TITLE: Propositional logic with categories
QUESTION [14 upvotes]: I have some vague sense that certain types of categories are related to certain types of logic.  I've been meaning to learn more about this, so I thought I'd ask about the simplest case, propositional logic.  In particular, I'd be interested in a statement of the completeness theorem for propositional logic using these ideas (something like a representation theorem for a certain class of categories).
Here's what I think is a start: a propositional calculus should, at least, behave like a bicartesian closed category $C$, possibly satisfying some extra conditions.  The product should be $\vee$, the coproduct should be $\wedge$, and the exponential object should be $\rightarrow$.  I think the terminal object is $\top$ and the initial object is $\bot$, and the exponential $p \rightarrow \bot$ should then be $\neg p$.  Maybe this means the same thing as "Heyting algebra."
We're specifically interested in the free bicartesian closed category $C_P$ on a set $P$ of primitive propositions, and in functors from $C_P$ to the truth-value category $\mathbf{2} = \{ \bot \rightarrow \top \}$.  If I've set up the definitions correctly, then a subset $S$ of $C_P$ semantically entails an object $t$ if and only if, for every functor $F : C_P \to \mathbf{2}$ such that $F(s) = \top \forall s \in S$, it is also true that $F(t) = \top$.  
If I've set up the definitions correctly, functors like $F$ are defined by what they send to $\top$.  If we want to characterize such sets syntactically, we write down some set of axioms and some set of inference rules telling us when objects sent to $\top$ let us construct other objects sent to $\top$, and then the completeness theorem tells us that these axioms and inference rules characterize the functors $F$.  
But I'm clearly missing something important: at some point we introduce the law of the excluded middle, and then our Heyting algebras should collapse to Boolean algebras.  But I'm not sure exactly when that point is.  Can someone help me out?  I suspect this is an exercise in a book on topos theory somewhere.

REPLY [2 votes]: This paper by Abramsky relates Joyal's Lemma on the collapse of a cartesian closed category with a dualizing object to no cloning theorems and no deleting theorems in quantum mechanics.