Labelled Natural Deduction for Substructural Logics

Logic Journal of the IGPL 7 (3):283-318 (1999)

In this paper a uniform methodology to perform natural\ndeduction over the family of linear, relevance and intuitionistic\nlogics is proposed. The methodology follows the Labelled\nDeductive Systems (LDS) discipline, where the deductive process\nmanipulates {\em declarative units} -- formulas {\em labelled}\naccording to a {\em labelling algebra}. In the system described\nhere, labels are either ground terms or variables of a given {\em\nlabelling language} and inference rules manipulate formulas and\nlabels simultaneously, generating (whenever necessary)\nconstraints on the labels used in the rules. A set of natural\ndeduction style inference rules is given, and the notion of a\n{\em derivation} is defined which associates a labelled natural\ndeduction style ``structural derivation'' with a set of generated\nconstraints. Algorithmic procedures, based on a technique called\n{\em resource abduction\/}, are defined to solve the constraints\ngenerated within a structural derivation, and their termination\nconditions discussed. A natural deduction derivation is then\ndefined to be {\em correct} with respect to a given substructural\nlogic, if, under the condition that the algorithmic procedures\nterminate, the associated set of constraints is satisfied with\nrespect to the underlying labelling algebra. Finally, soundness\nand completeness of the natural deduction system are proved with\nrespect to the LKE tableaux system \cite{daga:LAD}.\footnote{Full\nversion of a paper presented at the {\em 3rd Workshop on Logic,\nLanguage, Information and Computation\/} ({\em WoLLIC'96\/}), May\n8--10, Salvador (Bahia), Brazil, organised by Univ.\ Federal de\nPernambuco (UFPE) and Univ.\ Federal da Bahia (UFBA), and\nsponsored by IGPL, FoLLI, and ASL.}
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DOI 10.1093/jigpal/7.3.283
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.A. Russo - 2008 - In Unendlichkeit. Mohr. pp. 87--112.

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