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The issue is that some people believe in intuitionistic logic, so to them, there's a difference between assuming something is true and deriving absurdity in ord

The issue is that some people believe in intuitionistic logic, so to them, there's a difference between assuming something is true and deriving absurdity in order to prove its negation (which they think is perfectly fine, and which is what Euclid was doing), and assuming that its negation is false and deriving absurdity in order to prove that it's true (which they think is a logical error). They reserve the term "proof by contradiction" only for the latter type of proof, i.e. the one they think is not a valid type of proof. Whereas for believers in classical logic, to say that something is true is just to say that its negation is false, so they don't generally care about the distinction between these two types of proof---as far as they're concerned it's a valid proof, and a "proof by contradiction", either way.

puffoflogic 5y ago
> which they think is a logical error This is FUD designed to discredit intuitionistic logic. An instance of LEM is not "a logical error"; it is merely a claim which requires justification. For the claim at hand -- i.e. some natural number G is a member of a finite, constructed set, or it is not -- the corresponding LEM instance is perfectly true [and proven]. We say that the claim is decidable. So (given that we have a proof the claim is decidable) there would be no problem in intuitionistic logic with this part of Euclid's proof even if it took the form of a proof by contradiction (which of course it doesn't). To make a more general rebuttal of the FUD, proofs by contradiction are not automatically invalid in intuitionistic logic; they just require one extra piece of evidence. Furthermore the claim that whether something is a proof by contradiction depends on one's position on the logical validity of LEM is ridiculous on its face. There is a fact of the matter as to whether any given proof is by contradiction. There may be disagreements as to the definition of "proof by contradiction", but the definition which includes proof of negation is completely useless. In mathematics we usually treat useless definitions as in some sense "false" in order to facilitate communication - e.g., 1 could be prime depending on definitions, but that would be useless so we facilitate communication by treating "1 is prime" as false. Likewise here we should treat as false the claim that the quoted section of Euclid's proof is by contradiction. Source for claims about LEM/proof by contradiction: https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...
- housecarpenter 5y ago
  I am certainly not intending to intentionally spread FUD, though I guess you probably mean that I am unintentionally propagating FUD spread by other people. All I mean by saying it's a "logical error" is that it's not an argument which is always valid. Of course particular instances can still be correct. I think a definition of "proof by contradiction" which includes proof of negation is useful if you are reasoning classically (and this doesn't have to be something you always do, or always avoid; you can reason classically some of the time and intuitionistically some of the time). Although the ideal situation to avoid any confusion would be to have three separate terms, one covering both types of proof and two for the individual types.

puffoflogic 5y ago

> which they think is a logical error This is FUD designed to discredit intuitionistic logic. An instance of LEM is not "a logical error"; it is merely a claim which requires justification. For the claim at hand -- i.e. some natural number G is a member of a finite, constructed set, or it is not -- the corresponding LEM instance is perfectly true [and proven]. We say that the claim is decidable. So (given that we have a proof the claim is decidable) there would be no problem in intuitionistic logic with this part of Euclid's proof even if it took the form of a proof by contradiction (which of course it doesn't). To make a more general rebuttal of the FUD, proofs by contradiction are not automatically invalid in intuitionistic logic; they just require one extra piece of evidence. Furthermore the claim that whether something is a proof by contradiction depends on one's position on the logical validity of LEM is ridiculous on its face. There is a fact of the matter as to whether any given proof is by contradiction. There may be disagreements as to the definition of "proof by contradiction", but the definition which includes proof of negation is completely useless. In mathematics we usually treat useless definitions as in some sense "false" in order to facilitate communication - e.g., 1 could be prime depending on definitions, but that would be useless so we facilitate communication by treating "1 is prime" as false. Likewise here we should treat as false the claim that the quoted section of Euclid's proof is by contradiction. Source for claims about LEM/proof by contradiction: https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016...

housecarpenter 5y ago
I am certainly not intending to intentionally spread FUD, though I guess you probably mean that I am unintentionally propagating FUD spread by other people. All I mean by saying it's a "logical error" is that it's not an argument which is always valid. Of course particular instances can still be correct. I think a definition of "proof by contradiction" which includes proof of negation is useful if you are reasoning classically (and this doesn't have to be something you always do, or always avoid; you can reason classically some of the time and intuitionistically some of the time). Although the ideal situation to avoid any confusion would be to have three separate terms, one covering both types of proof and two for the individual types.