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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 9th 2017
• (edited May 9th 2017)

I have added to the References at double negation pointer to Andrej’s exposition:

which is really good. I have also added this to double negation transformation, but clearly that entry needs some real references, too.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 1st 2017

I have added a sentence mentioning forcing to the Idea-section at double negation,

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJun 1st 2017

That is a nice post, but I have to take issue with

classical logic is happy with lack of negative evidence.

I would say that what classical logic is happy with is the impossibility of negative evidence, which is rather stronger than the present lack thereof.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 23rd 2019

The article claims

Classically, we have L=L¬¬ if and only if L is the discrete locale on some set S of points. In constructive mathematics, S must also have decidable equality.


But any complete Boolean algebra is a frame for which the corresponding locale satisfies L=L_¬¬ because ¬¬=id. There are plenty of nonatomic complete Boolean algebras.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeJul 23rd 2019

Yes, it seems that statement had been there since Day 1. I suspect Toby was thinking along roughly the following lines: a topological space $X$ in which every open is regular open is (classically) discrete. If you allow the luxury of $T_1$ spaces, and if $x$ is any non-isolated point, then its set-theoretic complement $U$ has $U \cup \{x\} = X$ as its closure, so $\neg \neg U = X$. So maybe with a sufficiently generous interpretation of “classical”, the statement is defensible – but I agree the statement is confusing as it stands.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 23rd 2019

Maybe it should be “If $L$ is spatial, then $L=L_{\neg\neg}$ if and only if …”?

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeJul 23rd 2019

Yes, spatiality is certainly necessary here.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 23rd 2019

Add spatiality condition to the characterization of $L=L_{\neg\neg}$.