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, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. 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. In this paper we present a new kind of natural deduction system for lattice logic and extensions of lattice logic with negation. This natural deduction system has a natural symmetry. The rules for lattice disjunction are exactly the reverse of the rules for lattice conjunction. In this way, the natural deduction system shares some features with Girard’s proofnets for multiplicative linear logic [12]. Our proofs differ from proofnets, however, in that they are oriented graphs in which the flow of information from premise to conclusion in inference is directly represented in the graph structure. Nonetheless, our proof graphs can be given a global graph-theoretical criterion for correcteness, just as can be done for Girard’s proofnets. Readers will also note that our proof structures bear more than a passing resemblence to Shoesmith and Smiley’s multiple conclusion proofs for classical logic, which are oriented graphs allowing both divergent and convergent branching, just like ours. Our lattice logic proofs differ from Shoesmith and Smiley’s multiple conclusion proofs in a number of important respects, not the least of which is that lattice logic proofs have