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The way I used to be getting it was that circular reasoning occurs when a proof contains its thesis within its assumptions. Then, everything such a proof "proves" is that this particular statement entails itself; which is trivial since any statement entails itself.

But I witnessed a conversation that made me think I'm not getting this at all.

In short, Bob accused Alice of circular reasoning. But Alice responded in a way that perplexed me:

Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved. Thus each set of axioms implicitly contains all theses that can be proven from this set of axioms. As we know, each theorem in mathematics and logic is little more than a tautology: so is mine.

Not sure what should I think? On the one hand, Alice's reasoning seems correct. I, at least, can't find any error there. On the other hand, this entails that... Every valid proof must be circular! Which is absurd.

What is a circular proof? And what is wrong with the reasoning above?

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    $\begingroup$ Your understanding, prior to reading about Alice and Bob, is correct. $\endgroup$ – Namaste Jul 28 '18 at 23:40
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    $\begingroup$ I agree with @amWhy. I deny that each set of axioms implicitly contain all theorems that can be proven from this set of axioms. To say such a thing would be to confuse truth utterly. Otherwise, we'd have to say that the statement, "$1+1=2$" is the same as $\frac{d}{dx}\,x^2=2x.$ They're not the same thing at all, even though they're both true. $\endgroup$ – Adrian Keister Jul 28 '18 at 23:44
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    $\begingroup$ @PoonLevi: I don't think we're using the word "contain" the same way. Surely any mathematician would agree that any given set of axioms would uniquely determine a set of theorems provable from those axioms. If that's how you're using the term "contain", then I'd agree. But if "contain" means that the meaning of the theorems is contained in the meaning of the axioms, I can't go there. Different statements mean different things, or there'd be no use for language at all and no one could distinguish one true statement from another. $\endgroup$ – Adrian Keister Jul 29 '18 at 1:45
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    $\begingroup$ "Thus each set of axioms implicite contains all thesis" That's weaslely but not nesc. false. However if Alice wishes to take that viewpoint then the job of a proof is to demonstrate how her thesis is contained specifically from the axioms. Bob's complaint, in this viewpoint, isn't that Alice's thesis in contained via the axioms but that her demonstration failed to show how or why it is so contained in a satisfactory manner. $\endgroup$ – fleablood Jul 29 '18 at 2:07
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    $\begingroup$ @AdrianKeister "They're not the same thing at all, even though they're both true." Although I agree with you, I feel if you make such a claim, you are behoven to explain why you think $1+1=2$ and $\frac d{dx}x^2 =2x$ are different things. (Because $1+1=2$ is an axiom/definition and the other is a result? According to Alice those are the same. I disagree with her but I see her point. Why are you claiming they are different.) " But if "contain" means that the meaning of the theorems is contained in the meaning of the axioms, I can't go there." I don't think that is Alice's viewpoint. $\endgroup$ – fleablood Jul 29 '18 at 2:14

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Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved.

Hold it right there, Alice. These specific axioms are to be accepted without proof but nothing else is. For anything that is true that is not one of these axioms, the role of proof must be to demonstrate that such a truth can be derived from these axioms and how it would be so derived.

Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms.

Implicit. But the role of a proof is to make the implicit explicit. I can claim that Fermat's last theorem is true. That is a true statement. But merely claiming it is not the same as a proof. I can claim the axioms of mathematics imply Fermat's last theorem and that would be true. But that's still not a proof. To prove it, I must demonstrate how the axioms imply it. And in doing so I can not base any of my demonstration implications upon the knowledge that I know it to be true.

As we know, each theorem in mathematics and logics is little more than a tautology:

That's not actually what a tautology is. But I'll assume you mean a true statement.

so is mine.

No one cares if your statement is true. We care if you can demonstrate how it is true. You did not do that.

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    $\begingroup$ To quote Google, a tautology (in logic) is "a statement that is true by necessity or by virtue of its logical form". That describes mathematics in a nutshell. $\endgroup$ – Theo Bendit Jul 29 '18 at 3:49
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    $\begingroup$ "Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms." No, that's not true. At least not if rules of inference are not axioms, and axioms are not rules of inference. Axioms don't imply anything at all without rules of inference. $\endgroup$ – Doug Spoonwood Jul 29 '18 at 4:18
  • $\begingroup$ @DougSpoonwood I'd agree with that. But Alice wouldn't. Alice would say rules of inference and all potentially inferred statements (even those which haven't yet been inferred) are in the .... whatever it is that Alice is calling th body of math. This can be one way of looking at it, but it has nothing to do with proofs and how we know things are true or not. I'd say Alice is wrong but, I can't just say "she's stupid; ignore her" $\endgroup$ – fleablood Jul 29 '18 at 5:06
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    $\begingroup$ @TheoBendit I think google isn't correct about that, or maybe they are using some colloquial definition. Most people who study formal logic use "tautology" to mean exactly that a statement is true under all possible assignments to the variables. What "all possible assignments" and "variables" means depends on the logic. So $a + b = b + a$ is a tautology in quantified arithmetic logic, but not if you are reasoning about strings with $+$ being string concatenation. $\forall x. P(x) \to P(x)$ is a tautology in First Order Logic, letting $P$ range over all possible relations. $\endgroup$ – DanielV Jul 30 '18 at 0:18
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    $\begingroup$ In my head, $1+1= 2$ is the definition of $2$. Not a theorem, not an axiom. Simply stating that 'whenever I use the squggle "$2$", take that to be shorthand for "$1+1$"'. $\endgroup$ – Arthur Jul 30 '18 at 9:31
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I believe this has a simple resolution:

When we say informally that Alice is required to prove a result, it is sloppy language; she is actually required to prove the implication axioms $\implies$ result. So of course, she can have the axioms in her premises. However, she cannon have axioms $\implies$ result as one of her premises; that would be circular reasoning.

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  • $\begingroup$ Are you defining circular reasoning, as making a statement that a certain conclusion (result) follows from a set of axioms, without proof? $\endgroup$ – Pieter Rousseau Jul 30 '18 at 6:45
  • $\begingroup$ @PieterRousseau I would say yes, at least as a practical definition that we use informally. However, if you back me into a corner, I would retreat to saying that a circular argument is any argument which has anything else other than axioms as premises (I would have to phrase it much more carefully m of course since that definition clearly encompasses more arguments than just the ones we would like to call circular. $\endgroup$ – Ovi Jul 30 '18 at 7:48
  • $\begingroup$ " she is actually required to prove the implication axioms ⟹ result. " For clarification, you're saying that basically Alice has to show "axioms $\vdash$ result" instead of "$\vdash$ (axioms $\rightarrow$ result)", correct? Also, you said "However, she cannon..." I think you meant 'cannot'... though maybe Alice is a cannon shrugs and laughs. $\endgroup$ – Doug Spoonwood Jul 30 '18 at 7:52
  • $\begingroup$ @Ovi. Thanks for the clarification, I think I can agree with your more formal definition, because the premise (axioms ⟹ result) without proof is simply that: an unproven premise, nothing circular (I think). Unless I really misunderstood your answer. $\endgroup$ – Pieter Rousseau Jul 30 '18 at 8:19
  • $\begingroup$ @DougSpoonwood Yes, I think so. I'm just a student so honestly I wasn't even familiar with the sideways T. I looked it up and it seems to mean "RHS is provable from LHS", but then I'm not sure what just "$\vdash P$ means without a LHS $\endgroup$ – Ovi Jul 30 '18 at 9:42
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All reasoning (whether formal or informal, mathematical, scientific, every-day-life, etc.) needs to satisfy two basic criteria in order to be considered good (sound) reasoning:

  1. The steps in the argument need to be logical (valid .. the conclusion follows from the premises)

  2. The assumptions (premises) need to be acceptable (true or at least agreed upon by the parties involved in the debate within which the argument is offered)

Now, what Alice is pointing out is that in the domain of deductive reasoning (which includes mathematical reasoning), the information contained in the conclusion is already contained in the premises ... in a way, the conclusion thus 'merely' pulls this out. .. Alice thus seems to be saying: "all mathematical reasoning is circular .. so why attack my argument on being circular?"

However, this is not a good defense against the charge of circular reasoning. First of all, there is a big difference between 'pulling out', say, some complicated theorem of arithmetic out of the Peano Axioms on the one hand, and simply assuming that very theorem as an assumption proven on the other:

In the former scenario, contrary to Alice's claim, we really do not say that circular reasoning is taking place: as long as the assumptions of the argument are nothing more than the agreed upon Peano Axioms, and as long as each inference leading up the the theorem is logically valid, then such an argument satisfied the two forementioned criteria, and is therefore perfectly acceptable.

In the latter case, however, circular reasoning is taking place: if all we agreed upon were the Peano axioms, but if the argument uses the conclusion (which is not part of those axioms) as an assumption, then that argument violates the second criterion. It can be said to 'beg the question' ... as it 'begs' the answer to the very question (is the theorem true?) we had in the first place.

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  • $\begingroup$ "Alice is correct that every proof relies on a set of axioms to which Alice and Bob have to agree beforehand." Alice said: "As we know, each theorem in mathematics and logics is little more than a tautology" Since she says all theorems are contained in the axioms, it seems that she would say that all axioms are tautologies. But, natural deduction proofs exist which don't use any tautologies at all, nor any axioms. Why need Alice and Bob agree on axioms instead of using a pure natural deduction system? $\endgroup$ – Doug Spoonwood Jul 29 '18 at 5:45
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    $\begingroup$ @DougSpoonwood, while your point is quite sensible in non-mathematical contexts, the mathematical answer is: natural deductions from what? Whatever starting point for reasoning that you have, must, from a mathematical standpoint, be an assumption. (In logical reasoning applied to life this is not so: your starting point can be an assumption, or a concrete observation. Mathematics has no concrete observations not based on other assumptions.) $\endgroup$ – Wildcard Jul 29 '18 at 8:26
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    $\begingroup$ @DougSpoonwood Natural deduction systems tend to minimize or even eliminate logical axioms. The non-logical axioms that discuss the actual domain subject matter, e.g. real numbers, remain unless you go full type theory. In that latter case (and the former case to a lesser extent), instead of agreeing about axioms, you have to agree on what constitutes a valid rule of inference/construction. At any rate, the (usual, semantic) definition of "tautology" doesn't depend on the deductive system. $\endgroup$ – Derek Elkins Jul 29 '18 at 22:48
  • $\begingroup$ I feel it's important (albeit probably obvious) to note that, while we mustn't use our thesis as assumption in our proof, we must assume the negation of our thesis if we want to proof our thesis via contradiction. $\endgroup$ – Sudix Jul 29 '18 at 23:10
  • $\begingroup$ It seems to me though that in defining good reasoning, you are using good reasoning. Is this not circular? $\endgroup$ – Pieter Rousseau Jul 30 '18 at 6:35
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The fallacious version of circular reasoning is an appeal to the proposition

$$(A \to A) \to A \tag{C}$$

That is, to establish a proposition $A$ from no premises, you first establish the proposition $A$ under an assumption of $A$. Example:

  • Accuser: "You stole that."
  • Defendant: "No I didn't, it was mine."
  • Accuser: "It couldn't have been yours because you can't steal something you own."

In that case the accuser is correct as far as pointing out that "stealing" implies "not yours" which implies "stealing", the $A \to A$ part, but the fallacy comes from dropping the assumption and concluding "stealing" under no assumptions, which is the final $\to A$.

Guessing what Bob's side of the argument was : "Because the semantic meaning of a theorem is contained in the semantic meaning of the assumptions, you used circular reasoning". Here Bob is making 2 mistakes. First, he is equating circular reasoning with the proposition $A \to A$. But that isn't what circular reasoning is, because $A \to A$ always holds, how can you object to that?

Second, he is not addressing the argument that Alice made. Even if $(A \to A) \to A$ (for her specific claim) applies to the assumptions and conclusions of her argument: unless she appealed to that theorem as an inference, she hasn't made any mistake. $(A \to A) \to A$ does hold in the case that $A$ itself is provable, such as when $A$ is a tautology. Observing this isn't deductively equivalent to assuming $(A \to A) \to A$ holds in all cases and using that as an inference.

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If her thesis is true, you could argue she is somehow right. But we dont know that her thesis is true. Lets say A is the set of axioms, and S(A) all true statements derivable from this set, and t her thesis. Then she wants to show that t in S(A). Only if this were true she could use that t is in S(A). But she doesnt know it.

To make it clear to her: you could ask why not use also not t to show that the axiom system is contradictory. How does she know not t is not in S(A)

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  • $\begingroup$ This hits the nail right on the head. $\endgroup$ – Vincent Jul 30 '18 at 15:27
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(Not really an answer, but too long for a comment. Refer to amWhy's comment and Bram28's answer for a direct answer to your question.)

From what I understand, tautologies have a reputation in philosophy as being a waste of time. Things like "if I am cold and wet, then I am cold" don't seem to contain any new information, and have little value. But, the philosophers wonder, why is mathematics, which traditionally is 100% tautology, so non-trivial and produces unexpected results?

Well, I'm sure there are many answers to this, but the way I see it, it's finding a path from premises to conclusions that can be new information. Human beings aren't perfect; they cannot instantly account for every statement they hold to be true, and combine them in every possible way, to effortlessly see every consequence one can derive from them.

This is the point of proofs/arguments. I can read over and properly comprehend the axioms of ZFC, the definition of $\mathbb{R}^3$ and its Euclidean unit sphere, but that doesn't mean I instantly know how to prove the Banach-Tarski paradox!

Alice's reply seems to take the opposite stance. She seems to be assuming, implicitly, that there is no such gap between premises, consequences, and hence conclusions, rendering arguments useless. It's a view that I think most people would consider extreme (and wrong).

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Supposing that Alice tells the truth about using circular reasoning, it can get demonstrated that Alice has made an error. Alice says:

"Of course my proof contains its thesis within its assumptions. Each and every proof must be based on axioms, which are assumptions that are not to be proved. Thus each set of axioms implicite contains all thesis that can be proven from this set of axioms."

Alright, let's suppose the first two sentences correct. The third is false though. Why? Because without rules of inference we can't conclude anything from any axioms. So, we can't conclude that each set of axioms implies all of the theses, because axioms don't imply theses in the first place. Ever since Frege it has been clear that rules of inference and axioms are not the same thing. Only axiom and rules of inference taken together imply a thesis. But then, the set of axioms doesn't implicitly contain all theses, because it's axioms and rules of inference that imply all theses provable in the theory.

Edit: Also, it really doesn't work out that "logic is little more than a tautology" or "logic is little more than a collection of tautologies". Logic often involves showing something like A $\vdash$ B, where A is not a tautology, nor is B, and A $\vdash$ B is not a tautology.

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  • $\begingroup$ So what you mean is that a set of axioms does not contain the rules of inference, without which no conclusions will follow anyways (Rules of inference + Axioms -> theses)? $\endgroup$ – Pieter Rousseau Jul 30 '18 at 6:53
  • $\begingroup$ @PieterRousseau Yes, that's it. $\endgroup$ – Doug Spoonwood Jul 30 '18 at 6:57
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    $\begingroup$ So what if Alice rephrased her statement: "Of course my proof contains its thesis within its assumptions. Each and every proof must be based on rules of inference and axioms, which are assumptions that are not to be proved. Thus each set of rules and axioms implicitly contains all thesis that can be proven from this set." $\endgroup$ – Pieter Rousseau Jul 30 '18 at 7:00
  • $\begingroup$ @PieterRousseau I don't think that a rule of inference can accurately get called an assumption. Rules of inference don't bear truth, while assumptions do. You also have seemed to have dropped Alice's "implicite". $\endgroup$ – Doug Spoonwood Jul 30 '18 at 7:04
  • $\begingroup$ "Rules of inference don't bear truth, while assumptions do." What do mean? Point of clarification: I am assuming with rules of inference that you are talking about rules of logic, such as A is A; A is not not A and if A is B and B is C then A is C. (sorry about dropping the "implicite" - I actually thought it was a typo, or I am misunderstanding what you mean - would you mind if we get back to that once I understand what you mean about assumptions.) $\endgroup$ – Pieter Rousseau Jul 30 '18 at 7:35
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We can take the extreme case, where the axioms not only imply the conclusion, but contain the conclusion. Suppose I'm trying to prove it's raining outside. The following piece of reasoning is perfectly valid:

It's raining, therefore it's raining.

Nothing wrong with that, and no one would call it circular. It's just pointless. But suppose someone skeptical, without access to a window to look outside, were to challenge me to prove it's raining. If I now start my reasoning with "Well, we know that it's raining, so...", now I'm committing circular reasoning.

Notice that in both cases my proof and my axioms were the same, only the context changes. Circularity is not a property of a proof, it's a property of the context in which that proof is given. When we prove something from axioms, we should choose the axioms to be statements that are known to be true. If they're not known to be true, this is a Bad Thing. A special case of this kind of Bad Thing is where one of the axioms is the conclusion we're trying to prove (which is not known to be true): this special case is called circular reasoning.

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The apparent dilemma arises from two words which do not mean the same thing.

You are talking about proofs. Alice is talking about truth. These are not the same thing.

A statement which is always true is a tautology, so in a sense, every such statement, including a true theorem, is a tautology.

But truth is not a proof.

Proofs are simply the re-expression of statements as other statements without relying on other statements (i.e., no circular reasoning).

But the result of a correct proof is no more true than the original statements, so a proof is likewise not always the truth.

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    $\begingroup$ "So in a sense, every true statement, including a true theorem, is a tautology." The statement "it is raining withing 100 feet of me" is sometimes true and sometimes not true. In mathematics, the statement "a = b" is only sometimes true, such as where 'a' and 'b' have particular, known computations, such as a = (2 + 2) and b = (3 + 1), or in say the case where (a + c) = (b + c). So, I don't agree that every true statement is a tautology. $\endgroup$ – Doug Spoonwood Jul 30 '18 at 18:50
  • $\begingroup$ @DougSpoonwood You have proposed a sentence which depends on outside values to be true or false. This kind of statement is fine. It is called a conditional statement. I stated, "a statement which is always true is a tautology." I will edit my answer to make the phrase you quoted part of the same sentence so it is more clear. $\endgroup$ – Joseph Myers Aug 3 '18 at 23:17
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Circular reasoning is when two statements depend on each other in order to be true (consistent with the model).

  • The inside angles of a triangles are supplementary
  • The co-interior angles of a line intersecting parallel lines are supplementary

Both statements can be used to prove the other but neither of the above statements can be proven from the existing set of axioms in Geometry unless one of them is assumed as true without proof (an axiom). This "straightens" the argument so as to begin at this assumption. This is not fallacious at all unless it produces a contradiction with existing assumptions. It just limits or rather refines the set of statements that can follow as a logical consequences of the additional assumption.

Alice's Argument that Mathematical Proofs are Circular can be understood as:

  • Each and every proof must be based on axioms.

  • Axioms implicitly contains all theses that can be proven from the set of axioms.

To prove any one of these two statements will require the other one. This is circular. To "straighten" it we must assume one of the two to be true and the other can follow by the rules of inference/logic.

Is Mathematics then a tautology? If you define a tautology as a system of statements that is true by necessity or by virtue of its logical form, then I think it unashamedly is, and has to be based on the rules of inference/logic and the assumption just made: Each and every proof must be based on axioms or restated: All theses must follow by the rules of inference (proof) from the set of axioms.

Is Alice's argument valid that, in form, it is equivalent to that of axiomatic system of Mathematics? Is it consistent with the model produced by the assumption: all theses must follow by the rules of inference (proof) from the set of axioms.

Proof by contradiction: Assumed it is true that a proof contains its thesis within its assumptions.

Two situations, 1) The thesis is an existing axiom (assumption without proof).

If the thesis is an existing axiom, then of course her model is equivalent. A leads to A.

2) The thesis is not an existing axiom (assumption without proof).

If the thesis is not an existing axiom but it is an axiom because it is assumed without a proof, then is leads to a logical contradiction.

Which means her proof is only valid if her thesis is indeed an existing axiom.

Alice's solution: So if her argument is indeed circular, then her solution is to admit that one of the statements that she is making must be is an necessary assumption to develop the model/argument/worldview etc. Then it could be equivalent to the axiomatic system of Mathematics if it does not contradict any existing assumptions/axioms.

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