Fuzzy Arithmetic - Fuzzy Logic: An Introductory Course for Engineering Students - page 150

6.4 A Note on Fuzzy Quantifiers

In classical logic, only two quantifiers are considered. The universal quantifier ∀ (for

all), and ∃!

(exists only one). For example, given a sequence of real numbers ( a n ), it is said that
the real number a is its limit , when

∃ (it exists, or for some), the existential quantifier, with the addition of

∀ > 0 , ∃ k ∈ N , ∀ n ∈ N :[ ( n > k ) ₒ ( | a n − a | < ) ] .

Analogously, a function f :[ a , b ]ₒR is bounded , when

∈ R + , ∀ ∃ M x ∈[ a , b ]:[| f ( x ) | < M ] .

The importance of these two quantifiers to clearly write mathematical expressions
does not need to be stressed. Nevertheless, both in arithmetic computing and in natural
language more quantifiers are needed and used.

Examples of arithmetical quantifiers are the percentages. For example

• The 85% of the employees are married

• Between the 40 and the 70% of the employees are single.

For example, if it is known that

The 35% of the employees are married
The 25% of the married employees are young
What can be said on the employees that are young?

A question which answer is, obviously, 35 × 25 = 875, that is, at least the 8 . 75%

of the employees are young.

Another example is
Between 15 and 25 employees are married
Between 5 and 10 married employees are young
What can be said on the employees that are young?

in which what matters is the length of the two intervals [ 15 , 25 ] ( l = 10 ) , and

[ 5 , 10 ] ( l = 5 ) , that give at least, the interval [ 5 , 10 + ( 10 − 5 ) ]=[ 5 , 15 ] of

employees that are young.
In natural language imprecise quantifiers like 'about five', 'about half', 'most',
etc., appear and can be represented by means of fuzzy numbers. For example,