Ejecución de las políticas
REPÚBLICA DOMINICANA
Unless otherwise specified, for the rest of this section all the sets are abnormal, normal,
ri-stable, contingent or a priori w.r.t. Piw.
Succ:Biφφ
Proof. Pick a world w∈W.
• Case 1. ||φ|| is abnormal. Then Biφφ holds at w, by Corollary 5.2.
• Case 2. ||φ|| is normal. Take W to be the whole space of possible worlds. Now W is normal and 1−stable.
Moreover, we have that: W ∩ ||φ||=||φ||. ThereforeBiφφ holds atw.
IE(a):Biφψ ⇒(Biφ∧ψχ⇔Biφχ).
Proof. Pick w∈W and assume that Biφψ holds atw.
• Case 1. ||φ|| is abnormal. Then ||φ|| ∩ ||ψ|| ⊆ ||φ|| and therefore ||φ|| ∩ ||ψ|| is abnormal as well.
Hence Biφ∧ψχ⇔Biφχ holds atw, by Corollary 5.2.
• Case 2. ||φ|| is normal. Assume that Biφ∧ψχ holds atw.
Then by Biφψ, we have that ∃S :ri-stable set such that: ∅ 6=S∩ ||φ|| ⊆ ||ψ||.
Moreover, by Biφ∧ψχ, we have that ∃S0 : ri-stable set such that ∅ 6= S0∩ ||φ|| ∩
||ψ|| ⊆ ||χ||.
Now consider the set: S00 =S0 ∩S.
By Observation 4.7,S00 isri-stable as well.
Now take the setS00∩ ||φ||.
SinceS∩ ||φ|| 6=∅, S0∩ ||φ|| 6=∅ and S006=∅, we have that S00∩ ||φ|| 6=∅. Now pick x∈S00∩ ||φ||.
Thenx∈S∩ ||φ|| and therefore x∈ ||ψ||.
Now x∈S0∩ ||φ|| ∩ ||ψ|| as well. Thereforex∈ ||χ||. Hence: ∅ 6=S00∩ ||φ|| ⊆ ||χ||. Thus Biφχ holds at w. Now for the other direction, assume that Biφχ holds atw.
• Case 1. ||φ|| ∩ ||ψ|| is abnormal. Then Biφ∧ψχ holds at was well.
• Case 2. ||φ|| ∩ ||ψ|| is normal. Then:
byBiφψ, we have that ∃S:ri-stable set , such that: ∅ 6=S∩ ||φ|| ⊆ ||ψ||.
Also, byBiφχ, we have that ∃S0 :ri-stable set, such that: ∅ 6=S0∩ ||φ|| ⊆ ||χ||.
Once again consider the set S00 =S0 ∩S. We know that S00 is ri-stable as well,
by Property 4.7.
Take the set S00∩ ||φ|| ∩ ||ψ||. Now S00∩ ||φ|| is non-empty. Pick x∈S00∩ ||φ||.
Thenx∈S∩ ||φ||. Therefore x∈ ||ψ||. Hence S00∩ ||φ|| ∩ ||ψ|| 6=∅.
46 7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
Thereforex∈S0∩ ||φ||. Hence x∈ ||χ||.
Thus∅ 6=S00∩ ||φ|| ∩ ||ψ|| ⊆ ||χ|| and hence Biφ∧ψχholds at w.
IE(b):¬Biφ¬ψ ⇒(Biφ∧ψχ⇔Biφ(¬ψ∨χ)).
Proof. Pick a world w∈W.
Assume that ¬Biφ¬ψ holds at world w. If ||φ|| is abnormal, then ¬Biφ¬ψ can not hold since by Corollary 5.2 everything is believed when conditioning on abnormal sets. Therefore assume that ||φ||is normal and that Biφ∧ψχ holds at w.
Now we need to consider the following cases:
• Case 1. ||ψ||is abnormal. Then||ψ||cis a priori. Now if||φ|| ∩ ||ψ||cis abnormal,
then by 4.13 we have that||φ||is abnormal, which contradicts our assumptions. Hence ||φ|| ∩ ||ψ||c is normal and ||ψ||c is r
i-stable as a priori.
Moreover, ||φ|| ∩ ||ψ||c⊆ ||ψ||c, hence Bφ
i¬ψ holds. Contradiction.
• Case 2. ||ψ|| is normal. Then we have two more cases:
– First case: ||φ∩ψ|| is normal. Then since ¬Biφ¬ψ holds at w, we get that
6 ∃S :ri-stable set such that ∅ 6=S∩ ||φ|| ⊆ ||¬ψ||.
Therefore, for all S :ri-stable sets, we have that if S∩ ||φ|| is normal, then
S∩ ||φ|| ∩ ||ψ|| 6=∅.
Now we also have that ∃S0 :ri-stable set such that: ∅ 6=S0∩ ||φ|| ∩ ||ψ|| ⊆
||χ||.
Now pick x∈S0∩ ||φ||.
Then either x∈ ||ψ|| orx∈ ||ψ||c.
If x∈ ||ψ|| then x∈ ||χ||.
Hence ∅ 6=S∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ|| and Biφ(¬ψ∨χ) holds at w.
– Second case: ||φ∩ψ|| is abnormal.
Then (||φ|| ∩ ||ψ||)c= (||φ||c∪ ||ψ||c) =K is a priori and 1−stable.
Now K∩ ||φ||=||φ|| − ||ψ||. If ||φ|| − ||ψ||=∅, then||φ|| ∩ ||ψ||=||φ|| and hence ||φ|| is abnormal, contradicting our assumption.
Therefore ||φ|| − ||ψ||=K∩ ||φ|| 6=∅. But now we have that ¬Biφ¬ψ holds atw.
Hence K∩ ||φ|| ∩(||¬ψ||)c=K ∩ ||φ|| ∩ ||ψ|| 6=∅.
But this means that (||φ||∩||ψ||)c∩(||φ||∩||ψ||)6=∅, which is a contradiction. Now for the other direction assume that Biφ(¬ψ∨χ) holds at w.
Since¬Biφ¬ψ holds at w, we have that: ∀S :ri-stable sets ifS∩ ||φ|| 6=∅then S∩ ||φ|| ∩
||ψ|| 6=∅.
Now since Biφ(¬ψ ∨χ) holds at w, we have that: ∃S0 : ri-stable set such that: ∅ 6=
S0∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||.
Since S0∩ ||φ|| 6=∅, thenS0∩ ||φ|| ∩ ||ψ|| 6=∅.
Now: S0∩ ||φ|| ∩ ||ψ|| ⊆S0∩ ||φ|| and we have thatS0∩ ||φ|| ⊆ ||¬ψ|| ∪ ||χ||. Therefore:
S0∩ ||φ|| ∩ ||ψ|| ⊆ ||χ||.
3. SOUNDNESS 47
RE: from ψ infer Biφψ.
Proof. Pick w∈W.
If ψ holds, then ||ψ|| is an 1-stable set.
• Case 1. ||φ|| is abnormal. Then Biφφ holds at w, by Corollary 5.2.
• Case 2. ||φ|| is normal. Then ||φ|| ∩ ||ψ|| 6=∅.
For assume otherwise. Then ∃x∈ ||φ|| such thatx /∈ ||ψ||.
Butψ holds everywhere in W and therefore this can not be the case. Therefore: ∅ 6=||φ|| ∩ ||ψ|| ⊆ ||ψ||.
And hence Biφφ holds atw.
LE: from φ ⇔ψ inferBiφχ⇔Biψχ.
Proof. Pick w∈W. Assume that φ⇔ψ holds. Assume that Biφχholds at w.
• Case 1. ||φ||is abnormal. Then so is||ψ||and henceBψi χholds atw, by Corollary 5.2.
• Case 2. ||φ|| is normal. Then ∃S :ri-stable set such that: ∅ 6=S∩ ||φ|| ⊆ ||χ||.
But sinceφ ⇔ψ holds, then||φ||=||ψ||.
Therefore∅ 6=S∩ ||ψ|| ⊆ ||χ|| and Biψχ holds atw. Assume that Biψχ holds atw.
Analogous.
PI:Biφψ ⇒BiχBiφψ
Proof. Pick w∈W. Assume that Biφψ holds inw.
• Case 1. ||χ|| is abnormal. ThenBχiBiφψ holds atw, by Corollary 5.2.
• Case 2. ||χ|| is normal.
Consider the setw(i), which is 1−stable.
Then∅ 6=w(i)∩ ||χ|| ⊆ ||Biφψ||, since Biφψ holds at w. ThereforeBiχBiφψ holds at w.
NI:¬Biφψ ⇒Biχ¬Biφψ
Proof. Pick w∈W. Assume that ¬Biφψ holds inw.
48 7. LOGIC OF r-STABLE CONDITIONAL BELIEFS
• Case 1. ||χ|| is abnormal. ThenBχi¬Biφψ holds, by Corollary 5.2.
• Case 2. ||χ|| is normal. Then consider the set w(i) which is 1−stable. Then∅ 6=w(i)∩ ||χ|| ⊆ ||¬Biφψ||, since ¬Biφψ holds at w.
ThereforeBiχ¬Biφψ holds at w.
D:¬B>i ⊥
Proof. Follows directly from the condition of probabilistic frames that (W,F, Piw)
is a non-trivial conditional probability space.
4. Completeness
In this section we prove thatBRSID is a complete axiomatization w.r.t. our proba- bilistic models.
The idea is to prove a truth preserving lemma that will connect our probabilistic models with Board’s belief revision structures and use Board’s completeness result.
So we will begin by presenting Board’s semantics ([18]).
Consider a set of agents: Ag ={1, ..., n} and a set of atomic propositions At.
Definition 7.5. The structure
M =hW,4,|| · ||i, such that: • W a set of possible worlds,
• 4 a vector of binary relations over W,
• || · || is a valuation function, assigning sets of worlds to each atomic proposition will be called a belief revision structure.
4w
i is the plausibility ordering of agent i at world w.
x4w
i y denotes that world y is at least as possible as world x for agent i while at world
w.
Definition7.6. Define: Wiw :={x|y 4wi x for some y}, the set of all the conceivable
worlds for agent i at world w.
We assume moreover that:
R1 ∀i, w:4w
i is complete and transitive on Wiw
R2 ∀i, w:4w