Abstract
In this series of papers we set out to generalize the notion of classical analytic deduction (i.e., deduction via elimination rules) by combining the methodology of labelled deductive systems (LDS) with the classical systemKE. LDS is a unifying framework for the study of logics and of their interactions. In the LDS approach the basic units of logical derivation are not just formulae butlabelled formulae, where the labels belong to a given “labelling algebra”. The derivation rules act on the labels as well as on the formulae, according to certain fixed rules of propagation. By virtue of the extra power of the labelling algebras, standard (classical or intuitionistic) proof systems can be extended to cover a much wider territory without modifying their structure. The systemKE is a new tree method for classical analytic deduction based on “analytic cut”.KE is a refutation system, like analytic tableaux and resolution, but it is essentially more efficient than tableaux and, unlike resolution, does not require any reduction to normal form.
We start our investigation with the family of substructural logics. These are logical systems (such as Lambek's calculus, Anderson and Belnap's relevance logic, and Girard's linear logic) which arise from disallowing some or all of the usual structural properties of the notion of logical consequence. This extension of traditional logic yields a subtle analysis of the logical operators which is more in tune with the needs of applications. In this paper we generalize the classicalKE system via the LDS methodology to provide a uniform refutation system for the family of substructural logics.
The main features of this generalized method are the following: (a) each logic in the family is associated with a “labelling algebra”; (b) the tree-expansion rules (for labelled formulae) are the same for all the logics in the family; (c) the difference between one logic and the other is captured by the conditions under which a branch is declared closed; (d) such conditions depend only on the labelling algebra associated with each logic; and (e) classical and intuitionistic negations are characterized uniformly, by means of the same tree-expansion rules, and their difference is reduced to a difference in the labelling algebra used in closing a branch. In this first part we lay the theoretical foundations of our method. In the second part we shall continue our investigation of substructural logics and discuss the algorithmic aspects of our approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abrusci, V. M.: Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic,Journal of Symbolic Logic 56 (1991), 1403–1451.
Anderson, A. R. and Belnap, N. D., Jr.:Entailment: The Logic of Relevance and Necessity, volume 1, Princeton University Press, Princeton, 1975.
Allwein, G. and Dunn, J. M.: Kripke models for linear logic,Journal of Symbolic Logic 58 (1993), 514–545.
Avron, A.: The semantics and proof theory of linear logic,Theoretical Computer Science 57 (1988), 161–184.
Avron, A.: Simple consequence relations,Journal of Information and Computation 92 (1991), 105–139.
Beckert, Bernhard, Hähnle, Reiner and Schmitt, Peter: The even more liberalized δ-rule in free-variable semantic tableaux, in Gottlob, G., Leitsch, A. and Mundici, D. (eds)Proc. 3rd Kurt Gödel Colloquium, Brno, Czech Republic, Springer-Verlag, 1993, pp. 108–118.
D'Agostino, Marcello: Are tableaux an improvement on truth-tables?Journal of Logic, Language and Information 1 (1992), 235–252.
D'Agostino, Marcello: On the efficiency of proof search in classical analytic deduction, to appear. (Preliminary version presented at theWorkshop on Theorem Proving with Analytic Tableaux and Related Methods, Marseille, 28–30 April 1993.)
D'Agostino, Marcello and Mondadori, Marco: The taming of the cut,Journal of Logic and Computation 4(3) (1994), 285–319.
Dôsen, Kosta: Sequent systems and groupoid models, I,Studia Logica 47 (1988), 353–385.
Dôsen, Kosta: Sequent systems and groupoid models, II,Studia Logica 48 (1989), 41–65.
Dunn, J. M.: Intuitive semantics for first-degree entailment and coupled trees,Philosophical Studies 29 (1976), 149–168.
Fitting, M.:Proof Methods for Modal and Intuitionistic Logics, Reidel, Dordrecht, 1983.
Gabbay, D.M.: Labelled deductive systems, part I. Technical Report CIS-Bericht-90-22, CIS-Universität München, February 1991. Preliminary partial draft of a book intended for Oxford University Press.
Gabbay, Dov M.: Fibred semantics and the weaving of logics, I. Technical Report, University of Stuttgart, 1993.
Gabbay, Dov M.: Classical versus non-classical logics, in Dov Gabbay, Chris Hogger, and J. H. Robinson (eds),Handbook of Logic in AI and Logic Programming, Oxford University Press (to appear).
Gabbay, Dov M.: General theory of structured consequence relations, in P. Schroeder Heister and Kosta Dôsen (eds),Substructural Logics, Oxford University Press, 1993, pp. 109–151.
Gabbay, Dov M. and Ohlbach, Hans J.: An algebraic fine structure for logical systems, in Dov Gabbay and Franz Guenthner (eds),What is a Logical System? Oxford University Press (to appear).
Gabbay, Dov M. and Ohlbach, Hans J.: From Hilbert calculus to possible-world semantics, in Krysia Broda (ed.),Proceedings of ALPUK Logic Programming Conference 1992, Lecture Notes in Computer Science. Springer 1993, pp. 219–252.
Hähnle, Reiner: Automated Deduction in Multiple-Valued Logics, inMonographs on Computer Science, Oxford University Press,10 (1994).
Maslov, S. Yu.: Invertible sequent variant of constructive predicate calculus,Seminar in Mathematics, V. A. Steklov Mathematical Institute, Leningrad,4 (1969), 36–42.
McRobbie, Michael A. and Belnap, Nuel D.: Relevant analytic tableaux,Studia Logica 38 (1979), 187–200.
Moortgat, Michael, Labelled deductive systems for categorial theorem proving, in Dekker and Stokhof (eds),Proc. 8th Amsterdam Colloquium, 1992.
Ono, H.: Semantics for substructural logics, in Peter Schroeder-Heister (ed.),Substructural Logics, Oxford University Press, 1993, pp. 259–291.
Rosenthal, Kimmo, I.:Quantales and their applications, Longman, 1990.
Routley, R. and Meyer, R. K.: The semantics of entailment, I, in H. Leblanc (ed.),Truth, Syntax and Semantics, North-Holland, 1973.
Routley, R. and Routley, V.: The semantics of first degree entailment,Noûs VI (1972), 335–359.
Sambin, G.: The semantics of pretopologies, in Peter Schroeder-Heister (ed.),Substructural Logics, Oxford University Press, 1993, pp. 293–307.
Thistlewaite, P. B., McRobbie, M. A. and Meyer, B. K.:Automated Theorem Proving in Non Classical Logics, Pitman, 1988.
Urquhart, Alasdair: Semantics for relevant logic,Journal of Symbolic Logic 37 (1972), 159–170.
Wallen, L. A.:Automated Deduction in Non-Classical Logics, The MIT Press, Cambridge, Mass., 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
D'Agostino, M., Gabbay, D.M. A generalization of analytic deduction via labelled deductive systems. Part I: Basic substructural logics. J Autom Reasoning 13, 243–281 (1994). https://doi.org/10.1007/BF00881958
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00881958