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 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.
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.