**3.3 Soundness of** *PBLf*

14 Will-be-set-by-IN-TECH

**Definition 3.1** An inner probabilistic model *PM* of *PBLf* is a tuple (*S*, *π*, *P*1, ..., *Pn*), where

(3) *Pi* is a map, it maps every possible world *s* to a *PBLf* -probability space *Pi*(*s*)=(*S*, *X*, *μi*,*s*).

*μi*,*<sup>s</sup>* is a *PBLf* -inner probability measure assigned to the set *X*, which means *μi*,*<sup>s</sup>* satisfies the

Remark: Since *X* = ℘(*S*), therefore *X* is a constant set, and we omit the subscript of *Xi*,*<sup>s</sup>* as

It is easy to see that the conditions (d) and (e) in Definition 3.1 are weaker than the finite additivity condition in Definition 2.2. One can check that if *μ* is a probability measure, then inner measure *μ*∗ induced by *μ* obeys the conditions (d) and (e) in Definition 3.1, i.e., the

The notation Λ*i*,*<sup>s</sup>* in the condition (f) represents the set of states whose probability space is same as the probability space of state *s*. Therefore the condition (f) means that for any state *s*,

The inference system of *PBLf* is the same as *PBL�* except without *Axiom* 6 and *Rule* 6. *Axiom* 6 corresponds to the finite additivity property of probability. Since the inner probabilistic measure in the model of *PBLf* does not obey the finite additivity property, therefore *Axiom* 6


) |= *ϕ*}.

(1) *S* is a nonempty finite set whose elements are called possible worlds or states;

(2) *π* is a map: *S* × *Prop* → {*true*, *f alse*}, where *Prop* is an atomic formulas set;

(d) If *A*1, *A*<sup>2</sup> ∈ *X* and *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) ≥ *μi*,*s*(*A*1) + *μi*,*s*(*A*2);

the probability space of almost all states is same as the probability space of *s*.

)}, then *μi*,*s*(Λ*i*,*s*) = 1.

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

**3.1 Semantics of** *PBLf*

Here *X* = ℘(*S*).

(f) Let Λ*i*,*<sup>s</sup>* = {*s*�

was used in Definition 2.2.

(*PM*,*s*) |= ¬*ϕ* iff (*PM*,*s*) �|= *ϕ*;

**3.2 Inference system of** *PBLf*

fails with respect to the semantics of *PBLf* .

following conditions:

(a) 0 ≤ *μi*,*s*(*A*) ≤ 1 for all *A* ∈ *X*. (b) *μi*,*s*(∅) = 0 and *μi*,*s*(*S*) = 1.

(c) If *A*1, *A*<sup>2</sup> ∈ *X* and *A*<sup>1</sup> ⊆ *A*2, then *μi*,*s*(*A*1) ≤ *μi*,*s*(*A*2);


reason we call *μi*,*<sup>s</sup>* inner probability measure.

**Definition 3.2** Inner probabilistic semantics of *PBLf*

(*PM*,*s*) |= *ϕ*<sup>1</sup> ∧ *ϕ*<sup>2</sup> iff (*PM*,*s*) |= *ϕ*<sup>1</sup> and (*PM*,*s*) |= *ϕ*2;

(*PM*,*s*) |= *p* iff *π*(*s*, *p*) = *true*, where *p* is an atomic formula;

(*PM*,*s*) |= *Bi*(*a*, *ϕ*) iff *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, where *evPM*(*ϕ*) = {*s*�

(e) If *A*1, *A*<sup>2</sup> ∈ *X*, then *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) ≥ *μi*,*s*(*A*1) + *μi*,*s*(*A*2) − 1;

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; we apply the property *μi*,*s*(Λ*i*,*s*) = 1 (where Λ*i*,*<sup>s</sup>* = {*s*� |*Pi*(*s*) = *Pi*(*s*� )}) in the proof, which also differs from the last property of *PBLω*-probabilistic model (If *A* ∈ *Xi*,*<sup>s</sup>* and *μi*,*s*(*A*) ≥ *a*, then *μi*,*s*({*s*� |*μi*,*s*�(*A*) ≥ *a*}) = 1; if *A* ∈ *Xi*,*<sup>s</sup>* and *μi*,*s*(*A*) < *b*, then *μi*,*s*({*s*� |*μi*,*s*�(*A*) < *b*}) = 1.).

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

*Proo f* . We only discuss *Axiom* 2, *Axiom* 3 and *Axiom* 4 of *PBLf* , other cases can be proved similarly as in Proposition 2.1.

*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* − 1, 0), *ϕ* ∧ *ψ*).

*Axiom* 3: Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), therefore *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*. Let Λ*i*,*<sup>s</sup>* = {*s*� |*Pi*(*s*) = *Pi*(*s*� )}, then Λ*i*,*<sup>s</sup>* ∈ *X* and *μi*,*s*(Λ*i*,*s*) = 1. Let Ξ = {*s*� |*μi*,*s*�(*evPM*(*ϕ*)) ≥ *a*}. Since *s*� ∈ Λ*i*,*<sup>s</sup>* 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 desired.

*Axiom* 4: Suppose (*PM*,*s*) |= ¬*Bi*(*a*, *ϕ*), so *μi*,*s*(*evPM*(*ϕ*)) < *a*. Let Λ*i*,*<sup>s</sup>* = {*s*� |*Pi*(*s*) = *Pi*(*s*� )}, then Λ*i*,*<sup>s</sup>* ∈ *X* and *μi*,*s*(Λ*i*,*s*) = 1. Let Ξ = {*s*� |*μi*,*s*�(*evPM*(*ϕ*)) < *a*} , for *s*� ∈ Λ*i*,*<sup>s</sup>* 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 desired.
