Non-reductionism: lowering the bar

As we mentioned above, Leitgeb’s stability theory of belief admits a variant of a strongly non-reductionistic flavour. The theory amounts to keeping the Stability Principle (SP) and abandoning the Relativised Lockean Principle (RLP) which gives rise to theτ-rule. Thus, givenµ andt, selectingany (µ, t)-stable set constitutes a reasonable translation from the probabilistic credal state: in other words, the pair (µ, K) – withK∈Aas strongest accepted proposition – is considered coherent, as long asKis (µ, t)-stable. In this way, qualitative belief states are not seen as emerging deterministically from subjective probabilities. Accordingly, if one accepts only (SP) as a criterion of acceptance, this also yields a weaker criterion of dynamic coherence between conditioning and a qualitative revision∗: one requires only that, given a pair (µ, K) that is coherent in the sense above, the pair (µX, K∗X) be coherent for any X withµ(X)>0. In this sense, we achieve coherence between AGM and Bayesian conditioning by taking ∗ to be the τ-generated revision: as we have seen, K∗X is then (µX, t)-stable and coherence is preserved at every revision step.

This is, however, a very weak criterion of dynamic coherence. It is simply due to the fact that (SP) is a rather weak constraint. For example, the following revision process would pass the coherence test: begin by accepting only the least probability-1 proposition

S∞ and, for any Bayesian update by X, accept as strongest S∞∩X=X, iterating this

process as needed by taking intersections. This makes acceptance trivial: at every step, the strongest accepted proposition simply represents the weakest proposition that has not been yet ruled out: the plausibility ordering given by the corresponding system of spheres forgets all other information provided by the probability distribution. One can hardly say

that such a procedure ‘approximates’ tracking: it simply consists in ignoring virtually any detail provided by the probabilistic description of the agent’s doxastic state. Nothing in this weakened coherence criterion allows to rule out such cases as degenerate.

Can we do any better? Here is another idea: use theτ-rule as long as tracking works, and ‘force’ commutativity when it does not. This amounts to setting another acceptance mapτ0

as follows: starting at some prior measure µand some fixed threshold, we setτ0(µ) =τ(µ). For the next revision step byX∈A, simplydefine the mapτ0 as pickingτ(µ)∗X as strongest accepted proposition. Thus, if the revision τ(µ)7→τ(µX) coincides with the τ-generated revision, we haveτ0(µX) =τ(µX); but whenever tracking fails and we haveτ(µX)⊂τ(µ)∗X, the τ0 map selects τ(µ)∗X. So we forceτ0 to coincide withτ in all revision cases except the problematic ones, by setting it to respect the AGM revision generated by the sphere system of (µ, t)-stable sets. It is important to note thatτ0 isnot an acceptance rule in the technical sense: it is clearly not a function from ∆A to A, as the value ofτ0(µ) for some measuresµ–

indeed, the problematic ones, of the form µX, for which τ(µX)⊂τ(µ)∗X – will depend on the particular revision history that has taken place so far9.

Leitgeb himself has suggested a solution similar to the above [31]. An acceptance principle behaving like the one just described yields a much closer connection between AGM revision and Bayesian dynamics. It respects (SP) at all times, and imitates the behaviour of the

τ-rule, except when theτ-rule fails to respect commutativity. In this sense, it can be said to approximate tracking.

However, on one level, this connection remains unsatisfactory. To begin with, the acceptance principle above lacks an independent motivation: it is simply designed to satisfy AGM, and is defined directly in terms of the AGM operator generated by the τ-rule. That one can use this principle to approximate tracking without violating (SP) is, of course, a step towards reconciliating AGM with Bayesian models of reasoning – particularly so since (SP) can be motivated in a manner entirely independent of the tracking problem for AGM revision. But an obvious difficulty is that this solution is in direct conflict with (RLP). Consider any case where tracking fails: we have some ruleτ, thresholdt∈(0.5,1], andX∈A such thatτ(µX)⊂τ(µ)

∗_{X. After conditioning onX, the probability of τ(µ}

X) is above the threshold t: thus, Lockean intuitions – as expressed in (RLP) – dictate thatτ(µX) should be believed. So picking τ(µ)∗X as strongest accepted proposition goes against (RLP).

This violation of (RLP) calls for a justification. After all, an important motivation for 9

For instance, to identify the strongest accepted propositionτ0(µX) given a distributionµX, theτ0 map needs to be provided at least enough information to identifyµandX, in order to know whatτ(µ)∗X is. We may well haveρ=µX=νX for different measuresµandνsuch that commutativity holds when updatingµby

X(in which case we setτ0(ρ) =τ(ρ)), but fails when updatingνbyX (in which case we wantτ0(ρ) =τ(ν)∗X). Theτ0map must be able to differentiate between those cases. One way to ensure this is to provide it with (1) the initial prior distributionµand (2) the propositionT

i≤nXi, wherehXi|i≤niis the sequence of all revision inputs provided so far.

the stability-based account – and one of its strengths – was to remain as close as possible to the Lockean thesis whilst avoiding paradoxes of the Lottery kind. Indeed, theτ-rule, which selects the minimal (µ, t)-stable set, owes its specificity (among all rules satisfying (SP)) to the fact that it satisfies the greatest number of instances of the (←)-direction of the Lockean thesis. But we see that selecting τ(µ)∗X as strongest accepted proposition, in cases where commutativity fails, cannot be done for free. Not only does a Bayesian reasoner have no positive incentives to obey the AGM requirements in those cases, she can do so onlyat the cost of violating the Lockean principle (RLP). Thus, she is confronted with a true trade-off between AGM and Lockean intuitions.

One ideal of compatibility could be described as follows: under an independently estab- lished acceptance principle10, feasible AGM revisions can be probabilistically represented by a Bayesian update, and any Bayesian update translates into an AGM revision. This corresponds to a two-way agreement between the Bayesian and AGM-compliant reasoner, mediated by an acceptance principle that is acceptable to both. But under this proposal, only the latter side can be content: the Bayesian, on the other hand, may naturally still demand a rationale for privileging AGM dynamics over the (←)-direction of the Lockean thesis.

So if, in Leitgeb’s words, a “peace project” [30, p.70] between AGM and Bayesian models of reasoning is to be carried out successfully, it would be reasonable to require a more principled reason to ‘force’ commutativity: justifying the commutative choice by a simple desire to preserve AGM revision gets us only half of the way there.