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 Dec 22 awarded Good Answer Oct 6 awarded Good Answer Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? I think the above is because, viewing this problem from the perspective of natural deduction, if there's a proof of $A \lor B$ then there must be a proof of $A$ or a proof of $B$ somewhere above. That's in intuitionistic logic. But with the LEM, in classical logic, I guess this argument doesn't work because $A \lor B$ could be deduced via the LEM and not directly. Right? Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? How odd! I'm so used to intuitionistic logic, where you can conclude this, no problem. Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? I think my confusion here comes from conflating 'sentences' and 'propositions'... I'm still not sure on the exact difference, mind you. Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? @MauroALLEGRANZA: Ah, thanks for clarifying. If $p$ is a sentence though, surely $\vdash p \lor \lnot p$ implies $\vdash p$ or $\vdash \lnot p$. (BTW I don't see an answer from you, only a couple of comments.) Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? @mmw Also, I think I see what you mean by your first point. I was thinking in terms of constructive (at least intuitionistic) logic, where that statement about provability of the components of a disjunction must hold. In classical logic though, I suppose it doesn't necessarily hold! (I'm not sure where exactly the crucial difference lies though.) Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? @mmw Can LEM hold but not be derivable though? Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? @MauroALLEGRANZA: Sure, I understand this (perhaps you were explaining to @tetori?). I'm not sure where that's relevant to the question though. Sep 20 comment Does the Law of the Excluded Middle imply syntactical completeness? @tetori: I specifically referred to syntactic (negation) completeness here. Semantic completeness does not come into it anywhere. And in fact, there are even more than 2 notions of completeness. Sep 20 asked Does the Law of the Excluded Middle imply syntactical completeness? Sep 17 awarded Famous Question Jul 20 awarded Yearling Jul 10 awarded Popular Question Feb 22 awarded Peer Pressure Dec 26 awarded Nice Question Sep 26 comment Is Aluffi's “Algebra. Chapter 0” a good introduction to algebra? Not particularly challenging?! The exercises in Aluffi are well-known to be incredibly difficult. Sep 11 comment Is Aluffi's “Algebra. Chapter 0” a good introduction to algebra? Care to elaborate which you prefer? Aug 7 comment A finite abelian group whose order is divisible by 10 contains an element of order 10 Okay, thanks. Unfortunately I have a few similar question to the asker but do not have the FTAG yet however. Aug 5 comment A finite abelian group whose order is divisible by 10 contains an element of order 10 How on earth does this help? Surely not all abelian groups are isomorphic to $Z_n$? Or perhaps I just haven't covered this yet in my book...