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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?
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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...