Deformación de tubería

To express the notion of triviality precisely, we take a modal propositional language L. Assume that we have a group of n agents named 1,…,n. For simplicity, we assume that these agents wish to reason about a world that can be described in terms of a nonempty set Φ of primitive propositions p, p’, q, q’,… As is standard, these propositions stand for basic facts about the world such as “Munich is a city” and “Sam is hungry”. To express statements like “Agent i knows that ϕ” and “It is trivial for agent i that ϕ” respectively, we enrich the language by the modal operators K1,…,Kn and T1,…,Tn (one for each agent), respectively. L

is just the set of formulas which contains the primitive propositions and is closed under negation, entailment and the modal operators. L0 is the fragment of L that does not contain any modal operators. LK is the modal fragment of L without triviality operators.

120 _{The prime number example goes back to a suggestion of Godehard Link’s in the Logik Oberseminar. Link} also applies this test in his (1998) to show that plural expression do not denote any special plural objects. However, I doubt that the argument establishes this.

For the semantics we use the familiar Kripke structures. Those structures were first introduced in Kripke’s seminal (1959). A Kripke structure M for n agents over Φ is a tuple (S,π,K1,…,

Kn), where S is a non-empty set of possible worlds, π is an interpretation which associates with each world in S a truth assignment to the primitive propositions in Φ (i.e.

Φ:π(s)→{True, False}) for each world s∈S, and Ki is a binary relation on S, that is a set of pairs of S.

The binary relation Ki is intended to capture the possibility relation according to agent i: (s,t)∈Ki iff agent i considers world t possible, given his information in world s. Let Ki be an equivalence relation.

So here is our proposal for an explication of knowledge and triviality in terms of possible worlds:

(1) (M,s) |= Kiϕ if and only if (M,t)|=ϕ for all t such that (s,t)∈Ki.

(2) (M,s) |= Tiϕ if and only if (M,t)|=ϕ for all and only all t such that (s,t)∈Ki

(M,s) |= Ki(ϕ∧ψ) if and only if (M,t)|=ϕ for all t such that (s,t)∈Ki and (M,t)|=ψ for all t such

that (s,t)∈Ki. Obviously, if ϕ is trivial for agent i, then i knows that ϕ.

In the text, the triviality explication was sketched by means of possibilities among which agents want to distinguish and which they can iteratively eliminate. Formally, this idea is best captured by means of Aumann structures. The name of these structures traces back to Robert Aumann, who introduced them in his (1976).121 Now Aumann structures are defined as:

(S,P 1,…,Pn).

Aumann structures are like Kripke structures, with two differences: S is still a set of possible worlds, but, first, there is no analogue to the π function, since in the event-based approach, there are no primitive propositions. The second difference is that to define what worlds agent i considers possible, in Aumann structures there is a partition Pi of S for each agent I, rather than using a binary relation Ki. If Pi = {Si1,…,Sin}, then the sets Sij are called the cells of the

partition Pi, or the information sets of agent i. The intuition is that if Sij is an information set

of agent i, and if s∈Sij, then the set of states the agent i considers possible (which corresponds

to the information of agent i) is precisely Sij. P is the row vector (P1,…,Pn)T, consisting of the

column vectors Pi = (Si1,…,Sin). The initial information set I0 = S.

Formally, given an Aumann structure (S,P1,…,Pn), we define knowledge and triviality operators Ki: 2S→2S, and Ti: 2S→2S, for agent i=1,…,n, respectively, as follows:

(12) Ki(e) = {s∈S: Pi(s) ⊆ e}and (13) Ti(e) = {s∈S: Pi(s) = e}.

Ki(e) and Ti(e) are itself events, namely the event of agent i knowing e and the event e being

trivially true for agent i. Thus, if event e is trivial for agent i, i knows that e.

We can now define an event evM(ϕ) for each formula ϕ by induction on the structure of ϕ:

1. evM(p) = epM, where epM, is the event that p is true, for each primitive proposition p (i.e.

epM = {s∈S: (M,s)|=p}).

2. evM(ϕ∧ψ)=evM(ϕ)∩ evM(ψ)

3. evM(¬ϕ) = _S-evM(ϕ)

4. evM(Kiϕ) = Ki(evM(ϕ))

5. evM(Tiϕ) = Ti(evM(ϕ))

Now we will show that we can go from any Kripke structure to a corresponding Aumann structure. We get the corresponding Aumann structure AM = (S,P 1,…,Pn) of a Kripke

structure M (with the same set S of states) by taking P i to be the partition corresponding to

Ki.

PROPOSITION I Let M be a Kripke structure where each possibility relation Ki is an equivalence relation, and let AM be the corresponding Aumann structure. Then for every formula ϕ, we have ϕM_{= ev}

M(ϕ), where ϕM={s∈S: (M,s)|=ϕ}.

PROOF by induction on the structure of the formulas 1. p∈Φ: pM = {s∈S: (M,s)|=p} = epM = evM(p).

2. (ϕ∧ψ)M = {s∈S: (M,s)|=ϕ∧ψ} = {t∈S: (M,t)|=ϕ}∩{u∈S: (M,u)|=ψ}= (by induction hypothesis) evM(ϕ) ∩ evM(ψ) = evM(ϕ∧ψ)

3. (¬ϕ)M = {s∈S: (M,s)|=¬ϕ} = {s∈S: (M,s)|=ϕ} = S-{t∈S: (M,t)|=ϕ} = (by induction

hypothesis) S-evM(ϕ) = evM(¬ϕ)

To prove the desired condition for the Ki and Tioperator, we use the following lemma (where

O ∈ {K,T}).

LEMMA: s∈Oi(evM(ϕ)) holds in A ⇔ (M,s)|=Oiϕ.

PROOF

s∈Oi(evM(ϕ)) holds in A ⇔ s∈evM(Oiϕ)holds in A ⇔ (by correspondence) s∈{x∈S:

(M,x)|=Oiϕ} ⇔ (M,s) |= Oiϕ.

Q.E.D

PROOF

4. (Oiϕ)M = {s∈S: (M,s)|=Oiϕ} = (by LEMMA) = evM(Oiϕ).

Q.E.D

We have just shown that for any Kripke structure there is a corresponding Aumann structure to which we can go to from the Kripke structure. What about the other direction? Is there for any Aumann structure a corresponding Kripke structure? Yes.

To see this, assume that we are given not only an Aumann structure A but also an arbitrary set

Φ of primitive propositions and an arbitrary function π that associates with each state in S a truth assignment to the primitive propositions in Φ. We need these additional constraints since Aumann structures normally do no presuppose some set of propositions and no truth assignment to some subset of the set of propositions. However, the events that are worked with in Aumann structures can be given names. These names correspond to the primitive propositions. By the assumption that it is clear when these events hold, we get the truth assignment.

So let A be the Aumann structure (S, P1,…, Pn), and define MA,π= (S, π, K1,…, Kn) as the Kripke structure, where Ki is the partition corresponding to Pi, for i=1,…, n. (Generally, it holds that R is an equivalence relation that we obtain from partition P if and only if P is a partition that we obtain from equivalence relation R .)

PROPOSITION II The Aumann structure corresponding to the Kripke structure MA,π is A. PROOF

Let A = (S, P1,…, Pn) be an Aumann structure. Suppose that MA,π and M2A,π= (S, π, K’1,…, K‘n) are Kripke structures corresponding to A. Then M2A,π= MA,π,since M2A,πand MA,π have

the same set S of states, the same function π, and Ki = K’i for all i∈{1,…,n}, since Ki and K’i both correspond to Pi for all i∈{1,…,n}. Q.E.D

Thus by PROPOSITION 1, the intensions of formulas in MA,πand the events corresponding to these formulas in A coincide.

PROPOSITION 1 and PROPOSITION 2 establish the close connection between the logic- based and event-based approaches that was claimed previously.

Together with a calculus, we could now prove completeness for the Aumann structure by proving completeness for the Kripke structures. However, to give the rules making up this calculus is not really easy since triviality turns out to be very instable and elusive when inferences are involved (see Sec. 5.2.1).