The Geometry of Non-Distributive Logics

In this paper, we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness. We show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic, and we indicate how proofs in this system may be labelled with terms exhibiting a kind of Curry-Howard isomorphism. This natural deduction system is inspired both by Shoesmith and Smiley’s multiple conclusion systems for classical logic and Girard’s proofnets for linear logic.

This paper is joint work with Francesco Paoli.

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