On the Roots of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 2

What is it meant by the predicate odd? In principle, it depends on where it is
used. With natural numbers n ,' n is odd ' is used accordingly with the mathematical
definition/rule ' n is odd if and only if once divided by 2 the rest is 1'. With people,
things, or situations, odd does coincide with the meaning of 'strange', 'separated',
and 'not often'. In the first context, the predicate odd is precise or crisp, since natural
numbers are only odd or not odd, but in other contexts the predicate is imprecise or
fuzzy, since for example, 'this is an odd book', or 'that is an odd event', could admit
degrees depending on to what extent the topic or the event can be qualified as odd.
When the predicate is imprecise there is not a perfect classification of the objects to
which it refers to.
Only after a predicate acquires meaning, concepts like 'tallness', 'oddity',
'heaviness', etc., appear in the corresponding language. Predicates do appear in
language after being used, and only after being used they can evolve in new con-
texts and give birth to concepts. Like it happened with 'high' and 'highness', and
'royal highness'. Concepts like uncertainty come from a mother-predicate as it is
'uncertain' in this case.
Notice that there are not natural numbers that can be qualified as 'very odd', or
'more or less odd', but there are buildings that can be called to be so. If the predicate
is crisp, that is, it names just an either yes, or not, property, the application to it of a
linguistic modifier needs to be newly defined, but if the predicate is imprecise it does
not, since people immediately understand what it is meant by, for instance, 'very odd'.
In what follows we will only deal with predicates P , the name of a property on
a previously given set X

={ , , ,... }

, and considering the use of P through the
elemental statements ' x is P ', for all x in X , and accepting (à la Wittgenstein) that
the meaning of P is its use in the current language. Hence, the first problem to tackle
is placed by the following question, How the use , or meaning, of P on X can be
mathematically represented or modeled ?
First, and to distinguish between two statements ' x is P ' and ' y is P ', for x

x y z

y
in X , let us suppose it is possible to decide when is ' x is less P than y ', where x
shows the property named P less than y shows it.
Let us call

=

P the relation in X given by

x P y ⃔ x is less P than y ,

and suppose P is a preorder (enjoys the reflexive and transitive properties). That is,

• x P x , for all x in X

• If x P y , and y P z , then x P z .

The preordered set ( X , P ) reflects the organization P induces in X , and the

− 1
P

− 1
P

preorder P is the primary use of P in X . The relation is defined by ' x

P is, usually, empirically perceived; it could
also be called the perceptive meaning of P in X .
We will say that ' x is equally P than y ' whenever x

y ⃔ y P x '. Note that the relation

P y and y P x , and write

( = P ) = ( P ∩ − 1

x = P y , with ) . Since, obviously,

P