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
Before proving a few facts about communication, I shall introduce some new terminology. Basically, I define terms to qualify sentences for which certain kinds of
theorems can be proved in the logic. The main reason for doing so is to identify a number of concepts that play an important role in the treatment of communication.
Let us consider the theorems of the form £
c . B w.
E
If such a theorem can be proved for given sentences c and w, we can think of c as
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. Colombetti Mathematical Social Sciences 38 1999 171 –196
describing an event that makes w evident to all agents. In such a case, I say that c
displays w. When c 5w, that is when
£ w . B w.
E
I follow the existing practice and call w public. When w is public, it follows immediately
from RI with c 5w that
B
£ w . B w.
Let us now consider the theorems of the form £
w . I w.
a
If such a theorem can be proved for a given sentence w, I say that w is intentional.
Interestingly, when w is intentional we can regard it as describing an action performed
by agent a: in fact, actions are basically those propositions for which being intentional is a constitutive property. Therefore, when
w .I w is a theorem I shall also say that w
a
describes an action by a. Finally, I want to define the concept of a communicative act. This is simply an action
that implies the communication that w, for some given sentence w. More precisely, c
describes a w-communicative act by a if and only if:
£ c . I c
∧ C
w.
a a
6.1. An alternative characterization of communication As defined so far, communication has a number of interesting properties. The first
one, which is an immediate consequence of FF , is that communication implies common
C
belief: C1
£ C
w . B w.
a
This is certainly reasonable when communication is of the assertive type and probably also in a more general context: see Section 2. Instead, the converse implication,
B w .C w, is not a theorem of BI , as can be shown by a countermodel; this guarantees
a 2
that communication does not merely coincide with common belief. The second important property is that communication is intentional. From FF , FF ,
C B
and E we derive that £C
w .B I C w, from which, by the theorem £B I w .I w
B a
a a a
a a a
Section 3, we derive: C2
£ C
w . I C w.
a a
a
The converse implication, I C w .C w, is not a theorem of BI ; hence, the intention to
a a
a 2
communicate, by itself, is not sufficient to realize communication. A third property is that communication is public; that is:
C3 £
C w . B C w.
a E
a
This is a consequence of F , through the theorem £B w .B B w. From C3 we then
C E
derive by induction that:
M . Colombetti Mathematical Social Sciences 38 1999 171 –196
187
C39 £
C w . B C w.
a a
In this case, the converse implications hold: C4
£ B C
w . C w.
E a
a
C49 £
B C w . C w.
a a
Theorem C4 is a consequence of F , through the theorem £B B w .B w, and C49 is a
C E
direct consequence of C4. Property C4 is very interesting, because it shows that there is nothing really ‘objective’ in communication: if all agents individually believe that
communication takes place, then communication does actually take place. It is also possible to show that C
w is the weakest statement in BI that implies
a 2
mutual belief that w, and is both intentional and public. That is, if there is a statement c
for which the following theorems can be proved: £
c . B w, £
c . I c,
a
£ c . B c,
E
then it follows that: £
c . C w.
a
We can restate this condition in the form of a derived inference rule: c . B w, c . I c, c . B c
a E
]]]]]]]] C5
. c . C w
a
To prove the validity of C5, it is sufficient to remark that if c satisfies the assumption of
C5, then it also satisfies the assumption of RI apply RI
to c .B c, and the
C B
E
monotonicity property of B to
c .I c. We have now an interesting alternative
a
characterization of communication: in BI , C w is the weakest intentional and public
2 a
statement that implies the common belief that w.
The fact that communication is intentional suggests that the statement C w describes
a
some kind of action. However, a sentence of the form C w need not by itself describe a
a 12
basic action. In general, to communicate that
w one has to perform some lower-level action
c that realizes C w; typically, c will be some action conventionally expressing a
a
meaning, like raising one’s hand, or ringing a bell, or uttering a sentence in a natural language. We now face a fundamental problem: under what conditions does a lower-
level action realize a communicative act? One such a set of conditions is shown by rule C5: an intentional statement describes a
w-communicative act if it is public and it implies the common belief that w. However,
12
A basic action is an action that can be performed immediately, with no need of a lower-level action to realize it. Examples of basic actions are raising one’s arm and emitting a sequence of vocal sounds.
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we can prove the validity of a more interesting inference rule, which I shall call the rule of communication:
c . I c, c . B w ∧
c
a E
]]]]]]] RC
. c . C w
a
The derivation of RC is immediate: From c .B w
∧ c we derive through RI that
E C
c .B w ∧
c. From this and c .I c, we then derive that c .B w ∧
I c, and from this
a a
we derive by RI that
c .C w.
C a
The rule of communication can be paraphrased as follows: a sufficient condition for an intentional statement
c to describe a w-communicative act is that c is public and displays
w. Now we can go back to the problem I have put forward in Section 1: how can agents infer communication from individual beliefs? The answer comes by
combining RC with C4 exploiting the monotonicity of B ; we thus derive the inference
E
rule: c . I c, c . B w
∧ c
a E
]]]]]]] RC9
, B
c . C w
E a
which says that if an intentional statement c is public and displays w, for w to be
communicated it is sufficient that the speaker and all agents in the audience individually believe that
c is performed. This rule gives a possible solution to the problem of inferring communication from individual beliefs.
7. Discussion
