Fuzzy Logic - Quantification

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In modeling natural language statements, quantified statements play an important role. It means that NL heavily depends on quantifying construction which often includes fuzzy concepts like “almost all”, “many”, etc. Following are a few examples of quantifying propositions −

• Every student passed the exam.
• Every sport car is expensive.
• Many students passed the exam.
• Many sports cars are expensive.

In the above examples, the quantifiers “Every” and “Many” are applied to the crisp restrictions “students” as well as crisp scope “(person who)passed the exam” and “cars” as well as crisp scope ”sports”.

Fuzzy Events, Fuzzy Means and Fuzzy Variances

With the help of an example, we can understand the above concepts. Let us assume that we are a shareholder of a company named ABC. And at present the company is selling each of its share for ₹40. There are three different companies whose business is similar to ABC but these are offering their shares at different rates - ₹100 a share, ₹85 a share and ₹60 a share respectively.

Now the probability distribution of this price takeover is as follows −

Now, from the standard probability theory, the above distribution gives a mean of expected price as below −

$100 × 0.3 + 85 × 0.5 + 60 × 0.2 = 84.5$

And, from the standard probability theory, the above distribution gives a variance of expected price as below −

$(100 − 84.5)2 × 0.3 + (85 − 84.5)2 × 0.5 + (60 − 84.5)2 × 0.2 = 124.825$

Suppose the degree of membership of 100 in this set is 0.7, that of 85 is 1, and the degree of membership is 0.5 for the value 60. These can be reflected in the following fuzzy set −

$$\left \{ \frac{0.7}{100}, \: \frac{1}{85}, \: \frac{0.5}{60}, \right \}$$

The fuzzy set obtained in this manner is called a fuzzy event.

We want the probability of the fuzzy event for which our calculation gives −

$0.7 × 0.3 + 1 × 0.5 + 0.5 × 0.2 = 0.21 + 0.5 + 0.1 = 0.81$

Now, we need to calculate the fuzzy mean and the fuzzy variance, the calculation is as follows −

Fuzzy_mean $= \left ( \frac{1}{0.81} \right ) × (100 × 0.7 × 0.3 + 85 × 1 × 0.5 + 60 × 0.5 × 0.2)$

$= 85.8$

Fuzzy_Variance $= 7496.91 − 7361.91 = 135.27$