# Is there a name for an argument like "A implies B; B implies A; therefore A"?

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.

• Similar to en.wikipedia.org/wiki/Transposition_(logic) - though this is where `not B implies not A`, and is valid rather than a fallacy. Jan 8, 2016 at 11:56

It seems some variance of the Circular Reasoning.

• Thanks, this is what I was looking for. It's kind of obvious that it would be called that, but I hadn't thought of it. Jun 26, 2012 at 15:24

## "Begging the Question" or petitio principii

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.
• It's not begging the question. In the argument form in my question, the conclusion is not contained in either premise but is obtained through a false inference. Jun 26, 2012 at 17:02
• Okay, I think I see what you mean. Would you agree that the fallacious argument is can be summed up as: `(A <=> B) => A`? Jun 26, 2012 at 17:58
• @Nathaniel, Please take a look at these descriptions for a Vicious Circle fallacy and a Vacuous Explanation fallacy. they both seem related to what you are describing. Jun 26, 2012 at 18:10

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:

1. A: Unicorns exist. B: All unicorns have horns.
2. A: Unicorns exist. B: At least one unicorn with a horn exists.

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

• A implies not (not A)
• not (not A) implies A

Therefore A.

EDIT: I see by your comment below that you are interested in question-begging arguments in general.

• Within cognitive science I occasionally find arguments where someone (usually somewhat implicitly) defines an action as that which an agent does, and then tries to show that something is not an agent because it doesn't act. I consider this to be a case of the fallacy in question. As with most fallacies it's obviously dumb when written out directly, but when wrapped up in rhetoric it can be surprisingly hard to spot. Jun 26, 2012 at 14:15
• I think you're right that the argument I described regarding agency is a case of question-begging. However, I still think the syllogism I described should have a name :) Potentially some versions of the ontological argument could be classified as examples of this fallacy: if God exists then God is perfect; that which is perfect must exist; therefore God exists. Jun 26, 2012 at 14:44
• By the way: I don't get automatically notified when you edit your answer, so if you want to guarantee that I'll see the change then it's best to add a comment tagged with @Nathaniel as well. Jun 26, 2012 at 14:44
• Precisely by the same token it is not circular reasoning, if we stick by the Wikipedia entry that Yuri cites: "the reasoner begins with what he or she is trying to end up with". You are right though that your original argument is not question begging. Editing again! Jun 26, 2012 at 15:55
• The "Transcendental Argument for God" (in some guises) is the most common argument of this form I've seen. Nathaniel gave that example in his comment before last, though it's usually stated more like: God is by definition perfect; it is more perfect to exist than not; therefore God exists. Jun 26, 2012 at 20:30

The conclusion does not necessarily follow from the premises :

• Premise 1: If A then B ( is equivalent to A implies B)
• Premise 2: If B then A (is equivalent to B implies A).
• Conclusion : therefore A

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 :

• Premise 1: A if and only if B
• Premise 2: B
• Conclusion : Therefore A