# Re: Ways to evaluate fuzzy predicates

Frank O'Dwyer (fod@brd.ie)
Wed, 16 Dec 1998 23:04:01 +0100 (MET)

WSiler wrote:
(I wrote:)
> >I was wondering what systems people use to evaluate fuzzy predicates. I
> >have seen definitions such as:
> >
> > A AND B=min(A,B)
> > A OR B=max(A,B)
> >
> >and am wondering what alternative definitions exist.
>
> There is a whole family of alternatives, but max-min dominates; with anty other
> AND/OR definitions, the AND of several clauses in a rule with a complex
> antecedent drizzles down to zero, and the OR of several clauses drizzles up to
> one. One alternative is the "probabilistic" a AND b = a*b, a OR b = a+b -
> a*b.
>
> > Are there any means of assigning different weights to A and B, for example?
>
> Not that I know of.

The reason I ask is I am trying to weight different truth values based
on the authority (or relevance) of their sources. So for example if Fred
says the truth value of a proposition P is 0.9 and Joe says it is 0.1,
but I trust Joe more than Fred, then I want to assign P a truth value
nearer 0.1 than 0.9. My thinking was to assign different weights to each
source. Numerically, this would look like:

Suppose Pi={Wi, Vi} where Wi is a weight and Vi is a truth value (both
0..1), all referring to the same proposition.

Then the result P={W,V}=(OR(Wi), SUM(Wi.Vi)/SUM(Wi)}

Where OR is defined probabilistically as you state above, i.e. a OR
b=a+b-a*b. (This is necessary because the various sources may not be
independent--summing or maxing the weights doesn't seem to give good
results.)

Thus, if I want to require 4 independent sources in order to have a
reliable idea of V, I might assign weights of 0.2 to each independent
source and a threshold weight of 0.8. This seems to behave in
accordance with intuition, e.g.:

4 fully independent sources, roughly agreeing (sufficient evidence):
{0.2, 0.9},{0.2,0.8},{0.2,0.85},{0.2, 0.91}=> {0.8,0.865}
4 fully dependent sources, roughly agreeing (insufficient evidence):
{0.2, 0.9},{0.2, 0.9},{0.2,0.85},{0.2, 0.91}=> {0.2,0.865}
4 independent sources, contradicting (sufficient evidence, but vague
conclusion for P):
{0.2,0.9},{0.2,0.9},{0.2,0.1},{0.2,0.1} => {0.4,0.5}

Anyone got any comments on this formulation or know of anything better
that might be used instead? Pointers to any theory in this area would be
greatly appreciated.

> >Can fuzzy implication be dealt with in analogous terms to normal Boolean
> >implication? i.e. does this hold: A->B == NOT(A) OR B == max(1-A,B)
>
> There is a whole slew of fuzzy implications, of which you list one; Klir and
> Yuan list over a dozen. However, their utility in practice is up for grabs.
> This is to complicated to address here - I've got to walk my dogs!

OK :) Thanks for your response.

Cheers,
Frank O'Dwyer.

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