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
I now extend BI to the logic BI
of belief, intention, common belief and
1 2
communication. The starting point to define communication is the fixpoint axiom F
C w ; B w
∧ I C
w,
C a
a a
conceptually justified in Section 2. First we have to augment our formal language. 5.1. The formal language L
2
Let us extend language L by adding one more family of modal operators, C ,
1 a a [ A
with the following intended interpretation: •
C w means that agent a, the speaker, communicates that w to all agents in A2haj,
a
the audience. The language thus obtained will be denoted by L .
2
5.2. Semantics of L
2 10
I want now to define the semantics of L according to the following requirements:
2
1. if possible, C should be given a Kripkean semantics within the class of models ;
a
2. the semantics of C should enforce the validity of the fixpoint axiom F ;
a C
3. if more than one definition satisfies points 1 and 2 above, then the semantics of C
a
should be made as weak as possible i.e., C w should be true at the largest set of
a
possible worlds, compatibly with points 1 and 2. Requirement 1 is met by defining the semantics of C in terms of an accessibility
a
relation in the usual way:
a
nC wm 5 hw [ W:w w9
⇒ w9 [ n
wm j,
a M
a M
provided we can find a relation that meets the other requirements. To satisfy
a
requirement 2, let us define the function:
10
A similar approach could be adopted to define the semantics of common belief. I have not followed it because it would just lead to the well known definition of the accessibility relation for common belief in
terms of transitive closure.
184 M
. Colombetti Mathematical Social Sciences 38 1999 171 –196
2 2
g : 3W →
3W ,
a
g 5 5 + + 5 ,
a a
11
where + denotes the composition of binary relations. The function
g maps binary relations on W onto binary relations on W. It is easy to
a
verify that for the fixpoint axiom F to be valid in a class of frames, it is necessary and
C
sufficient that the accessibility relation be a fixpoint of g :
a a
g 5 .
a a
a
The function g is monotonic; that is:
a
5 6 ⇒
g 5 g 6 .
A a
2
Given that k3W ,l is a complete lattice, it follows from Tarski’s fixpoint theorem Tarski, 1955 that the set of fixpoints of
g , fixg , is not empty, and is itself a
a a
complete lattice with respect to . In general, fix g will contain more than one
a
element; this means that the fixpoint axiom F , by itself, is not sufficient to characterize
C
communication unambiguously, as anticipated in Section 1. We are now left with requirement 3. Given that k fix
g ,l is a complete lattice, g
a a
has a least fixpoint, fix g , which is the smallest relation satisfying g 5 . The
a a
a a
three requirements for C are thus met if we define as:
a a
5 fix g .
a a
It remains now to see whether we can characterize in a more constructive way. In
a
fact, the function g is upward continuous; that is, for any increasing chain
a
5 5 . . . 5 . . . ,
1 n
we have g 5 5 g 5 .
a n
n n a
n
Therefore, the least fixpoint of g is the limit of the chain:
a
5 5 [,
5 5
g 5 .
n 11 a
n
Such a limit is given by the relation: 5 + + + + + + . . . 5 + 1 + ,
a a
a a
a
where 1 denotes the identity relation. We now have a full Kripkean semantics for BI .
2
The reference class of models for BI is still supplemented with the above
2
definition of . I continue to denote valid sentences of L by w.
a 2
11
I remind the reader that the composition 5+6 of two binary relations 5 and 6 is defined as follows: x 5 + 6 y
⇔ z.x5z z6y.
M . Colombetti Mathematical Social Sciences 38 1999 171 –196
185
5.3. The system BI
2
The modal system BI is defined by adding to BI the forward fixpoint axiom for
2 1
communication, FF
C w . B w
∧ I C
w,
C a
a a
and the rule of induction for communication, c . B w
∧ I
c
a
]]]]] RI
.
C
c . C w
a
Theorems of BI will still be denoted by £ w.
2
Again, it is easy to check that BI is sound with respect to : FF is true in all
2 C
models such that the accessibility relation of C contains the relation defined in the
a a
previous subsection; and RI is valid if the accessibility relation of C is contained in
C a
such relation . In Appendix A, I prove that BI is also complete with respect to .
a 2
As we should expect, F is a theorem of BI : the implication from left to right is FF ,
C 2
C
and its converse can be proved by RI . The repeated use of FF allows us to derive the
C C
following theorems: £
C w . B w,
a
£ C
w . B I B w,
a a
£ C
w . B I B I B w,
a a
a
£ . . . .
As one can observe, the first theorem states that the main effect the speaker wants to obtain through communication i.e., that
w be common belief has been achieved; the second theorem represents the achievement of a Gricean condition i.e., that the
intention to obtain the main effect also be common belief; the third theorem represents the achievement of a Strawsonian condition; and the further theorems represent the
achievement of analogous conditions, for all higher orders.
In the next section, I prove further properties of communication.
6. Some properties of communication
