Talk:Intuitionistic logic
From Academic Kids
Very nice start to the article. A bit POV, though, it's kind of obvious it's written by an advocate of intuitionism. But still good start.
Ummm ... isn't it a bit narrow-based around the assumption of that old chestnut, intuitionism as one of the three main philosophies of mathematics? The Kolmogorov interpretation may have been the first step clear of that. Certainly the Brouwer not not not = not result needs a mention, too. Goedel used IL at one juncture, and after that, historically speaking, wasn't IL respectable in a technical sense? What happened then in relation to Markov was perhaps confusing.
Anyway, there are several strands to disentangle here: this is a bit ahistorical for me.
Charles Matthews 10:36, 15 Oct 2003 (UTC)
my take
Terms should not be equivocated. Speakers end up suggesting that not isn't not, so classical logicians should use a term meaning the exact opposite: un. lysdexia 00:16, 13 Nov 2004 (UTC)
I don't get it. How is this an example?
In the "law of the excluded middle" example under "Heyting algebra", <math>R^2<math> is divvied up into a set A and its complement, the first comprising <math>\{(x,y) : y>0\}<math>. Now, it would seem to me that the complement of A would be <math>\{(x,y) : y\le0\}<math>. However, it's given as <math>\{(x,y) : y<0\}<math>, which is then used to show that the <math>y=0<math> plane isn't included in the union of A and not-A. How can this be? grendel|khan 03:45, 2005 Jan 14 (UTC)
- It's really badly explained. But the idea is that if you consider subsets of <math>R^2<math>, they form a Heyting algebra if you consider "or" and "and" to be set union and intersection, and (this is they key point) the complement of X to be the interior of <math>R^2 - X<math>. Then the complement of the open upper half-plane is the open lower half-plane. The axioms of the Heyting algebra are satisfied in this case, but the law that <math>X\vee\not X = R^2<math> doesn't hold. -- Dominus 12:30, 14 Jan 2005 (UTC)
