The Geometry of Non-Distributive Logics

Journal of Symbolic Logic 70 (4):1108 - 1126 (2005)
Abstract
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
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DOI 10.2178/jsl/1129642117
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The Structure of Multiplicatives.Vincent Danos & Laurent Regnier - 1989 - Archive for Mathematical Logic 28 (3):181-203.
Duplication of Directed Graphs and Exponential Blow Up of Proofs.A. Carbone - 1999 - Annals of Pure and Applied Logic 100 (1-3):1-67.
The Additive Multiboxes.Lorenzo Tortora de Falco - 2003 - Annals of Pure and Applied Logic 120 (1-3):65-102.

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