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of a
sound
semantics. Two semantics will be taken as reference in our attempt:
probabilistic semantics and fuzzy (
t-norm
-based) semantics -see for example [10]
or [12] for more on
t-norms
and some other concepts mentioned below that are re-
lated to them-. The use of probabilistic semantics is motivated by the natural iden-
tification of the degrees of confirmation in the rules of the system with probabilities
(in principle with frequencies, as suggested in [3], estimated from medical databases
and patient records, although not all degrees of confirmation were obtained in this
way) and the rules themselves with probabilistic conditional statements. The use of
a fuzzy semantics is mostly motivated by the natural identification of the degrees of
presence of symptoms in the patient with
membership degrees
in fuzzy set theory
(also called
truth degrees
) and by some inferential methodology derived from fuzzy
set theory. It is common practice in the field to choose a
t-norm
as the interpre-
tation of the conjunction and its
residuum
as the interpretation of the implication
with which we will characterize the rules of the system: rules in this context will be
formalized as graded implications in the context of many-valued logics, in a sense
that will be made clear later.
The outcome of such an attempt can be at least partially anticipated. The infer-
ence mechanism in CADIAG2 is partially based on methodology from fuzzy set
theory and thus it is bound to be unsound with respect to probabilistic semantics.
Some aspects of the probabilistic unsoundness of the inference mechanism in CA-
DIAG2 were soon observed in earlier studies concerning the celebrated rule-based
expert system MYCIN (that shares some background methodology with CADIAG2)
-see [4] or [21] for a description of MYCIN, [8] for a comparison of CADIAG2 and
MYCIN-like systems and [11], [13], [14], [22] for some probabilistic approaches to
it-. However, it remains to be seen how far the system CADIAG2 is from proba-
bilistic soundness and thus how much of its inference process could be interpreted
probabilistically. A better match of the inference process may be expected with re-
spect to some
t-norm
-based fuzzy semantics (in particular, as will be clear later, with
the semantics based on the indentification of the
t-norm
with the
minimum
operator)
yet, as with probabilistic semantics, it needs to be seen how much of the inference
process can be interpreted on the grounds of this semantics.
It is worth mentioning here that, although the interest among theoretical AI re-
searchers in rule-based expert systems seems to be lesser today than some years ago,
rule-based expert systems are very popular among AI engineers. Many CADIAG2-
like systems are in use and more are being built for future implementation. It is
mainly for this reason that we believe that further analysis and understanding of
CADIAG2-like systems is of relevance (CADIAG2 is presented in some mono-
graphs as an example of a fuzzy expert system -for example in [15] or [25]- and
thus is used as a reference for some newly developed knowledge-based systems).
The paper is structured as follows: in Section 22.2 we introduce some notation
and give some preliminary definitions necessary for the description of the inference
mechanism of the medical system CADIAG2, which is done in Section 22.3, where
we also describe the knowledge base of the system. In Section 22.4 we introduce
the logical system
CadL
that consists of a collection of rules that formalize the steps
