  We study group knowledge taking into account that members of a group may not
be perfect reasoners. Our focus is on distributed knowledge, which is what
privately informed agents can come to know by communicating freely with one
another and sharing everything they know. In descending order of reasoning
skills, agents can be fully introspective, positively introspective or
non-introspective. A fully introspective agent satisfies both positive and
negative introspection: When she knows a thing, she knows that she knows it;
when she does not know a thing, she knows that she does not know it. A
positively introspective agent may fail to satisfy negative introspection. A
non-introspective agent may fail to satisfy both properties.
  We introduce revision operators and revision types to model the inference
making process that leads to distributed knowledge. Since group members may be
differently introspective, inference making turns out to be order dependent.
Then we produce two equivalent characterizations of distributed knowledge: one
in terms of knowledge operators and the other in terms of possibility
relations, i.e., binary relations. We show that there are two qualitatively
different cases of how distributed knowledge is attained. In the first,
distributed knowledge is determined by any group member who can replicate all
the inferences that anyone else in the group makes. In the second case, no
member can replicate all the inferences that are made in the group. This is due
to different levels of introspection across group members and to the order
dependence of inference making. As a result, distributed knowledge is
determined by any two group members who can jointly replicate what anyone else
infers. This case is a type of wisdom of crowd effect, in which the group knows
more than what any of its members can conceivably know by having access to all
the information available in the group.
