# Intuitionistic logic or classical logic

I've noticed that most of questions about logic are either asked or answered in a way that quietly assumes classic logic, and whereas this might be appropriate at an high-school level (I still maintain it isn't, but that's a whole different can of worms) I think it may be worthwhile to make the distinction clear in here, asking for more rigour instead of seeing logic as a blind and empty dance of symbols.

I'll make an example to help clarify (I hope) my point.

This question's answers use classical logic in a pretty evident way (both use the law of the excluded middle). In fact only $(\mathrm p \wedge \mathrm q) \vee (\neg\mathrm p \wedge \neg\mathrm q) \triangleright \mathrm p \equiv \mathrm q$ whereas the opposite can't be proved in N.

Now, the question specifically talks about a Discrete Mathematics text, which means that classical logic is implied, but there may be questions in which the distinction is actually useful.

Your thoughts on dealing with this?

• Nothing keeps anyone here from asking questions about nonclassical logic. – Michael Greinecker Jan 13 '14 at 22:16
• Yeah, but I feel an urge to comment "this proof only works in classical logic" to a lot of answers... – miniBill Jan 13 '14 at 22:18
• Unless the context indicates otherwise, one can safely assume questions about formal logic are about classical logic. – Michael Greinecker Jan 13 '14 at 22:20
• Mumble... while that may be true... I don't think it's satisfactory... – miniBill Jan 13 '14 at 22:24
• For a hyperfinitist, most answers here are unsatisfactory. And there are of course a lot of answers outside logic that make use of nonconstructive methods. I don't think MSE would be a better place if people start attacking answers for philosophical reasons. – Michael Greinecker Jan 13 '14 at 22:31
• Ok, I see your point. Can you make it an answer? – miniBill Jan 13 '14 at 22:32
• Uhm, I'm not familiar with meta, does the downvote mean "bad question" or does it mean "I don't agree"? – miniBill Jan 13 '14 at 22:53
• Downvotes on meta mean disagreement. Voting on meta has no effect on ones "reputation". – Michael Greinecker Jan 13 '14 at 23:02
• Thank you for the explanation :) – miniBill Jan 13 '14 at 23:09
• @miniBill: Do you also feel compelled to add "... with respect to the usual Euclidean topology" to virtually any question talking about continuous functions on $\mathbb{R}$? – user642796 Jan 13 '14 at 23:14
• Much less, but I have to admit I'm probably biased toward what my personal experience with math has been – miniBill Jan 13 '14 at 23:19
• If the proposition the OP wishes to prove does hold in intuitionistic logic, you are perfectly free to offer an answer demonstrating that. – dfeuer Jan 14 '14 at 0:40

Classical logic is the implicit background logic for most mathematics and is the mot popular logic in use. Unless otherwise specified, one can usually assume that the logic discussed is classic. Answers based on classical logic may not be useful to those not accepting classical logic, but that should not be relevant as long as an answer is correct and relevant to the one who asked the question or those who might have the same question.

It is of course legitimate to have views on the foundations of mathematics that differ from the mainstream and ask questions from that perspective. But M.SE. is not a discussion site, so one should not use questions as a platform for arguing over foundations and to express disagreement with the mainstream.

Personally, I very much enjoy questions related to different foundational perspectives (constructivism, finitism etc.) and think asking good question from or on alternative foundational perspectives, and providing good answers for such questions, is a better way to increase awareness for the alternatives.

• You write "Answers based on classical logic may not be useful to those not accepting classical logic", which sounds like a misunderstanding of intuitionistic logic to me. Preferring a more general theory over a more restricted one where possible has nothing to do with not accepting the more restricted theory. You don't even need to add more axioms to intuitionistic logic to make sense of classical logic, you just have to interpret "x or y" as "not(not x and not y)" and "exist x P" as "not (forall x not P)". – Thomas Klimpel Jan 14 '14 at 8:17
• @ThomasKlimpel That one can interpret a classical proof of someting intuitionistically as a proof "of something" does in general not imply that this "something" is what a question asked for, even though it classically is. – Michael Greinecker Jan 14 '14 at 8:46
• When asking questions with non-standard logics, carefully specify what logic is being used. Do not assume readers understand code-words like "effective" or "constructive" in the same way you do. – GEdgar Jan 14 '14 at 15:04
• @MichaelGreinecker I still have the impression that you misunderstand intuitionistic logic. How can intuitionistic logic not accept classical logic, if classical logic is an organic part of intuitionistic logic? Of course a classical proof based on classical logic is also useful for intuitionistic logic, because it can be mechanically translated into an intuitionistic proof proofing an intuitionistic formula, which is trivially equivalent for classical logic to the formula proofed by the classical proof. – Thomas Klimpel Jan 14 '14 at 21:24
• @ThomasKlimpel I still don't see what I'm supposed to be confused about. There are a lot of results that have constructive proofs (say, in Bishop style analysis) and their classical counterparts have another proof that does not imply the constructive version. – Michael Greinecker Jan 14 '14 at 21:29
• @MichaelGreinecker Who do you think are the people that you describe by "those not accepting classical logic"? I have seen Andrej Bauer defend intuitionism on FOM, but why do you want to claim that he does not accept classical logic? As most mathematics is based on classical logic, do you want to imply that he doesn't accept most of mathematics either? Even so I only recently started to appreciate intuitionistic logic, I always interpreted non-constructive proofs (using Zorn's lemma) as "nobody can disprove this", i.e. I used the double-negation translation without being an intuitionist. – Thomas Klimpel Jan 14 '14 at 22:01
• @ThomasKlimpel An example of these people is Fred Richman who says "When I think about real numbers, for example, I believe that I am talking about the same objects that a classical mathematician is. The difference is that I don't accept certain methods of reasoning about real numbers which obliterate distinctions that I think are important." Andrej Bauer does not identify himself positively as constructivist. – Michael Greinecker Jan 14 '14 at 22:17