On the Ratio Challenge for Comparativism

ABSTRACT

This paper discusses a challenge for comparativists about belief, who hold that numerical degree of belief (in particular, subjective probability) is a useful fiction, unlike comparative belief, which they regard as real. The challenge is to make sense of claims like ‘I am twice as confident in A as in B’ in terms of comparative belief only. After showing that at least some comparativists can meet this challenge, I discuss implications for Zynda's [2000] and Stefánsson's [2017] defences of comparativism.

KEYWORDS:

• comparativism
• ratios
• degrees of belief
• comparative belief

Notes

¹ That is, from an epistemic point of view, the choice of numerical model is arbitrary, comparativists think, since we can have no reason for thinking that, of two numerical models that represent the same comparative relation, one model more accurately describes the agent's beliefs than the other. However, from a practical point of view, the choice may be far from arbitrary, since we may find one model easier to use and understand than another.

² I will use ‘belief’ and ‘confidence’ interchangeably throughout this paper.

³ A preference, as I shall be using the term (following a convention in decision theory), is a comparative instrumental desire, that corresponds to the agent's non-instrumental desires and her beliefs about how these desires are best satisfied. An instrumentally rational person's choices are determined by her preferences.

⁴ Zynda approvingly quotes Maher's [1993: 9] view that ‘we understand attributions of probability and utility [i.e. respectively, numerical degree of belief and desire] as essentially a device for interpreting a person's preferences.’ Maher continues: ‘an attribution of probabilities and utilities is correct just in case it is part of an overall interpretation of the person's preferences that makes sufficiently good sense of them and better sense than any competing interpretation does.’ The founders of Bayesian decision theory, Frank Ramsey [1926] and Leonard Savage [1954], also endorsed such a pragmatism about belief. For a more recent (but qualified) endorsement and discussion of pragmatism, see Bradley [forthcoming: sec. 4.1].

⁵ Note also that Zynda's valuation rule ensures that any preference ordering that can be represented as maximizing valuation is insensitive to ‘empty events’ (i.e. the events of whose occurrence the agent is least confident).

⁶ According to such primitivism, belief cannot be reduced to some other attitude, such as preference, and our interest in correctly attributing beliefs to people goes beyond our interest in interpreting their preferences.

⁷ The symbol .=. should be read as ‘is equally likely as’, and is defined by E .=. F ↔ (E .≤. F) & (F .≤. E).

⁸ In short, the reason why comparativists who accept these constraints on the comparative relation need not sacrifice cardinal facts is that a relation with such a structure satisfies the conditions of the following principle (in the right circumstances), where ‘(A;B)’ denotes the difference between the agent's in confidence A and in B and ‘≤≤’ orders such pairs in terms of this difference:

Comparison of Confidence Intervals (CCI). For all A, B, C, D, E, F ∈ Ω such that (i) B∩E = Ø = D∩F, and (ii) B∪E .=. A, D∪F .=. C:

(A;B) ≤≤ (C;D) ↔ E .≤. F

The reason why comparativists need not relinquish interpersonal facts is that, if the comparative relation satisfies the above axioms, then such facts are not dependent on any particular numerical model of the relation: If the distance between Ann's confidence in A and in the tautology is greater than the corresponding distance for Bob according to one model (e.g. a probability model or believability model) that numerically represents their comparative belief relation, then the same is true for any such model.

⁹ Thanks to Greg Restall for suggesting to me a principle like this and for graciously giving me the permission to discuss it in print.

¹⁰ I thank two referees for AJP for comments that made me realize the need to clarify this issue.

¹¹ The strict comparative belief relation, .<., is defined from the weak comparative belief relation, .≤., by E .<. F ↔ (E .≤. F) & ¬(F .≤. E), where ¬A should be read as ‘not-A’.

¹² Note that each cell is an event (i.e. a set of states). Hence, the axiom entails that there are no atomic events (so, S is infinite). Savage [1954: 38–9] justified this assumption by pointing out that, for any partition, we can always construct a finer one by considering the outcome of an additional toss of some coin.

¹³ It may be worth emphasizing that this does not mean that Savage Continuity entails an indifference principle like the one that Suppes Continuity entails. For Savage Continuity does not entail that there is some partition such that any confidence relation (that satisfies the axiom) must be indifferent between cells in that partition. Rather, it entails that, for any confidence relation (that satisfies the axiom), there is a partition such that the relation is indifferent between cells in that partition.

¹⁴ This also shows that a qualitative probability that satisfies Savage Continuity directly entails cardinal facts (without relying on any numerical representation).

¹⁵ Moreover, this points to a flaw in my earlier paper [2017]. Rather than saying that numerical degrees of belief are positive affine transformations of probabilities, I should have said—given the axioms on comparative belief that I endorsed—that they are similarity transformations of probabilities, which means that, e.g., a believability function is not an acceptable representation of degrees of belief. (A similarity transformation is a special case of a positive affine transformation, so what I said was not strictly false, but at least misleading.)

¹⁶ I would like to thank Arif Ahmed, Richard Bradley, Bernhard Salow, and Folke Termsman for helpful comments on early versions of this paper. Special thanks to Greg Restall, for comments that inspired me to write this paper. Finally, thanks to two referees for AJP, for very helpful comments and suggestions.