**2.3 Inference system of** *PBL<sup>ω</sup>*

Now we list a number of valid properties of probabilistic belief, which form the inference system of *PBLω*.

*Axioms and in f erence rules o f proposition logic*

*Axiom* 1. *Bi*(0, *ϕ*) (For any proposition *ϕ*, agent *i* believes that the probability of *ϕ* is no less than 0.)

*Axiom* 2. *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) (For any *ϕ* and *ψ*, if agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than *b*, then agent *i* believes that the probability of *ϕ* ∧ *ψ* is no less than *max*(*a* + *b* − 1, 0).)

*Axiom* 3. *Bi*(*a*, *ϕ*) → *Bi*(1, *Bi*(*a*, *ϕ*)) (If agent *i* believes that the probability of *ϕ* is no less than *a*, then agent *i* believes that the probability of his belief being true is no less than 1.)

*Axiom* 4. ¬*Bi*(*a*, *ϕ*) → *Bi*(1, ¬*Bi*(*a*, *ϕ*)) (If agent *i* believes that the probability of *ϕ* is less than *a*, then agent *i* believes that the probability of his belief being true is no less than 1.)

*Axiom* 5. *Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0. (If agent *i* believes that the probability of *ϕ* is no less than *a*, and 1 ≥ *a* ≥ *b* ≥ 0, then agent *i* believes that the probability of *ϕ* is no less than *b*.)

*Axiom* 6. *Bi*(*a* + *b*, *ϕ* ∨ *ψ*) → (*Bi*(*a*, *ϕ*) ∨ *Bi*(*b*, *ψ*)), where 1 ≥ *a* + *b* ≥ 0. (If agent *i* believes that the probability of *ϕ* ∨ *ψ* is no less than *a* + *b*, then agent *i* believes that the probability of *ϕ* is no less than *a* or believes that the probability of *ψ* is no less than *b*.)

*Rule* 1. � *ϕ* ⇒� *Bi*(1, *ϕ*) (If *ϕ* is a tautology proposition, then agent *i* believes that the probability of *ϕ* is no less than 1.)

*Rule* 2. � *ϕ* → *ψ* ⇒� *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*) (If *ϕ* → *ψ* is a tautology proposition, and agent *i* believes that the probability of *ϕ* is no less than *a*, then agent *i* believes that the probability of *ψ* is no less than *a*.)

*Rule* 3. � ¬(*ϕ* ∧ *ψ*) ⇒� ¬(*Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*)) for any *a*, *b* ∈ [0, 1] such that *a* + *b* > 1. (If *ϕ* and *ψ* are incompatible propositions, then it is impossible that agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than *b*, where *a* + *b* > 1.)

*Rule* 4. � ¬(*ϕ* ∧ *ψ*) ⇒� *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*a* + *b*, *ϕ* ∨ *ψ*), where *a* + *b* ≤ 1. (If *ϕ* and *ψ* are incompatible propositions, agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than *b*, where *a* + *b* ≤ 1, then agent *i* believes that the probability of *ϕ* ∨ *ψ* is no less than *a* + *b*.)

6 Will-be-set-by-IN-TECH

In the above example, according to Definition 2.3, we have (*PM*,*s*1) |= *B*1(1/2, *q*), (*PM*,*s*2) |=

In order to characterize the properties of probabilistic belief, we will characterize the formulas that are always true. More formally, given a probabilistic model *PM*, we say that *ϕ* is valid in *PM*, and write *PM* |= *ϕ*, if (*PM*,*s*) |= *ϕ* for every state *s* in *S*, and we say that *ϕ* is satisfiable in *PM* if (*PM*,*s*) |= *ϕ* for some *s* in *S*. We say that *ϕ* is valid, and write |= *ϕ*, if *ϕ* is valid in all probabilistic models, and that *ϕ* is satisfiable if it is satisfiable in some probabilistic model.

Now we list a number of valid properties of probabilistic belief, which form the inference

*Axiom* 1. *Bi*(0, *ϕ*) (For any proposition *ϕ*, agent *i* believes that the probability of *ϕ* is no less

*Axiom* 2. *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) (For any *ϕ* and *ψ*, if agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than

*Axiom* 3. *Bi*(*a*, *ϕ*) → *Bi*(1, *Bi*(*a*, *ϕ*)) (If agent *i* believes that the probability of *ϕ* is no less than

*Axiom* 4. ¬*Bi*(*a*, *ϕ*) → *Bi*(1, ¬*Bi*(*a*, *ϕ*)) (If agent *i* believes that the probability of *ϕ* is less than

*Axiom* 5. *Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0. (If agent *i* believes that the probability of *ϕ* is no less than *a*, and 1 ≥ *a* ≥ *b* ≥ 0, then agent *i* believes that the probability of *ϕ* is no less

*Axiom* 6. *Bi*(*a* + *b*, *ϕ* ∨ *ψ*) → (*Bi*(*a*, *ϕ*) ∨ *Bi*(*b*, *ψ*)), where 1 ≥ *a* + *b* ≥ 0. (If agent *i* believes that the probability of *ϕ* ∨ *ψ* is no less than *a* + *b*, then agent *i* believes that the probability of *ϕ* is

*Rule* 1. � *ϕ* ⇒� *Bi*(1, *ϕ*) (If *ϕ* is a tautology proposition, then agent *i* believes that the

*Rule* 2. � *ϕ* → *ψ* ⇒� *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*) (If *ϕ* → *ψ* is a tautology proposition, and agent *i* believes that the probability of *ϕ* is no less than *a*, then agent *i* believes that the probability of

*Rule* 3. � ¬(*ϕ* ∧ *ψ*) ⇒� ¬(*Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*)) for any *a*, *b* ∈ [0, 1] such that *a* + *b* > 1. (If *ϕ* and *ψ* are incompatible propositions, then it is impossible that agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than *b*, where *a* + *b* > 1.) *Rule* 4. � ¬(*ϕ* ∧ *ψ*) ⇒� *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*a* + *b*, *ϕ* ∨ *ψ*), where *a* + *b* ≤ 1. (If *ϕ* and *ψ* are incompatible propositions, agent *i* believes that the probability of *ϕ* is no less than *a*, and believes that the probability of *ψ* is no less than *b*, where *a* + *b* ≤ 1, then agent *i* believes that

*b*, then agent *i* believes that the probability of *ϕ* ∧ *ψ* is no less than *max*(*a* + *b* − 1, 0).)

*a*, then agent *i* believes that the probability of his belief being true is no less than 1.)

*a*, then agent *i* believes that the probability of his belief being true is no less than 1.)

no less than *a* or believes that the probability of *ψ* is no less than *b*.)

We write Γ |= *ϕ*, if *ϕ* is valid in all probabilistic models in which Γ is satisfiable.

*B*1(0, *p* ∧ *q*), (*PM*,*s*3) |= *B*1(1, *p* ∧ *q*), etc.

*Axioms and in f erence rules o f proposition logic*

**2.3 Inference system of** *PBL<sup>ω</sup>*

probability of *ϕ* is no less than 1.)

the probability of *ϕ* ∨ *ψ* is no less than *a* + *b*.)

*ψ* is no less than *a*.)

system of *PBLω*.

than 0.)

than *b*.)

*Rule* 5. Γ � *Bi*(*an*, *ϕ*) for all *n* ∈ *M* ⇒ Γ � *Bi*(*a*, *ϕ*), where *a* = *supn*∈*M*({*an*}). (If agent *i* believes that the probability of *ϕ* is no less than *an*, where *n* is any element in the index set *M*, then agent *i* believes that the probability of *ϕ* is no less than *a*, where *a* = *supn*∈*M*({*an*}).)

*Rule* 6. Given a set of formulas Σ, Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*})) � *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1] ⇒ Γ � *ψ*. (If *ψ* can be proved from Γ with any possible probabilistic belief of agent *i* for Σ, then *ψ* can be merely proved from Γ.)

Remark: In *Rule* 5, the index set *M* may be an infinite set, therefore we call *Rule* 5 an infinite inference rule. For example, let Γ = {*Bi*(1/2, *ϕ*), *Bi*(2/3, *ϕ*), ..., *Bi*(*n*/*n* + 1, *ϕ*), ...}, we have Γ � *Bi*(*n*/*n* + 1, *ϕ*) for all *n* ∈ *M* = {1, 2, ..., *k*, ...}, by *Rule* 5, we get Γ � *Bi*(1, *ϕ*) since 1 = *supn*∈*M*({*n*/*n* + 1}).

In *Rule* 6, {*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*} means that agent *i* believes the probability of *ϕ* is exactly *ai*,*ϕ*. Therefore Γ ∪ {*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*} � *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1] means that under any possible probabilistic belief of agent *i* for *ϕ*, *ψ* can be proved from Γ. Intuitively, in this case, the correctness of *ψ* is independent of the exact probability of *ϕ* that agent *i* believes, so we can get *ψ* from Γ. In *Rule* 6, formula *ϕ* here is generalized to arbitrary set Σ of formulas. Since the premises of *Rule* 6 are infinite, it is also an infinite inference rule.

We will show that in a precise sense these properties completely characterize the formulas of *PBL<sup>ω</sup>* that are valid with respect to probabilistic model. To do so, we have to consider the notion of provability. Inference system *PBL<sup>ω</sup>* consists of a collection of axioms and inference rules. We are actually interested in (substitution) instances of axioms and inference rules (so we in fact think of axioms and inference rules as schemes). For example, the formula *Bi*(0.7, *ϕ*) ∧ *Bi*(0.8, *ψ*) → *Bi*(0.5, *ϕ* ∧ *ψ*) is an instances of the propositional tautology *Bi*(*a*, *ϕ*)∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* −1, 0), *ϕ*∧*ψ*), obtained by substituting *Bi*(0.7, *ϕ*), *Bi*(0.8, *ψ*) and *Bi*(0.5, *ϕ* ∧ *ψ*) for *Bi*(*a*, *ϕ*), *Bi*(*b*, *ψ*) and *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) respectively. A proof in *PBL<sup>ω</sup>* consists of a sequence of formulas, each of which is either an instance of an axiom in *PBL<sup>ω</sup>* or follows from an application of an inference rule. (If "*ϕ*1, ..., *ϕ<sup>n</sup>* infer *ψ*" is an instance of an inference rule, and if the formulas *ϕ*1, ..., *ϕ<sup>n</sup>* have appeared earlier in the proof, then we say that *ψ* follows from an application of an inference rule.) A proof is said to be from Γ to *ϕ* if the premise is Γ and the last formula is *ϕ* in the proof. We say *ϕ* is provable from Γ in *PBLω*, and write Γ �*PBL<sup>ω</sup> ϕ*, if there is a proof from Γ to *ϕ* in *PBLω*.

#### **2.4 Soundness of** *PBL<sup>ω</sup>*

We will prove that *PBL<sup>ω</sup>* characterizes the set of formulas that are valid with respect to probabilistic model. Inference system of *PBL<sup>ω</sup>* is said to be sound with respect to probabilistic models if every formula provable in *PBL<sup>ω</sup>* is valid with respect to probabilistic models. The system *PBL<sup>ω</sup>* is complete with respect to probabilistic models if every formula valid with respect to probabilistic models is provable in *PBLω*. We think of *PBL<sup>ω</sup>* as characterizing probabilistic models if it provides a sound and complete axiomatization of that class; notationally, this amounts to saying that for all formulas set Γ and all formula *ϕ*, we have Γ �*PBL<sup>ω</sup> ϕ* if and only if Γ |=*PBL<sup>ω</sup> ϕ*. The following soundness and completeness provide a tight connection between the syntactic notion of provability and the semantic notion of validity.

Firstly, we need the following obvious lemmas.

*evPM*(*ψ*)) = *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) and *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≤ 1, therefore *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≤ 1. Assume (*PM*,*s*) |= (*Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*)) where *a* + *b* > 1,

Probabilistic Belief Logics for Uncertain Agents 25

*Rule* 4: Suppose |= ¬(*ϕ* ∧ *ψ*) and for possible world *s*, (*PM*,*s*) |= *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*), so *evPM*(*ϕ*) ∩ *evPM*(*ψ*) = ∅, *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, and *μi*,*s*(*evPM*(*ψ*)) ≥ *b*. By the property of *PBLω*-probability space and Lemma 2.3, for any possible world *s*, we get *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) = *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)). Hence, *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≥ *a* + *b*

*Rule* 5: Suppose Γ |= *Bi*(*an*, *ϕ*) for all *n* ∈ *M*, therefore for every *s*, if (*PM*,*s*) |= Γ, then (*PM*,*s*) |= *Bi*(*an*, *ϕ*) for all *n* ∈ *M*, so *μi*,*s*(*evPM*(*ϕ*)) ≥ *an* for all *n* ∈ *M*. We get *μi*,*s*(*evPM*(*ϕ*)) ≥ *supn*∈*M*({*an*}). Therefore, (*PM*,*s*) |= *Bi*(*a*, *ϕ*) and *a* = *supn*∈*M*({*an*}), we

*Rule* 6: Suppose Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*})) |= *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1], let (*PM*,*s*) |= Γ and *ci*,*<sup>ϕ</sup>* = *μi*,*s*(*evPM*(*ϕ*)), it is clear that *ci*,*<sup>ϕ</sup>* ∈ [0, 1] and (*PM*,*s*) |= Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ci*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ci*,*ϕ*)). Since Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*})) |= *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1], we have (*PM*,*s*) |= *ψ*,

We shall show that the inference system of *PBL<sup>ω</sup>* provides a complete axiomatization for probabilistic belief with respect to a probabilistic model. To achieve this aim, it suffices to prove that every *PBLω*-consistent set is satisfiable with respect to a probabilistic model. We prove this by using a general technique that works for a wide variety of probabilistic modal logic. We construct a special structure *PM* called a canonical structure for *PBLω*. *PM* has a state *sV* corresponding to every maximal *PBLω*-consistent set *V* and the following property

We need some definitions before giving the proof of the completeness. Given an inference system of *PBLω*, we say a set of formulas Γ is a consistent set with respect to *LPBL<sup>ω</sup>* exactly if false is not provable from Γ. A set of formulas Γ is a maximal consistent set with respect to *<sup>L</sup>PBL<sup>ω</sup>* if (1) it is *PBLω*-consistent, and (2) for all *<sup>ϕ</sup>* in *<sup>L</sup>PBL<sup>ω</sup>* but not in <sup>Γ</sup>, the set <sup>Γ</sup> ∪ {*ϕ*} is not

(2) *Pi* maps every element of *S* to a probability space: *Pi*(Γ)=(*S*, *Xi*,Γ, *μi*,Γ), where *Xi*,<sup>Γ</sup> =

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


}; *μi*,<sup>Γ</sup> is a probability assignment:

**Definition 2.7** The probabilistic model *PM* with respect to *PBL<sup>ω</sup>* is (*S*, *P*1, ..., *Pn*, *π*).

(1) *S* = {Γ|Γ is a maximal consistent set with respect to *PBLω*};

*Proo f* . Since the rules and axioms of *PBL<sup>ω</sup>* are consistent, *S* is nonempty.

{*X*(*ϕ*)|*ϕ* is a formula of *PBLω*}, here *X*(*ϕ*) = {Γ�

*Xi*,<sup>Γ</sup> → [0, 1], and *μi*,Γ(*X*(*ϕ*)) = *sup*({*a*|*Bi*(*a*, *ϕ*) ∈ Γ});

**Lemma 2.5** *Xi*,<sup>Γ</sup> satisfies the conditions of Definition 2.2.

then *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, *μi*,*s*(*evPM*(*ψ*)) ≥ *b*, but *a* + *b* > 1, it is a contradiction.

and *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≥ *a* + *b*, which means (*PM*,*s*) |= *Bi*(*a* + *b*, *ϕ* ∨ *ψ*).

get Γ |= *Bi*(*a*, *ϕ*) and *a* = *supn*∈*M*({*an*}) as desired.

therefore Γ |= *ψ*.

*PBLω*-consistent.

**2.5 Completeness of** *PBL<sup>ω</sup>*

holds: (*PM*,*sV*) |= *ϕ* iff *ϕ* ∈ *V*.

**Lemma 2.4** *S* is a nonempty set.

**Lemma 2.1** *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2), here *A*<sup>1</sup> and *A*<sup>2</sup> are any disjoint members of *Xi*,*<sup>s</sup>* ⇒ for any members *A*<sup>1</sup> and *A*<sup>2</sup> of *Xi*,*s*, *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2).

*Proo f* . *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*<sup>1</sup> ∪ (*A*<sup>2</sup> − *A*1)) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) + *μi*,*s*(*A*<sup>2</sup> − *A*1) = *μi*,*s*(*A*1) + *μi*,*s*((*A*<sup>1</sup> ∩ *A*2) ∪ (*A*<sup>2</sup> − *A*1)) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2).

**Lemma 2.2** *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) ≥ *μi*,*s*(*A*1) + *μi*,*s*(*A*2) − 1.

*Proo f* . It follows from Lemma 2.1 immediately.

**Lemma 2.3** *μi*,*s*(*A*1) + *μi*,*s*(*A*2) ≥ *μi*,*s*(*A*<sup>1</sup> ∪ *A*2). If *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μi*,*s*(*A*1) + *μi*,*s*(*A*2) = *μi*,*s*(*A*<sup>1</sup> ∪ *A*2).

*Proo f* . It follows from Lemma 2.1.

Now, we can prove the following proposition:

**Proposition 2.1** (Soundness of *PBLω*) If Γ �*PBL<sup>ω</sup> ϕ*, then Γ |=*PBL<sup>ω</sup> ϕ*.

*Proo f* . We show each axiom and each rule of *PBL<sup>ω</sup>* is sound, respectively.

*Axiom* 1: By the definition of *PBLω*-probability measure, for any *s*, if *A* ∈ *Xi*,*<sup>s</sup>* then *μi*,*s*(*A*) ≥ 0. Since by the definition of *Xi*,*s*, for any *ϕ*, *evPM*(*ϕ*) = {*s*� |(*PM*,*s*� ) |= *ϕ*} ∈ *Xi*,*s*, we have *μi*,*s*(*evPM*(*ϕ*)) ≥ 0, therefore *Bi*(0, *ϕ*) holds.

*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 *PBL<sup>ω</sup>* - probability measure, by Lemma 2.2, 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*) <sup>|</sup><sup>=</sup> *Bi*(*a*, *<sup>ϕ</sup>*), so *<sup>μ</sup>i*,*s*(*evPM*(*ϕ*)) <sup>≥</sup> *<sup>a</sup>*. Let <sup>Λ</sup>*<sup>a</sup> <sup>i</sup>* (*ϕ*) = {*s*� <sup>|</sup>*μi*,*s*�(*evPM*(*ϕ*)) <sup>≥</sup> *<sup>a</sup>*}, by the definition of *<sup>μ</sup>i*,*s*, we get *<sup>μ</sup>i*,*s*(Λ*<sup>a</sup> <sup>i</sup>* (*ϕ*)) = 1. Assume *<sup>s</sup>*� <sup>∈</sup> <sup>Λ</sup>*<sup>a</sup> <sup>i</sup>* (*ϕ*), then *<sup>s</sup>*� <sup>∈</sup> *evPM*(*Bi*(*a*, *<sup>ϕ</sup>*)), hence <sup>Λ</sup>*<sup>a</sup> <sup>i</sup>* (*ϕ*) ⊆ *evPM*(*Bi*(*a*, *ϕ*)), so *μi*,*s*(*evPM*(*Bi*(*a*, *ϕ*))) = 1, which implies (*PM*,*s*) |= *Bi*(1, *Bi*(*a*, *ϕ*)).

*Axiom* 4: Suppose (*PM*,*s*) <sup>|</sup><sup>=</sup> <sup>¬</sup>*Bi*(*a*, *<sup>ϕ</sup>*), so *<sup>μ</sup>i*,*s*(*evPM*(*ϕ*)) <sup>&</sup>lt; *<sup>a</sup>*. Let <sup>Θ</sup>*<sup>a</sup> <sup>i</sup>* (*ϕ*) = {*s*� <sup>|</sup>*μi*,*s*�(*evPM*(*ϕ*)) <sup>&</sup>lt; *<sup>a</sup>*}, by the definition of *<sup>μ</sup>i*,*s*, we get *<sup>μ</sup>i*,*s*(Θ*<sup>a</sup> <sup>i</sup>* (*ϕ*)) = 1. Assume *<sup>s</sup>*� <sup>∈</sup> <sup>Θ</sup>*<sup>a</sup> <sup>i</sup>* (*ϕ*), then *s*� ∈ *evPM*(¬*Bi*(*a*, *ϕ*)), hence *μi*,*s*(*evPM*(¬*Bi*(*a*, *ϕ*))) = 1, which implies (*PM*,*s*) |= *Bi*(1, ¬*Bi*(*a*, *ϕ*)).

*Axiom* 5: Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), so *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*. If 1 ≥ *a* ≥ *b* ≥ 0, then *μi*,*s*(*evPM*(*ϕ*)) ≥ *b*, so (*PM*,*s*) |= *Bi*(*b*, *ϕ*), therefore *Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*) holds.

*Axiom* 6: Suppose (*PM*,*s*) |= *Bi*(*a* + *b*, *ϕ* ∨ *ψ*), then *μi*,*s*(*evPM*(*ϕ* ∨ *ψ*)) ≥ *a* + *b*. Since *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≥ *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≥ *a* + *b*, we have *μi*,*s*(*evPM*(*ϕ*)) ≥ *a* or *μi*,*s*(*evPM*(*ψ*)) ≥ *b*. Hence (*PM*,*s*) |= *Bi*(*a*, *ϕ*) ∨ *Bi*(*b*, *ψ*).

*Rule* 1: Since |= *ϕ*, so for any possible world *s*, *μi*,*s*(*evPM*(*ϕ*)) ≥ 1, therefore |= *Bi*(1, *ϕ*) holds.

*Rule* 2: Since |= *ϕ* → *ψ*, so *evPM*(*ϕ*) ⊆ *evPM*(*ψ*). Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), therefore *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, by the property of *PBLω*-probability space, we get *μi*,*s*(*evPM*(*ϕ*)) ≤ *μi*,*s*(*evPM*(*ψ*)). So *μi*,*s*(*evPM*(*ψ*)) ≥ *a*. Therefore (*PM*,*s*)| = *Bi*(*a*, *ψ*), and *Rule* 2 of *PBL<sup>ω</sup>* holds.

*Rule* 3: Suppose |= ¬(*ϕ* ∧ *ψ*), so *evPM*(*ϕ*) ∩ *evPM*(*ψ*) = ∅. By the property of *PBLω*-probability space and Lemma 2.3, for any possible world *s*, we get *μi*,*s*(*evPM*(*ϕ*) ∪ 8 Will-be-set-by-IN-TECH

**Lemma 2.1** *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2), here *A*<sup>1</sup> and *A*<sup>2</sup> are any disjoint members of *Xi*,*<sup>s</sup>* ⇒ for any members *A*<sup>1</sup> and *A*<sup>2</sup> of *Xi*,*s*, *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2). *Proo f* . *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*<sup>1</sup> ∪ (*A*<sup>2</sup> − *A*1)) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) + *μi*,*s*(*A*<sup>2</sup> − *A*1) = *μi*,*s*(*A*1) + *μi*,*s*((*A*<sup>1</sup> ∩ *A*2) ∪ (*A*<sup>2</sup> − *A*1)) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2).

**Lemma 2.3** *μi*,*s*(*A*1) + *μi*,*s*(*A*2) ≥ *μi*,*s*(*A*<sup>1</sup> ∪ *A*2). If *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μi*,*s*(*A*1) + *μi*,*s*(*A*2) =

*Axiom* 1: By the definition of *PBLω*-probability measure, for any *s*, if *A* ∈ *Xi*,*<sup>s</sup>* then *μi*,*s*(*A*) ≥ 0.

*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 *PBL<sup>ω</sup>* - probability measure, by Lemma 2.2, 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

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

*Axiom* 4: Suppose (*PM*,*s*) <sup>|</sup><sup>=</sup> <sup>¬</sup>*Bi*(*a*, *<sup>ϕ</sup>*), so *<sup>μ</sup>i*,*s*(*evPM*(*ϕ*)) <sup>&</sup>lt; *<sup>a</sup>*. Let <sup>Θ</sup>*<sup>a</sup>*

*μi*,*s*(*evPM*(*ϕ*)) ≥ *b*, so (*PM*,*s*) |= *Bi*(*b*, *ϕ*), therefore *Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*) holds.

then *s*� ∈ *evPM*(¬*Bi*(*a*, *ϕ*)), hence *μi*,*s*(*evPM*(¬*Bi*(*a*, *ϕ*))) = 1, which implies (*PM*,*s*) |=

*Axiom* 5: Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), so *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*. If 1 ≥ *a* ≥ *b* ≥ 0, then

*Axiom* 6: Suppose (*PM*,*s*) |= *Bi*(*a* + *b*, *ϕ* ∨ *ψ*), then *μi*,*s*(*evPM*(*ϕ* ∨ *ψ*)) ≥ *a* + *b*. Since *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≥ *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≥ *a* + *b*, we have *μi*,*s*(*evPM*(*ϕ*)) ≥

*Rule* 1: Since |= *ϕ*, so for any possible world *s*, *μi*,*s*(*evPM*(*ϕ*)) ≥ 1, therefore |= *Bi*(1, *ϕ*) holds. *Rule* 2: Since |= *ϕ* → *ψ*, so *evPM*(*ϕ*) ⊆ *evPM*(*ψ*). Suppose (*PM*,*s*) |= *Bi*(*a*, *ϕ*), therefore *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, by the property of *PBLω*-probability space, we get *μi*,*s*(*evPM*(*ϕ*)) ≤ *μi*,*s*(*evPM*(*ψ*)). So *μi*,*s*(*evPM*(*ψ*)) ≥ *a*. Therefore (*PM*,*s*)| = *Bi*(*a*, *ψ*), and *Rule* 2 of *PBL<sup>ω</sup>*

*Rule* 3: Suppose |= ¬(*ϕ* ∧ *ψ*), so *evPM*(*ϕ*) ∩ *evPM*(*ψ*) = ∅. By the property of *PBLω*-probability space and Lemma 2.3, for any possible world *s*, we get *μi*,*s*(*evPM*(*ϕ*) ∪


*<sup>i</sup>* (*ϕ*) ⊆ *evPM*(*Bi*(*a*, *ϕ*)), so *μi*,*s*(*evPM*(*Bi*(*a*, *ϕ*))) = 1, which

) |= *ϕ*} ∈ *Xi*,*s*, we have

*<sup>i</sup>* (*ϕ*)) = 1. Assume *<sup>s</sup>*� <sup>∈</sup> <sup>Λ</sup>*<sup>a</sup>*

*<sup>i</sup>* (*ϕ*)) = 1. Assume *<sup>s</sup>*� <sup>∈</sup> <sup>Θ</sup>*<sup>a</sup>*

*<sup>i</sup>* (*ϕ*) =

*<sup>i</sup>* (*ϕ*) =

*<sup>i</sup>* (*ϕ*),

*<sup>i</sup>* (*ϕ*),

**Lemma 2.2** *μi*,*s*(*A*<sup>1</sup> ∩ *A*2) ≥ *μi*,*s*(*A*1) + *μi*,*s*(*A*2) − 1. *Proo f* . It follows from Lemma 2.1 immediately.

Now, we can prove the following proposition:

*μi*,*s*(*evPM*(*ϕ*)) ≥ 0, therefore *Bi*(0, *ϕ*) holds.

(*PM*,*s*) |= *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*).

then *<sup>s</sup>*� <sup>∈</sup> *evPM*(*Bi*(*a*, *<sup>ϕ</sup>*)), hence <sup>Λ</sup>*<sup>a</sup>*

implies (*PM*,*s*) |= *Bi*(1, *Bi*(*a*, *ϕ*)).

**Proposition 2.1** (Soundness of *PBLω*) If Γ �*PBL<sup>ω</sup> ϕ*, then Γ |=*PBL<sup>ω</sup> ϕ*.

Since by the definition of *Xi*,*s*, for any *ϕ*, *evPM*(*ϕ*) = {*s*�

<sup>|</sup>*μi*,*s*�(*evPM*(*ϕ*)) <sup>≥</sup> *<sup>a</sup>*}, by the definition of *<sup>μ</sup>i*,*s*, we get *<sup>μ</sup>i*,*s*(Λ*<sup>a</sup>*

<sup>|</sup>*μi*,*s*�(*evPM*(*ϕ*)) <sup>&</sup>lt; *<sup>a</sup>*}, by the definition of *<sup>μ</sup>i*,*s*, we get *<sup>μ</sup>i*,*s*(Θ*<sup>a</sup>*

*a* or *μi*,*s*(*evPM*(*ψ*)) ≥ *b*. Hence (*PM*,*s*) |= *Bi*(*a*, *ϕ*) ∨ *Bi*(*b*, *ψ*).

*Proo f* . We show each axiom and each rule of *PBL<sup>ω</sup>* is sound, respectively.

*μi*,*s*(*A*<sup>1</sup> ∪ *A*2).

{*s*�

{*s*�

holds.

*Bi*(1, ¬*Bi*(*a*, *ϕ*)).

*Proo f* . It follows from Lemma 2.1.

*evPM*(*ψ*)) = *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) and *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≤ 1, therefore *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≤ 1. Assume (*PM*,*s*) |= (*Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*)) where *a* + *b* > 1, then *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, *μi*,*s*(*evPM*(*ψ*)) ≥ *b*, but *a* + *b* > 1, it is a contradiction.

*Rule* 4: Suppose |= ¬(*ϕ* ∧ *ψ*) and for possible world *s*, (*PM*,*s*) |= *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*), so *evPM*(*ϕ*) ∩ *evPM*(*ψ*) = ∅, *μi*,*s*(*evPM*(*ϕ*)) ≥ *a*, and *μi*,*s*(*evPM*(*ψ*)) ≥ *b*. By the property of *PBLω*-probability space and Lemma 2.3, for any possible world *s*, we get *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) = *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)). Hence, *μi*,*s*(*evPM*(*ϕ*)) + *μi*,*s*(*evPM*(*ψ*)) ≥ *a* + *b* and *μi*,*s*(*evPM*(*ϕ*) ∪ *evPM*(*ψ*)) ≥ *a* + *b*, which means (*PM*,*s*) |= *Bi*(*a* + *b*, *ϕ* ∨ *ψ*).

*Rule* 5: Suppose Γ |= *Bi*(*an*, *ϕ*) for all *n* ∈ *M*, therefore for every *s*, if (*PM*,*s*) |= Γ, then (*PM*,*s*) |= *Bi*(*an*, *ϕ*) for all *n* ∈ *M*, so *μi*,*s*(*evPM*(*ϕ*)) ≥ *an* for all *n* ∈ *M*. We get *μi*,*s*(*evPM*(*ϕ*)) ≥ *supn*∈*M*({*an*}). Therefore, (*PM*,*s*) |= *Bi*(*a*, *ϕ*) and *a* = *supn*∈*M*({*an*}), we get Γ |= *Bi*(*a*, *ϕ*) and *a* = *supn*∈*M*({*an*}) as desired.

*Rule* 6: Suppose Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*})) |= *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1], let (*PM*,*s*) |= Γ and *ci*,*<sup>ϕ</sup>* = *μi*,*s*(*evPM*(*ϕ*)), it is clear that *ci*,*<sup>ϕ</sup>* ∈ [0, 1] and (*PM*,*s*) |= Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ci*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ci*,*ϕ*)). Since Γ ∪ (∪*ϕ*∈Σ({*Bi*(*a*, *ϕ*)|0 ≤ *a* ≤ *ai*,*ϕ*} ∪ {¬*Bi*(*b*, *ϕ*)|1 ≥ *b* > *ai*,*ϕ*})) |= *ψ* for any *ai*,*<sup>ϕ</sup>* ∈ [0, 1], we have (*PM*,*s*) |= *ψ*, therefore Γ |= *ψ*.
