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Coherent Systems

Karl Schlechta

Elsevier

STUDIES IN LOGIC

AND

PRACTICAL REASONING

VOLUME 2

Editors:

D.M. GABBAY, London

P. GARDENFORS, Lund

J. SIEKMANN, Saarbrucken

J. VAN BENTHEM, Amsterdam & Stanford

M. VARDI, Houston

J. WOODS, Vancouver & London

ELSEVIER

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS

SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

COHERENT SYSTEMS

Karl SCHLECHTA

Universite de Provence and

Laboratoire d'Informatique Fondamentalle (CNRS UMR 6166)

Marseille, France

ELSEVIER

2004

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD • PARIS

SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

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First edition 2004

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Foreword

(by David Makinson)

This book contains the deepest and most comprehensive study yet under-

taken of preferential and related models for nonmonotonic inference.

It is based on a series of papers and monographs by the author, published

over the last fifteen years. It goes beyond them, bringing separate results

into a coherent overall framework, accompanied by discussion of the gen-

eral concepts underlying the field. At the same time, the author solves

new problems that rose to the surface as he carried out this conceptual

organization.

The subject is logic, specifically nonmonotonic logic. The focus is on rep-

resentation theorems for such systems, particularly in the tradition of pref-

erential models. These were introduced by Shoham, and brought to wide

attention when Kraus, Lehmann and Magidor obtained the first deep results

in a classic paper. Whereas classical logic takes the conclusions obtainable

from a set of premises to be those formulas that are true in all the models

satisfying them, the driving idea of the preferential tradition is to permit

as conclusions all those formulas that are true in all the models that are

minimal, under a given ordering, among the models satisfying the premises.

A simple idea — but one that opened the door to the exploration of new

worlds, as well as revealing links with investigations in several other ar-

eas of logic such as belief change, update, counterfactual conditionals, and

conditional directives.

The author's approach is resolutely semantic, in the sense that as much

work as possible is done on the level of the models themselves, indepen-

dently of the logical superstructure. The representation problem is thus

transformed into one of determining when certain kinds of model using

selection functions can have that selection function determined by other

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