: But how would you define logical consequence in the fuzzy case????
: My first desperate attempt resulted in the following:
: B' is GENERALIZED LOGICAL CONSEQUENCE of P if and only if B' is a
: fuzzy equivalence with B, and that B is true in every model M of P.
: Note that there are no truth values (possibility values) connected to
: A' or A->B.
: ANY BETTER IDEAS??
You may want to see my "Fuzziness and Probability", in which there
is a chapter on Deductive Inference. The approach there
taken involves first distinguishing logical implication from
material implication. The first is a matter of semantic necessity,
the second a matter of semantic contingency. Or put another way,
the first is a necessary consequence of semantic form, the second
a contingency of semantic content. As an example, modus
ponendo ponens is a rule of form, a special tautologically reliable
way of a deriving a semantic consequence purely as a result of the
semantic form of the premises. For example, the two premises
All rich men are happy (Premise 1)
John is rich (Premise 2)
yields tautologically the conclusion
John is happy
without one having to know the meanings (fuzzy or otherwise) of
the terms "rich" and "happy" (also without making any assertions
as to the truth of the premises, only that the premises entail
the conclusion as a purely semantic matter). As Suppes has said,
"only a meager theory of meaning is required to apply the axiomatic
method."
Now, the approximate reasoning pioneered by Zadeh allows us to
draw semantic inference on a different basis. Instead of semantic
form -- and rules that rely on tautological truth irrespective of
underlying content -- we now may rely directly on the semantic
content of the terms involved. The semantic premise "rich"
may be elaborated in the form of a possibility distribution, as
may the other premise "rich materially implies happy". Their
combination, followed by projection onto the happiness domain
yields a possibility distribution on the happiness universe of
discourse, "happy-2", say, that presumably approximates "happy".
That is a result based entirely on notions of semantic content.
Now, form is but a special case of content.
So ask whether, arguing from content as embodied in elaborated
possibility distributions of semantic premises, one may obtain
a result that is true as to form, regardless of the underlying
possibility distributions. This would vindicate the rules of
form arguing now from a fuller theory of meaning as compared
with the meager one of which Suppes spoke.
That is, ask whether modus ponendo ponens (among other tautologies
based on semantic form) may be *derived* as a tautological
consequence, starting now with the possibility distribution as
the bearer of semantic content. Ask whether, for any terms "A",
"B" and "A->B", the composition and projection operations on
the possibility distributions associated with "A" and "A->B"
will yield a possibility distribution which is identical to that
corresponding independently to the term "B".
So, the short answer to your question
: But how would you define logical consequence in the fuzzy case????
is: logical equivalence implies and is implied by equivalence
of semantic *content*, which operationally means strict equality
of possibility distributions. There is also a weaker form of
inference, eg. inferring "not short" from "tall" where the former
in some sense contains, but is not identical to, the latter.
Likewise, "rich and happy" semantically entails "<however wealthy>
and happy", since any possibility distribution for the latter
will presumably contain the former. This is the notion of
"logical consequence", grounded in notions now of semantic
content, which is what is taken as fundamental, as explicated
by possibility theory.
(By the way, Zadeh's possibility theory runs into difficulty as
long as an absolute scale is maintained for the notion of degree
of possibility. In particular, any possibility distribution that
is everywhere less than 1/2 must logically entail its own
negation, which violates at least my intuition. This difficulty
disappears when the notion of relative possibility is put in
its place. There are additional benefits to this reworking as well,
which I won't go into here.)
"Fuzziness and Probability" develops a so-called "fundamental
rule of deduction" from these broad ideas. Some basic theorems
are developed. But modus ponendo ponens alas remains
unproven. (The required algebra looks forbiddingly tedious,
rather than difficult, per se.)
: - Harry
Regards,
S. F. Thomas