Online research in philosophy

AGM-theory, named after its founders Carlos Alchourrón, Peter Gärdenfors and David Makinson, is the leading contemporary paradigm in the theory of belief-revision. The theory is reformulated here so as to deal with the central relational notions ‘J is a contraction of K with respect to A’ and ‘J is a revision of K with respect to A’. The new theory is based on a principal-case analysis of the domains of definition of the three main kinds of theory-change (expansion, contraction and revision). The new theory is stated by means of introduction and elimination rules for the relational notions. In this new setting one can re-examine the relationship between contraction and revision, using the appropriate versions of the so-called Levi and Harper identities. Among the positive results are the following. One can derive the extensionality of contraction and revision, rather than merely postulating it. Moreover, one can demonstrate the existence of revision-functions satisfying a principle of monotonicity. The full set of AGM-postulates for revision-functions allow for completely bizarre revisions. This motivates a Principle of Minimal Bloating, which needs to be stated as a separate postulate for revision. Moreover, contractions obtained in the usual way from the bizarre revisions, by using the Harper identity, satisfy Recovery. This provides a new reason (in addition to several others already adduced in the literature) for thinking that the contraction postulate of Recovery fails to capture the Principle of Minimal Mutilation. So the search is still on for a proper explication of the notion of minimal mutilation, to do service in both the theory of contraction and the theory of revision. The new relational formulation of AGM-theory, based on principal-case analysis, shares with the original, functional form of AGM-theory the idealizing assumption that the belief-sets of rational agents are to be modelled as consistent, logically closed sets of sentences. The upshot of the results presented here is that the new relational theory does a better job of making important matters clear than does the original functional theory. A new setting has been provided within which one can profitably address two pressing questions for AGM-theory: (1) how is the notion of minimal mutilation (by both contractions and revisions) best analyzed? and (2) how is one to rule out unnecessary bloating by revisions?

Specified meet contraction is the operation defined by the identity where ∼ is full meet contraction and f is a sentential selector, a function from sentences to sentences. With suitable conditions on the sentential selector, specified meet contraction coincides with the partial meet contractions that yield a finite-based contraction outcome if the original belief set is finite-based. In terms of cognitive realism, specified meet contraction has an advantage over partial meet contraction in that the selection mechanism operates on sentences rather than on temporary infinite structures (remainders) that are cognitively inaccessible. Specified meet contraction provides a versatile framework in which other types of contraction, such as severe withdrawal and base-generated contraction, can be expressed with suitably chosen properties of the sentential selector.

This paper is concerned with the construction of a base contraction (revision) operation such that the theory contraction (revision) operation generated by it will be fully AGM-rational. It is shown that the theory contraction operation generated by Fuhrmann'sminimal base contraction operation, even under quite strong restrictions, fails to satisfy the supplementary postulates of belief contraction. Finally Fuhrmann's construction is appropriately modified so as to yield the desired properties. The new construction may be described as involving a modification of safe (base) contraction so as to make it maxichoice.

By replacement is meant an operation that replaces one sentence by another in a belief set. Replacement can be used as a kind of Sheffer stroke for belief change, since contraction, revision, and expansion can all be defined in terms of it. Replacement can also be defined either in terms of contraction or in terms of revision. Close connections are shown to hold between axioms for replacement and axioms for contraction and revision. Partial meet replacement is axiomatically characterized. It is shown that this operation can have outcomes that are not obtainable through either partial meet contraction or partial meet revision.

The postulate of Recovery, among the six postulates for theory contraction, formulated and studied by Alchourrón, Gärdenfors and Makinson is the one that has provoked most controversy. In this article we construct withdrawal functions that do not satisfy Recovery, but try to preserve minimal change, and relate these withdrawal functions with the AGM contraction functions.

This paper is concerned with formal aspects of the logic of theory change, and in particular with the process of shrinking or contracting a theory to eliminate a proposition. It continues work in the area by the authors and Peter Gärdenfors. The paper defines a notion of safe contraction of a set of propositions, shows that it satisfies the Gärdenfors postulates for contraction and thus can be represented as a partial meet contraction, and studies its properties both in general and under various natural constraints.

In some recent papers, the authors and Peter Gärdenfors have defined and studied two different kinds of formal operation, conceived as possible representations of the intuitive process of contracting a theory to eliminate a proposition. These are partial meet contraction (including as limiting cases full meet contraction and maxichoice contraction) and safe contraction. It is known, via the representation theorem for the former, that every safe contraction operation over a theory is a partial meet contraction over that theory. The purpose of the present paper is to study the relationship more finely, by seeking an explicit map between the component orderings involved in each of the two kinds of contraction. It is shown that at least in the finite case a suitable map exists, with the consequence that the relational, transitively relational, and antisymmetrically relational partial meet contraction functions form identifiable subclasses of the safe contraction functions, over any theory finite modulo logical equivalence. In the process of constructing the map, as the composition of four simple transformations, mediating notions of bottom and top contraction are introduced. The study of the infinite case remains open.

The paper surveys some recent work on formal aspects of the logic of theory change. It begins with a general discussion of the intuitive processes of contraction and revision of a theory, and of differing strategies for their formal study. Specific work is then described, notably Gärdenfors' postulates for contraction and revision, maxichoice contraction and revision functions and the condition of orderliness, partial meet contraction and revision functions and the condition of relationality, and finally the operations of safe contraction and revision. Verifications and proofs are omitted, with references given to the literature but definitions and principal results are presented with rigour along with discussion of their significance.

The paper surveys some recent work on formal aspects of the logic of theory change. It begins with a general discussion of the intuitive processes of contraction and revision of a theory, and of differing strategies for their formal study. Specific work is then described, notably Gärdenfors'' postulates for contraction and revision, maxichoice contraction and revision functions and the condition of orderliness, partial meet contraction and revision functions and the condition of relationality, and finally the operations of safe contraction and revision. Verifications and proofs are omitted, with references given to the literature, but definitions and principal results are presented with rigour, along with discussion of their significance.