[FOM] primer on vagueness

Lawrence Stout lstout at iwu.edu
Tue May 24 10:05:22 EDT 2005

Some comments on the fuzzy analysis of the baldness problem given by 
Stewart Shapiro.

  The use of the connectives max(a,b), min(a,b) and 1-a for "a or b", "a 
and b", and "not a" in fuzzy set theory goes back to the beginning of 
the subject, but is not what is frequently used in fuzzy circles (at 
least in some fuzzy circles) today.  Because the 1-a negation does not 
come from min using the residuation (an intuitionistic negation does 
instead) we often use the Łukaciewicz "and " instead a & b = 
max(0,a+b-1), together with the implication given by residuation a->b = 
min(1, b-a+1).  This gives 1-a as the negation a->0.  The Łukaciewcz 
connectives are certainly not the only ones used in Fuzzy sets (so is 
the Gödel conjunction using product; in general any t-norm may be used 
for a conjunction).  The Łukaciewicz & is nilpotent:  if you take any 
truth a with truth value strictly less than 1 and take its conjunction 
with itself enough times you get a truth value of 0.  The Gödel 
conjunction will make the truth value of multiple conjunctions 
arbitrarily small, though non-zero.

The inference from a and a->b to b becomes a statement of the form |a| 
& |a->b|≤|b| where I use |a| for the truth value of a.

Suppose that the truth value of mk being bald is Bmk=1-k/1000.
The chain of reasoning which starts with m0 being bald with truth value 
1 and then uses a chain of inferences

Bm0			with truth value 1
Bm0->Bm1		with truth value min(1,.999-1+1)=.999
Bmk->Bm(k+1) with truth value min(1, 1-(k+1)/1000 -(1-k/1000)+1)=.999
Bm999->Bm1000 with truth value .999

The inference chain then becomes 1&.999&.999&... &.999 ≤ Bm1000.
Now that includes 1000 terms of size .999 and, as it happens, 
.999&.999& ... &.999 k times (for k≤1000) gives 1-.001k.  Thus what we 
have is the conclusion that 0≤0, which is non paradoxical.

The use of the Łukaciewcz connectives allows us to have a loss of 
information at each inference about vague propositions.

Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University
Bloomington, IL 61702-2900

On Thursday, May 19, 2005, at 11:21 PM, Stewart Shapiro wrote:

> Recall the two versions of the sorites argument:
> ....
> Bm1
> Bm1 ->  Bm2
> Bm2 ->  Bm3
>       . . .
> Bm999 ->  Bm1000
> therefore Bm1000
> In both cases, the first premise has truth value 1, and the conclusion 
> has
> truth value 0.  The first few conditionals and the last few 
> conditionals
> each have truth value 1.  Given the assumption about the differences
> between adjacent formulas, each of the conditionals has truth value of 
> at
> least .998, and some are not fully true.  So the second premise of the
> first argument has truth value somewhere between .998 and .999.
> On the first definition of validity (as the necessary preservation of
> complete truth), both arguments are valid.  The only inferences used 
> are
> modus ponens and (for the first argument) universal elimination, and 
> both
> are valid on that conception of validity.  But neither argument is 
> sound,
> since it has premises that are less than fully true.
> One puzzling feature is that we can have a valid argument whose 
> premises
> are almost completely true (at least .998), but which has a completely
> false conclusion.

More information about the FOM mailing list