Two Approaches to Belief Revision

Abstract

In this paper, we compare and contrast two methods for the revision of qualitative (viz., “full”) beliefs. The first (“Bayesian”) method is generated by a simplistic diachronic Lockean thesis requiring coherence with the agent’s posterior credences after conditionalization. The second (“Logical”) method is the orthodox AGM approach to belief revision. Our primary aim is to determine when the two methods may disagree in their recommendations and when they must agree. We establish a number of novel results about their relative behavior. Our most notable (and mysterious) finding is that the inverse of the golden ratio emerges as a non-arbitrary bound on the Bayesian method’s free-parameter—the Lockean threshold. This “golden threshold” surfaces in two of our results and turns out to be crucial for understanding the relation between the two methods.

1 Setup

We will be considering a very simple and highly idealized type of epistemic agent who possesses both credences (viz. numerical degrees of confidence) and ordinary (qualitative) beliefs.¹ The objects of these attitudes will be (classical, possible worlds) propositions in a finite propositional language whose logic is classical. On the credence side, we adopt a naïve Bayesian account of credences where the agent’s credences are represented by a classical probability function, $b(\cdot )$. When it comes to the qualitative attitudes of our idealized agent, we will attend only to the beliefs of the agent (i.e., we will neither discuss disbelief nor suspension of judgment) and the agent’s belief state will be represented by a set, ${\mathbf {B}}$, comprising the set of propositions the agent believes. At times, it will be helpful to refer specifically to individual beliefs of the agent and we will write ${\mathsf {B}}(X)$ to denote that $X\in {\mathbf {B}}$.

The primary purpose of this paper is to investigate the diachronic norms governing the ways in which agents may revise their beliefs in light of new information. To do so, we will contrast the orthodox AGM approach with a novel, broadly Bayesian, one. So-called belief revision operators are functions mapping a prior belief set together with a proposition to a posterior belief set. More carefully, where $\star$ is an arbitrary belief revision operator and ${\mathbf {B}}$ is the agent’s prior belief set, when an agent learns the proposition E, her posterior belief set is ${\mathbf {B}}'={\mathbf {B}}\star E$.² While AGM’s belief revision operator ($*$) is defined in wholly qualitative terms, the Lockean belief revision operator ($\divideontimes$) is defined by way of a diachronic version of Foley’s (1992) Lockean thesis. That is, Lockean revision requires that when an agent learns E, she adopts the posterior belief set, ${\mathbf {B}}'={\mathbf {B}}\divideontimes E$, such that ${\mathsf {B}}'(X)$ just in case her posterior credence in X is no less than the Lockean threshold, t. So, for an agent to perform Lockean revision, she will require some procedure for updating her prior credence function, b, to a posterior credence function, $b'$. As it will facilitate the simplest and most straightforward presentation, we will assume that credences are updated via conditionalization. So, when an agent with the prior credence function b learns E, she adopts the posterior $b'(\cdot )=b(\cdot {\mathbin {|}}E)$.³ However, this choice is made largely for convenience, since most (if not all) of our results will hold for any update of the prior b by E to the posterior $b'$ that satisfies the following two constraints: (1) $b'(E) > b(E)$ and (2) if $b'(X) \ge t$, then $b(E \supset X) \ge t$.⁴

In the next section, we will introduce Lockean revision, and provide some initial results designed to clarify how the dialectic will unfold. In section three, we will discuss AGM revision and some related issues. In the subsequent four sections, we will provide our novel results, primarily concerning the precise similarities and differences between these two approaches to belief revision. Finally, we will close with some remarks about open questions and future work. In a brief epilogue, we contrast Lockean revision with Leitgeb’s (2016) recent account of belief revision, which satisfies both the Lockean thesis and AGM’s revision postulates.

2 Lockeanism and Its Revision Operator

While we are presently concerned with diachronic rational norms on belief, Lockean and AGM revision both presuppose their own synchronic constraints as well. Accordingly, we will begin our discussions of each with brief descriptions of their synchronic presuppositions. For the Lockean, the underlying synchronic constraint on an agent’s beliefs is that they display a certain coherence with her credences. This provides the core synchronic constraints of Lockeanism and was first dubbed the Lockean thesis by Foley (1992). Intuitively, the constraint requires that an agent believe all and only those propositions to which she assigns “sufficiently high” credence. This may be more carefully captured by treating “sufficiently high” credence as credence above some (Lockean) threshold t.⁵ Then, we may rewrite the intuitive principle as requiring that an agent believe X iff her credence in X is at least t. Formally:

${(\hbox {LT}^t)}$ :

$\hbox {For some } t\in (\frac{1}{2},1]: {\mathsf {B}}(X)\hbox { iff }b(X)\ge t.$

The Lockean thesis offers an intuitively plausible normative joint-constraint on credences and beliefs, since it may be observed that: (1) it seems irrational for an agent to believe a proposition which she takes to be (sufficiently) improbable; (2) it appears to be a rational shortcoming if an agent fails to believe a proposition that she thinks is (sufficiently) likely; and (3) it is never rationally permissible for an agent to concurrently belief both X and $\lnot X$ (as would be permissible if $t\le \frac{1}{2}$).

Despite its intuitive plausibility (and the fact that it can be given a very elegant justification via epistemic utility theory as discussed in Sect. 7), the Lockean thesis remains rather controversial. There are a variety of objections that have been raised against it. The earliest offerings stem from the fact that it permits (indeed, in some cases, requires) agents to possess belief sets that are logically inconsistent and are not closed under logical consequence. Kyburg’s (1961) lottery paradox provides the most well-known example of this phenomenon. As we will see in the next section, AGM requires both consistency and closure of all qualitative belief sets. So, this is a key difference between our two paradigms of belief revision. We don’t have much to add to this well trodden aspect of the debate between “Bayesian” and “Logical” approaches.⁶ Instead, our analysis will primarily focus on deductively cogent agents (i.e. agents with logically closed and consistent beliefs). In doing so, we will uncover some crucial differences and interesting connections between the two approaches.

Beyond objections to Lockeanism based on failure of closure and consistency, it is also commonly complained that there appear to be only two non-arbitrary (viz., non-context-dependent) candidate values for the Lockean threshold: $\frac{1}{2}$ and 1. As alluded to above, given our current formulation of the thesis, were an agent to assign equal credence to a proposition and its negation, a Lockean threshold of $\frac{1}{2}$ would permit belief in both. Nonetheless, even if the thesis were stated with a strict inequality (viz. ${\mathsf {B}}(X)$ iff $b(X)>t$), a $\frac{1}{2}$-threshold would be too permissive. Surely, rationality does not (always) require belief when an agent is only slightly more confident in a proposition than its negation. Similarly, the extremal Lockean threshold ($t=1$) is not permissive enough, since this would make certainty a rational requirement for belief.⁷ For these reasons, many are inclined to think that appropriate, particular Lockean thresholds are determined in a context-dependent way (although, in Sect. 7, we will show how to derive rational Lockean thresholds from an agent’s epistemic utility function).

One of the novel and important contributions of this paper will be to establish a new non-arbitrary and non-context-dependent bound on the Lockean threshold: the inverse of the golden ratio ($\phi ^{-1}\approx 0.618$), which we will refer to as the golden threshold. We will arrive at this result in two different ways. The first will come later in this section when we will demonstrate that, on pain of violating an intuitively well-motivated and purely qualitative constraint on belief revision, Lockean revision must rely on a threshold strictly greater than the golden threshold, so $t\in (\phi ^{-1},1]$. The second will emerge in Sect. 6 as a result of the comparison of Lockean revision with AGM.

But, first, we need to define our Lockean belief revision operator. Lockean revision requires that an agent revise her beliefs so as to satisfy the Lockean thesis, relative to her posterior credences.

• Lockean revision Where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), the Lockean revision of ${\mathbf {B}}$ by the proposition E (${\mathbf {B}}\divideontimes E$) is defined as:

  $$\begin{aligned} {\mathbf {B}}\divideontimes E:= \left\{ X : b(X {\mathbin {|}}E) \ge t\right\} . \end{aligned}$$

Lockean revision offers the approach to belief revision that would be required for an agent who wishes to satisfy the Lockean thesis at any given time and updates her credences using conditionalization.

Since Lockean revision requires diachronic coherence with the agent’s credences, the procedure is sensitive to certain changes in the agent’s non-qualitative information. As we will see, this means that Lockean revision will sometimes lead agents to give up some beliefs after acquiring some non-definitive counter-evidence to their prior beliefs. By contrast, AGM revision only permits agents to give up a prior belief when they learn some proposition that is logically inconsistent with their prior belief set. As we will see shortly, this fact about Lockean revision will ultimately prove to be at the heart of its divergence with AGM revision.

2.1 Weak Preservation and the Golden Threshold

Belief revision operators are often defined by way of qualitative axioms on possible revisions to an agent’s prior beliefs. In the next section, we will present one standard axiomatization of AGM’s revision operator and, in the subsequent section, show which of these axioms are satisfied and which may be violated by Lockean revision. But first, as a preview of how the dialectic will unfold, we will discuss two constraints on belief revision operators, which are both entailed by AGM’s axioms. Not only will these results aid in understanding how we will progress, but they will expose some interesting and surprising features of Lockean revision that will resurface later in the paper.

The first principle that we will consider is known as Weak Preservation⁸ and requires that the posterior belief set contains all of the beliefs in the prior belief set subsequent to revision by some prior belief.

${(^\star 4^w)}$ :

$\hbox { If }E\in {\mathbf {B}}, \hbox {then }{\mathbf {B}}\subseteq {\mathbf {B}}\star E\qquad \qquad \qquad \qquad {\mathbf{{Weak \;Preservation}}}$

In words: learning something you already believe should never cause you to stop believing anything you previously believed. At first pass, this principle may seem indubitable. After all, when an agent already believes E, learning E would not appear to provide her with any new information. Thus, there should be no basis for any change to her beliefs. Although Weak Preservationis satisfied by AGM revision,⁹ it is not satisfied by Lockean revision in full generality.

If a Lockean agent revises by a previously believed proposition, then it is possible for her posterior credence in other previously believed propositions to fall (significantly) below the Lockean threshold. This will then result in the loss of a belief. But, given a Lockean threshold, there is a bound on the “degree” to which Lockean revision can violate Weak Preservation. That is, there is a precise bound on how far below the threshold an agent’s posterior credence in a previously believed proposition may be after having revised by some other previously believed proposition. This bound can be provided as a function of the Lockean threshold as established in the following lemma.

Lemma 1

Where $t\in (\frac{1}{2},1]$, if $b(X) \ge t$ and $b(E) \ge t$, then $b(X {\mathbin {|}}E) \ge \frac{2t-1}{t}$.

Proof

Let $x:=b(E\wedge X)$, $y:=b(E\wedge \lnot X)$, and $z:=b(\lnot E\wedge X)$ and assume $b(E)\ge t$ and $b(X)\ge t$. This implies that:

$$\begin{aligned} b(E)&=x+y\ge t; \end{aligned}$$

(1)

$$\begin{aligned} b(X)&=x+z\ge t;{\text { and}}\end{aligned}$$

(2)

$$\begin{aligned} b(X{\mathbin {|}}E)&=\frac{x}{x+y}. \end{aligned}$$

(3)

Suppose, for reductio, that

$$\begin{aligned} b(X{\mathbin {|}}E)&=\frac{x}{x+y}<\frac{2t-1}{t}. \end{aligned}$$

(4)

Because $t>0$ and $x \ge 0$, we may cross-multiply and simplify (4), yielding

$$\begin{aligned} t&> \frac{x+y}{x+2y}. \end{aligned}$$

(5)

Combining (1) and (5), we may infer

$$\begin{aligned} x+y&> \frac{x+y}{x+2y}, \end{aligned}$$

which simplifies to

$$\begin{aligned} x&> 1-2y. \end{aligned}$$

(6)

Because $x+y+z \le 1$, we know that $1-y \ge x+z$. Combining this with (2) and (5) yields

$$\begin{aligned} 1-y&> \frac{x+y}{x+2y}, \end{aligned}$$

which simplifies to the following quadratic inequality

$$\begin{aligned} y - xy - 2y^2&> 0. \end{aligned}$$

(7)

Since $x \le 1$, this quadratic inequality can be true only if

$$\begin{aligned} x&< 1-2y, \end{aligned}$$

(8)

which contradicts (6). This completes the reductio of our Lemma. $\square$

A visual explanation of Lemma 1 is provided by Fig. 1, which plots the lower bound on $b(X{\mathbin {|}}E)$, provided that $b(E)\ge t$ and $b(X)\ge t$.

Fig. 1

Lower bound on $b(X{\mathbin {|}}E)\ge \frac{2t-1}{t}$, given $b(X)\ge t$ and $b(E)\ge t$

It follows straightforwardly from this result that, indeed, Lockean revision can violate Weak Preservation.¹⁰

Proposition 1

Lockean revision can violate Weak Preservation. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), it is possible that:

$$\begin{aligned} E\in {\mathbf {B}}\,\,{\text { and }}\,\,{\mathbf {B}}\nsubseteq {\mathbf {B}}\divideontimes E. \end{aligned}$$

However, notice that as the Lockean threshold increases, the “degree” to which Lockean revision can violate Weak Preservationdecreases. Moreover, in the limit (when $t=1$), Lockean revision actually satisfies the principle. So, if full beliefs are assigned sufficiently high credence, Weak Preservation is “approximately” true; and it is exactly true in the extremal case (or when the agent’s priors are sufficiently far above the Lockean threshold).

Additionally, note that the initial motivation for Weak Preservation provided above is actually consistent with this result. We suggested that the principle was compelling since learning something you already believe would not appear to provide you with any new information. But, for a Lockean agent, so long as she was not previously certain that E, learning Ecan provide her with new information. Although she has not acquired any new qualitative information, Lockean revision is sensitive to the finer-grained information reflected by her credences.

Next, we report two further results concerning Lockean revision by a previously believed proposition. First, we find that Weak Preservation can only be violated by Lockean revision when the agent’s prior beliefs failed to be deductively cogent. Indeed, if the agent’s prior beliefs are deductively closed, then Lockean revision always satisfies Weak Preservation.¹¹ To wit:

Proposition 2

If ${\mathbf {B}}$ is deductively closed, then Lockean revision satisfies Weak Preservation. That is, where ${\mathbf {B}}$ satisfies ($\hbox{LT}^t$), the following is a theorem:

$$\begin{aligned} {\text {If\;}}{\mathbf {B}}{\text {\;is\;deductively\;closed\;and\;}}E\in {\mathbf {B}},{\text {\;then\;}}{\mathbf {B}}\subseteq {\mathbf {B}}\divideontimes E. \end{aligned}$$

Proof

Suppose for reductio that (a) ${\mathbf {B}}$ is deductively closed, (b) $E,X \in {\mathbf {B}}$, but (c) $X \notin {\mathbf {B}} \divideontimes E$. From (c), it follows that $b(X {\mathbin {|}}E) < t$. Moreover, by the definition of the conditional probability, this implies $b(X \wedge E) < t \cdot b(E) \le t$. So, $X \wedge E \notin {\mathbf {B}}$. But, by (a) and (b), we have $X\wedge E\in {\mathbf {B}}$. $\square$

The second additional result concerning these sorts of revisions involves a further weakening of Weak Preservation, which we will call Very Weak Preservation. This principle only requires that an agent not come to dis-believe any previously believed proposition after revision by some previously believed proposition.¹²

${(^\star 4^{v})}$ :

$\hbox {If } E,X\in {\mathbf {B}},\hbox {\;then\;} \lnot X\notin {\mathbf {B}}\star E \qquad \qquad \qquad \qquad {\mathbf{{Very\;Weak\;Preservation}}}$

Although, in full generality, Lockean revision does not satisfy Very Weak Preservation, it does so long as the Lockean threshold is set sufficiently high. Specifically, if we require that the Lockean threshold be greater than the inverse of the Golden ratio ($\phi ^{-1}\approx 0.618$), then Lockean revision will never require the agent to believe the negation of a previously believed proposition as a result of revising by another proposition previously believed. The following proposition confirms that requiring that $t\in (\phi ^{-1},1]$ rules out the recalcitrant cases.¹³

Proposition 3

Lockean revision satisfies Very Weak Preservation if the Lockean threshold $t\in (\phi ^{-1},1]$. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), the following is a theorem:

$$\begin{aligned} {\text {If}}\,\, t\in (\phi ^{-1},1],\quad {\text {then}}\,\,E,X\in {\mathbf {B}}\,\,{\text { implies\;}}\lnot X\notin {\mathbf {B}}\divideontimes E. \end{aligned}$$

Proof

Assume that $t\in (\phi ^{-1},1]$ and $E,X,\in {\mathbf {B}}$. It follows immediately that $b(E)\ge t$ and $b(X)\ge t$. Applying Lemma 1 gives us $b(X{\mathbin {|}}E)\ge \frac{2t-1}{t}$. But, since $t\in (\phi ^{-1},1]$, we have $\frac{2t-1}{t}>1-t$, so we know $b(X{\mathbin {|}}E)> 1-t$. Thus, $\lnot X\notin {\mathbf {B}}\divideontimes E$. $\square$

The proposition can also easily be confirmed by the graphical demonstration provided in Fig. 2. To see this, notice that the lower bound on $b(X{\mathbin {|}}E)$ is greater than the threshold for disbelief, $1-t$, when considering only Lockean thresholds greater than $\phi ^{-1}$.

Fig. 2

If $t\in (\phi ^{-1},1]$, $b(E)\ge t$, and $b(X)\ge t$, then $b(\lnot X)<t$

While the appearance of $\phi ^{-1}$ in our constraint may seem surprising, we will see shortly that there is another, independent route to $\phi ^{-1}$ that is revealed by a deeper analysis of the relationship between Lockean and AGM revision.

3 AGM and Its Revision Operator

The AGM theory of belief revision—first proposed by Carlos Alchourrón, Peter Gärdenfors, and David Makinson in their seminal 1985 paper (Alchourron et al. 1985)—has served as the basis for nearly all subsequent research into the dynamics of qualitative belief. In its raw form, the AGM theory offers a characterization of theory revision and is often presented in terms of the (so-called) Gärdenfors postulates. These postulates are stated in logical and set theoretic terms and are taken as axioms characterizing AGM revision. When understood as postulates for belief revision, they serve to constrain the ways in which an agent may change her belief set upon the receipt of new information. Accordingly, belief sets are taken to be theories in the mathematical sense, i.e. deductively closed sets of sentences.¹⁴ Just as Lockean revision does not uniquely identify a single revision operator (since it is consistent with a range of Lockean thresholds and probability functions), AGM’s axioms define a family of operators. Further constraints would be required to generate specific, individual AGM revision operators.

The primary conceptual principle underlying AGM revision is known as the principle of conservativity (also called the principle of informational economy or minimal mutilation).

• Conservativity. When an agent learns E, she should adopt a posterior belief set, ${\mathbf {B}}'$, such that (1) ${\mathbf {B}}'$ is deductively cogent, (2) ${\mathbf {B}}'$includesE, and (3) ${\mathbf {B}}'$ is the closest belief set to her prior belief set ${\mathbf {B}}$, which satisfies (1) and (2).¹⁵

This motivating principle provides the normative basis for the coherence requirements imposed by the AGM axioms. Condition (1) requires that the agent revises to a belief set that is both deductively consistent and closed under logical consequence. Condition (2) simply requires that learned propositions are adopted as beliefs. Finally, condition (3) poses AGM’s characteristic restriction on belief sets and specifies that accommodating the learned proposition into the posterior belief set should be accomplished by making as few changes as possible to the prior.¹⁶

We will begin our overview with a discussion of AGM’s synchronic coherence requirements by contrasting them with those of Lockeanism. Subsequently, we will complete our overview by briefly explaining each of the Gärdenfors postulates.

3.1 The Synchronic Requirements of AGM

At first pass, one might (mistakenly) be led to think that the AGM theory only imposes diachronic norms on full belief and has no synchronic presuppositions. Not only is this suggested by the statement of Conservativity provided above (which only constrains admissible posteriors), but also by the fact that AGM’s explicit constraints are put forth in the Gärdenfors postulates that govern its revision operator. Nonetheless, there are compelling reasons to think that the rational requirements of AGM are not exhausted by its diachronic requirements. We will argue—in similar fashion to arguments found in Rott (1999, 2001)—that AGM must adopt deductive cogency as a standing synchronic requirement. To wit:

• Cogency. An agent’s belief set should (always) be both deductively consistent and closed under logical consequence, i.e., ${\mathbf {B}}\nvdash \bot$ and ${\mathbf {B}}= {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}})$, where ${{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {X}})$ is the deductive closure of the set ${\mathbf {X}}$.

This requirement goes beyond Conservativity’s insistence that, following revision, agents’ posterior belief sets must be deductively cogent. Indeed, it requires that all belief sets (both prior and posterior) must be deductively cogent. To appreciate why this strengthening is needed, consider the following attractive principle, which maintains that revising by a tautology should not result in any change to an agent’s beliefs.

${({\text{*}}{\top})}$ :

${\mathbf {B}}*\top ={\mathbf {B}} \qquad \qquad {\mathbf{{Idempotence}}}$

It is intuitive to think that there would be no rational basis for an agent to change her beliefs if she has simply revised by a proposition that expresses no information at all about the way the world is. Clearly, Idempotence is imminently reasonable and should be satisfied by any adequate belief revision operator.

As expected, both AGM and Lockean revision are guaranteed to satisfy Idempotence.¹⁷ However, if AGM permitted prior belief sets that were either inconsistent or not deductively closed, then Idempotence would expose an inconsistency in the theory.¹⁸ Thus, given AGM’s satisfaction of Idempotence and its commitment to Conservativity, if AGM is to offer an internally coherent account, it must presuppose Cogency as a standing synchronic requirement.¹⁹

In order to consider the fundamental requirements of AGM, it is useful to observe that Cogency is just the conjunction of two requirements. The first requires an agent’s belief set to be logically consistent.

• Consistency. At any given time, an agent’s belief set ${\mathbf {B}}$ should be such that there is some possible world w in which every member ${\mathbf {B}}$ is true, i.e., ${\mathbf {B}}\nvdash \bot$.

The second requires an agent’s belief set to be closed under logical consequence.

• Closure. At any given time, an agent’s belief set, ${\mathbf {B}}$, should be such that if ${\mathbf {B}}$ logically implies X, then ${\mathsf {B}}(X)$, i.e., ${\mathbf {B}}= {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}})$.

Although both Consistency and Closure are standing synchronic requirements of AGM, we might wonder whether one ought to be regarded as more epistemically fundamental than the other. As convincingly argued by Steinberger (2016), if failures of Consistency are permitted, then Closure will lose much (if not all) of its normative force. For this reason, we suggest that Consistency should be seen as the fundamental synchronic requirement of AGM.

3.2 Contrasting Synchronic Requirements of AGM and Lockeanism

Although our primary interest is in comparing the diachronic requirements of AGM and Lockean revision, we will first briefly discuss how their synchronic requirements relate. We will argue that AGM’s synchronic requirements are more demanding than those of Lockeanism. For reasons exemplified by Kyburg’s lottery paradox (Kyburg 1961), the Lockean thesis permits belief sets satisfying neither Closure nor Consistency. To see why Lockean revision permits violations of Closure, observe that it can be probabilistically coherent for an agent to have credences in X and Y such that $b(X) \ge t$, $b(Y) \ge t$, but $b(X\wedge Y)<t$. As such, the Lockean thesis may require belief in two propositions, but not require belief in their conjunction. So, the Lockean thesis permits violations of Closure. Moreover, it can be probabilistically coherent for the agent to have credences in $X_1,\dots , X_n$ such that $b(X_i) \ge t$ for each $i:1\le i\le n$, but $b(\lnot (X_1\wedge \dots \wedge X_n)) \ge t$. In this case, the Lockean thesis permits not only a violation of Closure, but also of Consistency, since it would require the agent to believe each proposition individually, but also believe the negation of their conjunction.

Nonetheless, it is important to recognize that both Consistency and Closure are consistent with the Lockean thesis. After all, they are actually entailed by an extreme form of Lockeanism with a maximal threshold ($t = 1$). While this is consistent with Lockeanism, we view Cogency as far too demanding to be adopted as a universal requirement of rationality. As Foley (1992, p. 186) explains,

[...] if the avoidance of recognizable inconsistency were an absolute prerequisite of rational belief, we could not rationally believe each member of a set of propositions and also rationally believe of this set that at least one of its members is false. But this in turn pressures us to be unduly cautious. It pressures us to believe only those propositions that are certain or at least close to certain for us, since otherwise we are likely to have reasons to believe that at least one of these propositions is false. At first glance, the requirement that we avoid recognizable inconsistency seems little enough to ask in the name of rationality. It asks only that we avoid certain error. It turns out, however, that this is far too much to ask.

One might try to resist Foley’s suggestion that inconsistency pressures us to believe only near certainties, while maintaining Consistency as a rational requirement [e.g., Leitgeb (2013) has recently defended just such a view]. However, even if one has abandoned the extreme version of Lockeanism, there remains reason to think that Consistency is (still) too demanding, epistemically. Pettigrew (2016) has recently shown—using the tools of epistemic utility theory—that agents who satisfy Consistency are exhibiting epistemically risk-seeking behavior. Stated intuitively, he shows that requiring consistency involves disproportionately weighting the epistemic best-cases scenarios over the epistemic worst-cases ones. So, in addition to rejecting the extreme version of Lockeanism which entails Consistency, we maintain that Consistency is too strong to serve as a universal rational requirement. We will return to this point as well as other applications of epistemic utility theory in the final section of this paper. However, first we will rehearse AGM’s Gärdenfors postulates before exhaustively comparing them with Lockean revision.

3.3 The Gärdenfors Postulates

AGM theory can be presented in a variety of equivalent ways.²⁰ But, for our purposes, it will be most perspicuous to present the approach in terms of the axioms provided by the Gärdenfors postulates mentioned earlier in this section. Here are the six basic postulates of AGM.²¹

• (*1) ${\mathbf {B}}*E={{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}*E)\qquad \qquad \qquad \qquad {\mathbf{{Closure}}}$

  In words, (*1) says that if an agent revises by E, then her posterior belief set ${\mathbf {B}}' = {\mathbf {B}}*E$ should satisfy AGM’s standing synchronic requirement, Closure. As suggested earlier in this section, we take it that (*1) is grounded in AGM’s synchronic requirements and does not independently impose any genuinely diachronic requirements.²²

• (*2) $E\in {\mathbf {B}}*E \qquad \qquad \qquad \qquad {\mathbf{{Success}}}$

  In words, (*2) says that if an agent revises by E, then E should be included in her posterior belief set ${\mathbf {B}}' = {\mathbf {B}}*E$. (*2) directly encodes Conservativity’s constraint (2).

• (*3) ${\mathbf {B}}*E\subseteq {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}\cup \{E\})\qquad \qquad \qquad \qquad {\mathbf{{Inclusion}}}$

  In words, (*3) says that if an agent revises by E, then her posterior belief set ${\mathbf {B}}' = {\mathbf {B}}*E$ should contain no more propositions than the logical consequences of E together with her prior belief set ${\mathbf {B}}$. This may be understood as the conjunction of two principles: the first governs cases in which E is consistent with ${\mathbf {B}}$, and the second governs cases in which it is not. The former places an upper-bound on the agent’s posterior beliefs ensuring that she does not adopt belief in propositions that are logically independent of her priors and the new evidence, while the second places no restrictions on the posterior.

• (*4) $\hbox {If } {\mathbf {B}}\nvdash \lnot E, \hbox {then} \,\,{\mathbf {B}}\subseteq {\mathbf {B}}*E \qquad \qquad \qquad \qquad {\mathbf{{Preservation}}}$

  In words, (*4)²³ says that if an agent revises by E and E is consistent with her prior belief set, then all of her prior beliefs should be retained in her posterior belief set. When the agent is revising by a proposition consistent with her priors, this places a lower-bound on her posterior and guarantees that she does not lose any beliefs as a result. Note: Taken together, (*1), (*3) and (*4) imply that if E is consistent with the agent’s prior belief set ${\mathbf {B}}$, then her posterior belief set ${\mathbf {B}}' = {\mathbf {B}}*E$ will be identical to ${{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}\cup \{E\})$.

• (*5) $\hbox {If }E\nvdash \bot ,\hbox { then}\,\, {\mathbf {B}}*E\hbox { is consistent}.\qquad \qquad \qquad \qquad {\mathbf{{Consistency}}}$

  In words, (*5) says that if an agent revises by E and E is not itself a contradictory proposition, then her posterior belief set ${\mathbf {B}}' = {\mathbf {B}}*E$ should satisfy AGM’s synchronic bfConsistency principle. Much like (*1), we think that (*5) is really grounded in AGM’s synchronic requirements and does not independently impose any diachronic requirements.²⁴

• (*6) $\hbox {If} \vdash X\equiv Y, \hbox {then}\,\, {\mathbf {B}}*X={\mathbf {B}}*Y\qquad \qquad \qquad \qquad {\mathbf{{Extensionality}}}$

  In words, (*6) says that if X and Y are tautologically equivalent, then updating on X should have exactly the same effect as updating on Y.

Each of these postulate places a restriction on which posteriors are admissible under AGM revision and, thereby, constrains the outputs of individual AGM revisions. It should be noted that the theory is often presented as also including the two supplementary postulates, (*7) and (*8), which generalize (*3) and (*4) respectively constraining iterated revisions.²⁵ However, since we are interested in the diachronic requirements governing single revisions rather than the dispositional ones governing iterated revisions (mentioned in fn. 19), we will focus exclusively on (*1)–(*6).

4 Convergences Between Lockean Revision and AGM Revision

Now that we have presented the basics of these two approaches to belief revision, we will direct our attention to their relative behavior. In this section, we will begin our exploration by reporting some of the general convergences between Lockean revision and AGM revision.

4.1 Extremal Lockean Revision is AGM Revision

In the case of the extremal Lockean threshold, where $t=1$, our agent believes every proposition to which she assigns maximal credence. It is easy to see that this entails that the Lockean agent’s prior and posterior belief sets ${\mathbf {B}}$ and ${\mathbf {B}}'$ will both satisfy Cogency. As a result, extremal Lockean revision must satisfy both Closure and Consistency. Furthermore, it has been known for some time that extremal Lockean agents must satisfy all of the other AGM postulates as well. To wit, we report the following classic theorem.

Theorem

(Gärdenfors 1986) Given a Lockean threshold of $t=1$, for every E such that $b(E)>0$, ${\mathbf {B}}\divideontimes E$satisfies (*1)–(*6).²⁶

The situation is much more interesting when our agent’s Lockean threshold is non-extremal. As we will see, the relationship between Lockean revision and AGM revision in these cases is more nuanced.

4.2 General Convergences Between Lockean Revision and AGM Revision

In addition to fully converging in the special case described above, the following three of AGM’s postulates are satisfied by Lockean revision in full generality.

Proposition 4

Lockean revision satisfiesSuccess. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), the following is a theorem:

$$\begin{aligned} E\in {\mathbf {B}}\divideontimes E. \end{aligned}$$

Proof

It is a theorem of probability calculus that $b(E {\mathbin {|}}E) = 1$.²⁷ Therefore, $b(E {\mathbin {|}}E) = 1 \ge t$, for any Lockean threshold t. So, $E \in {\mathbf {B}} \divideontimes E$. $\square$

Proposition 5

Lockean revision satisfiesInclusion. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), the following is a theorem:

$$\begin{aligned} {\mathbf {B}}\divideontimes E\subseteq {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}\cup \{E\}). \end{aligned}$$

Proof

Suppose $X \in {\mathbf {B}} \divideontimes E$. Then, $b(X {\mathbin {|}}E) \ge t$. And, it is a theorem of probability calculus that $b(E \supset X) \ge b(X {\mathbin {|}}E)$. Therefore, $b(E \supset X) \ge t$. So, $E \supset X \in {\mathbf {B}}$. Hence, by modus ponens (for material implication), $X \in {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\})$.²⁸$\square$

Proposition 6

Lockean revision satisfiesExtensionality. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), the following is a theorem:

$$\begin{aligned} {\text {If }}\vdash X\equiv Y,\quad {\text {then\;}}{\mathbf {B}}\divideontimes X={\mathbf {B}}\divideontimes Y. \end{aligned}$$

Proof

Suppose X and Y are tautologically equivalent. Then, X and Y are probabilistically indistinguishable (under every probability function). Therefore, Lockean revisions on X are indistinguishable from Lockean revisions on Y. $\square$

These three positive results exhaust the set of AGM’s postulates that Lockean revision is guaranteed (in full generality) to satisfy.

Lockean revision’s satisfaction of Inclusion is of particular interest, since at first sight it may not have been so obvious that this would obtain. Since Lockean revision is driven by the Bayesian apparatus, we might have been inclined to think that there may be cases in which an agent may acquire a new belief in some proposition X, which is probabilistically (but not logically) dependent on the learned proposition E. However, this is not so. Inspecting the proof of Proposition 5 exposes that Lockeanism’s synchronic requirements ensure that any time such a new belief is acquired, it might have equally well been acquired through modus ponens.²⁹

Later in this paper, we will examine whether a convergence between Lockean revision’s and AGM’s new beliefs holds more generally. Ultimately, we will find that sometimes AGM will require the agent to form (strictly) more new beliefs than Lockean revision.

5 Divergences Between Lockean Revision and AGM Revision

In the general, non-extremal case, Lockean revision and AGM revision may diverge significantly. In this section, we explore this divergence and discuss the ways in which Lockean revision may violate the remaining three postulates: Closure, Consistency, and Preservation. As previously noted, Lockean revision’s violation of Consistency and Closure has been widely discussed (Christensen 2004; Foley 1992; Easwaran and Fitelson 2015) and so they will only receive a cursory treatment. Instead, we will focus on the more interesting case of the possibility that Lockean revision may violate Preservation and include an instructive counter-example.

First, we know that Lockeanism, in general, does not require Cogency. Indeed, since Lockean revision is driven entirely by the Lockean apparatus, we see that it admits counter-examples to both Closure and Consistency. For brevity, we omit the proofs.

Proposition 7

Lockean revision violatesClosure and Consistency. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), it is possible that:

• 1. ${\mathbf {B}}\divideontimes E\ne {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}\divideontimes E)$ , and

• 2. $E\nvdash \bot$ and ${\mathbf {B}}\divideontimes E$ is not consistent.

It has been widely known since the early 1960’s (Kyburg 1961) that non-extremal Lockean representability is compatible with failures of Consistency (e.g., the lottery paradox). And, of course, if Consistency fails, then Closure must also fail (on pain of epistemic triviality). So, the well-known paradoxes of consistency will (inevitably) yield examples of non-extremal Lockean revision which violate both Consistency and Closure. As mentioned in Sect. 2, we are not so interested in this well-known divergence between the two approaches. Rather, we will focus our attention on cases where the agent’s prior and posterior belief sets satisfyCogency, but Lockean revision and AGM revision still manage to disagree.

This leads us to the central disagreement between the two approaches provided by Lockean revision’s failure to generally satisfy AGM’s characteristic postulate: Preservation. The counter-example provided in the proof of the next proposition highlights a deeper (and hitherto not fully understood) possible divergence between these two approaches.

Proposition 8

Lockean revision can violatePreservationeven if${\mathbf {B}}$ satisfies Cogency. That is, where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$), it is possible that:

$$\begin{aligned}{\mathbf {B}}{\text {\;satisfies\;}} \mathbf {Cogency}, {\mathbf {B}}\nvdash \lnot E,\quad {\text {and\;}}{\mathbf {B}}\nsubseteq {\mathbf {B}}\divideontimes E.\end{aligned}$$

Proof

The proof strategy involves constructing a case in which Lockean revision recommends that an agent—whose priors are synchronically coherent by the lights of both Lockeanism and AGM—gives up one of her beliefs after revising by a proposition that is consistent with her prior belief set.

Let Table 1 describe the distribution of our agent’s credences over the algebra generated by the two atomic sentences, E and X. And, suppose the agent has a Lockean threshold $t=0.85$. $\square$

Given the Lockean threshold of 0.85 and the prior credence $b(\lnot E\vee X)=0.9$, it follows that (aside from the tautology) the Lockean agent will only have one belief: ${\mathsf {B}}(\lnot E\vee X)$. However, learning E would leave her with the posterior credence $b(\lnot E\vee X\mathbin {|}E)=\frac{2}{3}$. Thus, $\lnot E\vee X\notin {\mathbf {B}}\divideontimes E$ even though E is consistent with ${\mathbf {B}}$ and ${\mathsf {B}}(\lnot E\vee X)$. $\square$

Table 1 Proof of Proposition 8

The proof above is more illustratively explained using a simple urn case. Suppose that we are tasked with taking a random sample from an urn containing a total of ten objects. The objects in the urn—as represented in Fig. 3a—include four black circles, three black squares, one red square and two red circles.

Fig. 3

Visualization of counter-example to Preservation for Lockean revision. a Prior distribution, b Posterior (given E)

Let the atomic sentences E and X be assigned the following interpretations:

$$\begin{aligned} E&:={\hbox{`The object sampled from the urn is red', and}}\\ X&:={\hbox{`The object sampled from the urn is a circle'}}. \end{aligned}$$

Finally, assume that some Lockean agent knows the prior distribution of the objects and, as such, has credences in propositions about the shapes and colors of the objects in the urn that are calibrated to this distribution. In this case, the only proposition (aside from the tautology $\top$) that receives a credence higher than the Lockean threshold of 0.85 is $\lnot E\vee X$. Thus, our agent’s prior belief set will be the singleton ${\mathbf {B}}=\{\lnot E\vee X\}$.

Now, suppose the agent learns only that the object drawn from the urn was red (i.e. she learns E). Upon conditionalizing on her new evidence, the agent’s credence in the previously believed $\lnot E\vee X$ will drop from 0.9 to $\frac{2}{3}$, as represented in Fig. 3b. So, when the Lockean agent revises her beliefs, she will be led to give up her belief that $\lnot E\vee X$ and only be left with belief in the newly learned E and its logical consequences, which are all assigned maximal credence. That is, after learning E, the agent’s posterior belief set is ${\mathbf {B}}' = {\mathbf {B}} \divideontimes E=\{E, E\vee X, E \vee \lnot X\}$.

To see how the described case serves as a counter-example to Preservation, observe that the following four crucial facts obtain:

• both the prior and posterior belief sets, ${\mathbf {B}}$ and ${\mathbf {B}}\divideontimes E$, satisfy Cogency;

• E is consistent with ${\mathbf {B}}$;

• $\lnot E \vee X \in {\mathbf {B}}$; but,

• $\lnot E\vee X \notin {\mathbf {B}} \divideontimes E$.

Each of these facts can be easily verified using Table 1. So, this simple case offers a demonstration that Preservation need not be satisfied by Lockean revision, even given Cogency.

In this case, learning E would seem to suffice for her to rationally infer X on the basis of her belief $\lnot E\vee X$. In light of this, one may wonder why our Lockean agent is precluded from adopting belief in a deductive consequence of her prior beliefs and her new evidence. To see why, it will be illuminating to consider an analogy with the literature on epistemic closure. Hawthorne (2005, p. 29) defends a closure principle which grants agents knowledge of the conclusions of their logical inferences only if they retain knowledge of the premises throughout said inferences. While Hawthorne’s principle is aimed at the closure of knowledge, a similar “premise maintenance” caveat also seems reasonable for rational belief. In our example, the learned proposition, E, actually serves as counter-evidence to the (previously believed) second premise, $\lnot E\vee X$, required for her to have inferred X. Thus, learning E serves to make a premise crucial for the inference no longer sufficiently likely to warrant belief.

It is important to note that in our counter-example both the agent’s prior and posterior belief sets actually satisfyCogency. As such, this disagreement between Lockean revision and AGM revision is orthogonal to the traditional disputes between “Bayesian” and “logical” schools of thought in formal epistemology, which have tended to fixate on their disagreement over Cogency (Christensen 2004; Foley 1992; Easwaran and Fitelson 2015). In this sense, the Lockean counter-example to Preservation reveals a more fundamental disagreement between the diachronic requirements of the two approaches. At its core, this disagreement amounts to their differing on the question whether it is ever rational for an agent to give up belief in the face of non-definitive counter-evidence. The AGM theorist answers in the negative, only allowing for beliefs to be given up when the agent has learned something logically inconsistent with her prior beliefs. Whereas the Lockean responds in the affirmative, permitting beliefs to be dropped when the learned proposition causes her prior beliefs to fall below the Lockean threshold.³⁰

6 Lockeanism’s Golden Threshold

It may be observed that our counter-example to Preservation provided above relies on the rather high Lockean threshold of 0.85. We have already observed in our discussion of Very Weak Preservation that it is satisfied by Lockean revision when the Lockean threshold is restricted to the range $(\phi ^{-1},1]$. We might then wonder whether similar results are available for Preservation. Interestingly, we are able to provide a result that both offers an affirmative answer to this question and also offers another avenue to appreciating the theoretical importance of the golden threshold.

Recall that, when Cogency is assumed, Lockean revision satisfiesWeak Preservation. It immediately follows from this that when Cogency is assumed, Lockean revision will satisfy Very Weak Preservation in full generality—viz. not only when $t\in (\phi ^{-1},1]$. But, our counter-example from the previous section shows that Cogencyalone is not sufficient for Lockean revision to satisfy Preservation. However, when Cogency is assumed, we are able to establish that Lockean revision’s possible counter-examples to Preservation are limited to instances when the Lockean threshold is restricted to values in $(\phi ^{-1},1]$, as we will see shortly in Theorem 1. From this we then infer the immediate corollary that by assuming bothCogencyand that the Lockean threshold is no greater than the golden threshold, Lockean revision will actually satisfy Preservation.

To establish these results, we will begin with an insightful lemma³¹ that establishes some conditions that are jointly sufficient to infer that $t\in (\phi ^{-1},1]$. Note that in all of the remaining results, it will be assumed that the prior belief set ${\mathbf {B}}$ satisfies the Lockean thesis relative to the Lockean threshold t.

Lemma 2

Where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$) and Cogency: if $E\notin {\mathbf {B}}$, $E\supset \lnot X\notin {\mathbf {B}}$, and $X\notin {\mathbf {B}}\divideontimes E$, then $t\in (\phi ^{-1},1]$.

Proof

Let $x:=b(X\wedge E)$ and $y:=b(\lnot X\wedge E)$ and assume that $E\notin {\mathbf {B}}$, $E\supset \lnot X\notin {\mathbf {B}}$, and $X\notin {\mathbf {B}}\divideontimes E$. From this it follows that:

$$\begin{aligned} b(E)&=x+y<t; \end{aligned}$$

(1)

$$\begin{aligned} b(X\supset \lnot E)&=1-x<t;\quad {\text {and}} \end{aligned}$$

(2)

$$\begin{aligned} b(X\mathbin {|}E)&=\frac{x}{x+y}<t. \end{aligned}$$

(3)

Then from (1) and (3) it follows that:

$$x<t^2.$$

(4)

By combining (2) with (4), we may conclude that:

$$\begin{aligned} 0< t^2+t-1. \end{aligned}$$

(5)

Finally, note that the quadratic in (5) has two polynomial roots: $-\phi$ and $\phi ^{-1}$. Only the latter is an admissible Lockean threshold, so we conclude that $t\in (\phi ^{-1},1]$. $\square$

With the aid of this lemma, we may now establish our desired theorem.³²

Theorem 1

Where ${\mathbf {B}}$ is satisfies ($\hbox {LT}^t$) and Cogency: if${\mathbf {B}}\divideontimes E$ violatesPreservation, then $t\in (\phi ^{-1},1)$. That is:

$$\begin{aligned} {\text {If\;}}{\mathbf {B}}\nvdash \lnot E, \quad {\text {but\;}}{\mathbf {B}}\nsubseteq {\mathbf {B}}\divideontimes E, \quad {\text {then\;}}t\in (\phi ^{-1},1).\end{aligned}$$

Proof

Suppose that ${\mathbf {B}}$ is a deductively cogent Lockean belief set such that ${\mathbf {B}}\nvdash \lnot E$ and ${\mathbf {B}}\nsubseteq {\mathbf {B}}\divideontimes E$. By Proposition 2, we know that ${\mathbf {B}}\subseteq {\mathbf {B}}\divideontimes E$ when $E\in {\mathbf {B}}$, so it only remains to confirm the case where $E\notin {\mathbf {B}}$. For this case, because ${\mathbf {B}}\nsubseteq {\mathbf {B}}\divideontimes E$, we may select an arbitrary $X\in {\mathbf {B}}$ such that $X\notin {\mathbf {B}}\divideontimes E$. But, because ${\mathbf {B}}\nvdash \lnot E$ and ${\mathbf {B}}$ is cogent, we know that for any such X, $X\supset \lnot E\nvdash {\mathbf {B}}$. Then, by Lemma 2 we know that $t\in (\phi ^{-1},1]$. Finally, recall that in Sect. 4.1 we saw a theorem from Gärdenfors establishing that Lockean revision satisfies all of the AGM postulates when $t=1$. Thus, we may conclude that $t\in (\phi ^{-1},1)$. $\square$

In other words, a cogent Lockean agent can violate Preservationonly if her Lockean threshold is greater than the golden threshold.

Not only is this result surprising from a formal point of view, but it offers some philosophically important lessons as well. For one, the theorem provides a straightforward path to consistently endorsing the Lockean thesis in conjunction with AGM. As discussed earlier, it is well known that proponents of AGM may do so by treating their preferred theory as an account of belief revision under certainty.³³ In Lockean terms, proponents of AGM have previously viewed their theory as applying to cases in which ${\mathsf {B}}(X)$ iff $b(X)=1$. But, by examining some of the consequences of Theorem 1, we can see that AGM theorists may regard their account as relevant to a broader class of situations. To fully appreciate this fact, note the following immediate corollary of our theorem.

Corollary

If${\mathbf {B}}$satisfies ($\hbox {LT}^t$) andCogencyand$t\in (\frac{1}{2},\phi ^{-1}],$then Lockean revision satisfies (*1)–(*6).

In other words, where ${\mathbf {B}}$ is deductively cogent and $t\in (\frac{1}{2},\phi ^{-1}]$, AGM and Lockean revision will be (qualitatively) equivalent. So, if proponents of AGM additionally accept a Lockean thesis with a threshold$t\in (\frac{1}{2},\phi ^{-1}]$, then they may reasonably take their theory to hold more generally without violating any Bayesian/Lockean intuitions. This is because the result establishes that (in the presence of deductive cogency) AGM will never diverge from a Lockean account for thresholds in that interval.

A second philosophically interesting consequence of Theorem 1 is that it provides Lockeans a rebuttal to the standard challenge involving the arbitrariness/context-dependence of Lockean thresholds aside from$\frac{1}{2}$ and 1. This result, along with Proposition 3, establishes an additional non-arbitrary (and context-independent) Lockean threshold at $\phi ^{-1}$.

7 AGM is More Epistemically Risk-Seeking than Lockeanism

In this section, we will present two final theorems and argue that each provide reason to think of AGM revision is more epistemically risk-seeking than Lockean revision. Interestingly, this analysis of the diachronic requirements of the two approaches aligns with the argument from Pettigrew (2016) mentioned earlier in this paper, which shows that AGM is also more epistemically risk-seeking than Lockeanism in its synchronic requirements. As we saw earlier, AGM assumes Closure and Consistency as its core synchronic requirements, while Lockeanism assumes coherence under the synchronic Lockean thesis. Moreover—in so far as Closure and Consistency are understood to be synchronic requirements—we established that Preservation is the only genuinely diachronic requirement of AGM on which Lockean revision may diverge. Thus, if AGM is more epistemically risk-seeking than Lockeanism in its diachronic requirements, then it is due to Preservation. The first of our concluding theorems offers a purely qualitative demonstration of the sort of epistemically risk-seeking behavior implied by Preservation. In our second theorem, we use the tools of epistemic utility theory to extend our argument.

We have seen that Lockean revision can sometime lead agents to give up beliefs that would be retained using AGM revision. But, because Lockean revision satisfies Inclusion, it follows that that it can never require an agent to adopt more new beliefs than AGM revision would have generated. More precisely, because both approaches satisfy Inclusion, they both rule out posterior belief sets ${\mathbf {B}}' = {\mathbf {B}} \star E$ that are proper supersets of ${{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\})$. In other words, neither approach will ever require an agent to be committed to new beliefs that go beyond the logical consequences of their prior belief set together with their new evidence.

Still further, it turns out that when the two approaches diverge, AGM will require the agent to have strictly more new beliefs than would be mandated by Lockean revision. For example, recall our counter-example to Preservation from above (as provided in Table 1). There, we saw that Lockean revision requires the agent to give up belief in $\lnot E\vee X$, while AGM revision does not. When E is learned, AGM revision will result in the agent believing X (along with a variety of other things), while Lockean revision will preclude the adoption of these new beliefs. But, this feature is not unique to our case. In fact, whenever Lockean revision and AGM revision disagree in an interesting way (i.e., not as a result of failures of Closure or Consistency), the former will require the agent to adopt strictly fewer new beliefs than the latter. And, the converse holds as well. So, we arrive at our final theorem, which confirms that Lockean revision violates Preservation just in case performing an AGM revision by a proposition consistent with the agent’s prior belief set leaves the agent with all the beliefs implied by Lockean revision—plus some additional beliefs.

Theorem 2

Where ${\mathbf {B}}$ satisfies ($\hbox {LT}^t$) and Cogency: ${\mathbf {B}}\divideontimes E$ violates Preservationiff ($\Leftrightarrow$) E is consistent with ${\mathbf {B}}$ and ${\mathbf {B}}\divideontimes E \subset {\mathbf {B}} * E$.

Proof

($\Rightarrow$) Suppose ${\mathbf {B}}$ is deductively cogent and ${\mathbf {B}}\divideontimes E$ violates Preservation. Then, (a) E is consistent with ${\mathbf {B}}$; and, (b) ${\mathbf {B}}\nsubseteq {\mathbf {B}} \divideontimes E$. By (b), there is some $X\in {\mathbf {B}}$ such that $X \notin {\mathbf {B}}\divideontimes E$. It follows from (a), Preservation and Inclusion that ${\mathbf {B}} * E={{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\})$. Therefore, $X \in {\mathbf {B}}* E$ and $X \notin {\mathbf {B}} \divideontimes E$. Finally, recalling from Proposition 5 that Lockean revision satisfies Inclusion, it follows that ${\mathbf {B}} \divideontimes E \subseteq {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\}) = {\mathbf {B}} * E$. ($\Leftarrow$) Suppose E is consistent with ${\mathbf {B}}$ and ${\mathbf {B}}\divideontimes E \subset {\mathbf {B}} * E$. Then, there exists an X such that $X \in {\mathbf {B}} * E$ but $X \notin {\mathbf {B}} \divideontimes E$. Because E is consistent with ${\mathbf {B}}$, Closure, Preservation, and Inclusion imply that ${\mathbf {B}} * E = {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\})$. Therefore, $X \in {{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}} \cup \{E\})$, but $X \notin {\mathbf {B}} \divideontimes E$. Finally, once more appealing to Lockean revision’s satisfaction of Inclusion, it follows that $X\in {\mathbf {B}}$. $\square$

In other words, when Lockean revision and AGM revision (interestingly) diverge, AGM will be more demanding on an agent’s posterior beliefs, since the Lockean agent’s posterior will be a strict subset of the AGM agent’s posterior. Because AGM will require agents to maintain beliefs in the face of non-definitive counter-evidence, it may be aptly viewed as an epistemically risk-seeking policy for belief revision. So, since Lockean revision will recommend that agents suspend belief in many cases when AGM revision recommends belief, it can be rightly viewed as the more epistemically risk-averse approach.

In Sect. 3, we mentioned that Pettigrew uses tools from epistemic utility theory to argue that AGM’s synchronic requirements are the more epistemically risk-seeking of the two. But, the same tools can be used in conjunction with our result from Theorem 1 to further argue that the diachronic requirements of AGM imply an epistemically risk-seeking approach to belief revision.

To do so, we rely on Dorst’s (2014) representation theorem revealing that the Lockean thesis can be derived using epistemic utility theory. To see how Dorst’s result works, we first equip our Bayesian agent with a (naïve) epistemic utility function for individual beliefs. Let $u({\mathsf {B}}(X),w)$ refer to the epistemic utility of believing X in world w, and suppose that u is provided the following simple, piecewise definition:

$$\begin{aligned} u({\mathsf {B}}(X),w):={\left\{ \begin{array}{ll} {\mathfrak {r}} &{} {\text {if}} \,\,{X}\,\, {\text {is\;true\;at}} \,\,{w}\\ -{\mathfrak {w}} &{} {\text {if}} \,\,{X}\,\, {\text {is\;false\;at}}\,\, {w} \end{array}\right. } \end{aligned}$$

That is, when an agents believes that X and X is true, then the utility function rewards her accuracy with the “epistemic credit” ${\mathfrak {r}}$; on the other hand, if her belief is false, then the utility function penalizes her with the “epistemic debit” $-{\mathfrak {w}}$.³⁴ The value yielded by this function is wholly determined by the truth of the agent’s belief and is insensitive to its content and other considerations. So, the epistemic utility theory approach supposes a veritistic and value monistic account of the epistemic worth of beliefs. This treatment directly aligns with the those offered by so-called accuracy-first epistemologists and has been motivated on (broadly) (James 1896) grounds that belief has the simultaneous aims of attaining truth, while avoiding error. Accordingly, we take the epistemic utility of individual beliefs to contribute equally to the overall epistemic utility of an agent’s total belief state.

For the moment, we impose only the following single constraint on the value ranges of the utility function’s parameters ${\mathfrak {r}}$ and ${\mathfrak {w}}$:

$$\begin{aligned} {\mathfrak {w}}> {\mathfrak {r}} > 0. \end{aligned}$$

The justification for these minimal restrictions is straightforward. If the epistemic benefit of believing a truth were not greater than zero, then there would never be incentive for belief over suspension. Of course, that would be an unwelcome result and so we have justification for the constraint that ${\mathfrak {r}}>0$. A similar reason can be given to justify the restriction that the epistemic harm of false belief must be greater than the epistemic benefit of true belief. If they were the same, then it would be no better to suspend judgment on the outcome of a fair coin flip than to believe both that it will come up heads and that it will come up tails, since both would have an expected utility of zero. Finally, if the epistemic harm of false belief were less than the epistemic benefit of true belief, then suspension would actually have a worse expected value than the inconsistent alternative. Thus, ${\mathfrak {w}}>{\mathfrak {r}}$.

On the basis of this simple accuracy-centered utility function, it is straightforward to define the expected epistemic utility (EEU) for an agent’s belief that X (relative to her credence function):

$$\begin{aligned} EEU\left( {\mathsf {B}}(X),b\right) :=\sum _{w\in W} b(w) \cdot u\left( {\mathsf {B}}(X),w\right) . \end{aligned}$$

Then, we define the overall EEU of an agent’s total belief set ${\mathbf {B}}$ as simply the sum of the EEUs of all of her individual beliefs.

$$\begin{aligned} EEU({\mathbf {B}},b):=\sum _{X\in {\mathbf {B}}}EEU\left( {\mathsf {B}}(X),b\right) . \end{aligned}$$

This basic apparatus is all that is needed to generate Dorst’s theorem, which establishes that a belief set maximizes EEU relative to a credence function b just in case it satisfies the following precise (normative) Lockean thesis.³⁵

Theorem

(Dorst 2014) Where b is an agent’s credence function, her belief set ${\mathbf {B}}$maximizes EEU just in case

$$\begin{aligned} \begin{array}{ll}{ \text {if\;}}{\mathsf {B}}(X),&\quad{ \text {then\;}}b(X) \ge \dfrac{\mathfrak {w}}{\mathfrak {r+w}},\text { and}\\ {\text {if\;}}b(X)> \dfrac{\mathfrak {w}}{\mathfrak {r+w}},&\quad {\text {then\;}}{\mathsf {B}}(X). \end{array} \end{aligned}$$

It is straightforward to see that this implies that if an agent’s beliefs and credences jointly satisfy the Lockean thesis, then she will maximize EEU. Thus, Lockean revision, as we have explored it, is entailed by the more general norm requiring that agents have belief sets that maximize EEU at any given time. Assuming conditionalization³⁶ as the rational procedure for credal update, the norm entails that Lockean revision is the unique procedure that will guarantee that agents maximize overall EEU.

The definition of the Lockean revision operator can now be equivalently restated with the aid of this new apparatus using the free-parameter values ${\mathfrak {r}}$ and ${\mathfrak {w}}$:

$$\begin{aligned} {\mathbf {B}} \divideontimes E = \left\{ X \,\,\bigg |\,\, b(X\mathbin {|}E) \ge \dfrac{{\mathfrak {w}}}{{\mathfrak {r}}+{\mathfrak {w}}}\right\} . \end{aligned}$$

Now, notice that the greater $\mathfrak {w}$ is relative to ${\mathfrak {r}}$, the greater the resulting Lockean threshold will be. That is, the larger the debit incurred by the agent for believing a falsehood relative to the credit for her believing a truth, the larger the Lockean threshold will be. In the limiting case, we see that a maximal Lockean threshold is established by letting there be no benefit at all for believing truths (i.e. letting ${\mathfrak {r}}=0$).

In the corollary to Theorem 1, we saw that AGM coheres with (cogent) Lockeanism when $t\in (\frac{1}{2},\phi ^{-1}]$. Using these new tools from epistemic utility theory, we can see that this restriction on t is equivalent to requiring that $\mathfrak {w} \le \phi \cdot {\mathfrak {r}}$. Thus, the adoption of AGM can be seen as placing more weight on the epistemically best-case scenarios, which corresponds to a kind of epistemically risk-seeking behavior.³⁷ So, not only are the synchronic requirements of AGM more epistemically risk-seeking than those of Lockeanism—as argued by Pettigrew (2016)—but its diachronic requirements are as well.

8 Conclusion and Future Work

We have pinpointed the precise ways in which a (broadly Bayesian) Lockean approach to belief revision agrees (and disagrees) with the more traditional AGM theory. Setting aside issues surrounding Cogency, Lockean revision and AGM revision exhibit a surprising degree of convergence. Our analysis reveals that, holding Cogency fixed, the two approaches to belief revision disagree only regarding the universal validity of Preservation. Intuitively, this simply results from the fact that Lockean revision is sensitive to non-definitive counter-evidence, while AGM revision is not.

In this paper, we have chosen to focus on the diachronic coherence requirements of the two theories. However, there are further issues relating to their dispositional coherence requirements, which govern iterated revision. Critics of AGM have often complained that it does not easily generalize to offering an account of how iterated revision should proceed. On the other hand, Lockean revision has no special problem with iterated revision.³⁸ In future work, we plan to compare Lockean revision to other systems of belief revision beyond AGM. In doing so, it will be of particular interest to consider systems whose specific aim is the accommodation of iterated revision. More specifically, we plan to investigate the dispositional norms of Lockean revision as contrasted with the Darwiche and Pearl postulates for iterated revision (Darwiche and Pearl 1996).

Another interesting next step in the exploration of Bayesian qualitative revision is to investigate how Lockean revision changes when the agent’s credence function is a non-classical probability function (thus permitting conditionalization by propositions with zero unconditional credence) or when combined with other, more general, credal update procedures in place of conditionalization.³⁹ One especially interesting application along these lines would be to the problem of explicating a Bayesian notion of contraction. We have some preliminary ideas about “Bayesian contraction,” which we plan to explore in a sequel to this paper.⁴⁰

Finally, we would (ideally) like to have a purely qualitative axiomatization of the Lockean revision operator. Some progress toward such an axiomatization has recently been made—e.g. see Makinson and Hawthorne (2015). Of particular interest, van Eijck and Renne (2014) recently provided an axiomatization for the modal logic of belief with a Lockean threshold of $\frac{1}{2}$. We think that our results involving the non-arbitrary Lockean threshold at $\phi ^{-1}$ suggest that a fruitful next step may be to investigate the logic of belief satisfying this threshold. However, there remains significant theoretical work to be done in order to determine precisely which axioms would be needed to characterize Lockean revision.

9 Epilogue: A (Third) Approach to Belief Revision

Those well versed in the recent literature may wonder how our results relate to Leitgeb’s (2014, 2016) results concerning his stability theory of belief. The fundamental synchronic requirement of the stability theory is provided by the Humean thesis, which requires that an agent believe X just in case she has a sufficiently high credence in Xthat would remain sufficiently high were she to learn any proposition logically consistent with her prior beliefs. Leitgeb establishes a remarkable representation theorem for this single synchronic requirement proving its equivalence to jointly requiring the synchronic requirements of both Lockeanism and AGM. That is, the Humean thesis turns out to be equivalent to adopting probabilism and the Lockean thesis along with Cogency. Leitgeb offers a set of diachronic requirements as well that can be used to define a Humean belief revision operator ($\circ$).⁴¹ In a second representation theorem, Leitgeb establishes that Humean revision satisfies all of AGM’s postulates. Since Leitgeb’s Humeanism combines core principles from both Lockeanism and AGM, we might wonder how this squares with the results that we have established in this paper, which seem to drive a wedge between Lockeanism and AGM? In this brief epilogue, we will seek to answer this question and will establish a potentially problematic result for Humean revision.

We begin by explaining Humeanism’s synchronic requirement in more detail. The Humean thesis says that an agent should believe all and only those propositions to which she assigns “resiliently high” credence. A more careful formulation of the requirement is provided below.

($\hbox {HT}^r$)    ${\mathsf {B}}(X)$ iff $b(X\mathbin {|}Y)>r$ for all Y such that $\lnot {\mathsf {B}}(\lnot Y)$ and $b(Y)>0$.

As with the Lockean thesis, the Humean thesis relies on the probability threshold, r, to capture the notion of sufficient likelihood mentioned in the intuitive statement of the principle. As mentioned above, Leitgeb proves that the Humean thesis is equivalent to probabilism, the Lockean thesis, and Cogency. With the aid of this definition, we can now provide a precise statement of Leitgeb’s central representation theorem:

Theorem

(Leitgeb 2016) Where ${\mathbf {B}}$ is an agent’s belief set and b is her credence function, b and ${\mathbf {B}}$ jointly satisfy ($\hbox {HT}^r$) for some r just in case:

• 1. b is a probability function;

• 2. ${\mathbf {B}}$ is satisfies Cogency ; and

• 3. ${\mathbf {B}}$ and b jointly satisfy the Lockean thesis for some t.

Thus, we see that Humeanism’s univocal synchronic requirement ($\hbox {HT}^r$) is equivalent to combining all of the synchronic requirements of both Lockeanism and AGM. At this point, the reader is invited to note that the Humean threshold, r, provided by ($\hbox {HT}^r$) is distinct from the Lockean threshold, t, found in the representation theorem. Although these two thresholds may converge, they need not. We will see shortly that, in many cases, the greatest Humean threshold satisfied by a given belief set and credence function may be significantly lower than the greatest Lockean threshold that they satisfy.

In addition these synchronic coherence requirements, Leitgeb (2016), Ch. 4 proposes a set of diachronic requirements for Humean agents. These requirements yield a characterization of Humean revision and begin with the two following bridge principles:

($^\circ \hbox {BP}1^r$):

If ${\mathbf {B}}\nvdash \lnot E$ and $b(E)>0$, then $X\in {\mathbf {B}}\circ E$ only if $b(X\mathbin {|}E)>r$

($^\circ \hbox {BP}2$):

${\mathbf {B}}\circ E$ is inconsistent iff $b(E)=0$

The first principle, ($^\circ \hbox {BP}1^r$), requires that the left-to-right direction of the Lockean thesis is satisfied relative to propositions consistent with the agent’s prior beliefs and the threshold r. The second, ($^\circ \hbox {BP}2$), requires that Humean revision treats logically impossible propositions and propositions assigned zero credence in the same manner.

In addition to these bridge principles, Leitgeb offers the following AGM-like axioms:

($^\circ$1)	$X\in {\mathbf {B}}\circ X$	Reflexivity
($^\circ$2)	$\hbox {If }Y\in {\mathbf {B}}\circ X \hbox {and }Y\vdash Z,\hbox { then }Z\in {\mathbf {B}}\circ X$	Single Premise Closure
($^\circ$3)	$\hbox { If }Y\in {\mathbf {B}}\circ X \hbox {and }Z\in {\mathbf {B}}\circ X,\hbox { then }Y\wedge Z\in {\mathbf {B}}\circ X$	Finite Conjunction
($^\circ$4)	$\hbox {For any} \mathcal {Y}=\{Y\mathbin {|}Y\in {\mathbf {B}}\circ X\}, \wedge \mathcal {Y}\in {\mathbf {B}}\circ X$	General Conjunction
($^\circ$5)	${\mathbf {B}}\circ \top \nvdash \bot$	Consistency
($^\circ$6)	$\hbox { If }{\mathbf {B}}\circ X\nvdash \lnot Y,\hbox { then }{\mathbf {B}}\circ (X\wedge Y)={{\mathsf {C}}}{{\mathsf {n}}}({\mathbf {B}}\circ X\cup \{Y\})$	General Revision

In conjunction with ($^\circ \hbox {BP}1^r$) and ($^\circ \hbox {BP}2$), these requirements suffice to guarantee that Humean revision satisfies all of AGM’s postulates.⁴² The inclusion of General Revision reveals that Humean revision is constructed so as to guarantee the satisfaction of Preservation.⁴³

Not only does Humean revision satisfy the axioms of AGM, as we have said, but it is guaranteed to yield posteriors satisfying the Humean thesis. In order to explain this result, we first define P-stability$^r$, which is similar to ($\hbox {HT}^r$), but applies to individual propositions rather than belief sets.

Definition 1

A proposition, X, is P-stable$^r$ iff $b(X\mathbin {|}Y)>r$, for any Y such that $X\nvdash \lnot Y$ and $b(Y)>0$

Clearly, if a belief set ${\mathbf {B}}$ satisfies ($\hbox {HT}^r$), then the strongest proposition in ${\mathbf {B}}$ will be P-stable$^r$. With this new notion in hand, we may now state Leitgeb’s second representation theorem (modified only for coherence with current notational conventions).

Theorem

(Leitgeb 2016) Provided a deductively cogent belief set ${\mathbf {B}}$and a probabilistic credence function b, the revision operator $\circ$ satisfies ($^\circ \hbox {BP}1^r$), ($^\circ \hbox {BP}2$), and $(^\circ 1)-(^\circ 6)$ relative to ${\mathbf {B}}$ and b iff there exists a class$\mathcal {X}$ of non-empty P-stable$^r$ propositions such that:

• $\mathcal {X}$ contains the least set of probability 1 in the algebra,

• all other members of $\mathcal {X}$ have probability less than 1,

• for any Y, such that $b(Y)>0$, if X is the strongest proposition in $\mathcal {X}$ such that $Y\cap X\ne \varnothing$, then for all Z:

  $$\begin{aligned}Z\in {\mathbf {B}}\circ Y{\text {\;iff\;}}Z\supseteq Y\cap X,\end{aligned}$$

  and

• for all Y, if $b(Y)=0$ , then ${\mathbf {B}}\circ Y$ is inconsistent.

Intuitively, the result establishes an equivalence between the satisfaction of his principles and the existence of some P-stable$^r$ set. Leitgeb suggests that the right-to-left direction offers the benefit of providing a recipe for building models for his revision postulates by finding some P-stable$^r$ set $\mathcal {X}$ and imposing the restrictions listed.

At first pass, this might seem to suggest that an agent whose prior belief set ${\mathbf {B}}$ satisfies ($\hbox {HT}^r$) and revises by E in such a way that satisfies the right side of the theorem relative to a set $\mathcal {X}$ whose members are P-stable$^r$, her revision must be representable by ${\mathbf {B}}\circ E={\mathbf {B}}*E$. Nonetheless, we will demonstrate that this need not be so. A reconsideration of our counter-example from the proof of Proposition 8 will demonstrate that the agent’s Lockean revision also satisfies these conditions as well despite violating Preservation. Table 2, below, includes the the probability distribution across the strongest propositions from earlier.

Table 2 Counter-example to Preservation for Leitgeb’s stability theory

As we saw, the agent’s belief set, ${\mathbf {B}}$, is deductively closed and satisfies the Lockean thesis for $0.8<t<0.9$. However, a close examination of the distribution confirms that the prior belief set ${\mathbf {B}}=\{\lnot E\vee X\}$ also (uniquely) satisfies ($\hbox {HT}^{.65}$).

Now, we compare the two diverging recommendations the stability theory and Lockean revision and show that both are revisions that lead to posteriors including only P-stable$^{.65}$ propositions. First, consider the result of Humean revision:

$$\begin{aligned} {\mathbf {B}}\circ E=\{E\wedge X,E,X,E\vee X,E\vee \lnot X,\lnot E\vee X\}. \end{aligned}$$

In this case, the class of P-stable$^{.65}$ propositions used to generate the posterior was $\mathcal {X}=\{\lnot E\vee X,\top \}$. Now recall the Lockean revision from earlier, which generated the following posterior.

$$\begin{aligned} {\mathbf {B}}\divideontimes E=\{E,E\vee X,E\vee \lnot X\} \end{aligned}$$

Notice that this is the revision that would follow from the right side of the theorem if we choose $\mathcal {X}'=\{\top \}$ as the class of P-stable$^{.65}$ propositions. Not only does this satisfy the required conditions, but ${\mathbf {B}}\divideontimes E$ also satisfies ($^\circ \hbox {BP}1$), ($^\circ \hbox {BP}2$), and $(^\circ 1)$–$(^\circ 5)$. The Lockean revision only violates $(^\circ 6)$ (which is, of course, just the conjunction of the generalizations of Inclusion and Preservation).

Naturally, this observation does not show that Leitgeb’s theorem was mistaken. After all, his theorem merely required that there is someP-stable$^r$ set that can be used to construct an AGM revision that satisfies his bridge principles. Indeed, there is some such set (as demonstrated above). But, it does show that further information is required to determine which class of P-stable$^r$ sets is the appropriate one.⁴⁴