This short contribution presents some results on a characterization of those belief sets that are definable by the Lockean thesis and that are deductively closed. Furthermore, results on minimal revision and credibility-limited iterated revision are presented. The Lockean thesis is that philosophical principle according to which beliefs of rational agents are definable in terms of confidence and, more precisely, within the (subjective) probabilistic setting. The principle was formulated by Foley in [1] as follows: fix a finite language ℒ over the signature of classical propositional logic; it is rational for an agent to believe in a statement ∈ ℒ provided that the agent's subjective probability () of overcomes a certain threshold that is strictly greater than 1/2. Lockean thesis: For any ∈ ℒ , it is rational to believe in provided that its probability () overcomes a certain threshold .
Lockean belief sets can hence be used to represent the epistemic states of rational agents, and they have
been already considered as a basis for theories of probabilistic belief change. The idea of grounding belief
change on Lockean belief sets is in fact not new and it has been proposed, explored and investigated by
others. Among them, it is worth recalling the following ones, whose ideas inspired the present paper:
[
The elementary observation that if a rational agent believes then it is reasonable to assume that she does not believe its negation ¬ forces the threshold to be strictly above 1/2. Indeed, this requirement implies that belief sets defined as in (1) do not contain pairs of contradictory formulas, in the sense that it is not the case that there exists a formula such that and its negation ¬ both belong to a Lockean belief set B, .
Although they are not contradictory in the above sense, Lockean belief sets lack some properties that
are usually required in the classical approaches to belief change as developed by Alchourrón, Gärdenfors,
and Makinson [
A characterization of those Lockean belief sets that are closed under logical deduction has been
recently presented in the paper [
The basic ground of our investigation is classical propositional logic (CPL) and our language ℒ, up to redundancy, is built from a finite set of propositional variables, say 1, . . . , , connectives ∧ conjunction, disjunction, and negation respectively, and constants ⊤, ⊥ for true and false. Formulas in that , ∨, ¬ for language will be denoted by lower case Greek letters , , . . . with possible subscripts. The connective of implication is defined as → = ¬ ∨ , and double implication is ↔ = ( → ) ∧ ( → ) view, every formula can be identified with the set As anticipated above, the consequence relation of CPL is denoted by ⊢ : ℒ → {0, 1} such that () = 1 , also written |= . By Ω JK of its models, i.e., those logical valuations we will henceforth denote the finite set . From the semantic point of to say, ↑JK is the filter of 2 of logical valuations of ℒ. For all ∈ ℒ , Ω, the algebra of parts of Ω, principally generated by .
↑JK
denotes the set of subsets of Ω that contains JK , that is
Probability distributions will be defined on Ω as usual and hence the corresponding probability measures, that are denoted by the same symbol, will be defined additively on JK 2Ω. For every formula ∈ ℒ
we adopt the convention of defining the probability of as the probability of the set of models of , i.e. we set () = ( ) = ∑︀ mean a distribution (resp. measure) s∈uJcKh th(at). B(y)a>po0siftoivrealplrob∈abΩ i(lit(y)d>ist0rifbourtiaolnl (̸=m⊥ea)s.ure), we
JK
The first result introduces a notational convention that will be often used throughout the paper. Lemma 1. For a positive probability on 2Ω and > 1/2 , there exist a minimal and a maximal such that: = () for some ∈ 2
Ω, < ≤ We say that a belief set B, is trivial when B, , and B ′, = B, for every ′ ∈ ( , ].
= ↑J⊤K, that is to say, B, is trivial if it only contains logical theorems. Note that if is a positive probability and B, is non-trivial, then < 1.
The following result is a characterization of those Lockean belief sets that are deductively closed.
→ [
= ↑JK.
2. there exists ∈ B , such that () = and 1 − < min{ () | ∈ JK} . In such a Moreover, under the further hypothesis that < 1, the above conditions are also equivalent to 3. there exists ∈ Ω such that () > ∑︀
′: (′)< () (′) > 0.
Probabilities satisfying the above point 3 have been called to have an -step (or to have a step at )
in [
A probability function might have more than one step as shown in the previous example.
The probabilistic setting of our approach suggests a natural and straightforward way to revise a Lockean belief set B, by a formula . This is relying on the conditional probability (· | ) = (·) , and hence defining
B, * = { ∈ ℒ |
() ≥ }
several authors, and notably in [
, , that is to say, no revision is produced by those formulas a priori () < , then a posteriori
() = . 2. If ̸∈ B , , then we include in B, by minimally changing its probability; in other words, if In order to fulfill the above requests, we will define the following method to update an initial positive probability function. by and is the probability function
Given a positive probability on Ω and ∈ ℒ whose distribution on Ω is defined by:
and > 1/2 , the revised probability of () = ⎧ ⎨ () · max ⎩ () · min {︁ {︁ 1, () }︁ 1, 1(− ¬) }︁ if ∈ JK if ̸∈ JK It also satisfies desideratum 2 because, if () < , () ≥
Notice that the above definition accommodates the above desideratum 1 because, if ∈ B , , then and hence () ≤ 1 . Thus () = () for all ∈ JK and hence () = () ≥
() = as we required above.
One can easily check that ∑︀
∈Ω () = 1, therefore, is a probability distribution on Ω that
on ℒ, ∈ [
Proposition 1. For every probability , > 1/2 and such that () > 0, we have for all ∈ ℒ, () = ︂{ (), if () ≤
(), otherwise.
Thus, when () ≤ ,
is the Jefrey conditionalization of by .
We now formally introduce our revision operator * for Lockean belief sets.
Given a Lockean belief set B, , and a proposition , the minimal Lockean revision of
B, * = { | () ≥ } = B , Next result fully describes the situations in which B, * is deductively closed. 2–7 (2) Theorem 2. Let B, be a Lockean belief set, a proposition and = min{ () | ∈ JK}. Then: (i) B, * is deductively closed whenever > 1− (). In that case, B , * = ↑JK. (ii) Conversely, assuming ̸∈ B , , if B, * is deductively closed then > 1− (). by putting ∘ = . Note that B, * = ( ∘ ).
Actually, the present setting may be read within the framework introduced in [
Some of the basic AGM postulates for revision (K*1-K*6) from [
K*1 (Closure): B, *
is logically closed. This is not generally satisfied. However, it is satisfied if either ∈ B , and B, is closed or ̸∈ B , and satisfies the condition (i) of Theorem 2. K*2 (Success): ∈ B , * . This holds by definition of .
K*3 (Inclusion): B, * ⊆ (B ,
∪ {}) , where denotes the deductive closure operator.
This postulate holds. Essentially, this is due to the fact that if ∈ B , * then ( → ) ∈ B , K*4 (Preservation): If B, ̸⊢ ¬
then B, ⊆ B , * . This postulate is not satisfied in general
K*5 (Consistency): If is consistent then B, *
is consistent. This postulate is not satisfied in
general (see [
the revision is defined semantically.
K*6 (Extensionality): If ⊢ ↔ then B, * = B , * . This postulate clearly holds because
It is worth noticing that the distribution used to define our revision operator * is the one minimally difering from
, according to the Kullback-Leibler divergence, among all the probability
distributions satisfying that the probability of is equal to , see also [
consider the iterated revision (B, * ) * iterated revision and deductive closure holds.
Regarding the updated probability introduced above, notice that, unless = 1 , we have () > 0 for ∈ JK above expression for Jefrey conditionalization is definable and hence so is the further revision, by say , of the positive probability function . Therefore, if and are two formulas, it makes sense to and this fact makes the resulting revision operator * amenable to be iterated because the of a Lockean belief set B, . The following result on (B, * ) *
is deductively closed and moreover Proposition 2. If B, * , with ̸∈ B , , is deductively closed and is such that ⊢ , then (B, * ) *
= B , * .
The last claim of the above proposition ensures that the first postulate of iterated revision, as proposed
by Darwiche and Pearl in [
Now, let us consider the other postulates, namely DP2 If ⊢ ¬ then (B express this extra hypothesis, let us consider a set C, of credible formulas, where and are as above and that is defined as follows:
C, = B, ∪ | > ︂{ 1 − according to the present definition. where, as in the above section, = min{ () | ∈ JK} . Notice that, as it is reasonable and intuitive to assume, B, ⊆ C , , i.e., formulas that are believed according to the Lockean thesis, are credible
More formally, in order to consider iterated revision, we consider now the epistemic space ℰ =
(P , ) in which a credibility-limited revision operator [
is closed and, as before, for every ∈ P we put ( ) = B, . Then, on this epistemic space, we define an operator ∘ : P × ℒ * → P by putting ∘ = ︂{ ,
if ∈ C , ,
∘ = , otherwise
Note that, by Theorem 2, this operator is well-defined, and it can be proved that the following postulates
for iterated revision proposed in [
( ∘ ), , and ̸∈ ( ), then (( ∘ ) ∘ ) =
( ∘ ), , and ̸∈ ( ), then (( ∘ ) ∘ ) ⊢ ,
, and ̸∈ ( ), then (( ∘ ) ∘ ) ̸⊢ ¬.
As a matter of fact, we have that if we define
B, * = ( ∘ ) where ∘ is the above operator ℰ defined on
, the postulates (K*1), (K*3), (K*5), (K*6) are satisfied, although the postulate (K*4) is not
satisfied (the same counterexample in [
B, * ⊢ or B B, * ̸⊢ ⊥ , , * = B
, , If B, * ⊢ and ⊢ then B , * ⊢ . plus the above postulates, and thus showing a good iterative behavior.
In summary, the operator ∘ satisfies the majority of the basic postulates of credibility-limited revision
We have provided two characterizations of those probability functions on formulas ensuring the existence of suitable thresholds for which the corresponding Lockean set of beliefs B, is closed under logical deduction. Then we have discussed how to revise a Lockean belief set in a way that is compatible with an intuitive principle of minimal change. We have shown that this principle univocally leads to a revision operator * for Lockean belief sets closely related (but not equal) to Jefrey conditionalization, satisfying several AGM postulates. Finally, we have introduced a credibility-limited revision operator satisfying adapted versions of all Darwiche and Pearl postulates for iterated revision.
As for future work, we plan to further investigate and characterize our revision operators and compare
them to the ones proposed in the related literature. In particular, we plan to study in more detail the
links of our approach with that of Leitgeb based on -stable sets [
The authors thank Vanina M. Martinez and Ricardo O. Rodriguez for initial interesting discussions on the topic of the paper. Flaminio and Godo acknowledge partial support from the Spanish projects SHORE (PID2022-141529NB-C22) and LINEXSYS (PID2022-139835NB-C21) respectively, both funded by the MCIN/AEI/10.13039/501100011033. They also acknowledge partial support from the H2020MSCA-RISE-2020 project MOSAIC (GA number 101007627). Ramón Pino Pérez has benefited from the support of the AI Chair BE4musIA of the French National Research Agency (ANR-20-CHIA-0028) and the support of I. Bloch’s chair in AI (Sorbonne Université and SCAI).
The authors have not employed any Generative AI tools.