Collective knowledge andpotential individual knowledge are types of implicit knowledge. They are meant to capture the knowledge that (group) agents would know, were the members of the group to share all their knowledge with them. In this section, I consider the action of truthful public announcement through which agents share their knowl- edge with their group members in an epistemic model. Learning from truthful public announcement is certainly an idealized form of learning, though the possibility of testi- monial knowledge also presupposes it. By showing thatcollective knowledge can indeed become individual knowledge–even common knowledge–through public announcement, and similarly for potential individual knowledge, considering such actions can provide further justification for their characterizations.
Condition P1 is a static condition: it states that if agent a knows that p, then it follows thatpis entailed by answers to her questions. From the perspective of knowledge acquisition, it must also hold that agents only acquire knowledge that is relevant to their issues. This condition can be viewed as the dynamic version of P1, as it describes the corresponding property of knowledge acquisition.
The above condition, however, does not specify how to handle the case where an agent is confronted with testimony that is partially relevant to her issue. Suppose that an agent is informed that some propositionP is true, but thatP distinguishes between states that are conceptually indistinguishable for her. The above condition can be interpreted as implying either that a will ignore P altogether or that she will come to know only the agent-relevant truths from P (even though P contains information that she cannot
conceptualize). I adopt the latter interpretation. This is based on the assumption that epistemic agents are able to extract relevant information. I formalize this as Pa, where
P ⊆S and a∈A:
Pa:={s∈S :s0 ≈asfor somes0 ∈P}=
\
{P0 ⊆S:P0 is agent relevant andP ⊆P0}
In words,Pa is the strongest agent-relevant consequence of P.
In Dynamic Epistemic Logic, the action of publicly (and truthfully) announcing P
is usually denoted by !P – where !P takes any given model S to a new model SP
(Baltag et al. 2008). However, in order to ensure that agents only update their model with Pa (rather than P), the standard semantics for this action has to be modified. Moreover, the action !P concerns a single proposition. Yet, both collective knowledge andpotential individual knowledgeare characterized as the knowledge that would become common knowledge and individual knowledge (respectively) were all group members to share all their knowledge. This latter action is denoted by !G. That is, !G denotes the action by which all agents in G publicly announce all their knowledge. At state
s, this corresponds to them announcing (together) P = {w : s →G w}–that is, the conjunction T
b∈G{w : s →b w} of the knowledge of each b ∈ G at s. Thus !G is the public announcement of P. From this announcement, each agent will only learn its strongest agent-relevant consequence: Pa={t:s→G≈at}. As such,a’s new knowledge →!G
a after action !G is given by the combination of her old knowledge with Pa: s→!aGt iffs→atand s→G≈at. The action !G modifies the models as follows:
Definition 17 (Sharing all knowledge in epistemic group models). Given an epistemic group modelS= (S,→,≈,k • k) over (A,Φ) and a groupG⊆A, a new epistemic model S!G= (S!G,→!G,≈!G,k • k!G), is obtained as follows:
1. the set of possible worlds stays the same: S!G =S; 2. s→!G
a t iff boths→at ands→G≈at. 3. the issue-relations stay the same: s≈!G
a tiff s≈a t. 4. the valuation stays the same: kpk!G=kpk, forp∈Φ.
When P ⊆S is a set of states, the following two propositions describe the dynamics of
KG and KaG:
Proposition 9. KGp ⇔ [!G]CkGp (for atomic sentencesp). Proof. To show: w|=S KGp if and only if w|=S[!G]CkGp. By definitionw|=SKGp ⇔ ∀w0∈S: (w(Sa∈G→G;≈a) ∗w0 ⇒w0 |= S p ⇔ ∀w0 ∈ S!G : (w(S a∈G →!aG)∗w0 ⇒ w0 |=S!G p) ⇔ w |=S!G Ckap ⇔ w |=S [!G]KGp.
Proof. To show: w|=S KaGpif and only if w|=S [!G]Kap.
By definitionw|=S KaGp ⇔ ∀w0 ∈S: (w(→G;≈a)w0 ⇒w0 |=Sp) ⇔ ∀w0 ∈
S!G: (w→!G
a w0 ⇒w0 |=S!G p)⇔w|=S!G Kap⇔w|=S [!G]Kap.
It should be noted that the new modelsS!Gare not necessarily epistemic group models, as condition P20 is not preserved by the transformation. Still, they do satisfy all the other semantic conditions of epistemic group models. In fact, after action !G the new model is a testimonial model, thus satisfying conditionP2 (cf. chapter 5).