**3.1 Semantics of** *PBLf*

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

(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;

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

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

(a) 0 ≤ *μi*,*s*(*A*) ≤ 1 for all *A* ∈ *X*.

$$\text{(b) }\mu\_{i,s}(\mathbb{Q}) = 0 \text{ and } \mu\_{i,s}(\mathbb{S}) = 1.$$

(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);

(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);

(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;

(f) Let Λ*i*,*<sup>s</sup>* = {*s*� |*Pi*(*s*) = *Pi*(*s*� )}, then *μi*,*s*(Λ*i*,*s*) = 1.

Remark: Since *X* = ℘(*S*), therefore *X* is a constant set, and we omit the subscript of *Xi*,*<sup>s</sup>* as was used in Definition 2.2.

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

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

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

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

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

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

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

#### **3.2 Inference system of** *PBLf*

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 fails with respect to the semantics of *PBLf* .
