Putting the Building-Blocks Together

• Sven Ove Hansson
Chapter
Part of the Trends in Logic book series (TREN, volume 46)

Abstract

This is the first of four chapters that form the central part of the book, introducing and developing a new approach to belief change, descriptor revision. In this chapter, the new model is constructed from its basic components. We begin with a skeletal input-output model that contains no sentences but only primitive (i.e., unstructured) belief states and inputs, together with a revision function $$\circledcirc$$ that takes us from any belief state $$\mathcal {K}$$ and input $$\text {\i }$$ to a new belief state $$\mathcal {K} \circledcirc \text {\i }$$. This model has the advantage of making few controversial assumptions but also the disadvantage of low expressive power. It is used as a starting-point to which more structure is added successively, allowing us to see what assumptions are needed to obtain the resulting increase in expressive power. Sentences are added to the framework with the help of a support function that takes us from any belief state $$\mathcal {K}$$ to the set of sentences representing the beliefs that it supports. After that, the two major components of the new framework are introduced. The first is belief descriptors, a versatile construct for describing belief states. The metalinguistic expression $$\mathfrak {B}p$$ denotes that p is believed in the belief state under consideration. Truth-functional combinations are interpreted in the usual way, thus $$\lnot \mathfrak {B}p$$ denotes that p is not believed and $$\mathfrak {B}p \vee \mathfrak {B}\lnot p$$ that either p or $$\lnot p$$ is believed. Sets of such expressions are used to denote combined properties, hence $$\{\lnot \mathfrak {B}p, \lnot \mathfrak {B}q\}$$ denotes that neither p nor q is believed. Belief descriptors (either single sentences or sets of such sentences) are used as inputs for belief change (replacing primitive inputs such as $$\text {\i }$$). Due to their versatility, all changes can be performed with a single, uniform change operation $$\circ$$. In order to revise the belief set K by a sentence p we use the input (success condition) $$\mathfrak {B}p$$, and the outcome is $$K \circ \mathfrak {B}p$$. In order to remove the sentence q we perform the operation $$K \circ \lnot \mathfrak {B}q$$, etc. In order to perform these operations, the second major component of the new framework is introduced, namely a selection mechanism (a monoselective choice function) that directly selects the output from those among a given set of potential outcomes (the “outcome set”) that satisfy the success condition.