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In-Depth Information
22.5.2
Fuzzy (t-norm-Based) Semantics
The motivation for an interpretation of the inference in CADIAG2 on the grounds
of a
t-norm
-based semantics is motivated by the (fuzzy) methodology on which it
is based and by the interpretation of the degree of presence
η
in a graded statement
of the form
in the input of CADIAG2 in the natural, most intuitive way: as
a membership degree (i.e., truth degree) in the context of fuzzy set theory. How-
ever, in our attempt to provide a fuzzy interpretation of the inference in CADIAG2,
we also need an interpretation of the rules in the system in those same terms. Al-
though, as mentioned in previous sections, degrees of confirmation are intended to
represent degrees of certainty about the presence of the corresponding diagnoses in
the patient and are better characterized by means of uncertainty measures such as
probability functions, in this section we will consider a characterization of the rules
of the system and the corresponding degrees of confirmation in terms of truth de-
grees, arguably more suitable from the point of view of the intended fuzzy semantics
(although the use of some fuzzy semantics to model uncertainty is not rare in the
literature).
Graded statements in our settings become in this context what have been called
graded formulas
in [12]. Truth degrees in them will now be regarded as lower-bound
thresholds (i.e.,
(
φ
,
η
)
η
in a graded statement of the form
(
φ
,
η
)
on
L
will now be regarded
as a lower-bound threshold for the degree of truth of
). Such an interpretation is
not only motivated by the fact that it constitutes the common one to fuzzy logics but
also by the inference in CADIAG2 itself when interpreted on fuzzy grounds: the
choice of the maximal value with respect to the ordering
φ
generated in relation to
a certain diagnose as the output value for it goes well (i.e., is consistent) with the
characterization of any values generated at each step in the inference as lower-bound
thresholds.
3
For our fuzzy semantics, the interpretation for conjunction (
∧
), disjunction (
∨
)
and negation (
) suggests itself by the values (degrees of truth in this context) that
the system assigns to compound medical entities in
SL
along the inference process.
Therefore, for
v
:
L
∼
−→
[
0
,
1
]
a fuzzy valuation on
L
, we will have the following
constraints, for
φ
,
θ
∈
SL
:
•
v
(
φ
∧
θ
)=
min
(
v
(
φ
)
,
v
(
θ
))
.
•
v
(
φ
∨
θ
)=
max
(
v
(
φ
)
,
v
(
θ
))
.
•
v
(
∼
φ
)=
1
−
v
(
φ
)
.
It is common in the field of fuzzy logic to interpret the conjunction by a
t-norm
(based on some natural, desirable properties that such interpretation should satisfy)
3
This is not so in our probabilistic interpretation of the rules and graded statements involved
in the inference process, as seen in the previous subsection. Recall that, in our probabilis-
tic characterization, distinct values generated along the inference for the same diagnose
(or, in general, medical entity) were intended to represent distinct degrees of certainty
about the presence of such diagnose in the patient due mostly to differing amounts of ev-
idence (i.e, subsets of what we denoted by Ω, explicitly formalized in our probabilistic
characterization).
