golden@FRODO.STANFORD.EDU (Richard Golden) (02/16/88)
I am not an expert in Fuzzy Logic or Probability Theory but I have examined
the literature regarding the foundations of Probability Theory and the
derivation of these foundations from basic principles of deductive logic.
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
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.
The reference from which these arguments are based is given
by Cox (1946). Probability Frequency and reasonable expectation.
American Journal of Statistical Physics, 14, 1-13.
The argument is based upon the following hypotheses:
(i) The belief of the event B given A may be represented by a
real-valued function F(B,A).
(ii) F(~B,A) may be computed from F(B,A)
(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.
(iv) Assumptions (i), (ii), and (iii) must be consistent with the
laws of Boolean Algebra (i.e., deductive/symbolic logic).
From these assumptions one can prove that F(B,A) must be equivalent
to the conditional probability of B given A. That is, F(B,A) must
lie between a maximum and minimum value (say 1 and 0) and the sum of
all possible values for B for a particular value of A must equal the
maximum value (1). Note that we are taking the subjectivist view of
probability theory and we are NOT interpreting the probability of
an event as the limiting value of the relative frequency of an event.
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?!!!
Richard Golden
Psychology Department
Stanford University
Stanford, CA 94305 GOLDEN@PSYCH.STANFORD.EDU
Cc: emneufeld@watdragon.waterloo.EDU (Eric Neufeld) (02/19/88)
In article <8802180658.AA11175@ucbvax.Berkeley.EDU> golden@FRODO.STANFORD.EDU (Richard Golden) writes: >I am not an expert in Fuzzy Logic or Probability Theory but I have examined >the literature regarding the foundations of Probability Theory and the >derivation of these foundations from basic principles of deductive logic. >[...] >The reference from which these arguments are based is given >by Cox (1946). Probability Frequency and reasonable expectation. >American Journal of Statistical Physics, 14, 1-13. >[...] >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?!!! The interest in reasoning with and about uncertainty in AI has sparked a re-investigation into foundations of Prob. Theory. Cox's theorem has become an important result for those interested in prob. theory as a measure of belief. You may be interested in the following references: Proceedings of the AAAI Workshop on Uncertainty and AI: 1985, 1986, 1987. A number of articles investigate the relationship between formal prob. theory and the various alternate formalisms: Fuzzy, Certainty Factors, Dempster-Shafer. Note articles by Cheeseman, Grosof, Heckerman and Horvitz. Kyburg, Henry E- Bayesian and Non-Bayesian Evidential Updating, AI Journal, Vol 31, 1987. Investigates probabilistic assumptions underlying Dempster-Shafer Theory. Heckerman et al: AAAI-86: "A Framework for comparing alternate formalisms..." Has been described as a presentation of Cox's result to the AI community. Computational Intelligence: Upcoming issue (delayed in printing) Contains a polemic article by Peter Cheeseman on probability theory (versus everything in the world) and responses by various researchers, etc. Aleliunas, Romas: "Mathematical Models of Reasoning". Contains a generalization of Cox's result to topologies other than real-valued continuous [0,1] probability. University of Waterloo Tech Report. Eric Neufeld Dept. Computer Science University of Waterloo Waterloo Canada
ST401843@BROWNVM.BITNET (02/21/88)
Here is my two bits about fuzzy logic: Richard Golden writes: >...Rational selection in this case meaning consistency with the classic >deductive/symbolic logic - boolean algebra... But here's the rub! In Boolean Algebra you have clear cut True or False truth values.In Crisp Set Theory (to a subclass of which theory Boolean Algebra is isomorphic) you have clear cut "x belongs A" relationships. Write this as B(x,A)=1 , where x is an element, A is a set and B(.,A) takes values (for "belongs" and 0 for "not belongs". In other words B(.,A) is the "belongingness" function of A. In Fuzzy Set Theory on the other hand, B(.,A) can take any value between 0 and 1. There are no clear cut answers to questions such as: "does a person 5 ft. 10 in. tall belong to the set of tall persons?". Cisp Set Theory is what we traditionally call Set Theory. It is a subset of Fuzzy Set Theory, in that in CST B can take only the extreme values 0 and 1. Of course in all of FST we use the reasoning, syllogisms etc. of the Boolean Algebra. I don't know if anyone does Fuzzy Mathematics. In FM, things such as reasoning by Reductio ad Absurdum would not be valid. Why? Well, in RaA we usually want to prove something for x, call it P(x). And we begin by saying: "assume NOT P(x)..." But in FST P(x) and NOT P(x) are just two of uncountably many possibilities. But maybe ther is a Fuzzy generalization to RaA. Does anyone know? Anyways, I think FST is still an alternative way to reasoning about uncertain events, different from Probability Theory. In fact there has been work done on Fuzzy Probability, postulating fuzzy probability measures. I did not have time to check before I send this, but I suppose they would drop things like the countable additivity hypothesis. And classical PT is by no means the only way to reason about uncertain events- see von Mises and deFinneti, among others. One last observation. I do not want to make much of it, but is it not remarkable that FST, an alternative to the Aristotelian logic, was invented by Zadeh, an Indian? Thanasis Kehagias
gaines@calgary.UUCP (Brian Gaines) (02/23/88)
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