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How can you be reasonably assured that you are not engaging in circular reasoning when you invoke a theorem, lemma, etc.?

For instance, what if you accidentally "prove" a theorem using a consequence of that theorem itself? Such a procedure was proposed by a classmate in an abstract algebra class when the professor asked for suggestions.

Steve
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  • I don't think there's any other answer than "be careful about what you have proved and what you haven't". In principle one could decide that proofs should all be written in formal computer-checkable format that would allow keeping precise track of assumptions right back to whichever axioms you start from -- but that is a lot of work, and the minutiae of making it clear to the computer what is going on tend to overshadow the understanding that in the end is the real purpose of looking at how things are proved. – hmakholm left over Monica Apr 15 '14 at 00:59
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    I quite nearly did this in a paper I was working on. It happens to the best of us. You just have to be very methodical about your work and very thorough. My paper was about $18$ pages and it's easy to somewhat "stream of consciousness" write it up without critiquing your own work at a deep level at that point. I ultimately fixed it (just had to rearrange the order of things in the paper). I think the easiest way is (if you are generalizing something) to make a flow diagram of the things you need to prove and attack each thing a step at a time. It keeps human error to a minimum. – Cameron Williams Apr 15 '14 at 01:07
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    You can avoid circular reasoning by not using circular reasoning. – Hayden Apr 15 '14 at 01:10
  • @cameron Flow diagram seems like a rather nice way to organize things. Thanks for the suggestion. – Steve Apr 15 '14 at 01:30

1 Answers1

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Formally, a demonstration is a finite list of formulas $\phi_1,\ldots,\phi_n$ such that:

$\phi_1$ is an axiom or a hypothesis

$\phi_k$ is obtained by Modus Ponens of $\phi_i \colon \phi_j \rightarrow \phi_k$ and $\phi_j$, with $j,i <k$

There is no risk of circular reasoning in a formal proof.

In everyday mathematics we don't write formal demonstrations, but we can never assume something we want to prove. If your demonstration only uses theorems/lemmas you have already proved, there is no risk of circular reasoning, as long as the demonstration of the theorem/lemmas you are evoking have not used themselves something that wasn't proved before.

Jonas Gomes
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  • This is a nice way to put it. But, what do we do if we are faced with a theorem in a book that has been conjured up without proof? Such things appear often in texts geared towards all levels. – Steve Apr 15 '14 at 01:35
  • If you wish to use theorems that you don't know the demonstration (and all the chain of demonstrations that came before the theorem: all the lemmas and previous results), there is always a risk of circular reasoning. Suppose you use theorem A to prove theorem B. But theorem A assumes lemma C, which was proved as a corolary of theorem B, clearly you have circular reasoning, because you didn't know the demonstration of theorem A. – Jonas Gomes Apr 15 '14 at 01:48