Lockean Belief Sets: A Probabilistic Framework

• Lockean belief sets are probabilistically defined models where a proposition is believed if its probability exceeds a threshold (typically >1/2), highlighting rational preference but often lacking full deductive closure.
• The framework employs formal definitions, algebraic filters, and step-probability analyses to characterize when beliefs are deductively closed and to manage belief revision through minimal-change operators.
• Applications include modeling belief dynamics, addressing paradoxes like the lottery and preface paradox, and contrasting probabilistic belief revision with orthodox AGM-style approaches.

Lockean belief sets are probabilistically defined belief sets associated with the Lockean thesis: for a probability function $P$ on a propositional language $\mathcal{L}$ and a threshold $\lambda$, an agent believes exactly those formulas whose probability meets or exceeds $\lambda$. In the formalization studied in recent work, the associated Lockean belief set is

$$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$

typically with $\lambda>1/2$, so that rational belief favors $\varphi$ over $\neg\varphi$ and no formula and its negation can both belong to the belief set. The modern literature treats Lockean belief sets as a natural probabilistic model of belief, but also emphasizes two persistent difficulties: they are usually not deductively closed, and their dynamics do not in general coincide with orthodox AGM-style belief revision (Flaminio et al., 8 Jul 2025).

1. Formal definition and semantic setting

Within the formal setting of the Lockean thesis, belief is defined in terms of degrees of confidence described probabilistically. Foley’s Lockean thesis is stated as follows: for any $\varphi\in\mathcal{L}$, it is rational to believe $\varphi$ provided that its probability $\mathcal{L}$0 overcomes a certain threshold $\mathcal{L}$1. The resulting Lockean belief set is

$\mathcal{L}$2

The semantic backdrop in the 2025 treatment is classical propositional logic over a finite language, with a finite set $\mathcal{L}$3 of valuations. A probability function $\mathcal{L}$4 is called positive if $\mathcal{L}$5 for every valuation $\mathcal{L}$6, and the standing assumption $\mathcal{L}$7 guarantees that no sentence and its negation are both accepted (Flaminio et al., 8 Jul 2025).

A related formulation appears in the 2023 probabilistic-dynamic literature, where orthodox Lockeanism is presented as the thesis that one believes $\mathcal{L}$8 iff $\mathcal{L}$9 for some threshold $\lambda$0. That formulation is treated as the canonical threshold view against which alternative probabilistic theories are compared. The central complaint is that threshold acceptance is too crude when belief must be closed under entailment and when dynamics should reflect the structure of a question under discussion rather than bare marginal probabilities (Goodman et al., 2023).

This core Lockean idea should be sharply distinguished from stricter notions of acceptance in other probabilistic belief-change frameworks. In particular, one influential contrastive account defines acceptance not by high probability but by certainty: $\lambda$1 That framework is explicitly non-Lockean, because it uses probability-$\lambda$2 acceptance rather than any threshold $\lambda$3 (Voorbraak, 2013).

2. Deductive closure and the structural problem

The standard objection to Lockean belief sets is that they are not generally closed under classical logical deduction. A belief set is deductively closed if

$\lambda$4

For Lockean belief sets, the difficulty is not upward closure. Since probability is monotone, every Lockean belief set is automatically upward closed: if $\lambda$5 and $\lambda$6, then $\lambda$7, so $\lambda$8. Consequently, deductive closure reduces to a stronger condition: deductive closure is equivalent to conjunctive closure. In other words, it is enough that whenever $\lambda$9, then

$\lambda$0

The general reason closure fails is that many formulas can each have probability at least $\lambda$1, while their conjunction falls below $\lambda$2. This is the source of familiar paradoxes such as the lottery paradox and preface paradox (Flaminio et al., 8 Jul 2025).

The algebraic reformulation of this problem uses filters in the Boolean algebra of propositions. A filter $\lambda$3 is a non-empty proper set closed under conjunction and upward closure. Every filter is principally generated by a unique formula $\lambda$4: $\lambda$5 Accordingly, if a Lockean belief set is deductively closed, it must be of the form $\lambda$6 for some generator $\lambda$7. This identifies the closure problem with the existence of a unique proposition whose upward closure exactly matches the threshold-induced belief set (Flaminio et al., 8 Jul 2025).

The contrast with AGM-style belief sets is instructive. In AGM and related semantic frameworks, belief sets are ordinarily stipulated to be deductively closed and consistent from the outset. For example, in Kripke-Lewis semantics the initial belief set at state $\lambda$8 is

$\lambda$9

and $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$0 is proved to be deductively closed and consistent. Lockean belief sets, by contrast, derive belief from probabilistic thresholds and therefore face closure as a substantive constraint rather than as part of the representation itself (Bonanno, 2023).

3. Characterizations of deductively closed Lockean belief sets

The 2025 analysis gives two characterizations of deductively closed Lockean belief sets. The first begins from the observation that, for a positive probability $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$1 and $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$2, the induced belief set is locally constant on an interval of thresholds. More precisely, there exist $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$3 such that

$$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$4

This allows one to replace $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$5 by the maximal threshold $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$6 without changing the belief set (Flaminio et al., 8 Jul 2025).

The first main characterization states that, for a positive probability $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$7 and $$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$8,

$$\mathscr{B}_{\lambda, P}=\{\varphi\in \mathcal{L}: P(\varphi)\geq \lambda\},$$9

and in that case

$\lambda>1/2$0

The accompanying intuition is that closure occurs when there exists a formula $\lambda>1/2$1 whose probability is exactly the relevant threshold and whose models are all individually heavy enough relative to the probability mass outside $\lambda>1/2$2. The same analysis yields the corollary that if $\lambda>1/2$3 and $\lambda>1/2$4, then

$\lambda>1/2$5

This shows that, as the generator $\lambda>1/2$6 has more models, the threshold must move correspondingly closer to $\lambda>1/2$7 (Flaminio et al., 8 Jul 2025).

The second characterization is expressed in terms of step probabilities. A probability $\lambda>1/2$8 has an $\lambda>1/2$9-step if

$\varphi$0

and $\varphi$1 is a step probability if it has an $\varphi$2-step for some $\varphi$3. From such a world one defines

$\varphi$4

If $\varphi$5 has an $\varphi$6-step, then

$\varphi$7

and

$\varphi$8

The resulting theorem states that for every positive probability $\varphi$9, there exists $\neg\varphi$0 such that $\neg\varphi$1 is deductively closed and non-trivial iff $\neg\varphi$2 has an $\neg\varphi$3-step. Equivalently, if $\neg\varphi$4 has a step at $\neg\varphi$5, then taking $\neg\varphi$6 yields

$\neg\varphi$7

and conversely every non-trivial deductively closed Lockean belief set arises from such a step. This establishes that deductive closure is possible exactly when the probability distribution contains a sufficiently strong step structure (Flaminio et al., 8 Jul 2025).

4. Minimal-change revision and deductive closure after update

Lockean belief sets are of established interest in belief change theory, but their use is troublesome precisely because closure is not automatic. The 2025 paper therefore proposes a probabilistic update rule intended to implement minimal revision of Lockean beliefs while preserving as much of the original belief set as possible. A naive strategy would use ordinary conditioning $\neg\varphi$8, but that rule does not always satisfy the intended minimal-change idea. The target behavior is twofold: if $\neg\varphi$9, revising by $\varphi\in\mathcal{L}$0 should do nothing,

$\varphi\in\mathcal{L}$1

whereas if $\varphi\in\mathcal{L}$2, revision should make $\varphi\in\mathcal{L}$3 just reach the threshold,

$\varphi\in\mathcal{L}$4

The revised probability is defined worldwise by

$\varphi\in\mathcal{L}$5

If $\varphi\in\mathcal{L}$6, nothing changes; if $\varphi\in\mathcal{L}$7, then $\varphi\in\mathcal{L}$8. The induced revision operator is

$\varphi\in\mathcal{L}$9

This is the minimal Lockean revision operator (Flaminio et al., 8 Jul 2025).

The revised probability is closely related to Jeffrey conditionalization: $\varphi$0 with

$\varphi$1

Thus the proposal coincides with Jeffrey conditionalization exactly when revision is needed and otherwise leaves the prior unchanged (Flaminio et al., 8 Jul 2025).

The same work gives a precise criterion for when minimal revision yields a deductively closed Lockean belief set. Let

$\varphi$2

Then:

$\varphi$3

the revised belief set is deductively closed and in fact

$\varphi$4

Conversely, assuming $\varphi$5, if $\varphi$6 is deductively closed, then necessarily

$\varphi$7

The revised set is therefore deductively closed exactly when $\varphi$8 itself becomes the generator of the new filter (Flaminio et al., 8 Jul 2025).

This operator is AGM-like but not fully AGM. The paper checks the postulates $\varphi$9–$\mathcal{L}$00 and reports the following pattern: $\mathcal{L}$01 (Success), $\mathcal{L}$02 (Inclusion), and $\mathcal{L}$03 (Extensionality) are satisfied; $\mathcal{L}$04 (Closure) is not generally satisfied but does hold in the deductively closed cases identified above; $\mathcal{L}$05 (Preservation) is not generally satisfied; and $\mathcal{L}$06 (Consistency) is not generally satisfied, though it holds under the closure condition just stated. Minimality is also made precise. Using

$\mathcal{L}$07

the revised distribution $\mathcal{L}$08 is closer to $\mathcal{L}$09 than ordinary conditioning $\mathcal{L}$10, and it minimizes Kullback–Leibler divergence

$\mathcal{L}$11

among all $\mathcal{L}$12 such that $\mathcal{L}$13 (Flaminio et al., 8 Jul 2025).

5. Question-relative probabilistic belief and the critique of orthodox Lockeanism

A distinct line of work develops a probabilistic account of belief that is explicitly question-relative and is presented as an improvement on standard Lockean views. In that framework, a probability structure has the form

$\mathcal{L}$14

where $\mathcal{L}$15 is a partition of $\mathcal{L}$16 representing the question under discussion. Beliefs given evidence $\mathcal{L}$17 are determined by a set of states $\mathcal{L}$18, and a proposition $\mathcal{L}$19 is believed iff

$\mathcal{L}$20

The set $\mathcal{L}$21 is defined by

$\mathcal{L}$22

where $\mathcal{L}$23. This makes belief depend not merely on whether a proposition has high probability, but on whether an answer to the question is sufficiently competitive among the answers (Goodman et al., 2023).

The framework is said to satisfy three conditions built into the account: closure under entailment, sufficiently high probability given evidence, and sensitivity to the relative probabilities of answers. It is therefore explicitly stronger than simple Lockeanism, while also weaker than AGM. Its dynamics are formulated through principles governing discovery of $\mathcal{L}$24 by moving from $\mathcal{L}$25 to $\mathcal{L}$26. The paper proves that $\mathcal{L}$27 and $\mathcal{L}$28 are valid for all probability structures, whereas $\mathcal{L}$29, $\mathcal{L}$30, and $\mathcal{L}$31 can fail. The resulting dynamic profile is weaker than AGM-style triviality of compatible belief change, but more constrained than the dynamics associated with orthodox Lockeanism (Goodman et al., 2023).

The comparison with Lockean belief sets is explicit. Orthodox Lockean belief revision is described as very weak: it validates only $\mathcal{L}$32 in the relevant range, and even that only under the right threshold conditions. By contrast, the question-relative framework validates both $\mathcal{L}$33 and $\mathcal{L}$34. The criticism of Lockeanism is twofold. First, threshold belief ignores the role of the question $\mathcal{L}$35 and the relative ordering of answers to $\mathcal{L}$36. Second, it does not by itself secure closure under entailment if belief is simply a threshold on probability. This suggests a broader lesson: merely assigning a fixed high-probability threshold to propositions is insufficient for capturing several structural and dynamic features often required of belief (Goodman et al., 2023).

The same account also identifies a restricted class of models satisfying orthogonality,

$\mathcal{L}$37

for the specified states and evidence sets. Under orthogonality, $\mathcal{L}$38 becomes valid, and a stronger principle $\mathcal{L}$39 is validated. However, orthogonality is explicitly not treated as plausible as a general constraint. This preserves the paper’s overall position that a plausible probabilistic theory of belief should remain below AGM strength while exceeding the weakness of orthodox Lockean revision (Goodman et al., 2023).

6. Lockean belief sets in relation to AGM, conditioning, and belief-set theories

Lockean belief sets occupy a specific position within the broader theory of belief change. Classical AGM and KM frameworks model belief states as deductively closed, consistent belief sets. In Kripke-Lewis semantics, the initial belief set is determined by universal truth across doxastically possible states: $\mathcal{L}$40 and updated or revised beliefs are characterized as consequents of believed conditionals with antecedent $\mathcal{L}$41. The difference between update and revision is then reduced to stronger or weaker doxastic-priority constraints on the Lewis selection function. Nothing in this framework is threshold-probabilistic: belief is classical, not Lockean (Bonanno, 2023).

A different contrast arises in probabilistic AGM-inspired work on belief change. There, a belief set $\mathcal{L}$42 is a logically closed, consistent set of propositional sentences,

$\mathcal{L}$43

while a probability function $\mathcal{L}$44 represents degrees of belief, and a sentence is accepted exactly when

$\mathcal{L}$45

The set of fully accepted sentences in $\mathcal{L}$46 is its top,

$\mathcal{L}$47

That framework generalizes to partial probabilistic models $\mathcal{L}$48 and distinguishes extended conditioning,

$\mathcal{L}$49

from constraining,

$\mathcal{L}$50

Its central thesis is that conditioning is not the probabilistic analogue of expansion; constraining is the better candidate. For Lockean belief sets, the important point is negative: the framework does not adopt a threshold-based acceptance rule and does not endorse a Lockean conception. Acceptance is certainty, not mere high probability (Voorbraak, 2013).

This contrast clarifies a recurrent misconception. Lockean belief sets are not just another notation for the “top” of a probability state, because the top uses exact acceptance at probability $\mathcal{L}$51, whereas the Lockean thesis uses a threshold $\mathcal{L}$52 and thereby permits acceptance short of certainty. The distinction matters both statically and dynamically. Once belief is defined by high probability rather than certainty, deductive closure becomes non-trivial, compatible evidence can alter beliefs in non-AGM ways, and special revision mechanisms such as minimal Lockean revision become necessary (Voorbraak, 2013).

A final related development concerns belief-set theories that preserve logical closure while rejecting recovery. In enriched latent belief theory, belief states remain logically closed: $\mathcal{L}$53 yet contraction and re-expansion need not recover all prior beliefs. The significance for Lockean discussion is indirect but clear: failure of recovery does not require abandoning belief sets as such, only enriching their internal structure. This suggests that some tensions often attributed to threshold belief may also reflect broader limitations of simple belief-set representations (Arisaka, 2015).