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be given as an outcome in a run of the inference process of CADIAG2 on input
Φ
(
φ
,
η
)
T
only if there exists a maximal proof of
in
CadL
-for more details
on this point see [6]-. In
CadL
a runtime inconsistency generated by the system
would imply the existence of maximal proofs of
from
(
θ
,
)
(
θ
,
)
T
0
and
1
from
,forsome
medical entity
θ
.
22.5
Towards a Semantics for CadL
In this section we look at the interpretation of the inference process in CADIAG2.
We consider two possible alternatives in our attempt: probabilistic semantics and
fuzzy semantics.
22.5.1
Probabilistic Semantics
The motivation for a probabilistic interpretation of the inference in CADIAG2
comes from the identification of the degrees of confirmation of rules in
KR
with
frequencies or, more generally, probabilities and the rules in
KR
themselves with
probabilistic conditional statements.
In this subsection we will assume that rules of the form
θ
,
φ
,
η
∈
KR
represent
probabilistic conditional statements, where
θ
is the conditioning event or evidence,
φ
is true or that it occurs.
In order to set the inference process on probabilistic grounds and analyze its ad-
equacy with probability theory we need also a suitable probabilistic interpretation
of the graded propositions taken as input and generated along the process by the
system. Recall that the value
the uncertain event and
η
the probability of
φ
given that
θ
(
φ
,
η
)
η
in a statement of the form
in the input of CA-
DIAG2 is intended to represent the degree of presence of
in the patient, normally
identified with a membership degree in the context of fuzzy set theory (i.e., with a
degree of truth). Here though we will adopt a probabilistic interpretation for these
values.
We will focus our analysis on the binary fragment of
KR
(i.e., on the binary
rules in
KR
), which we will denote by
KR
bin
. The vast majority of rules in
KR
are,
as mentioned earlier, binary and they constitute the most characteristic fragment of
CADIAG2 when seen as a representative example of a certain type of expert system.
This restriction means leaving the evaluation rules in
CadL
aside. We will focus our
analysis of the inference engine and thus of
CadL
on the manipulation rules.
Before going any further we need to introduce some preliminary notation and
definitions.
φ
Definition 31.
Let
ω
:
SL
−→
[
0
,
1
]
. We say that
ω
is a probability function on L if
the following two conditions hold, for all
θ
,
φ
∈
SL:
•
If
|
=
θ
then
ω
(
θ
)=
1
.
.
2
•
If
|
=
∼
(
θ
∧
φ
)
then
ω
(
θ
∨
φ
)=
ω
(
θ
)+
ω
(
φ
)
2
|
=
Here and throughout
represents classical entailment.
