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The first clause of the definition simply states that if
is always true (or if it always
occurs) then its probability must be 1 whereas the second one states that if
θ
φ
and
θ
are never true at once (or that they never occur together) then the probability of
θ
∨
φ
.
From Definition 31 the standard properties of probability functions on proposi-
tional languages follow. We give some without proof -for a proof and more details
on probability functions see for example [17]-. For
is equal to the sum of the probabilities of
θ
and
φ
ω
a probability function on
L
and
θ
,
φ
∈
SL
,
•
ω
(
θ
∨
φ
)=
ω
(
θ
)+
ω
(
φ
)
−
ω
(
θ
∧
φ
)
,
•
ω
(
∼
θ
)=
1
−
ω
(
θ
)
,
•
if
θ
|
=
φ
then
ω
(
θ
)
≤
ω
(
φ
)
.
For the next definition let us consider
θ
,
φ
,
η
to be a conditional probabilistic
statement, for
θ
,
φ
∈
SL
and
η
∈
[
0
,
1
]
.
Definition 32.
We say that a probability function
ω
on L satisfies
θ
,
φ
,
η
if
ω
(
θ
∧
φ
)
ω
(
θ
)
=
η
.
If there exists such a probability function we then say that
is
satisfiable
.
As seen in the previous section, the inference mechanism in CADIAG2 gets
started with a set of graded statements of the form
θ
,
φ
,
η
L
abasicmed-
ical entity present in the patient. Let us consider as an example the medical entity
'
reduced glucose in serum
'. Let us assume that the value assigned at the outset in
a run of the inference engine by the evaluation system in CADIAG2 to the state-
ment '
Patient A has reduced glucose in serum
' out of the evidence given by the
corresponding measurement of the amount of glucose in Patient A is
(
q
,
η
)
, with
q
∈
η
,forsome
η
∈
[
. As an example, we could interpret such value as the
degree of belief
that a
medical doctor has in the truth of the statement given the evidence. As such
0
,
1
]
could
be interpreted as a probability. The probabilistic interpretation is certainly favoured
by the discretization applied to medical concepts in CADIAG2 (for example, the
concept '
glucose in serum
' generates five distinct medical entities in CADIAG2:
'
highly reduced glucose in serum
', '
reduced glucose in serum
', '
normal glucose in
serum
', '
elevated glucose in serum
'and'
highly elevated glucose in serum
'). Notice
that such an interpretation places us within the subjective probabilistic frame and
thus, for the sake of coherence, the knowledge base
KR
should also be interpreted
subjectively. Other interpretations are also possible though. For example, one could
regard such values as the ratio given by the number of doctors that agree on the truth
of the statement out of all the doctors involved in the assessment. In order to accom-
modate such values into a coherent probabilistic frame along with the statements in
KR
one could justify them as being subjective probabilities assessed by a group of
experts -see [9] or [16] for an analysis and justification of such concept-.
Formally, let
q
η
∈
L
represent a basic medical entity present in the patient and
assume that
η
∈
[
0
,
1
]
is the initial value assigned to it by the evaluation system of
