Re: prob. vs. poss., again...

Ulf Nordlund (
Fri, 1 May 1998 12:43:12 +0200 (MET DST)

I got an e-mail asking me what the "half-eaten apple" example is all
about. So, here it is, modified from Kosko, 1993, p. 4:

Example.--- No one would question the fact that an apple belongs to the
(crisp) set <apples>. So, if a person displays an apple and asks: ³Is this
an apple?², most people would answer ³Yes². If the person chews a bite
from the apple and repeats the question, the answer would most likely
still be ³Yes². But what happens if the person keeps on eating so that, in
the end, only the core of the apple remains? Is it still an apple? Most
would now say ³No². The question then becomes: When did it pass from being
an apple (belonging to the set <apples>) to a non-apple (belonging to the
complementary set <not apples>)? Although we at all times have all the
facts about the apple --- its shape, its size, its weight, etc. --- and
although we can measure the change in these parameters with any precision,
we still cannot say exactly when it ceased to be an apple. Obviously the
problem cannot be solved using statistics, since the uncertainty involved
is non-random (chance is not involved).
If we instead assume that the person with the apple holds it behind his
back, and asks: ³Do I have the apple in my right or my left hand?², then
chance is indeed involved and we have a random uncertainty (p = 0.5 for
either hand).

(I used this example in an article "Formalizing geological knowledge -
with an example of modeling stratigraphy using Fuzzy Logic" in Journal of
Sedimentary Research, v. 66:4 (1996), p. 689-698.)


   Ulf Nordlund, Inst. of Earth Sciences
   Uppsala University, Sweden