**2.** *PBL<sup>ω</sup>* **and its probabilistic semantics**

In this section, we first review the standard belief logic system and the standard Kripke semantics. Some examples are given to illustrate why it is necessary to extend belief to probabilistic belief. Then we introduce a probabilistic belief logic *PBLω*.

In belief logic, the formula *Biϕ* says that agent *i* believes *ϕ*. Consider a system with *n* agents, say 1, ..., *n*, and we have a nonempty set Φ of primitive propositions about which we wish to reason. We construct formulas by closing off Φ under conjunction, negation and modal operators *Bi*, for *i* = 1, ..., *n* (where *Biϕ* is read as "agent *i* believes *ϕ*").

The semantics to these formulas is given by means of Kripke structure [19]. A Kripke structure for belief (for *n* agents) is a tuple (*S*, *π*, *R*1, ..., *Rn*), where *S* is a set of states, *π*(*s*) is a truth assignment to the primitive propositions of Φ for each state *s* ∈ *S*, and *Ri* is an accessible relation on *S*, which satisfies the following conditions: Euclideanness (∀*s*∀*s*� ∀*s*��(*sRis*� ∧ *sRis*�� → *s*� *Ris*��)), transitivity (∀*s*∀*s*� ∀*s*��(*sRis*� ∧ *s*� *Ris*�� → *sRis*��)) and definality (∀*s*∃*s*� (*sRis*� )).

We now assign truth values to formulas at each state in the structure. We write (*M*,*s*)|= *ϕ* if the formula *ϕ* is true at state *s* in Kripke structure *M*.

$$(M, \mathbf{s}) \vdash p \text{ (for } p \in \Phi\text{) iff } \pi(\mathbf{s})(p) = true$$

$$(M\_\prime s) \Vdash \neg \varphi \text{ iff } (M\_\prime s) \Vdash \varphi$$

(*M*,*s*) |= *ϕ* ∧ *ψ* iff (*M*,*s*) |= *ϕ* and (*M*,*s*) |= *ψ*

$$|(M, \mathbf{s})| = B\_l \boldsymbol{\varrho} \text{ iff } (M, t) \mid = \boldsymbol{\varrho} \text{ for all } t \in \mathbb{R}\_l \\
(\mathbf{s}) \text{ with } \mathbf{R}\_l(\mathbf{s}) = \{\mathbf{s}' | (\mathbf{s}, \mathbf{s}') \in \mathbb{R}\_l\}$$

The last clause in this definition captures the intuition that agent *i* believes *ϕ* in world (*M*,*s*) exactly if *ϕ* is true in all worlds that *i* considers possible.

It is well known that the following set of axioms and inference rules provides a sound and complete axiomatization for the logic of belief with respect to the class of Kripke structures for belief:

*All instances o f propositional tautologies and rules*.

$$\begin{aligned} (B\_i \,\varphi \wedge B\_i (\,\varphi \to \,\psi)) &\to B\_i \psi \\\\ B\_i \,\varphi &\to \neg B\_i \neg \varphi \\\\ B\_i \,\varphi &\to B\_i B\_i \,\varphi \end{aligned}$$

(a) If *p* is an atomic formula, then *evPM*(*p*) = {*s*�

(b) If *A* ∈ *Xi*,*s*, then *S* − *A* ∈ *Xi*,*s*;

the following conditions:

(b) *μi*,*s*(*S*) = 1;

members of *Xi*,*s*;

then *μi*,*s*({*s*�

*b*} = *S* − {*s*�

by *s*.

model *PM*.

(a) *μi*,*s*(*A*) ≥ 0 for all *A* ∈ *Xi*,*s*;

(c) If *A*1, *A*<sup>2</sup> ∈ *Xi*,*s*, then *A*<sup>1</sup> ∩ *A*<sup>2</sup> ∈ *Xi*,*s*; (d) If *A* ∈ *Xi*,*<sup>s</sup>* and *a* ∈ [0, 1], then {*s*�

(d) If *A* ∈ *Xi*,*<sup>s</sup>* and *μi*,*s*(*A*) ≥ *a*, then *μi*,*s*({*s*�


**Definition 2.3** Probabilistic semantics of *PBL<sup>ω</sup>*

(*PM*,*s*) |= *ϕ* ∧ *ψ* iff (*PM*,*s*) |= *ϕ* and (*PM*,*s*) |= *ψ*;

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

(*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*�

Notice that from the definition of *Xi*,*s*, we have {*s*�



Probabilistic Belief Logics for Uncertain Agents 21

(c) (finite additivity) *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2), where *A*<sup>1</sup> and *A*<sup>2</sup> are disjoint

probabilities on events, given that the state is *s*. *W* is the sample space, which is the set of states that agent *i* considers possible. *Xi*,*<sup>s</sup>* is the set of measurable sets. The measure *μi*,*<sup>s</sup>* does

As an example, we consider a *PBLω*-model such that *PM* = (*S*, *π*, *P*1). Here *S* = {*s*1,*s*2,*s*3}; *π*(*s*1, *p*) = *f alse*, *π*(*s*2, *p*) = *f alse*, *π*(*s*3, *p*) = *true*, *π*(*s*1, *q*) = *true*, *π*(*s*2, *q*) = *true*, *π*(*s*3, *q*) = *true*; *P*<sup>1</sup> is defined as follows: for every *s* ∈ *S*, *P*1(*s*)=(*S*, *X*1,*s*, *μ*1,*s*), where *X*1,*s*<sup>1</sup> = *X*1,*s*<sup>2</sup> = ℘(*S*), *X*1,*s*<sup>3</sup> = {∅, {*s*1,*s*2}, {*s*3}, *S*}, *μ*1,*s*<sup>1</sup> (∅) = *μ*1,*s*<sup>2</sup> (∅) = *μ*1,*s*<sup>3</sup> (∅) = 0, *μ*1,*s*<sup>1</sup> ({*s*1}) = *μ*1,*s*<sup>2</sup> ({*s*1}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*2}) = *μ*1,*s*<sup>2</sup> ({*s*2}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*3}) = 0, *μ*1,*s*<sup>3</sup> ({*s*3}) = 1, *μ*1,*s*<sup>1</sup> ({*s*1,*s*2}) = *μ*1,*s*<sup>2</sup> ({*s*1,*s*2}) = 1, *μ*1,*s*<sup>3</sup> ({*s*1,*s*2}) = 0, *μ*1,*s*<sup>1</sup> ({*s*1,*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*1,*s*3}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*2,*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*2,*s*3}) = 1/2, *μ*1,*s*<sup>1</sup> (*S*) = *μ*1,*s*<sup>2</sup> (*S*) = *μ*1,*s*<sup>3</sup> (*S*) = 1. It is easy to check that the above model satisfies the conditions in Definition 2.2. In this model, it is clear that the set of measurable sets *X*1,*<sup>s</sup>* and probability measure *μ*1,*<sup>s</sup>* varies with *s*, and consequently the probability space also varies with *s*, hence we index probability space

We now define what it means for a formula to be true at a given world *s* in a probabilistic

The intuitive meaning of the semantics of *Bi*(*a*, *ϕ*) is that agent *i* believes that the probability of *ϕ* is at least *a* in world (*PM*,*s*) if the measure of possible worlds satisfying *ϕ* is at least *a*.

not assign a probability to all subsets of *S* but only to the measurable sets.


, *p*) = *true*} ∈ *Xi*,*s*;





) |= *ϕ*}.

¬*Biϕ* → *Bi*¬*Biϕ* � *ϕ* ⇒ � *Biϕ*

There are examples of probabilistic belief in daily life. For example, one may believe that the probability of "it will rain tomorrow" is less than 0.4; in a football game, one may believe that the probability of "team *A* will win" is no less than 0.7 and so on. In distribute systems, there may be the cases that "agent *i* believes that the probability of 'agent *j* believes that the probability of *ϕ* is at least *a*' is no less than *b*". Suppose there are two persons communicating by email, agent *A* sends an email to agent *B*. Since the email may be lost in network, *A* does not know whether *B* has received the email. Therefore *A* may believe that the probability of "*B* has received my email" is less than 0.99, or may believe that the probability of "*B* has received my email" is at least 0.8, and so on. On the other hand, *B* may believe that the probability of "*A* believes that the probability of '*B* has received my email' is at least 0.9" is less than 0.8. In order to reply to *A*, *B* sends an acknowledgement email to *A*, *A* receives the email, and sends another acknowledgement email to *B*, now *B* believes that the probability of "*A* believes that the probability of '*B* has received my first email' is equal to 1" is equal to 1. In order to represent and reason with probabilistic belief, it is necessary to extend belief logic to probabilistic belief logic. In following, we propose a probabilistic belief logic *PBLω*, the basic formula in *PBL<sup>ω</sup>* is *Bi*(*a*, *ϕ*), which says agent *i* believes that the probability of *ϕ* is no less than *a*.

### **2.1 Language of** *PBL<sup>ω</sup>*

Throughout this chapter, we let *LPBL<sup>ω</sup>* be a language which is just the set of formulas of interest to us.

**Definition 2.1** The set of formulas in *PBLω*, called *LPBL<sup>ω</sup>* , is given by the following rules:

(1) If *<sup>ϕ</sup>* <sup>∈</sup>Atomic formulas set *Prop*, then *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(2) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , then <sup>¬</sup>*<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(3) If *<sup>ϕ</sup>*1,*ϕ*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , then *<sup>ϕ</sup>*<sup>1</sup> <sup>∧</sup> *<sup>ϕ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(4) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* and *<sup>a</sup>* <sup>∈</sup>[0,1], then *Bi*(*a*, *<sup>ϕ</sup>*) <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , where *<sup>i</sup>* belongs to the set of agents {1, ..., *n*}. Intuitively, *Bi*(*a*, *ϕ*) means that agent *i* believes the probability of *ϕ* is no less than *a*.

#### **2.2 Semantics of** *PBL<sup>ω</sup>*

We will describe the semantics of *PBLω*, i.e., a formal model that we can use to determine whether a given formula is true or false. We call the formal model probabilistic model, roughly speaking, at each state, each agent has a probability on a certain set of states.

**Definition 2.2** A probabilistic model *PM* of *PBL<sup>ω</sup>* is a tuple (*S*, *π*, *P*1, ..., *Pn*), where

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

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

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

Where *Xi*,*<sup>s</sup>* ∈ ℘(*S*), which satisfies the following conditions:

(a) If *p* is an atomic formula, then *evPM*(*p*) = {*s*� |*π*(*s*� , *p*) = *true*} ∈ *Xi*,*s*;


*μi*,*<sup>s</sup>* is a *PBLω*- finite additivity probability measure assigned to the set *Xi*,*s*, i.e., *μi*,*<sup>s</sup>* satisfies the following conditions:

(a) *μi*,*s*(*A*) ≥ 0 for all *A* ∈ *Xi*,*s*;

$$\text{(b)}\,\mu\_{i,s}(S) = 1;$$

4 Will-be-set-by-IN-TECH

There are examples of probabilistic belief in daily life. For example, one may believe that the probability of "it will rain tomorrow" is less than 0.4; in a football game, one may believe that the probability of "team *A* will win" is no less than 0.7 and so on. In distribute systems, there may be the cases that "agent *i* believes that the probability of 'agent *j* believes that the probability of *ϕ* is at least *a*' is no less than *b*". Suppose there are two persons communicating by email, agent *A* sends an email to agent *B*. Since the email may be lost in network, *A* does not know whether *B* has received the email. Therefore *A* may believe that the probability of "*B* has received my email" is less than 0.99, or may believe that the probability of "*B* has received my email" is at least 0.8, and so on. On the other hand, *B* may believe that the probability of "*A* believes that the probability of '*B* has received my email' is at least 0.9" is less than 0.8. In order to reply to *A*, *B* sends an acknowledgement email to *A*, *A* receives the email, and sends another acknowledgement email to *B*, now *B* believes that the probability of "*A* believes that the probability of '*B* has received my first email' is equal to 1" is equal to 1. In order to represent and reason with probabilistic belief, it is necessary to extend belief logic to probabilistic belief logic. In following, we propose a probabilistic belief logic *PBLω*, the basic formula in *PBL<sup>ω</sup>* is *Bi*(*a*, *ϕ*), which says agent *i* believes that the probability of *ϕ* is no less

Throughout this chapter, we let *LPBL<sup>ω</sup>* be a language which is just the set of formulas of interest

(4) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* and *<sup>a</sup>* <sup>∈</sup>[0,1], then *Bi*(*a*, *<sup>ϕ</sup>*) <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , where *<sup>i</sup>* belongs to the set of agents {1, ..., *n*}. Intuitively, *Bi*(*a*, *ϕ*) means that agent *i* believes the probability of *ϕ* is no less than *a*.

We will describe the semantics of *PBLω*, i.e., a formal model that we can use to determine whether a given formula is true or false. We call the formal model probabilistic model, roughly

(3) *Pi* is a map, it maps every possible world *s* to a *PBLω*-probability space *Pi*(*s*) =

speaking, at each state, each agent has a probability on a certain set of states.

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

Where *Xi*,*<sup>s</sup>* ∈ ℘(*S*), which satisfies the following conditions:

**Definition 2.2** A probabilistic model *PM* of *PBL<sup>ω</sup>* is a tuple (*S*, *π*, *P*1, ..., *Pn*), where

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

**Definition 2.1** The set of formulas in *PBLω*, called *LPBL<sup>ω</sup>* , is given by the following rules:

¬*Biϕ* → *Bi*¬*Biϕ* � *ϕ* ⇒ � *Biϕ*

than *a*.

to us.

**2.1 Language of** *PBL<sup>ω</sup>*

**2.2 Semantics of** *PBL<sup>ω</sup>*

(*S*, *Xi*,*s*, *μi*,*s*).

(1) If *<sup>ϕ</sup>* <sup>∈</sup>Atomic formulas set *Prop*, then *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(2) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , then <sup>¬</sup>*<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(3) If *<sup>ϕ</sup>*1,*ϕ*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* , then *<sup>ϕ</sup>*<sup>1</sup> <sup>∧</sup> *<sup>ϕ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBL<sup>ω</sup>* ;

(c) (finite additivity) *μi*,*s*(*A*<sup>1</sup> ∪ *A*2) = *μi*,*s*(*A*1) + *μi*,*s*(*A*2), where *A*<sup>1</sup> and *A*<sup>2</sup> are disjoint members of *Xi*,*s*;

(d) 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.

Notice that from the definition of *Xi*,*s*, we have {*s*� |*μi*,*s*�(*A*) ≥ *a*} ∈ *Xi*,*<sup>s</sup>* and {*s*� |*μi*,*s*�(*A*) < *b*} = *S* − {*s*� |*μi*,*s*�(*A*) ≥ *b*} ∈ *Xi*,*s*. Intuitively, the probability space *Pi*(*s*) describes agent *i*'s probabilities on events, given that the state is *s*. *W* is the sample space, which is the set of states that agent *i* considers possible. *Xi*,*<sup>s</sup>* is the set of measurable sets. The measure *μi*,*<sup>s</sup>* does not assign a probability to all subsets of *S* but only to the measurable sets.

As an example, we consider a *PBLω*-model such that *PM* = (*S*, *π*, *P*1). Here *S* = {*s*1,*s*2,*s*3}; *π*(*s*1, *p*) = *f alse*, *π*(*s*2, *p*) = *f alse*, *π*(*s*3, *p*) = *true*, *π*(*s*1, *q*) = *true*, *π*(*s*2, *q*) = *true*, *π*(*s*3, *q*) = *true*; *P*<sup>1</sup> is defined as follows: for every *s* ∈ *S*, *P*1(*s*)=(*S*, *X*1,*s*, *μ*1,*s*), where *X*1,*s*<sup>1</sup> = *X*1,*s*<sup>2</sup> = ℘(*S*), *X*1,*s*<sup>3</sup> = {∅, {*s*1,*s*2}, {*s*3}, *S*}, *μ*1,*s*<sup>1</sup> (∅) = *μ*1,*s*<sup>2</sup> (∅) = *μ*1,*s*<sup>3</sup> (∅) = 0, *μ*1,*s*<sup>1</sup> ({*s*1}) = *μ*1,*s*<sup>2</sup> ({*s*1}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*2}) = *μ*1,*s*<sup>2</sup> ({*s*2}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*3}) = 0, *μ*1,*s*<sup>3</sup> ({*s*3}) = 1, *μ*1,*s*<sup>1</sup> ({*s*1,*s*2}) = *μ*1,*s*<sup>2</sup> ({*s*1,*s*2}) = 1, *μ*1,*s*<sup>3</sup> ({*s*1,*s*2}) = 0, *μ*1,*s*<sup>1</sup> ({*s*1,*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*1,*s*3}) = 1/2, *μ*1,*s*<sup>1</sup> ({*s*2,*s*3}) = *μ*1,*s*<sup>2</sup> ({*s*2,*s*3}) = 1/2, *μ*1,*s*<sup>1</sup> (*S*) = *μ*1,*s*<sup>2</sup> (*S*) = *μ*1,*s*<sup>3</sup> (*S*) = 1. It is easy to check that the above model satisfies the conditions in Definition 2.2. In this model, it is clear that the set of measurable sets *X*1,*<sup>s</sup>* and probability measure *μ*1,*<sup>s</sup>* varies with *s*, and consequently the probability space also varies with *s*, hence we index probability space by *s*.

We now define what it means for a formula to be true at a given world *s* in a probabilistic model *PM*.

**Definition 2.3** Probabilistic semantics of *PBL<sup>ω</sup>*

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

$$(PM\text{\textquotedblleft}s) = \neg \varphi \text{ iff } (PM\text{\textquotedblright}s) \not\models \varphi\text{\textquotedblleft}s$$

(*PM*,*s*) |= *ϕ* ∧ *ψ* iff (*PM*,*s*) |= *ϕ* and (*PM*,*s*) |= *ψ*;

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

The intuitive meaning of the semantics of *Bi*(*a*, *ϕ*) is that agent *i* believes that the probability of *ϕ* is at least *a* in world (*PM*,*s*) if the measure of possible worlds satisfying *ϕ* is at least *a*.

*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*}).)

Probabilistic Belief Logics for Uncertain Agents 23

*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

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

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

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*,

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

probabilistic belief of agent *i* for Σ, then *ψ* can be merely proved from Γ.)

and write Γ �*PBL<sup>ω</sup> ϕ*, if there is a proof from Γ to *ϕ* in *PBLω*.

Firstly, we need the following obvious lemmas.

1 = *supn*∈*M*({*n*/*n* + 1}).

an infinite inference rule.

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

validity.

In the above example, according to Definition 2.3, we have (*PM*,*s*1) |= *B*1(1/2, *q*), (*PM*,*s*2) |= *B*1(0, *p* ∧ *q*), (*PM*,*s*3) |= *B*1(1, *p* ∧ *q*), etc.

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