M . Colombetti Mathematical Social Sciences 38 1999 171 –196
181
As usual, I rely on correspondences to define the reference class of models: the class of models for BI , denoted by , is made up by all models whose underlying frames
satisfy all the above conditions. Validity with respect to will be denoted by w. It is easy to see that the class of models determines the system BI . Given the
above correspondences BI is obviously sound. Moreover, van der Hoek 1993 has
proved that the six specific axioms I have adopted for BI are canonical; that is, they are
8
true in all models built by adding arbitrary valuations to the canonical frame. This is sufficient to guarantee that the canonical model of BI belongs to , and therefore that
BI is complete. 3.6. A few basic properties of belief and intention
Only a few theorems of BI are relevant to the rest of this paper. The first two such theorems tell us that an agent entertains a belief intention if the agent believes that he
or she entertains it; in symbols £
B B w . B w,
a a
a
£ B I
w . I w.
a a a
The proofs are elementary, and involve Axioms D and 5 for the first theorem, and
B B
Axioms D and 5 for the second theorem. Considering Axioms 4 and 4 , we then
B B
B IB
derive that: £
B w ; B B w,
a a
a
£ I
w ; B I w.
a a a
4. Common belief
I now define the logic BI of belief, intention, and common belief as an extension of
1
BI . 4.1. The formal language L
1
Let us extend language L by adding two more modal operators, B and B , with the
E
following intended interpretations: •
B w means that everybody i.e., each agent in A believes that w,
E
• B
w means that w is a common belief of A.
8
The canonical frame is the frame underlying the canonical model for details, see van der Hoek, 1993.
182 M
. Colombetti Mathematical Social Sciences 38 1999 171 –196
B is also known as the shared belief operator. The language thus obtained will be
E
denoted by L .
1
4.2. Semantics of L
1
I shall follow the most recent treatments of shared and common belief, by introducing the following relations between possible worlds:
• 5
,
E a [ A
a 9
• 5 , where the operator
denotes the transitive closure of a relation.
E
The interpretation of B w and B w is then defined in the obvious way:
E
nB wm 5 hw [ W:w w9
⇒ w9 [ n
wm j,
E M
E M
nB wm 5 hw [ W:w w9
⇒ w9 [ n
wm j.
M M
The reference class of models for BI is still supplemented with the above
1
definitions of and . Validity of sentences of L with respect to is again
E 1
denoted by w. 4.3. The system BI
1
I define BI as the system obtained by supplementing BI with the two axioms:
1
E B
w ; n B
w,
B E
a [ A a
FF B
w . B w ∧
B w,
B E
and the inference rule: c . B w
∧ c
E
]]]] RI
.
B
c . B w I call FF the forward fixpoint axiom for common belief, and RI the rule of induction
B B
for common belief. Theorems of BI are still denoted by £ w.
1
It is easy to check that BI is sound with respect to : E is obviously true in all
1 B
models of , given the definition of ; FF is true in all models such that the
E B
accessibility relation of B contains the transitive closure of ; and RI is valid if the
E B
accessibility relation of B is contained in the transitive closure of .
E
It is well known that various multi-agent epistemic logics are still sound and complete when they are augmented with E , FF , and RI Halpern and Moses, 1992; see Lismont
B B
B
and Mongin, 1995; Bonanno, 1996 for completeness proofs of similar systems. To my knowledge, however, a full completeness proof for such an extension of KD45 is not
n
9
The transitive closure of a binary relation 5 is defined as the least transitive extension of 5.
M . Colombetti Mathematical Social Sciences 38 1999 171 –196
183
available in the literature; in any case, no completeness proof exists for a logic of belief, common belief and intention. In Appendix A, I adapt a proof by Lismont and Mongin
1995 to a completeness proof of my logic of communication see next section; from this, a completeness proof for BI can be extracted.
1
5. Communication
