As we pointed out in Section3.2.3, strongest-stable-set operators cannot be represented as a mapX7→minR(X) for some order relation R⊆Ω2. In other words, probabilistically stable revision cannot be tracked using a minimisation operator for a plausibility relation. We also observed that strongest-stable-set operators, treated as selection functions, do not validate the (Or)rule. The former fact easily follows from the latter. As noted by e.g. van Benthem [60] and Rott [47] the following conditions are necessary and sufficient for representability as a minimisation operator.
Proposition 3.3.18 (van Benthem [60])
Let Ω a finite set. Given a function σ:P(Ω)→ P(Ω), the following are equivalent:
• σ satisfies the following properties for anyXi⊆Ω:
(1) σ(X)⊆X (2) σ(S i≤nXi)⊆ S i≤nσ(Xi) (3) T i≤nσ(Xi)⊆σ( S i≤nXi)
• There is an asymmetric binary relationR⊆Ω2 such that for all X:
σ(X) = min
R (X) :={ω∈X| ¬∃v∈X, R(v, ω)}
As a quick verification reveals, strongest-stable-set operators validate both (1) and (3), but fail (2). The failure of (2) is unsurprising: interpreting Xi |∼Xj as σ(Xi)⊆Xj, the property corresponds to the (Or) rule.
It is worth noting, however that a weaker form of the (Or) rule does obtain for proba- bilistically stable revision:
Observation 3.3.19
For any representable selection function σ on an algebraA, we have that the following holds for any finite collection of events Xi (i≤n) in A:
If Xi\Xj⊆σ(Xi) for alli6=j, then σ(
[
i≤n
Xi)⊆ [
i≤n
σ(Xi). (wO)
Proof. Let µbe a probability measure representing the selection functionσ, so thatσ(X) =
τ(µX) for all X∈A. Assume Xi\Xj⊆σ(Xi) for all i, j≤n with i6=j. It is enough to show that S
i≤nσ(Xi) is stable with respect to the measure µ(· |Si≤nXi): this suffices, since σ(S
i≤nXi) is the strongest stable set w.r.t µ(· |
S
i≤nXi). Let ω∈
S
i≤nσ(Xi)38, and consider the relative complementS
i≤nXi\Si≤nσ(Xi). SinceXi\Xj⊆σ(Xi) for all distincti, j, this means thatS
i6=j(Xi\Xj)⊆Si≤nσ(Xi). So we get S i≤nXi\Si≤nσ(Xi)⊆ (S i≤nXi)\ S i6=j(Xi\Xj) = T i≤nXi. This establishes that S
i≤nXi\Si≤nσ(Xi)⊆Ti≤nXi; we can then also write
[ i≤n Xi\ [ i≤n σ(Xi)⊆Xj\σ(Xj)
38Here again, we take a singleton to represent an atom for simplicity, but this is immaterial: the argument applies for any choice ofA-atom inS
for all j≤n. But evidently, sinceω∈S
i≤nσ(Xi) we haveω∈σ(Xj) for some j: this means thatµ(ω)> µ(Xj\σ(Xj). In particular, µ(ω)> µ [ i≤n Xi\ [ i≤n σ(Xi)
where ω was arbitrary inS
i≤nσ(Xi). This establishes thatSi≤nσ(Xi) is stable. Since the condition σ(S
i≤nXi)⊆
S
i≤nσ(Xi) captures the (Or) rule, we can see (wO) as a substantially weaker form of (Or)-style reasoning: it specifies that the (Or) rule can be applied to a set of antecedents X1, ..., Xn provided that they satisfy the side condition
Xi\Xj⊆σ(Xi) for alli6=j. This weak (wO) rule can be written in semantic form as follows (here we write its two-premise version):
Xi\Xj⊆σ(Xi) for i6=j X1 |∼A X2 |∼A
X1∪X2 |∼A
(wO2)
The side constraints Xi\Xj⊆σ(Xi) give conditions under which Or-type inferences are valid: all of theXi\Xj states must be typical givenXi, in the sense of being in the selected subset. It is perhaps more informative to see this as a constraint onatypical states: namely, all of theatypical Xi states – those inXi\σ(Xi) – must be inXj.
In addition to outlining a very modest extent to which probabilistic stability obeys a version of case reasoning (or the sure thing principle), this rule can be used to obtain an alternative characterisation of Leitgeb structures: it is enough to replace the axiomS4nby the property (wO). This is because the property already entailsS4n.
Observation 3.3.20
Suppose a selection function σ satisfies the property (wO). Then it also satisfies S4n.
Proof. Supposeσ has property (wO). Suppose we have setsA, Xi (i≤n), such that ∀i≤n,
σ(A∪Xi) =Xi. WritingDi:=A∪Xi, we have, for eachi6=j,Di\Dj=Xi\(A∪Xj). Now
σ(Di) =σ(A∪Xi) =Xi, and so we can writeDi\Dj ⊆σ(Di). By (wO), we can conclude
σ(S
i≤nDi)⊆Si≤nσ(Di), which is equivalent toσ(Si≤nA∪Xi)⊆Si≤nXi. We have shown (S4n).
This observation entails the following reformulation of our representation theorem (Theorem 3.3.8):
Proposition 3.3.21
Let (Ω,A, σ) a selection structure. The following conditions are necessary and sufficient for
σ to be a strongest-stable-set operator for some underlying (regular) probability measure on A:
(S1) σ(X) =∅ only if X=∅
(S2) σ(X)⊆X
(S3) If σ(A)∩B6=∅, then σ(A∩B)⊆σ(A)∩B
(wO) If Xi\Xj⊆σ(Xi) for all i6=j, then σ(Si≤nXi)⊆Si≤nσ(Xi).
(Scott) If (Ai)i6n≡0(Bi)i6n and ∀i6n, Ai<∗σBi, then ∀i6n,Ai4∗σBi.
This reformulation of the representation theorem is slightly more perspicuous: proba- bilistically stable revision operations are characterised by reflexivity, rational monotonicity, a weaker form of the (Or) rule, and the (Scott)-type axiom for representability (which guarantees that the ‘preference’ ordering between atomic and other events implied by σ
can be captured quantitatively). Next, we note an interesting connection between our representation problem and the theory of simple voting games.