A Method for the Evaluation and Selection of an Appropriate Fuzzy Implication by Using Statistical Data

By Botzoris, G. N.; Papadopoulos, K. et al. | Fuzzy Economic Review, November 2015 | Go to article overview

A Method for the Evaluation and Selection of an Appropriate Fuzzy Implication by Using Statistical Data

JEL Classification: C02, C60, C61

1.INTRODUCTION

We know that the implication in classic logic depends only on whether the premise is true or false. That is, whether the syllogism (reasoning) is true or false depends solely on if the premise and the conclusion is true or false.

Every proposition in classic logic has two values 0 or 1, true or false, holds or does not hold. Let us assume that we have two such propositions p and q. We symbolize the conjunction (AND) of the propositions with λ and the disjunction (OR) with v, while -p is used to symbolize the negation of p (i.e. NOT-p).

The conjunction pAq is true, if and only if both propositions p and q are true. In such a case, it holds that pAq = min {p, q}, (Table 1). Indeed, let p be the proposition "The number 2 is prime" (true) and q the proposition "The number 6 is a multiple of 2" (true). Then, the conjunction pAq: "The number 2 is a prime (true) and the number 6 is a multiple of 2 (true)" has truth value equal to 1.

The disjunction pvq is true, if one of the two propositions is true, that is if it holds that pvq = max {p, q}, (Table 1). Indeed, let p be the proposition "The number 3 is an integer" (true) and q the proposition "The number 16 is a multiple of 5" (false). Then, the disjunction pvq "The number 3 is an integer (true) and the number 16 is a multiple of 5 (false)" has truth value equal to 1.

Let us consider the proposition p "The population of Spain is less than that of China" which is true. The negation of proposition p, i.e. the proposition -p "The population of Spain is greater than or equal to that of China" is false.

For determining the truth value of an implication, (denoted as ^) between two propositions p and q (we assume the implication p^q, i.e. the proposition p implies the proposition q), it would be enough to determine the truth value of the conjunction -pvq (Table 2).

From the last column of Table 2, it is obvious that the implication p^q is always true, except in the case where the proposition p is true and the proposition q is false, i.e. the case where from a false premise, we arrive at a false conclusion.

From Table 2, we also see that whenever we start from a false premise (p=0), the reasoning, i.e. the implication, is true regardless of the conclusion that we arrive at (q=0 or q=1). Finally, another characteristic feature of the classical logic is that the property of symmetry does not hold i.e. the truth value of the implication p^q generally has a different value from the truth value of the implication q^p. Indeed, from Table 2, we can observe that the implication 1 ^0 has truth value of 0, while the symmetric equivalent implication 0^1, has a truth value of 1, [1], [2].

2.THE FUZZY IMPLICATION

The fuzzy implication assigns a truth value J(x,y) to the fuzzy proposition "If p then q"for every truth value x, y of the fuzzy propositions p, q. It is a function of the form J : [0,1]x [0,1] ^[0,1] which satisfies the following nine conditions, every one of which does not contain symmetry [3], [4], [5], [6]:

1. Vx,y,z e [0,1], x < z ^ J(x,y) > J(z,y),

2. Vx,y,z e [0,1], y < z ^ J(x,y) < J(x,z),

3. Vy e [0,1], J(0,y) = 1,

4. Vz e [0,1], J(1, z) = 1,

5. Vx e [0,1], J(x,x)=1,

6. Vx, y, z e[0,1], J(x,J(y,z))=J(y,J(x,z)),

7. Vx,y e [0,1], J(x,y) = 1 when x < y,

8. Vx,y e [0,1], J(x,y) = J(c(y),c(x)), for a fuzzy complement c,

9. J is a continuous function.

In many applications, e.g. in the fuzzy inference system of MATLAB (such an application can be found in [7]), the classic forms of implication, min (Mamdani) [8] and the prod (Larsen) [9] are used as first choices, where: These implications are symmetric, since x^y = y^x, and they are called engineering implications. These implications are widely used in the field of engineering, where the cause and effect are often intertwined, hence the symmetry is acceptable. …

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