In natural language, we often make frequent use of comparative
propositions. For example, *redder than*, *taller than*, *prettier
than*, *older than*. Comparatives are two-place relations of the form
*F-er than* where F denotes a property.
Comparatives of genuine properties will be asymmetric and transitive as
matters of logic.
So we have . . . . . . .
Asymmetry: If A is F-er than B, then it is false that B is F-er than A
Transitivity: If A is F-er than B, and B is F-er than C, then A is F-er
than C.
Some comparatives will also be complete, but some will not.
Complete: For all A, B in X either A is F-er than B, B is F-er than A,
or A and B are equally as F.
This is true, for example, when we have *older than*, *taller than*, but
is not true for things like *prettier than*, *better than*. It is often
indeterminate as to which object is prettier than which, and is also
often indeterminate as to which social policy is better than which.
Objects may be incommensurate in their prettiness and social policies
may
be incommensurate in their goodness.
Let's fix some notation. F-er than is a *partial ordering* if F-er than
is asymmetric and transitive, and F-er than is a *linear ordering* if
F-er than is a complete partial ordering.
My question is this: what is the logical form of comparatives like
these, and can fuzzy set theory supply it ??
On my reading of fuzzy set theory, an object is assigned a degree of
F-ness. Typically, this is a number between 0 and 1. So let's introduce
a function J that assigns to an object and a predicate a degree.
I shall write J(x, F) to denote the degree to which x is F. We can
assume, for the moment, that F is a fuzzy set like *tall*.
Zadeh tells us in his original paper that the higher J(x, F) is to 1 the
higher the grade of membership of x to F. The only way to read this in
my mind is that the higher J(x, F) is to one, the higher is x's degree
of F-ness. So a natural way to interpret comparatives using the
language of degrees is as follows:
(1) A is F-er than B if and only if J(A, F) > J(B, F)
where > denotes *greater than*. So if F is the fuzzy set tall, and if
Alice is 0.8 and Bob is 0.6 then the only possible interpretation is
that Alice is taller than Bob. I don't know how else to interpret
Zadeh's statement that the higher J(x, F) is to 1, then the higher the
grade of membership of x to F. Alice belongs to the set of tall people
more than Bob does, so Alice's degree of tallness is greater than Bob's,
and so Alice is plain taller than Bob. All of these last three things
seem to me to be equivalent in meaning. So (1) must be true.
The problem is that fuzzy set theory violates (1). Imagine that Jim and
Joe are both plain tall, Jim is 6ft 5in and Joe 6ft 8in. Joe is taller
than Jim, but both are plainly tall. So both are assigned a degree of
tallness of one. But this contradicts (1). Give that (1) is true, fuzzy
set theory cannot give a satisfactory account of natural language
comparatives in terms of degrees.
One possible solution, which retains the attractiveness of (1) is the
following. I'll sketch it. I'm not sure whether this solution takes us
outside fuzzy mathematics.
Again, through a function J assign to an object, x, and a predicate, F,
a degree. This degree is an element of a set, call it L. Note that F
need not be a vague property. Take *acidic*. Acidic is not a vague
property, there is a sharp borderline between acidic and non-acidic.
But one thing can be more acidic than another, so there are *degrees* of
acidity even amongst those things acidic. So the existence of degrees
is independent of whether a property is vague or not.
The important thing about the elements of L is that they are partially
ordered by some relation R. So the set of degrees need not be
numerical, thus it does not have to be [0,1]. We write J(x,F) R J(y,F)
meaning that the degree to which x is F is greater than the degree to
which y is F. The assignment of degrees through J is not arbitrary, but
is constrained by the structure of the comparative *F-er than*. So we
can accept (1) as an axiom
(1*) A is F-er than B iff J(A,F) R J(B,F)
What this does is to tie in the structure of *F-er than* with the
partial ordering of degrees R. If *F-er than* is a partial ordering then
so is R. If F-er than is a linear ordering (like *taller than*) then the
elements of L will be linearly ordered by R too. But as I said, many
comparatives, like *prettier than*, will not be complete in which case
objects are assigned incomparable degrees. Requiring L to be a
partially ordered set permits this generalisation. If L was [0,1] then
we would be forcing all comparatives to be linear orderings and we do
not want that.
Imagine now that we want to consider the fact that some things are F and
some things are not-F but we do not know where this threshold lies.
(2) There exists some A, such that A is F
This enables us to say the following
(3) There exists some J(A,F) such that any S in X for which J(S,F) R
J(A,F) implies that S is F
So, for example, the fact that some people are tall does not effect the
underlying degrees of tallness. If A is taller than B, and we know that
B is tall we infer from (3) that A is tall. A and B still have different
degrees of tallness. But the vagueness of *tall* is preserved by (2) -
it does not specify where the threshold lies.
So rather than assign objects degrees and trying then to construct
comparatives from them (as in my interpretation of fuzzy set theory), it
seems more fruitful to take the comparative as the primitive and
construct a set of degrees and a partial ordering of them that is
consistent with it. This makes things satisfy (1) and I think that any
plausible theory of degrees must be compatible with (1).
Does anyone have an opinion on this? I'm not sure whether this revised
approach is inside or outside fuzzy set theory. Also, I notice looking
through Zadeh's Collected Papers that he never mentions comparatives at
all. This is surprising since he applies his theories to almost all
other constructions in natural language. Is this because the theory was
never intended to be applied to comparatives, or perhaps he is aware of
the problems with doing so ?
All the best
Ashley
Department of Economics
University of Bristol
Bristol BS8 1TN
ecpg487@ssa.bris.ac.uk