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, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron 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 Gardenfors 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 Gardenfors investigated, with a further representation theorem established. (shrink)

Kernel contraction is a natural nonrelational generalization of safe contraction. All partial meet contractions are kernel contractions, but the converse relationship does not hold. Kernel contraction is axiomatically characterized. It is shown to be better suited than partial meet contraction for formal treatments of iterated belief change.

Multiple contraction (simultaneous contraction by several sentences) and iterated contraction are investigated in the framework of specified meet contraction (s.m.c.) that is extended for this purpose. Multiple contraction is axiomatized, and so is finitely multiple contraction (contraction by a finite set of sentences). Two ways to reduce finitely multiple contraction to contraction by single sentences are introduced. The reduced operations are axiomatically characterized and their properties are investigated. Furthermore, it is shown how iterated contraction can be reduced to single-step, single-sentence (...) contraction. However, in this framework the outcome of iterated contraction depends unavoidably on the order in which the inputs are received. This order-dependence makes it impossible to treat two inputs on an equal footing. Therefore it is often preferable to perform changes involving several pieces of information as multiple rather than iterated change. (shrink)

We propose a new class of multiple contraction operations — the system of spheres-based multiple contractions — which are a generalization of Grove’s system of spheres-based (singleton) contractions to the case of contractions by (possibly non-singleton) sets of sentences. Furthermore, we show that this new class of functions is a subclass of the class of the partial meet multiple contractions.

In the logic of theory change, the standard model is AGM, proposed by Alchourrón et al. (J Symb Log 50:510–530, 1985 ). This paper focuses on the extension of AGM that accounts for contractions of a theory by a set of sentences instead of only by a single sentence. Hansson (Theoria 55:114–132, 1989 ), Fuhrmann and Hansson (J Logic Lang Inf 3:39–74, 1994 ) generalized Partial Meet Contraction to the case of contractions by (possibly non-singleton) sets of sentences. In this (...) paper we present the possible worlds semantics for partial meet multiple contractions. (shrink)

The AGM theory of belief contraction is extended tomultiple contraction, i.e. to contraction by a set of sentences rather than by a single sentence. There are two major variants: Inpackage contraction all the sentences must be removed from the belief set, whereas inchoice contraction it is sufficient that at least one of them is removed. Constructions of both types of multiple contraction are offered and axiomatically characterized. Neither package nor choice contraction can in general be reduced to contractions by single (...) sentences; in the finite case choice contraction allows for reduction. (shrink)

This paper focuses on the extension of AGM that allows change for a belief base by a set of sentences instead of a single sentence. In [FH94], Fuhrmann and Hansson presented an axiomatic for Multiple Contraction and a construction based on the AGM Partial Meet Contraction. We propose for their model another way to construct functions: Multiple Kernel Contraction, that is a modification of Kernel Contraction, proposed by Hansson [Han94] to construct classical AGM contractions and belief base contractions. This construction (...) works out the unsolved problem pointed out by Hansson in [Han99, pp. 369]. (shrink)

Alchourrón, Gärdenfors and Makinson have developed and investigated a set of rationality postulates which appear to capture much of what is required of any rational system of theory revision. This set of postulates describes a class of revision functions, however it does not provide a constructive way of defining such a function. There are two principal constructions of revision functions, namely an epistemic entrenchment and a system of spheres. We refer to their approach as the AGM paradigm. We provide a (...) new constructive modeling for a revision function based on a nice preorder on models, and furthermore we give explicit conditions under which a nice preorder on models, an epistemic entrenchment, and a system of spheres yield the same revision function. Moreover, we provide an identity which captures the relationship between revision functions and update operators (as defined by Katsuno and Mendelzon). (shrink)