Change, Choice and Inference (Recap, Pt 1.)


Published March 24, 2019


Belief is not the only attitude that is relevant to the cognitive situation of inquirers. Inquirers make posits, presumptions, assumptions and presuppositions as well. These methodological attitudes may diverge from belief in various ways, giving us additional complexity in a representation of a doxastic situation. But, I suggested, the cluster of propositional attitudes which were grouped together under the label acceptance share a common structure with belief. (Stalnaker 1984, 99)

A nearby future version of myself is writing a literature review of nonmonotonic belief revision. To help them out, I am writing brief notes of the various things I am now reading to that end.

Rott (2001)’s Change, Choice and Inference is a book treating, as the overview has it, “two new perspectives on the formation and transformation of belief” (1). These perspectives are two ways of thinking about belief revision. In the first, static perspective, the problem of belief revision is a problem of taking a certain stock of information, or, perhaps, evidence, which is liable to be both incomplete and inconsistent, and coming up with a coherent system of beliefs on top of that. From this point of view, I take it, revising one’s beliefs on the basis of new evidence is fundamentally just taking one’s old evidence, adding the new piece, and redoing the derivation of a complete system. In the other, dynamic view, the main problem of belief revision is one of reconciling old and new information.

Doxastic Attitudes

There is the question, prior to all this, of what exactly it is one is revising when one is revising beliefs. Rott makes a few preliminary remarks. Rott accepts something like the radical interpretation thought that the point of ascribing beliefs is to make sense of someone as a rational agent, and, further, that belief-ascriptions are holistic.1 Even if this is right, it can perhaps be factored out of a theory of belief revision, as we can assume a fixed interpretive context. Beliefs are, at least under one notion, graded: one can believe something with a certain level of confidence. However, there are propositional attitudes like affirming that are binary rather than graded. Rott is, in general, interested in these binary, belief-like attitudes that include perhaps some belief states, but also, for example, default assumptions.

These states are binary rather than graded, but they are also more or less entrenched. Consider two beliefs, each of which have a high degree of confidence (enough to make one comfortable with asserting them) but with very different amounts of corresponding evidence. If push came to shove and one had to, rationally speaking give one up, one would give up the one with the worse evidential standing. (Degrees of entrenchment, Rott will go on to claim, are like revealed preferences: what it is for a state to be more entrenched than another is that the latter is given up first when there is a choice between them.) Rott thinks that the distinction between beliefs and default assumptions (and other provisional attitudes) comes down to a contextually determined difference in entrenchment (beliefs are more entrenched than default assumptions).

There is a brief discussion of doxastic voluntarism (17), where Rott maintains, I think plausibly, that while certain doxastic attitudes are not under our control (I suspect degrees of confidence are like this) we still have the power to suspend acceptance (that is, we have the power to think I just don’t have good enough evidence to support this claim). Besides, even if we don’t have voluntary control of our beliefs or certain acceptance states, we can still use the language of choice in speaking about belief revision.

So far, so fair. We now get on to a bit of formal stuff. We want to represent people’s beliefs (acceptance states). Rott takes it, and this is again plausible, that we should start with a representation of belief bases. We may think of belief bases as containing basic, underived beliefs. As Rott (23) puts it.

(Maxim B) The elements of a base are explicitly given beliefs. They comprise beliefs, and only beliefs, which have some kind of independent standing, i.e., which are not derived by processes of inference from other beliefs. They serve as a basis or foundation of the belief set that can be ‘built on’ the base.

What is built on the base is a theory, a logically closed, consistent set of sentences. The theory is what can be inferred from the base, where this is different than the logical consequences of a base. Here, Rott assumes that the appropriate logical consequence relation is at least consequence in classical logic.

Inference

So what’s this inference relation. Well, Rott takes a common line here. Your belief base is probably inconsistent. So you can’t just take its consequences. What you do is first make it consistent; this is called consolidation. (One way of representing this is to think that in consolidating you are “contracting by \(\bot\)”, where to contract by a sentence is to remove what is inconsistent with that sentence.) Once you’ve consolidated, then you can just take the logical consequence. How do you consolidate, or contract? Rott forgoes one common approach (the AGM approach) of giving a general or axiomatic treatment and rather offers a simple definition. The basic concept is this: in contracting a base by a sentence you remove as little as possible so that the base does not include or logically imply that sentence. How do you know you’ve removed as little as possible? Well, it’s that if you put any of the removed beliefs back in, the result would again imply that sentence. (Contracting by a sentence not implied by a base either does nothing or is undefined.) Note that this is not a singly defined operation: more than one resulting sets are going to fulfill this description.

As noted, in belief base we have more or less firmly held states (beliefs vs. assumptions). These are represented by a priority relation (where this is related, but prior to, an entrenchment relation; differing in that priority relations hold between elements of a base and have no logical constraints while entrenchment relations hold in a theory and are constrained by logic). We represent a priority relation in the following way: take the sentences in the base, call them \(\mathcal{H}\). Divide \(\mathcal{H}\) into indexed, not necessarily disjoint subsets \(\mathcal{H} = \langle H_1, H_2, \ldots, H_n\rangle\). Now say that for \(\phi, \psi \in \mathcal{H}\), \(\phi \prec \psi\) just in case there are \(i < j\) such that \(\phi \in H_i\) and \(\psi \in H_j\). Note, this makes \(\prec\) a weak ordering (it is not anti-reflexive).

This gives us a better treatment of contraction, though the basic idea is the same. To contract by a sentence, first start with the most prioritized sentences (those with the highest priority index). Take a maximal subset of those not implying those sentences. Go down a priority level. Add one as many as you can without implying the sentence. Continue on to the first level. This will get you contractions with the desired property (maximal without implying the contracted sentence) which get in as much prioritized stuff as possible. You preserve the indexes as you go, which gets you a type-match in input and output.

Conclusion

That’s the gist of the first chapter. The approach on its face is plausible enough. I’m not yet sure what to think of priority and entrenchment (I suspect they have something to do with justification, but it’s unclear).

References

Rott, Hans. 2001. Change, Choice and Inference: A Study of Belief Revision and Nonmonotonic Reasoning. Oxford University Press.

Stalnaker, Robert. 1984. Inquiry. Cambridge University Press.


  1. One might perhaps think, and I suspect this is right, that we should take this interpretivist view as at best a theory of belief-ascriptions of others. And I cannot have a theory of other’s beliefs without being already capable of having them myself (that is to make sense of a person having beliefs ascribed to them, we must already understand someone—in a first person sense—as already a believer capable of making ascriptions). In fact, I suspect Burge is right that the bounds of this view are even more limited. For I generally don’t come to ascribe beliefs to others by interpreting them (not even subconsciously), but rather because I already speak their language and assume that they say what they believe. There may be cases (e.g. in learning a new language or faced with the prospects of radical deception) where we do do interpretation, but it is not the paradigm case. So I think, for the purposes of a theory of belief revision, radical interpretation is likely to be a red herring.