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For example, I have heard of a topological one wherein negation means the interior of the complement (but still would like a reference).

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The wikipedia entry for intuitionistic logic en.wikipedia.org/wiki/Intuitionistic_logic gives some references and in particular has a little bit to say about the topological interpretation you aluded to. See the section "Heyting Algebra Semantics". – Adrián Barquero Oct 22 '10 at 16:51

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The example that you mention in you question is explained in Maclane and Moerdijk 's book Sheaves in Geom and Logic very near the beginning. They discuss this in relation to Heyting algebras for the purpose of Topos Theory. This book is very very nice, I highly recommend it.

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I love this book! Although, I've only read very small parts of it. – Matt Oct 23 '10 at 3:21
@Matt, It is very nice, I like it a lot also. – BBischof Oct 23 '10 at 4:11
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An excellent book is also http://www.amazon.com/Lectures-Curry-Howard-Isomorphism-Foundations-Mathematics/dp/0444520775. This book is more focused on the $\lambda$-Calculus but it has some really excellent sections on general intuitionist logic, including Heyting Algebras, Hilbert Proofs, the (Gentzen's) sequent calculus and things like that.

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