Knowledge, Awareness and Probabilistic Beliefs

In order to establish the inclusion, it is enough to show that for any φ ∈ L I K , p , A and ω ∈ S , if prop ( φ ) ⊆ Ψ , then:

The proof will use the following Lemma.

( φ = p i ≥ α χ , α > 0) We consider the same two cases as in the step for K _i operator, and the proof of each of them is also analogous.

( φ = p i ≥ 0 χ ) Again, consider the same two cases as in the step for K _i operator. The proof for the second case is as above. When M , ω ⊭ A i χ , M * , ( ω ,  Ψ ) ⊭ A i χ clearly M , ω ⊭ p i ≥ 0 χ . We also have M * , ( ω ,  Ψ ) ⊭ p i ≥ 0 χ : each of the two satisfaction clauses (see def. 5) fails due to Lemma 2 (or IA).

(Proof Lemma 2) The proof is also inductive over the complication of φ .

(at, ∧) Trivial.

(¬) It is enough to notice that the first clause of the satisfaction definition is not satisfied.

( A i χ ) We have, for ω ∈ S β :

In the second implication we use the fact that S β ≻ ‾ S β ′ , and so γ ( S β ) ⊇ γ ( S β ′ ) .

( K i χ ) We have, for ω ∈ S β :

( p i ≥ α χ ) The proof is analogous to the K i χ – case. ( □ Proof Lemma (2))  □

We proceed as follows. The main induction in the proof is over the size of prop ( ψ ) – see proof of claim 1. For a given consistent formula ψ we construct a canonical state space model M with heterogenous priors. Then we prove an auxiliary Lemma 5 and so called existence and truth lemmas, establishing that ψ is satisfied at some state in M. Lastly, we transform M into a state space model M* with syntactic awareness with extended prior that satisfies ψ at some state (compare Lipman Lipman 2003).

Consider a consistent formula ψ ∈ L I K , p , A . Let L ( ψ ) ⊂ L I K , p , A be the finite set of all the formulas that have modal depth^[20] lower or equal to the modal depth of ψ , contain only the atomic sentences from prop ( ψ ) and the probabilistic operators with indexing rational numbers being multiples of 1 / d , where d is the least common denominator of all indexes appearing in ψ . Let M : = 〈 S , P o w ( S ) , ( P i ) i ∈ I , ( K i ) i ∈ I , π , ( A i ) i ∈ I 〉 be the following state space model with syntactic awareness and heterogenous priors:

1. S consists of all the maximal consistent subsets of formulas from L ( ψ ) .

2. heterogenous priors P i – see below.

3. For any ω ∈ S we have that ω′ ∈ K i ( ω ) iff

1. π ( q ) = { ω | q ∈ ω } .

2. A i ( ω ) = { q | A i q ∈ ω } .

Notice that the set S is finite.

For every agent i ∈ I , state ω ∈ S we define a prior probability distribution P i over ( S , E ) as follows. Let E i , ω be the set of those subsets of S of which agent i “is aware” at ω intersected with her knowledge partition:^[21]

It easily follows from A ⊤ ,  A¬ and A∧ that E i , ω is an algebra of subsets: if e.g. ⌈ φ ⌉ , ⌈ φ ′ ⌉ ∈ E i , ω , then A i φ , A i φ ′ ∈ ω , then due to A∧ also A i ( φ ∧ φ ′ ) ∈ ω , and so ⌈ φ ⌉ ∩ ⌈ φ ′ ⌉ ∩ K i ( ω ) = ⌈ φ ∧ φ ′ ⌉ ∩ K i ( ω ) ∈ E i , ω , where the last equality follows form the fact that A X A x includes all the instances of propositional tautologies.

For any φ such that A i φ ∈ ω let α ‾ φ = max { α | p i ≥ α φ ∈ ω } and α ‾ φ = min { α | p i ≤ α φ ∈ ω } . Let also I φ = { α ‾ φ } when α ‾ φ = α ‾ φ , and I φ = ( α ‾ φ , α ‾ φ ) = ( α ‾ φ , α ‾ φ + 1 / d ) ⊂ ℝ otherwise, where the last equality follows from III’. For every E ∈ E i , ω choose a representative formula θ E , A i θ E ∈ ω , such that E = ⌈ θ E ⌉ ∩ K i ( ω ) and θ E has minimal complexity (see Lemma 7).

(The only difference with their proof is that there the atoms of the σ − algebra are exactly the maximally consistent sets, which is not necessarily the case here. The proof is completely analogous.)

Observe that due to the definition of information assignments, if ω′ ∈ K i ( ω ) , then E i , ω = E i , ω′ and P i , ω ≡ P i , ω′ . Define a function P i ′ : ∪ ω ∈ S E i , ω → [ 0 , 1 ] e.g. as P i ′ ( E ) = 1 | { K i ( ω ) : ω ∈ S } | * P i , ω′ ′ ( E ) for E ∈ E i , ω′ .

We might construct P i e.g. by requiring that ∀ ω ∈ S P i ( ω ) = 1 | { φ ∩ K i ( ω ) } | * P i ′ ( { φ ∩ K i ( ω ) } ) such that { φ ∩ K i ( ω ) } is an atom in E i , ω and ω ∈ { φ ∩ K i ( ω ) } . This ends our construction of the canonical model with heterogenous priors.

( □ Proof Lemma (Blume 2000))

Suppose then that ◇ i ¬ φ ∈ ω and { ¬ φ } ∪ X K ∪ X A ∪ X ¬ A ∪ X p ∪ X ¬ p is inconsistent, and so:

Due to P0’’ we have that:

where X ′ ¬ p = { ¬ p i ≥ α χ | ¬ p i ≥ α χ ∈ ω ,  A i χ ∈ ω }

(Proof Lemma (7)) Proof of this lemma, which goes inductively over the complexity of formulas, establishes that ψ is satisfied at some state in M. This is because if ψ is consistent, then it is included in some maximally consistent subset of L ( ψ ) (Lindenbaum lemma, see e.g. Blackburn et al. 2001), and so is satisfied at this state of our canonical model.

The cases for atomic sentences and the induction steps over conjunction, negation and awareness operators are straightforward.

Consider χ = K i φ . In one direction, if K i φ ∈ ω , then by AKii A i φ ∈ ω and by IA M , ω ⊨ A i φ . Moreover, by the definition of K i in M we know that ∀ ω′ ∈ K i ( ω ) φ ∈ ω′ , and so by IA M , ω′ ⊨ φ . Altogether we have that M , ω ⊨ K i φ . The other direction follows from the existence lemma: Suppose M , ω ⊨ K i φ . By IA we have A i φ ∈ ω . If K i φ ∉ ω then by the existence lemma we have ¬ φ ∈ ω′ , i.e. M , ω′ ⊨ ¬ φ by IA, for some ω′ ∈ K i ( ω ) , and so M , ω ⊭ K i φ , contradiction.

Let finally χ = p i ≥ α φ . Observe that for ⌈ θ E ⌉ ∩ K i ( ω ) = ⌈ φ ⌉ ∩ K i ( ω ) we have M , ω ⊨ K i ( φ ↔ θ E ) . By IA also K i ( φ ↔ θ E ) ∈ ω , and Lemma 5 guarantees that α ‾ φ = α ‾ θ E and similarly α ‾ φ = α ‾ θ E .

Now, if A i φ ∉ ω then M , ω ⊭ A i φ by IA, and so M , ω ⊭ p i ≥ α φ and also p i ≥ α φ ∉ ω , by PO”. If A i φ ∈ ω we have the following: M , ω ⊨ p i ≥ α φ implies that P i , ω ( φ ∩ K i ( ω ) ) = P i ( φ | K i ( ω ) ) ≥ α . From the remarks above and the definition of P i , ω we have that α ‾ φ = α ‾ θ E and α ‾ θ E ≥ α , respectively, and so p i ≥ α φ ∈ ω . Implication from p i ≥ α φ ∈ ω to M , ω ⊨ p i ≥ α φ is even simpler to prove (□Proof Lemma (7))

We have shown that ψ is satisfied at some state in M. The last step is a transformation of M into a state space model M ^* with an extended prior. This follows closely, and is actually easier then the construction in (Lipman 2003), with the “depth” of extended prior assignment obviating the need for weak consistency axiom (Lipman 2003, Def. 1). We just sketch the construction here. Let k be equal to the modal depth of ψ , and lets assume without loss of generality that I = { 1 , … , | I | } . For M let M * = 〈 S * , P o w ( S * ) , P * , ( K i * ) i ∈ I , π * , ( A i * ) i ∈ I 〉 with:

• S * = S × { 1 , … , k | I | } .

• – P * is such that for m = l ∗ | I | + i , l ∈ ℕ ,

• K i * ( ( ω , m ) ) = ( K i ( ω ) × { ( l − 1 ) * | I | + i + 1 , … , l ∗ | I | + i } ) ∩ S , m ∈ { ( l − 1 ) * | I | + i + 1 , … , l ∗ | I | + i } .

• π * ( q ) = { ( ω , m ) ∈ S * | ω ∈ π ( q ) } .

• A i * ( ( ω , k ) ) = A i ( ω ) .

We leave it to the reader to show that if φ is a sub-formula of ψ of modal depth r, then M , ω ⊨ φ iff M * , ( ω , s ) ⊨ φ , for s ≤ ( k − r ) * | I | + 1 , for all ω ∈ S . This establishes the proof of the theorem. □

Consider now the additional axioms for the semantic models. We will prove only soundness of the last scheme. Suppose the antecedent is satisfied at ω in some M, M ∈ M A h . It guarantees that M , ω β ′ ∧ β ″ ⊨ p i ≥ α ( φ ) , for β ′ , β ″ such that K i ( ω ) ⊆ S β ′ , K j ( ω ) ⊆ S β ″ and S β ′ ∧ β ″ : = S β ′ ∧ S β ″ . It follows that M , ω β ″ ⊨ p i ≥ α ( φ ) , and so M , ω ⊨ ◇ j ( p i ≥ α ( φ ) ) .

Consider now the additional axioms for the semantic* models. We will only prove soundness of K i φ → φ , if φ ∈ L I p o s K , A . Take any M ∈ M A s . Due to the generalized reflexivity of information assignments, it is enough to prove that for φ ∈ L I p o s K , A , ω ∈ S β if S β ′ ≺ ‾ S β and M , ω β ′ ⊨ φ , then M , ω ⊨ φ . The case for literals, i.e. when φ is of the form q or ¬ q , follows from the definition of valuation on “subjective” spaces. Induction steps over ∧, ∨ are straightforward. Let K j ( ω ) ⊆ S β ″ . If M , ω β ′ ⊨ A j φ then prop ( φ ) ⊆ γ ( S β ′ ∧ S β ″ ) ⊆ γ ( S β ″ ) , and so M , ω ⊨ A j φ . If M , ω β ′ ⊨ ¬ A j φ then prop ( φ ) ⊆ γ ( S β ′ ) and prop ( φ ) ⊊ γ ( S β ′ ∧ S β ″ ) = γ ( S β ′ ) ∩ γ ( S β ″ ) , and so prop ( φ ) ⊊ γ ( S β ″ ) , meaning that M , ω ⊨ ¬ A j φ . Suppose now that M , ω β ′ ⊨ K j φ . It follows that M , ω′ ⊨ φ ∀ ω′ ∈ K j ( ω β ′ ) . As K j ( ω ) β ′ ∧ β ″ ⊆ K j ( ω β ′ ∧ β ″ ) = K j ( ω β ′ ) , for S β ′ ∧ β ″ : = S β ′ ∧ S β ″ , and φ ∈ L I p o s K , A , we have by IA that M , ω ″ ⊨ φ ∀ ω ″ ∈ K j ( ω ) , i.e. M , ω ⊨ K j φ .  □