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Suppose that you want to prove the biconditional proposition $p\iff q.$ How do we do that? Well, the most common way to prove it has two steps: first, assume $p$ and use that assumption to prove $q$ then, assume $q$ and use that assumption to prove $p.$ Hmm, wait. Isn't this circular reasoning? After all, in one type of circular reasoning, we use $A$ to prove $B$ and use $B$ to prove $A.$ In the biconditional proof, we are doing the same thing.

Yet we know that proofs of biconditional statements are valid, and not considered circular reasoning. What's wrong with my reasoning?

ryang
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    Note that the conclusions of the two types of argument are different: in circular reasoning we "conclude" that $A$ and $B$ are each true, while in a biconditional argument we just conclude that they are each true or each false. – Noah Schweber Nov 25 '24 at 05:43
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    There's a subtle (or maybe not so subtle) difference between "If $P$ then $Q$" and "$P$, therefore $Q$." – David K Nov 25 '24 at 05:49
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    Try proving $x<2$ if and only if $x<5$ by first assuming $x<2$ and concluding $x<5$, and then trying to assume $x<5$ using it to prove $x<2$, and you should see what's actually happening. – Gerry Myerson Nov 25 '24 at 06:42
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    You are not proving that either of $p$ or $q$ is true. You are proving that whenever $p$ is true, so is $q$ and vice versa. – John Douma Nov 25 '24 at 06:49
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    It seems to me this is a completely different question from the "already answered" one. That question actually is much more subtle. This one demands a different answer. I'm surprised that I could not find a sufficiently similar question to link to, but I tried and I failed. The main defect of the question seems to be that it leads people to answer in comments, but it's not really fair to blame that on the question. – David K Nov 25 '24 at 14:25
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    "Circular reasoning" is when you try to hide the fact that you've assumed something. A biconditional makes it very clear what you are assuming, and exactly what a given conclusion depends on. – JonathanZ Nov 25 '24 at 14:42

2 Answers2

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To prove the biconditional $p \leftrightarrow q$ means to prove that the two propositions (statements) are equivalent, i.e. that if you prove one of them, this proof gives you also the other one.

But having done so, you have proved neither of them, taken individually.

Consider a "real life" mathematical example concerning the Parallel postulate: we know that there are many geometrical statements that are equivalent to Euclid's original postulate (assuming the other axioms), the best-known of which is Playfair's axiom.

But the fact that we are able to prove the equivalence does not mean that we have proved the Parallel postulate.

Mauro ALLEGRANZA
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  • "But the fact that we are able to prove the equivalence does not mean that we have proved the Parallel postulate." Why? This made me think. So if we prove an equivalence of the Parallel Postulate, this means that its equivalence is true, and therefore the Parallel Postulate is also true. Yes, we didn't prove Parallel Postulate directly, but doesn't proving the logical equivalence of a proposition mean that we have proved that proposition indirectly? (Maybe this is more appropriate for a separate question, but I'm just curious) –  Nov 26 '24 at 02:52
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What's wrong with my reasoning?

Your argument unsoundly contains the false premise/claim, “in one type of circular reasoning, we use $A$ to prove $B,$ and use $B$ to prove $A.\text”$

Now, circular reasoning just means that the premise and conclusion are the same. Your claim describes each of the two independent subproofs of A⟺B, which can be outlined as

  1. A⟹B

    In this subproof, the premise $A$ isn't the same as the conclusion $B.$

  2. B⟹A

    In this subproof, the premise $B$ isn't the same as the conclusion $A.$

  3. Hence, by definition, A⟺B.

Each of the two conjuncts of the main conclusion A⟺B has been independently proven, rather than merely assumed. The entire proof also contains no undischarged assumption. So, where exactly is the circular reasoning??

P.S. It's worth noting that although the sentence A; therefore, B implies the sentence A implies B (that is, $B$ is true if $A$ is), the converse is not true.

ryang
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  • So if we use $A$ to prove $B$, and $B$ to prove $A$, that's not circular reasoning, and it's actually a biconditional proof, right? So what exactly is circular reasoning between two propositions $A$ and $B$? Isn't it something like this: if statement $A$ asserts that "$A$ is true because $B$ is true", and statement $B$ asserts that "$B$ is true because $A$ is true", then neither provides independent evidence, and therefore it is circular reasoning, right? Because now we're not assuming $A$ or $B$, we're clearly saying that $B$ is true because $A$ is true, and vice versa. –  Nov 26 '24 at 10:44
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    @Just_A_User I've updated the answer. – ryang Nov 26 '24 at 11:55