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true
and
unverified false
, where verified options have been the subject of an inquiry di-
alogue (and so have been agreed by the agents) and unverified options have not (and so
could change in the light of information that could be elicited by an inquiry dialogue).
Definition 7.
τ
ij
= 1 if option
O
i
has attribute
a
j
. If this attribute for this option has
been the subject of an inquiry dialogue,
τ
ij
has been verified. Attributes of options for
which
τ
ij
= 1 has been verified form a set
VT
ag
; those for which
τ
ij
= 0 has been
verified form a set
VF
ag
, for both agents. For a particular agent, if for an unverified
attribute
KB
Ag
F
τ
ij
=1
, the attribute is unverified true and so an element of
UT
ag
:
otherwise the attribute is unverified false and so is an element of
UF
ag
.
VT
ag
∪
UT
ag
∪
VF
ag
∪
UF
ag
=
A
O
.
Finally, we are able to define the
utility
of an option for an agent, based on attributes of
the option that are true.
Definition 8.
Let
A
⊆
A
O
be a set of attributes true of
O
i
. Then the utility of
O
i
for
an Agent
Ag
with respect to these attributes,
U
i
(
A
)
,is
a
j
∈A
w
Ag
(
a
j
)
. Now the current
utility is
U
i
(
VT
i
∪
UT
i
)
. This we will sometimes abbreviate to
U
i
when there is no
ambiguity. The maximum utility is
U
i
(
VT
i
∪
UT
i
∪
UF
i
)
and the minimum utility is
U
i
(
VT
i
)
.
2.2
Inquiry and Persuasion Dialogues
We can now characterise the distinction between inquiry and persuasion. Suppose we
have two agents, Wilma and Bert, so that
AG
=
: following the conventions of
chess,
W
will initiate the dialogues. We may say that for an inquiry dialogue between
Wilma and Bert concerning a proposition
f
(inquiry dialogues concern
only
factual
propositions), the following two conditions should hold:
{
W, B
}
I1 Initially either Wilma does not believe that
f
:
KB
W
0
F
f
or Bert does not believe
that
f
:
KB
B
F
f
or both.
I2 At the end of the dialogue both believe that
f
if and only if
f
is a consequence of the
joint knowledge bases:
(
KB
W
1
F
(
KB
B
1
F
KB
W
0
F
KB
B
0
F
f
)
∧
f
)
↔
∪
f
.
An inquiry dialogue will always result in agreement, since, if explicit agreement does
not terminate the dialogue it will continue until Bert and Wilma have exchanged all
their knowledge. Thus, if the dialogue has taken
n
steps,
KB
W
n
F
F
=
KB
W
F
∪
KB
B
F
. Some have argued that neither should believe that
f
at the outset, but we wish
to allow Wilma to start an inquiry dialogue to confirm her beliefs, since, given the
defeasible reasoning mechanism we are using, it is always possible that Bert may supply
information resulting in Wilma revising her beliefs.
The second condition is plausible (since we are assuming that factual propositions
are objectively assessed by the agents) and is the condition used to show soundness and
completeness of the inquiry dialogues in Black and Hunter [2]. Since that paper shows
soundness and completeness for their inquiry dialogues, we shall suppose that where
=
KB
B
n
