On the standard Bayesian picture, agents rationally change their beliefs by becoming certain of more and more facts. If we were to read this normatively, we would have to contend that it is irrational to forget. Michael Titelbaum wants to cure Bayesianism of this counterintuitive consequence by providing a new formal framework which he dubs the Certainty Loss Framework (CLF). The CLF is meant to offer the Bayesian answer to problems involving memory loss and, what is more, to problems exhibiting context-sensitivity. In this review, I shall briefly present the main ingredients of the CLF and illustrate it by Titelbaum's analysis of the Sleeping Beauty Problem, that familiar ‘test bed’ for formal frameworks that model memory loss and context-sensitivity. I shall conclude by mentioning some of the other topics the CLF can help clarify.
But first, a general comment on style. Quitting Certainties is entirely self-contained. The formal machinery is introduced in detail and the reader is only assumed to be familiar with set theoretical notation and some introductory logic. The proofs are all in the Appendices, making the book easy to read for someone who is not interested in the formal nitty-gritty and is only looking for the philosophical import of the results. Moreover, all references to the wider literature are succinctly and clearly explained and all puzzles the book sets out to solve are introduced at length. Titelbaum is meticulous in laying out his framework and makes astute observations along the way with regards to how formal frameworks should be used in philosophy – these remarks alone make the book worth reading for any student of formal philosophy.
The CLF comprises a model together with some systematic and extrasystematic constraints and an interpretation. A model consists of a set of time points (T) and a modelling language (L). Over the set of time points and the set of sentences, Titelbaum defines a probability function capturing the agent's credences at time i (Pi).
The extrasystematic constraints stipulate that the model assigns an agent a credence of 1 in a claim iff the story implies that the agent is certain of the truth of that claim or if the claim is a consequence of some of the agent's certainties. The extrasystematic constraints are meant to be determined by each particular story, and the systematic constraints are meant to be the same in all CLF models. The first systematic constraint requires credences to be finitely additive. The second is the ratio formula (if Pi(¬y) < 1, then Pi(x|y) = Pi(x&y)/Pi(y)). The third and the fourth constraints do the most interesting work in the CLF: Generalized Conditionalization (GC) and the Proper Expansion Principle (PEP). These are supposed to offer a direct response to how learning context-sensitive facts can affect an agent's credences in context-insensitive ones. I shall discuss each in turn.
(GC) For any tj, tk ∈ T and any x ∈ L, if Pj (¬<Ck−Cj>)<1 and Pk (¬<Cj−Ck>) < 1 then Pj (x|<Ck−Cj>) = Pk (x|<Cj−Ck>)
(GC) is a clever modification of the usual conditionalization formula. First, Cj is the set of certainties an agent entertains at time point tj. Second, <Ck−Cj> is the conjunction of all the certainties the agent gains between tj and tk (j < k). If the agent's certainties do not change, then <Ck−Cj> = T (the set of tautologies). The final equality is supposed to capture the intuition that ‘when you lose information, your resulting doxastic state should be such that were you to regain that information you would return to the doxastic state in which you began’ (127). For instance, assume that from tj to tk the agent becomes certain that a and does not lose any of the certainties he entertained at tj. Then for any arbitrary sentence x, Pj(x|a) = Pk(x|T). If the agent does not learn any new information between tj and tk, but instead forgets b, then for any arbitrary x, Pj(x|T) = Pk(x|b).
Despite its wide scope, (GC) does not generate any verdicts in models in which context-sensitivity plays a crucial role (Ch. 8.1.1). The reason is that if an agent's belief in sentence x goes from 1 to 0 (or vice versa) between tj and tk, then Cj ∪ Ck is inconsistent and in consequence the antecedent of (GC) will be false (Appendix 6, Theorem C.9). Therefore, Titelbaum suggests a way of obtaining a context-insensitive reduction of a context-sensitive CLF model which can offer us information about the initial context-sensitive model. A reduction of a model M, is a model M− with the same time points modelled and with a reduced language L− ⊆ L such that the analogue of any extrasystematic constraint in M− is an extrasystematic constraint in M. If M is context sensitive, then we cannot apply (GC), but (GC) can be used on a context-insensitive reduction of M.
(PEP) tells us we can trust that any verdicts of the reduction of a model M apply to the original model if and only if (∀y∈ L)(∀ti∈ T)(∃x∈ L−)(Pi(x = y) = 1). According to (PEP), the reduction can inform the initial model if for every sentence in the original context-sensitive model and for every time point, there exists a (context-insensitive) sentence in the language of the reduction which is equivalent to it.
Lastly, Titelbaum is very precise in how CLF models should be interpreted: ‘[i]f a model's verdicts contradict each other, that model indicates that the agent's doxastic evolution violates the requirements of ideal rationality. If contradictory verdicts cannot be derived, no violation is indicated’ (56). In other words, if no contradiction can be derived from a model then the agent's credences are ideally rational. And more importantly, judgements of ideal rationality always have to be made within a particular CLF model.
The formal set-up of the CLF is very considerate to the reader, with many examples and explanations, and with all proofs deferred to the Appendix. Before moving to its application to the Sleeping Beauty problem, I would like to highlight a difference between the CLF and standard Bayesianism. In the usual Bayesian set-up, conditionalization functions as a guide to how beliefs are to be rationally changed in response to new information. In consequence, after learning new facts, an agent is prompted to conditionalize his old beliefs on the newly gathered information and derive his new credal function. This new function will assign certainty to some facts, while other facts will receive a probability of 0, and still others will be assigned an intermediate probability. However, this is not the case with (GC). (GC) can only be applied if the agent already knows what he is certain of. In other words, (GC) is no longer a guide to how to modify beliefs in response to new information, but a way of aligning your non-extreme beliefs given that you know how your certainty set has changed. But the certainty set itself does not change through (generalized) conditionalization.
How does Titelbaum handle the well-known Sleeping Beauty Problem? As a reminder, here is Elga (Reference Elga2000)'s presentation:
Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads? (143)
We are interested in how Beauty changes her belief in the coin landing Heads from before being put to sleep, i.e. Sunday evening, to when she is first awakened, i.e. Monday morning. However, a CLF model capturing ‘t 0: Sunday evening’ and ‘t 1: Monday morning immediately after waking up’ will not be able to generate any verdict about Beauty's Monday morning beliefs (221). The reason is that the language of such a model would have to contain a sentence, ‘s: Today is Sunday’, which over the time set {t 0, t 1} is context-sensitive. Moreover there is no context-insensitive proper reduction of such a model and hence (GC) cannot be applied to derive any verdicts about how Beauty should change her beliefs. To avoid this problem, Titelbaum's analysis proceeds in a roundabout way. Firstly he introduces, following Elga, a third time point in the story ‘t 3: Monday evening when Beauty is told what day it is’. He also follows Elga in suggesting that the coin flip happens on Monday evening instead of Sunday evening. This is an innocuous change as no matter how the Sunday flip goes, Beauty will still be awakened on Monday. So the coin flip makes no difference to the Monday awakening. Secondly he proposes we model the evolution of Beauty's beliefs from Monday morning to Monday evening (Model S12, L12 = {‘m: Today is Monday’, ‘h: The coin comes up heads’}, T12 = {t 1, t 2}). Lastly, he proposes we also model the evolution of Beauty's beliefs from Sunday evening to Monday evening (Model S02, L02 = {s, m, h}, T02 = {t 0, t 2}).
Model S12 does not contain any context-sensitive information (m is not context-sensitive over T12) so we can apply (GC). This yields that Beauty's degree of belief in Heads on Monday morning has to be strictly less than her degree of belief in Heads on Monday evening (Appendix F). Model S02, on the other hand, contains two context-sensitive sentences, i.e. s and m. So, Titelbaum constructs the reduction – S02− with L02− = {h}. S02− is a proper reduction of S02 since they share the same time set and moreover for every sentence in the language L02 there is a sentence in the language L02− which has the same truth value: P1(s = F) = 1, P1(m = T) = 1, P0(s = T) = 1, P0(m = F) = 1. (GC) can now be applied to S02−. Since Beauty's certainty set does not change from t 0 to t 2, then P0− (h|T) = P2− (h|T). So, Beauty's degree of belief in Heads on Monday evening has to be the same as her degree of belief on Sunday evening. Moreover, according to the Principal Principle (which is taken to be an extrasystematic constraint), this degree of belief has to be ½, and due to (PEP), this result is also a verdict of S02. Patching the verdicts of S02 and S12 together, we obtain that Beauty's belief on Monday morning has to be strictly less than ½. This is not yet a full answer to what her rational degrees of belief have to be on Monday morning, but it is already a rejection of the so-called ‘Halfer’ analysis of the Sleeping Beauty Problem (i.e. Lewis Reference Lewis2001). Titelbaum contends that in order to derive the solution to the Sleeping Beauty Problem, CLF needs to be directed towards a related puzzle, the Technicolor Beauty Problem (223). For reasons of brevity, I will only focus in this review on Titelbaum's rejection of the Halfer analysis.
Titelbaum's rejection of the Halfer's answer to Sleeping Beauty is clever, but it raises some worries about the applicability of the CLF. Firstly, the CLF fails to offer any useful insights if applied to the entirety of scenario. In order to be able to successfully apply it to these stories, we need to make a very careful choice with regards to which time points to model (t 1 and t 2, and t 0 and t 2).
Titelbaum's answer to this criticism (Ch. 8: fn. 23) is that the doxastic evolution of an agent is captured by (GC) and that this rule is spelled out only in terms of the difference of certainties in between time points. But, no matter how we model the different time points of a story, the agent's certainties at those time points are always the same and hence we will never obtain clashing verdicts. This is because, in the CLF, an agent's certainties at a time point are not obtained from conditioning on the information the agent had at the previous time point and the information received in between. They are directly inferred from the story.
I accept Titelbaum's point, but the situation is, nevertheless, surprising. Bayesians have argued convincingly that in order to obtain the correct results from their formal models we need to follow the principle of Total Evidence and plug into the model all the aspects of the story we seek to understand. Titelbaum, on the other hand, tells us that his quasi-Bayesian framework functions by restricting the evidence we take into consideration. I am willing to follow Titelbaum's cue, but, in contrast with the methodological precision of the rest of the book, he is not forthcoming with indications of when and how to restrict the evidence we should take into consideration. Given his analysis of the Sleeping Beauty, his proposed restrictions seem ad hoc.
Secondly, there is the question of how we derive the verdict that Beauty's Monday morning beliefs should be strictly less than ½. Titelbaum says that we simply patch together the verdicts of S12 and S02. But in order for something to be a requirement of ideal rationality in his framework, it has to be a verdict of a model. However, P1(h)< ½ is neither a verdict of S02 nor of S12. Titelbaum's way out would be to argue that the verdict of either S02 or S12 would serve as an extrasystematic constraint in the other model. Then P1(h)< ½ would be recovered as a verdict in that model. But the verdict of S12, namely P1(h)<P2(h) cannot be recovered as an extrasystematic constraint in S02, as S02 does not model time point t 1. On the other hand, the facts that P0(h) = P2(h) and P0(h) = ½ (which were verdicts in S02) can be recovered as S12's extrasystematic constraint that P2(h) = ½. In consequence, the Halfer solution to Sleeping Beauty is not rational as it is directly contradicted by a verdict of the CLF model of time points t 1 and t 2, viz. Model S12.
This solution suggests not only that we need to be very careful at how we group the different time points in a story when applying the CLF to it, but also that the order in which we analyse the resulting CLF models of the different time points of a story is relevant. Without S02, S12 cannot yield the rejection of the Halfer solution.
These two problems show that there is a lot of work to be done at an intuitive level when applying the CLF, and this gives the impression of ad hocness. Titelbaum tries to address the concern at different junctions in the book, but his attempts do not always appear successful.
The book ranges further than the Sleeping Beauty Problem. Chapter 10 compares the CLF to other solutions proposed for dealing with context sensitivity, like Halpern and Tuttle’s, Meacham’s, Sarah Moss's and Stalnaker’s. Titelbaum also discusses Arntzenius's Shangri-La (Ch. 6.1), John Collins's Prisoner example (Ch. 9.1.3), the Judy Benjamin problem (Ch. 5.2.1), the principle of indifference (Ch. 11.1), the reflection principle (Ch. 6), fission and cloning cases (Ch. 11.2) and the relationship between Everettian interpretation of quantum mechanics and Bayesian updating (Ch. 11.3). All in all, the book is a competent survey of some of the most debated topics in formal epistemology and builds a powerful case for the wide scope and usefulness of the CLF.
However, I was disappointed not to find any extensive discussion of Piccione and Rubinstein (Reference Piccione and Rubinstein1997)'s Absentminded Driver Problem,Footnote 1 as it seems to also fall in the scope of the CLF. The Absentminded Driver problem would have strengthened Titelbaum's case for the wide applicability of the CLF by extending it to decision-theoretical problems (which are missing in Quitting Certainties).
All in all, Quitting Certainties puts forward a stimulating and wide-ranging framework for analysing puzzles involving memory loss and context-sensitivity. However, as any attempt of similar breadth, it is not without its problems. The main disadvantage is that a significant part of the analysis is done outside the formal framework outlined in the book and appears ad hoc. I encourage anyone interested in formal modelling to read the book and decide for herself if the CLF can genuinely answer the philosophical questions raised by epistemic puzzles such as the Sleeping Beauty.Footnote 2
