On the Roots of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 27

= − 1 = =[ , ] P

For example, if P small on X 0 1 is with (the reverse linear

= − 1 = =[ , ] Q

order on the real line), and Q short on Y 0 10 is with (also the

reverse linear order on the real line), u ( x ) = 10 x gives

− 1 − 1 • x y ⃔ 10 x 10 y

Q ∩ − Q equal to the identity (=) on the real line. Then,
small and short can be considered a pair of u -primary synonyms.
If P acts on X with an

∩ − 1
P

taking P and

L − degree μ P , Q acts on Y and is a u -primary synonym

of P , from

u − 1 y 1 ) P u − 1 u − 1 u − 1 y 1 Q y 2 ⃔ ( ( y 2 ) ⃒ μ P ( ( y 1 )) μ P ( ( y 2 )),

it follows that

u − 1 μ Q = μ P ⓦ

is an L − degree for Q . In this situation it is

u − 1 y 1 ) μ P u − 1

y 1 μ Q y 2 ⃔ ( ( y 2 ),

or

x 1 μ P x 2 ⃔ u ( x 1 ) μ Q u ( x 2 ),

that are equivalent to

μ Q = μ P ⓦ ( × ).

u u

For example, with the before mentioned predicates short and small ,itis

μ short ( y ) = μ small ( y / 10 )

for all y in [ 0 , 10 ] , and results

y 1 μ Q y 2 ⃔ y 1 / 10 μ P y 2 / 10 .

Remark 1.4.4 Whenever P and Q are u -synonyms, it could be stated that

“ P means Q ”.

Remark 1.4.5 The definition of primary meaning is just a formal one trying to
approach an important aspect of the meaning of linguistic predicates, when act-
ing on a given universe of discourse. The same can be said about the definition of
u -primary synonyms with which it does not hold, in general, that a pair of linguistic
synonyms are necessarily u primary synonyms. Anyway, what can be said is that

Q is a migration of P to the universe Y .

−

Remark 1.4.6 In some way, the current meaning of a predicate, the form in which it
is used today in the plane language, partially inherits its past meanings.