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Journal of Logic and Computation Advance Access published online on April 22, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp023
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

A Graph-theoretic Account of Logics

Amilcar Sernadas, Cristina Sernadas and Joao Rasga

Department of Mathematics, Instituto Superior Técnico, TU Lisbon and SQIG, Instituto de Telecomunicações, Lisbon, Portugal.
E-mail: acs{at}math.ist.utl.pt; css{at}math.ist.utl.pt; jfr{at}math.ist.utl.pt

Marcelo Coniglio

Department of Philosophy and CLE, State University of Campinas, Brazil. E-mail: coniglio{at}cle.unicamp.br

Received 29 July 2008.


  Abstract

A graph-theoretic account of logics is explored based on the general notion of m-graph (i.e; a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as multi-graphs (m-graphs). After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with non-deterministic semantics, and subsume all logics endowed with an algebraic semantics.

Keywords: Graph-theoretic account of logics; non-deterministic semantics; diagrammatic reasoning via morphisms; completeness results


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