Path: utzoo!mnetor!uunet!seismo!sundc!pitstop!sun!decwrl!decvax!ucbvax!calgary.UUCP!gaines
From: gaines@calgary.UUCP (Brian Gaines)
Newsgroups: comp.ai.digest
Subject: Re: FUZZY LOGIC VS. PROBABILITY THEORY
Message-ID: <1382@vaxb.calgary.UUCP>
Date: 23 Feb 88 07:27:37 GMT
References: <8802180658.AA11175@ucbvax.Berkeley.EDU>
Sender: daemon@ucbvax.BERKELEY.EDU
Organization: U. of Calgary, Calgary, Ab.
Lines: 51
Summary: Relations between fuzzy logic and probability theory
Approved: ailist@kl.sri.com

In article <8802180658.AA11175@ucbvax.Berkeley.EDU>, golden@FRODO.STANFORD.EDU
(Richard Golden) writes:
> The basic theoretical result is that selecting a "most probable" conclusion
> for a given set of data is the ONLY RATIONAL selection one can make in 
> an environment characterized by uncertainty.  (Rational selection in this
> case meaning consistency with the classic deductive/symbolic logic - boolean

Boolean logics are not appropriate for knowledge representation if the
underlying domain is truly fuzzy, that is, has borderline case where either,
x and not x are both true, or, x and not x are both false. A classic example
is shades of color that grade into one another.

> algebra.)  Thus, one could argue that if one constrains the class of 
> possible inductive logics to be consistent with the laws of deductive logic
> then Probability Theory is the MOST GENERAL type of inductive logic.
> 
> (iii) F(C and B,A) may be computed from F(C,B and A) and F(B,A)
>       Note this assumption's similarity to Bayes Rule but the 
>       multiplicative property is not assumed.

This is an assumption of truth functionality that is excessively strong.
In general we cannot infer truth values for conjunctions in this way.

> 
> To my knowledge, the axioms of Fuzzy Logic can not be derived from
> consistency conditions generated from the deductive logic so I conclude
> that Fuzzy Logic is not appropriate for inferencing.  Any comments?!!!
> 

There is a strong relation between classical, probability and fuzzy logics.
If one starts with a lattice of propositions and an additive measure over it
such that p(a and b) + p (a or b) = p(a) + p(b), then one gets a general
logic of uncertainty that:
a) Becomes probability logic if you assume excluded middle, ie no borderline
   cases;
b) Becomes fuzzy logic if you assume strong truth functionality, ie p(a and b)
   can be inferred from p(a) and p(b);
c) Becomes standard propositional logic if you assume binary truth values.
Most of the useful results in fuzzy and probability logics can be derived
in the general logic and do not need the restrictive assumptions.
Theorem provers for any of the multivalued logics are essentially
constraint chasers that bound the truth values of propositions, more
like linear programming than classical resolution.

There is a wealth of literature on fuzzy and probability logics -
The journals Fuzzy Sets and Systems, Approximate Reasoning, and Man-Machine
Studies, all carry articles on these issues and applications to
knowledge-based systems.  Noth-Holland have published several books
edited by Gupta, Kandel, Yager, and others, on these topics.

Brian Gaines, gaines@calgary.cdn, (403) 220-5901