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On the logic of theory change: Partial meet contraction and revision functions

Published online by Cambridge University Press:  12 March 2014

Carlos E. Alchourrón
Affiliation:
Universidad de Buenos Aires, Buenos Aires, Argentina
Peter Gärdenfors
Affiliation:
Lunds Universitet, Lund, Sweden
David Makinson
Affiliation:
American University of Beirut, Beirut, Lebanon

Abstract

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated.

In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gärdenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gàrdenfors investigated, with a further representation theorem established.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1985

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References

[1]Alchourron, Carlos E. and Makinson, David, Hierarchies of regulations and their logic, New studies in deontic logic (Hilpinen, R., editor), Reidel, Dordrecht, 1982, pp. 125148.Google Scholar
[2]Alchourron, Carlos E. and Makinson, David, On the logic of theory change: Contraction functions and their associated revision functions, Theoria, vol. 48 (1982), pp. 1437.Google Scholar
[3]Gärdenfors, Peter, Conditionals and changes of belief, The logic and epistemology of scientific change (Niiniluoto, I. and Tuomela, R., editors), Acta Philosophica Fennica, vol. 30 (1978), pp. 381404.Google Scholar
[4]Gärdenfors, Peter, Rules for rational changes of belief, 320311: Philosophical essays dedicated to Lennart Åqvist on his fiftieth birthday (Pauli, T., editor), Philosophical Studies No. 34, Department of Philosophy, University of Uppsala, Uppsala, 1982, pp. 88101.Google Scholar
[5]Gärdenfors, Peter, Epistemic importance and minimal changes of belief, Australasian Journal of Philosophy, vol. 62 (1984), pp. 136157.Google Scholar
[6]Makinson, David, How to give it up: A survey of some formal aspects of the logic of theory change, Synthese (to appear).Google Scholar
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