**3. PBL***<sup>f</sup>* **and its inner probabilistic semantics**

As is often the case in modal logics, the ideas in our completeness proof can be extended to get a finite model property. Therefore the question arises whether finite model property holds for *PBLω*, i.e., for every consistent formula *ϕ*, whether there is a finite sates model satisfies *ϕ*. Unfortunately, we cannot give a positive or negative answer here. Therefore, we seek for some weak variant of *PBL<sup>ω</sup>* whose finite model property can be proved. We call the variant *PBLf* , its reasoning system is the result of deleting *Axiom* 6 and *Rule* 6 from *PBLω*. In the semantics of *PBLf* , we assign an inner probability space to every possible world in the model, here "inner" means the measure does not obey the additivity condition, but obeys some weak additivity conditions satisfied by inner probability measure.

**3.3 Soundness of** *PBLf*

similarly as in Proposition 2.1.

then *μi*,*s*({*s*�

1, 0), *ϕ* ∧ *ψ*).

*Pi*(*s*�

desired.

we apply the property *μi*,*s*(Λ*i*,*s*) = 1 (where Λ*i*,*<sup>s</sup>* = {*s*�

**Proposition 3.1** (Soundness of *PBLf* ) If Γ �*PBLf ϕ*, then Γ |=*PBLf ϕ*.

)}, then Λ*i*,*<sup>s</sup>* ∈ *X* and *μi*,*s*(Λ*i*,*s*) = 1. Let Ξ = {*s*�

formula *ϕ*, we restrict our attention to sets of subformulas of *ϕ*.

*Sub*∗(*ζ*) = *Sub*(*ζ*) ∪ {¬*ψ*|*ψ* ∈ *Sub*(*ζ*)}. It is clear that *Sub*∗(*ζ*) is finite.

(2) For any Γ ∈ *S<sup>ζ</sup>* , *Pi*,*<sup>ζ</sup>* (Γ)=(*S<sup>ζ</sup>* , *X<sup>ζ</sup>* , *μζ*,*i*,Γ), where *X<sup>ζ</sup>* = {*X*(*ϕ*)| *X*(*ϕ*) = {Γ�

*X<sup>ζ</sup>* → [0, 1], and *μζ*,*i*,Γ(*X*(*ϕ*)) = *sup*({*a*|*Bi*(*a*, *ϕ*) is provable from Γ in *PBLf* }).

then Λ*i*,*<sup>s</sup>* ∈ *X* and *μi*,*s*(Λ*i*,*s*) = 1. Let Ξ = {*s*�

**3.4 Finite model property of** *PBLf*

(*S<sup>ζ</sup>* , *P*1,*<sup>ζ</sup>* , ..., *Pn*,*<sup>ζ</sup>* , *πζ* ).

*Sub*∗(*ζ*)}.

The proof of soundness of *PBLf* is similar to the proof in Proposition 2.1, but because there are some differences between *PBLω*-probabilistic model and *PBLf* -probabilistic model, there are a few differences. For example, in the following proof, we can use the property *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) ≥ *μi*,*s*(*A*1) + *μi*,*s*(*A*2) − 1 directly, rather than as a corollary of finite additivity property;

Probabilistic Belief Logics for Uncertain Agents 31

also differs from the last property of *PBLω*-probabilistic model (If *A* ∈ *Xi*,*<sup>s</sup>* and *μi*,*s*(*A*) ≥ *a*,

*Proo f* . We only discuss *Axiom* 2, *Axiom* 3 and *Axiom* 4 of *PBLf* , other cases can be proved

*Axiom* 2: Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*), so *μi*,*s*(*evPM*(*ϕ*)) ≥ *a* and *μi*,*s*(*evPM*(*ψ*)) ≥ *b*. For *μi*,*<sup>s</sup>* is *PBLf* -probability measure, we get *μi*,*s*(*evPM*(*ϕ* ∧ *ψ*)) = *μi*,*s*(*evPM*(*ϕ*) ∩ *evPM*(*ψ*)) ≥ *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) − 1 ≥ *a* + *b* − 1, which implies (*PM*,*s*) |= *Bi*(*max*(*a* + *b* −

implies *s*� ∈ Ξ, it is clear Λ*i*,*<sup>s</sup>* ⊆ Ξ, since *μi*,*s*(Λ*i*,*s*) = 1, so *μi*,*s*(Ξ) = 1. If *s*� ∈ Ξ, then *s*� ∈ *evPM*(*Bi*(*a*, *ϕ*)), therefore *μi*,*s*(*evPM*(*Bi*(*a*, *ϕ*))) = 1, we get (*PM*,*s*) |= *Bi*(1, *Bi*(*a*, *ϕ*)) as

it is clear Λ*i*,*<sup>s</sup>* ⊆ Ξ, since *μi*,*s*(Λ*i*,*s*) = 1, so *μi*,*s*(Ξ) = 1. If *s*� ∈ Ξ, then *s*� ∈ *evPM*(*Bi*(*a*, *ϕ*)),

We now turn our attention to the finite model property of *PBLf* . It needs to show that if a formula is *PBLf* -consistent, then it is satisfiable in a finite structure. The idea is that rather than considering maximal consistent formulas set when trying to construct a structure satisfying a

**Definition 3.3** Suppose *ζ* is a consistent formula with respect to *PBLf* , *Sub*∗(*ζ*) is a set of formulas defined as follows: let *<sup>ζ</sup>* <sup>∈</sup> *<sup>L</sup>PBLf* , *Sub*(*ζ*) is the set of subformulas of *<sup>ζ</sup>*, then

**Definition 3.4** The inner probabilistic model *PM<sup>ζ</sup>* with respect to formula *ζ* is

(1) Here *S<sup>ζ</sup>* = {Γ|Γ is a maximal consistent formulas set with respect to *PBLf* and Γ ⊆

combination of formulas in *Sub*∗(*ζ*) and Γ �*PBLf ϕ*}}; *μζ*,*i*,<sup>Γ</sup> is an inner probability assignment:

(3) *πζ* is a truth assignment as follows: For any atomic formula *p*, *πζ* (*p*, Γ) = *true* ⇔ *p* ∈ Γ.


*Axiom* 3: Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), therefore *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*. Let Λ*i*,*<sup>s</sup>* = {*s*�

*Axiom* 4: Suppose (*PM*,*s*) |= ¬*Bi*(*a*, *ϕ*), so *μi*,*s*(*evPM*(*ϕ*)) < *a*. Let Λ*i*,*<sup>s</sup>* = {*s*�

therefore *μi*,*s*(*evPM*(¬*Bi*(*a*, *ϕ*))) = 1, we get (*PM*,*s*) |= *Bi*(1, ¬*Bi*(*a*, *ϕ*)) as desired.


)}) in the proof, which





)},



The well formed formulas set *LPBLf* of *PBLf* is the same as *LPBL<sup>ω</sup>* .
