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Given two sets of attitudes
A
and
A
0
of the same type
A
, we say that
A
is in
conflict
with
A
0
iff the set
A [ A
0
entails a
contradiction
in the formal
system
L
A
that represents those attitudes. That is, the two sets of attitudes are
inconsistent with respect to
L
A
. For instance, two conflicting judgment sets in
the sense of the previous section are simply two sets of propositions that are
inconsistent with respect to propositional logic, e.g.
fA;B;Cg
and
f:A;B;Cg
.
Moreover, two conflicting preferences are two sets of preferences that together entail
a contradiction, such as
f
Pab
;
Pac
;
Pbc
g
and
f
Pba
;
Pac
;
Pbc
g
: i.e.
Pab
and
Pba
entail
by transitivity
Paa
, which contradicts irreflexivity. Conflicting preferences and goals
entail that they cannot be satisfied at the same time.
Note that our notion of conflict applies to sets of attitudes of the
same
type
A
.
Thus, we do not say, for example, that an intention is inconsistent with a belief.
This is so because, in our view, a belief can contradict an intention, or an ought,
only with respect to a reasoning system that includes both attitudes and makes the
relationships between them explicit. Such a reasoning framework has to contain
a principle that links the different types of propositional attitudes that are matter
of discussion. An example of such a principle is (one version of) the means-end
principle of instrumental rationality
3
: “if I intend to
A
and I believe that
B
is a
sufficient means for achieving
A
, then I intend to
B
.” By means of such a principle,
we can see how a belief may contradict an intention as follows: suppose I intend to
A
, my belief that
B
is a sufficient means to get
A
would be inconsistent with the
fact that I do not intend to
B
. Situations of acrasia can be represented by a similar
argument. Our approach is motivated by the fact that in general we do not want to be
committed with a philosophically onerous claim that a belief per se can contradict a
preference or a desire or an ought.
In our modeling, the notion of contradiction has the following intuitive interpre-
tation: two inconsistent sets of attitudes cannot be satisfied at the same time, e.g. two
conflicting preferences entail that either one or the other can be accepted. We can
define the conflict between two agents by simply saying that agent
i
is in conflict
with agent
j
if the set of attitudes of
iA
i
is inconsistent with the set of attitudes
of
j
,
A
j
, namely
A
i
[ A
j
is inconsistent with respect to the formal system
L
A
.
This can be easily generalized to conflicts involving
m
agents:
A
1
[ [A
m
is
inconsistent with respect to
L
A
.
Note that our definition allows for an agent being in conflict with him/herself,
in case she/he maintains a set of inconsistent attitudes; For example, if an agent
i
has a set of judgments such as
fA;A ! B;:Bg
. We shall use this fact in the next
paragraph. The abstract notion of conflict that we have defined can be instantiated in
order to provide a representation of actual conflicts. For example, if we want to view
a chess match as a situation of conflict between two agents, we can represent the
conflicting aspect of the match by describing the agents' opposing goals of winning
by beating the other.
3
For a discussion on the status of instrumental rationality, see Nozick (
1993
).
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