exist if and only if they have horns (Premise 1), and since we know that they have horns by definition (premise 2), therefore they exist (conclusion).
I made a mistake. The answer I gave was for the question I asked that google directed me to this page for an answer.
not B implies not A, and is valid rather than a fallacy.