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A categorical approach to the semantics of argumentation

Published online by Cambridge University Press:  04 March 2009

Simon Ambler
Simon Ambler
Affiliation:
Department of Mathematics and Computer Science, University of Leicester, Leicester LE1 7RH, England.
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Abstract

Argumentation is a proof theoretic paradigm for reasoning under uncertainty. Whereas a ‘proof’ establishes its conclusion outright, an ‘argument’ can only lend a measure of support. Thus, the process of argumentation consists of identifying all the arguments for a particular hypothesis φ, and then calculating the support for φ from the weight attached to these individual arguments. Argumentation has been incorporated as the inference mechanism of a large scale medical expert system, the ‘Oxford System of Medicine’ (OSM), and it is therefore important to demonstrate that the approach is theoretically justified. This paper provides a formal semantics for the notion of argument embodied in the OSM. We present a categorical account in which arguments are the arrows of a semilattice enriched category. The axioms of a cartesian closed category are modified to give the notion of an ‘evidential closed category’, and we show that this provides the correct enriched setting in which to model the connectives of conjunction (&) and implication (⇒).

Finally, we develop a theory of ‘confidence measures’ over such categories, and relate this to the Dempster-Shafer theory of evidence.

Information

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 6 , Issue 2 , April 1996, pp. 167 - 188
Copyright
Copyright © Cambridge University Press 1996

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References

Aqvist, L. (1984) Deontic Logic. In: Gabbay, D. and Guenthner, F. (eds.) Handbook of Philosophical Logic Vol. II, Reidel, Dordrecht.Google Scholar
Carboni, A., Kelly, G. M. and Wood, R. J. (1990) A 2-categorical approach to change of base and geometric morphisms I, Sydney Category Seminar Report 90–1, Dept of Pure Mathematics, University of Sydney, Australia.Google Scholar
Crouch, R. and Pulman, S. (1991) A simple approach to belief revision, Technical Report, SRI International, Cambridge Computer Science Research Centre.Google Scholar
Carboni, A. and Walters, R. F. C. (1987) Cartesian bicategories I. J. Pure Appl. Algebra 49 11–32.CrossRefGoogle Scholar
Clarke, M. and Wilson, P. N. (1991) Efficient algorithms for belief functions based on the relationship between belief and probability. Proc. ECSQAU, Marseille, 1991. Springer-Verlag Lecture Notes in Computer Science 548.CrossRefGoogle Scholar
Dubois, D. and Prade, H. (1988) Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York.Google Scholar
Di Paola, R. A. and Heller, A. (1987) Dominical categories: Recursion theory without elements. J. Symbolic Logic 3 52.Google Scholar
Fox, J., Glowinski, A. J., Gordon, C., Hajnal, S. J. and O'Neil, M. J. (1990) Logic engineering for knowledge engineering: design and implementation of the Oxford System of Medicine. Artificial Intelligence in Medicine 2 323–339.CrossRefGoogle Scholar
Fox, J., Krause, P. J. and Ambler, S. J. (1992) Arguments, contradictions and practical reasoning. Proc. European Conference on A.I. ‘92, Wiley.Google Scholar
Fox, T. (1976) Coalgebras and cartesian categories. Comm. Algebra 7 (4) 665–667.CrossRefGoogle Scholar
Girard, J. Y. (1987) Linear logic. Theoretical Computer Science 50 1–102.CrossRefGoogle Scholar
Girard, J. Y., Taylor, P. and Lafont, Y. (1989) Proofs and Types, Cambridge University Press.Google Scholar
Johnstone, P. T. (1982) Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press.Google Scholar
Kelly, G. M. (1982) Basic Concepts of Enriched Category Theory. London Math. Soc. Lecture Note Series 64, Cambridge University Press.Google Scholar
Krause, P. J., Ambler, S. J., Elvang-Gøransson, M. and Fox, J. (1995) A logic of argumentation for reasoning under uncertainty. Computational Intelligence 1 (11) 113–131.CrossRefGoogle Scholar
Krause, P. J., Ambler, S. J. and Fox, J. (1993) The development of a logic of argumentation. In: Bouchon-Meunier, B., Valverde, L. and Yager, R. R. (eds.) IPMU ‘92, Advanced Methods in Artificial Intelligence. Springer-Verlag Lecture Notes in Computer Science 682 109–118.CrossRefGoogle Scholar
Lawvere, F. W. (1973) Metric spaces, generalized logic, and closed categories. Rend, del Sent. Mat.e Fis. di Milano 43 135–166.CrossRefGoogle Scholar
Lindley, D. V. (1985) Making Decisions (2nd ed.), Wiley.Google Scholar
Lambek, J. L. and Scott, P. J. (1986) Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics 7, Cambridge University Press.Google Scholar
Mac Lane, S. (1971) Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag.CrossRefGoogle Scholar
Moggi, E. (1989) Computational lambda-calculus and monads. In: Proc. 4th annual symposium on Logic In Computer Science, IEEE.Google Scholar
O’Neil, M. and Glowinski, A. (1990) Evaluating and validating very large knowledge-based systems. Med. Inform. 15 237–251.CrossRefGoogle ScholarPubMed
Pitts, A. M. (1988) Applications of sup-lattice enriched category theory to sheaf theory. Proc. London Math. Soc. 3 (57) 433–480.CrossRefGoogle Scholar
Shafer, G. (1976) A Mathematical Theory of Evidence, Princeton University Press, Princeton, New Jersey.CrossRefGoogle Scholar
Wilson, P. N. (1989) Justification, computational efficiency and generalisation of the Dempster-Shafer theory, Research report no. 15, Dept. of Computing and Math. Sciences, Oxford Polytechnic.Google Scholar