The Logic of Belief Change and Nonadditive Probability

Abstract

The present paper investigates the rationality conditions on belief change of a theory which has recently come to the forefront of philosophical logic and artificial intelligence — the Alchourrón-Gärdenfors-Makinson (AGM) theory of belief change.¹ In contradistinction with the well-established Bayesian approach to belief revision, this one never explicitly refers to the individual’s decisions. Nor does it formalize the individual’s beliefs in measure-theoretic — let alone probabilistic — terms. The building blocks of the AGM theory are propositions. The major mathematical constraint is that these propositions are expressed in a language which in an appropriate sense includes the sentential calculus. Epistemic states, or states of belief, are captured by deductively closed sets of propositions. Epistemic attitudes — belief, disbelief, and indeterminacy — are then described by means of the membership relation. The epistemic input, that is the incoming information, is normally restricted to be propositional. Epistemic changes are axiomatized in terms of the following items: the input which bring them about, the initial epistemic state and the resulting epistemic state. There are three such operations: contraction, expansion and revision. In the principal case at least, contraction may be viewed as a move from belief or disbelief to indeterminacy, expansion as a move from indeterminacy to either belief or disbelief, and finally revision as a move from either determinate attitude to the opposite one. This is all good commonsense in an initially barren mathematical framework. The soberness of the approach makes it the more remarkable that it leads to important results, one of which will occupy the center stage here.