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and the interpretation of the implication (
) to its residuum -for more details on
these notions see [12]-. Such identification places a further constraint on v :
v
( θ φ )=
sup
{
v
(
p
) |
v
( θ
p
)
v
( φ ) },
for
SL .
The identification of the interpretation of the conjunction (
θ , φ
) with the Gödel t-
norm (i.e., with the minimum operator) leads to the following interpretation of the
implication (
), for
φ , θ
SL :
1if v
( θ )
v
( φ )
v
( θ φ )=
otherwise
In this framework we can identify an inference rule of the form
v
( φ )
θ , φ , η
in the
knowledge base of CADIAG2 -for
θ , φ
SL and
η [
0
,
1
]
- with the graded state-
ment
.
Satisfiability of rules in KR and, in general, of any graded statements is defined
in our framework as expected.
Definition 33. The fuzzy valuation v on L is said to satisfy
( θ φ , η )
( φ , η )
,forsome
φ
SL
and
.
If such a valuation exists we say that
η [
0
,
1
]
,ifv
( φ ) η
( φ , η )
is satisfiable .
CadL and Fuzzy Soundness
Among the manipulation rules in CadL , soundness of the rule C under the intended
interpretation is clear (i.e., any fuzzy valuation v that satisfies
( θ , ζ )
and
( θ
) and so is soundness of ME and AO .Therule
C responds to what is basically called fuzzy modus ponens ,seeforexample[10].
As for the evaluation rules in CadL , AND and OR are sound with respect to the
intended semantics but NOT is not sound (due to the characterization of the truth
values in graded statements as lower-bound thresholds). As shown in [6], soundness
of CadL can be basically provided by restricting the use of the rule NOT along
the inference and by reinterpreting the intended role of the truth degrees in some
graded statements: truth degrees in the graded statements that constitute the input
in a run of the inference engine can be regarded as point values and also those in
graded statements that are obtained from them by the application of any rules in
CadL other than C .Therule NOT would only be applied to these statements (i.e.,
to graded statements where the truth degree is known to represent a point value).
Thus, graded statements obtained as a result of the application of the rule C would
not be used by the rule NOT -for more details on this point and, in general, on the
content of this section see [6]-.
φ , η )
also satisfies
( φ ,
min
( η , ζ ))
22.6
Conclusion
Two semantics have been taken as reference in our attempt to provide an interpre-
tation of the inference process and output of the medical expert system CADIAG2:
 
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