## Is the reiteration rule in formal logic begging the question?

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Wikipedia defines "begging the question" as

To "beg the question" is to put forward an argument whose validity requires that its own conclusion is true.

I assume this is something Aristotle's term logic does not permit. Wikipedia quotes Hugh Tredennick's translation of Prior Analytics II xvi 64b28–65a26:

...If, however, the relation of B to C is such that they are identical, or that they are clearly convertible, or that one applies to the other, then he is begging the point at issue.... [B]egging the question is proving what is not self-evident by means of itself...

I assume this means one cannot have the following in Aristotle's term logic:

Premise 1:  All B is C.
Premise 2:  All B is C.
Conclusion: All B is C.


However, using the proof checker associated with forall x: Calgary Remix I can construct a valid argument in truth-functional logic using "reiteration" (page 123-4) by repeating a line I already have.

This is derived from conjunction introduction using two identical conjuncts and then using conjunction elimination to get one of those identical conjuncts. (page 136)

Could this reiteration rule, or permitting conjunction introduction to use two identical conjuncts, be considered begging the question?

Edit 10/6/2018: I was reading Frederic Fitch's Symbolic Logic: An Introduction and noticed that I could have simplified the proof I gave above using reiteration by doing the following (page 26):

5.13. There is nothing that excludes a formal proof from possessing only a single item. The following single-item proof is a hypothetical proof of p on the hypothesis p. It is a proof of p in the sense that p is the last (and only) item of the proof.

This is how it looks in Klement's proof checker:

References

Fitch, F. B. (1953). Symbolic Logic; an Introduction.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Wikipedia, "Begging the question" https://en.wikipedia.org/wiki/Begging_the_question

I find that an odd definition. Why would the argument’s validity require the truth of the conclusion? In fact, your P ⸫ P is a counter-example: that’s valid, even if P is false. Also, isn’t ‘beg the question’ a notion from informal logic / critical thinking / rhetoric (or whatever you want to call it), whereas reiteration is a rule of formal logic. (Also, it’s a proof rule, not a model-theoretic thing, which makes the appeal to (the semantic notion of) truth even stranger.) – MarkOxford – 2018-08-12T19:13:24.370

@MarkOxford Maybe there's a better definition of "begging the question"? The informal fallacy of begging the question doesn't imply that the conclusion of the argument is true or false only that the way to reach that conclusion is questionable. That would be the case for any logical fallacy, at least as I see them. – Frank Hubeny – 2018-08-12T19:22:22.277

I do not think your argument scheme with three identical lines falls under any of the syllogism figures. But Aristotle's syllogistic was not fully formalized, if it was and he wanted to use the same sentence more than once in an argument he'd need a repetition rule. "Begging the question" is a circularity charge against a possibly valid argument when its conclusion is already contained in the premises. – Conifold – 2018-08-12T19:27:28.757

Instead of just ‘beg the question’, I wonder whether we should say ‘beg the question against X’. E.g., the person who rejects p will also reject p&q; so p&q ⸫ p begs the question against them – but that tells us little about the formal validity of that argument. @conifold speaks of conclusions that are ‘already contained in the premises’. That’s no doubt the gist of begging the question, but note that this is a fairly vague notion. Consider: p, p->q ⸫ q. Is q ‘contained in’ the premise? – MarkOxford – 2018-08-12T19:31:43.403

More formally, if φ⊧ψ, but φ and ψ have no sentence letters in common, then either φ is a contradiction or ψ is a tautology. So, unless you’re arguing from a contradiction or for a tautology, the conclusion is always ‘contained in’ the premises, at least in part. I think my point is that it's potentially misleading to use an informal notion like 'beg the question' and then apply it to formal logic. – MarkOxford – 2018-08-12T19:50:00.073

@MarkOxford There are formalizations of "contained" going back as far as Peirce. In modern terms conclusion is "contained" in the premises if it is derivable using monadic predicate calculus ("syllogistic") only, for example, similar notions are used to distinguish "trivial" and "non-trivial" proofs by Hintikka and others. I also noticed that "begging against" is typically used with a meaning distinct from circularity, namely arguing from premises or presuppositions that the opponent is known to reject. – Conifold – 2018-08-12T20:06:16.397

@Conifold Interesting. So, is the conclusion Rab contained in Pab&(Pab->Rab)? (I ask because you say monadic FOL, which these sentences aren’t.) – MarkOxford – 2018-08-12T20:17:31.937

@MarkOxford This inference is purely propositional despite the presence of dyadic predicates, so yes. Hintikka's extension is that one does not need to introduce new quantified variables in the course of the proof, see Hintikka'80. It was objected that some purely propositional proofs are intuitively "non-trivial" while they come out as trivial by Hintikka's lights. I think Jago came up with a more sophisticated version recently.

– Conifold – 2018-08-12T20:43:42.323

@Conifold I made up the bogus syllogism to suggest it was not something Aristotle would have accepted to my knowledge. – Frank Hubeny – 2018-08-13T00:32:24.207

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That definition you quote from Wikipedia is simply incorrect. An argument of the form "P; therefore P" is valid irrespective of whether P is true or not. It is a question-begging argument because the premise does not provide a reason to accept the conclusion. Begging the question is epistemic rather than logical in character. There have been attempts to formulate a purely formal criterion for it, but nothing covers all cases. It would be better to characterise a question-begging argument in one of the following ways:

1. The premises do not serve as grounds or reasons for accepting the conclusion.
2. There is no 'flow' of justification from the premises to the conclusion.
3. One would not accept the premises if you did not already accept the conclusion.

John Stuart Mill criticised deductive reasoning (or at least, some particular syllogistic forms) as being entirely question-begging because the premises would not be accepted unless we already believed the conclusion. One of his examples was: "All men are mortal; Socrates is a man; therefore, Socrates is mortal". His complaint is that we would not accept that all men are mortal unless we already believed that each individual man, including Socrates, is mortal. The point is debatable, however, since we might argue that the mortality of all people rests upon far better evidence than merely an inductive enumeration of the dead. There are good scientific reasons to accept many universal claims. Also, there are cases where accepting a universal premise does not depend at all on a prior acceptance of the individual instances. Consider for example: "All dollar bills are legal tender in the USA; this is a dollar bill; therefore, this is legal tender in the USA". This argument is not question-begging. I do not need to examine each dollar bill to discover that they are all legal tender. They are all legal tender by fiat, because of an act of an appropriately constituted authority. And knowing this fact gives me reason to believe that my dollar bill is legal tender.

So, not all valid arguments are question-begging. Rather, one must consider what the reasons are for accepting the premises and the conclusion. An important corollary of this is that I might consider a particular argument question-begging while you do not. We might disagree because we have different background beliefs and these affect our reasons. If A and B together entail C, you might believe A and B and claim that this provides an excellent reason to believe C, while I might consider C to be so absurd that I reject the argument as question-begging and prefer to disbelieve A. Sometimes this is expressed by saying, "One person's modus ponens is another person's modus tollens", or even "One person's proof is another person's reductio".

+1 Would you have a reference for "attempts to formulate a purely formal criterion for" begging the question that might be relevant? I wonder if there are any logical approaches that avoid reiteration or prevent conjunction introduction from joining identical conjuncts. Do all modern logical systems need these tools? – Frank Hubeny – 2018-08-13T00:27:08.373

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Some versions of relevance logic attempt to capture the idea of logic as providing reasons for a conclusion. One version of relevance logic even lacks the theorem that A proves A. See, for example, the SEP article: https://plato.stanford.edu/entries/logic-relevance/ For a paper on formal and informal characterisations of question begging, try "Logical Dimensions of Question-Begging Argument". Dale Jacquette. American Philosophical Quarterly Vol. 30, No. 4 (Oct., 1993), pp. 317-327.

– Bumble – 2018-08-13T09:27:15.853

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No, that is not begging the question. The proof shows that P follows from P ... it makes no claim that P is actually true ...

It would be begging the question if someone does want to claim that P is true, but then assumes P in order to 'prove' P.

But in logic we're not concerned about truth, just implication.

+1 I don't think Aristotle's syllogisms do anything different and yet they do not appear to accept reiteration and I suspect this is because of begging the question which appears to be originally noted by Aristotle. I do agree that with logic we are concerned with implication, not truth, but we do assume the premises are true. – Frank Hubeny – 2018-08-13T00:08:11.393

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Could this reiteration rule, or permitting conjunction introduction to use two identical conjuncts, be considered begging the question?

Absolutely. I would go even further and assert that every logically valid argument is begging the question; in Wittgenstein's words:

If a proposition follows from another, then the latter says more than the former, the former less than the latter. (Tr. 5.14) If p follows from q and q from p then they are one and the same proposition. (Tr. 5.141)

Put differently, every valid argument can be translated (one way or another) to either
p ⊢ p or p & q ⊢ p.

Therefore, "begging the question" is considered a fallacy only because the term "fallacy" is not used precisely: in everyday use, the word "fallacy" does not always imply an invalid argument. "Begging the question" often merely implies that the argument is not an interesting one, because the premises too obviously contain the conclusion (this is exactly what your quote from Aristotle entails).

Nonetheless: In formal logic, "begging the question" cannot be considered a shortcoming, because it is unavoidable if the argument is to be valid.

Further reading: "What is a fallacy?" in the Internet Encyclopedia of Philosophy

+1 I tend to agree. I am trying to see if there is some way to keep both the informal fallacy of begging the question and the formal rule or reiteration. Perhaps begging the question should be ignored as an informal fallacy? – Frank Hubeny – 2018-08-13T00:14:56.993