What Is the Problem with Proof Nets for Classical Logic?

There is a very close and well-understood relationship between proofs in intuitionistic logic, simply typed lambda-terms, and morphisms in Cartesian closed categories. The same relationship can be established for multiplicative linear logic (MLL), where proof nets take the role of the lambda-terms, and starautonomous categories the role of Cartesian closed categories. It is certainly desirable to have something similar for classical logic, which can be obtained from intuitionistic logic by adding the law of excluded middle, i.e., A∨ Ā, or equivalently, an involutive negation, i.e., Ā = A. Adding this to a Cartesian closed category C , means adding a contravariant functor (−) : C → C such that Ā ∼= A and (A ∧B) ∼= Ā∨ B̄ where A∨B = Ā⇒B. However, if we do this we get a collapse: all proofs of the same formula are identified, which leads to a rather boring proof theory. This observation is due to André Joyal, and a proof and discussion can be found in [1,2,3]. Here we will not show the category theoretic proof of the collapse, but will quickly explain the phenomenon in terms of the sequent calculus (the argumentation is due to Yves Lafont [4, Appendix B]). Suppose we have two proofs Π1 and Π2 of a formula B in some sequent calculus system. Then we can form, with the help of the rules weakening, contraction, and cut, the following proof of B: