In our daily life we often make inferences whose antecedents and consequents contain fuzzy concepts. Such an inference cannot be made adequately by the methods which are based either on classical two valued logic or on many valued logic. In order to make such an inference, Zadeh suggested an inference rule called “compositional rule of inference”. Using this inference rule, he, Mamdani, Mizumoto et al., R. Aliev and A. Tserkovny suggested several methods for fuzzy reasoning in which the antecedent contain a conditional proposition with fuzzy concepts:

Ant 1: If x is P then y is Q

Ant 2: x is P'

---------------------------------- (1.1)

Cons: y is Q'.

Those methods are based on an implication operator in various fuzzy logics. This matter has been under discussion for the last couple decades [1] - [46] .

In (1.1) x and y are the names of objects, and P, P', Q and Q' are fuzzy concepts represented by fuzzy sets in universe of discourse U, U, V and V, respectively. This form of inference may be viewed as a generalized modus ponens which reduces to modus ponens when and. Let P and Q be fuzzy sets in U and V respectively and correspondent fuzzy sets be represented as such, , where

And let and be Cartesian product, union, intersection, complement and bounded-sum for fuzzy sets, respectively. Then the following fuzzy relations in can be derived from fuzzy conditional proposition “If x is P then y is Q” in Ant 1 of (1.1). The fuzzy relations and were proposed by Zadeh [42] [43] [44] [45] , by Mamdani [29] , , are by Mizumoto [32] [33] , , , and are by Aliev and Tserkovny [2] [3] [4] [5] , which are

where

where

where

where

A necessary consideration for this discussion is that with the only few exceptions for S-logic (1.6) and G-logic (1.7), and L1-L4(1.3)-(1.6) all other known fuzzy logics don’t satisfy either the classical “modus-ponens” principal, or other criteria which fit human intuition and first formulated in [32] . The proposed fuzzy logic has an implication operator, which satisfies the “modus-ponens” principal and criteria, which fit human intuition.

The second section of the article will cover some initial fuzzy logic creation considerations. In third section a set of operations in proposed fuzzy logic is presented. The fourth section is devoted to an introduction of a t-norm as a graduated conjunction operator in presented fuzzy logic. The Section five will cover a power sets based features of proposed fuzzy logic. The statistical analysis of the fuzzy logic is completed in Section six. Section Seven covers the issue of fuzzy conditional inference rules based on proposed fuzzy logic and extended investigation of its features.

In order to start formulating of a fuzzy logic major implication operator, we are proposing the following function as a part of it:

Definition 1.

An implication function is a continuous function from into such that the following properties hold for:

(I1) If then;

(I2) If then;

(I3), (Falsity Principle);

(I4), (Neutrality Principle);

(I5), (Exchange Principle);

(I6). (Contra positive Symmetry Principle), where -is a negation, which could be defined for as.

Before proving that defined as

and is from (2.1) satisfies (I1)-(I6) axioms, let us show some basic operations in proposed fuzzy logic.

Let us designate the truth values of logical antecedent and consequent as and respectively. Then relevant set of proposed fuzzy logic operators are shown in Table 1.

In other words we propose a new many-valued system, characterized by the set of base union () and intersection () operations with relevant complement, defined as. In addition, the operators and are

expressed as negations of the and correspondingly. It is a well-known fact that the operation implication in a fuzzy logic was the foundation of decision making procedure for numerous approximate reasoning tasks. Therefore let us prove that proposed implication operation from (2.2) satisfies axioms (I1)- (I6). For this matter let us pose the problem very explicitly. We are working in many-valued system, which for present purposes is all or some of the real interval. As was mentioned in [1] , the rationales there are more than ample for our purposes in very much of practice, the following set of 11 values is quite sufficient, and we shall use this set V₁₁in our illustration. Table 2 shows the operation implication in proposed fuzzy logic.

Theorem 1. Let a continuous function is defined in (2.3) and its values are from a Table 2, i.e.

Where function is defined in (1), then axioms (I1)-(I6) are satisfied and, therefore it is an implication operation.

Where function is defined in (1), then axioms (I1)-(I6) are satisfied and, therefore it is an implication operation.

Proof:

(I1):

whereas

(I2):

whereas

(I3):;

(I4):;

(I5): Since, then

Whereas

(I6):

Q.E.D.

In addition proposed fuzzy logic is characterized by the following features:

Commutativity for both conjunction ()and disjunction () operations, i.e.: and;

Assotiativity for these operations:

and

Distributivity:

To prove the feature (3.4) note that

On the other hand

To prove the Feature (3.5) by using (3.7) we have

Therefore the Expression (3.8) equals (3.9) Q.E.D.

DeMorgan theorems, which are extrapolated over the:

To prove these theorems notice that

On the other hand

Therefore the Expression (3.10) equals (3.11) Q.E.D. By analogous

On the other hand

Therefore the Expression (3.12) equals (3.13) Q.E.D. It should be mentioned that proposed fuzzy logic could also be characterized by yet another featured. As a conclusion we should admit that all above features confirm that resulting system can be applied to V₁₁for every finite and infinite n up to that (V_11,) is then closed under all its operations.

Proposition.

In proposed fuzzy logic the operation of conjunction

is a t-norm.

Proof:

The function is a t-norm if the following is true

1) Commutativity:

2) Associativity:

3) Monotonity:

4) Neutrality:

5) Absorption

Commutativity:

and

therefore

(Q.E.D.).

Associativity:

Case:

From where we have that

For the case, where

Using (4.3) we are getting similar to (4.2) results

(Q.E.D.).

Monotonity:

If then given

and

we are getting the following

(Q.E.D.).

Neutrality:

(Q.E.D.).

Absorption:

(Q.E.D.).

Definition 2. [9]

Given a fuzzy implication operator and a fuzzy subset of a crisp universe, the fuzzy power set of is given by the membership function, with

The degree to which is subset of is

Definition 3. [9]

Where conditions are as in Definition 2, the degree to which the fuzzy sets and is the same, or their degree of sameness, is

The following is then immediate.

Proposition 1. [9]

Based on (5.1)-(5.3) and taken into account (3.1) we can formulate the following.

Proposition 2. [9] (Degree of possibility of set-inclusion)

Proposition 3. (Degree of possibility of set-equality)

From (4-4) is clear that in case when. As it was mentioned in [9] there seem to be two plausible ways to define the degree to which sets and may be said to be disjointed. One is the degree to which each is a subset of the other’s complement. The second is the degree to which their intersection is empty.

Definition 4. [9] The degree of disjointness of and, or degree to which and are disjointed, in the first and second sense, are

(2).

For the case (1)

Therefore from (5.5) and (5.6) definition (1) looks like

For the case (2)

Definition 5. [9] (Degree to which a set is a subset of its complement). The expression

Definition 6. [9] (Degree to which a set is disjointed from its complement, in the two senses). From (5.7), (5.8) the following is taking place

The value of, whereas.

In this chapter we discuss some properties of proposed fuzzy implication operator (3.1), assuming that the two propositions (antecedent/consequent) in a given compound proposition are independent of each other and the truth values of the propositions are uniformly distributed [20] on the interval [0,1]. In other words we assume that the propositions and are independent of each other and the truth values and are uniformly distributed across the interval [0, 1]. Let,. Then the value of the implication is some function.

Because and are assumed to be uniformly and independently distributed across [0, 1], the expected value of the implication is

And its variance is

where. From (6.1) and given Expression (3.1) and the fact that

we have the following

But from Table 2 it is clear that for we have

because of the following

and, therefore

Given (6.3) the following is true

Whereas

From (6.4) and (6.5). Whereas. Let us notice that for the most implications we have the following

From (6.2) we have

From (6.6) finally we have

Whereas.

Therefore From (6.2) we have

Both values of and demonstrate that the proposed fuzzy implication operator could be considered as a second of the fuzziest implication from the list [34] of known so far. In addition to that feature it satisfies the set of important Criteria I-IV, which is not the case for the most above mentioned implication operators.

As it was mentioned in [32] in the semantics of natural language there exist vast amounts of concepts and we humans very often make inferences antecedents and consequences of which contain fuzzy concepts. Therefore, from the standpoint of artificial intelligence, it seems that formalization of inference methods for such inferences is very important. Following a well-known pattern, established a couple of decades ago and the standard approaches toward such formalization, let and (from now on) be two universes of discourses and correspondent fuzzy sets be represented as such

,

where

and

Whereas given (7.1) and (7.2) a binary relationship for the fuzzy conditional proposition of the type: “If x is P then y is Q” for proposed fuzzy logic is defined as

Given (3.1) expression (7.3) looks like

It is well known that given a unary relationship one can obtain the consequence by applying compositional rule of inference (CRI) to and of Type (7.3):

In order that Criterion I is satisfied, that is from (7.5) the equality

must be satisfied for arbitrary in and in order that the equality (7.6) is satisfied, it is necessary that the inequality

holds for arbitrary and. Let us define new methods of fuzzy conditional inference of the following type:

Ant 1: If x is P then y is Q

Ant2: x is P'

---------------------------------- (7.8)

Cons: y is Q'.

Where and, which requires the satisfaction of Criteria I-IV from Appendix. It is clear that (6.8) is translated by Expression (7.5), and into (7.8).

Theorem 2.

If fuzzy sets and are defined as (7.1) and (7.2) respectively and is defined by the following

where

then Criteria I, II, III and IV-1 [32] are satisfied.

Proof:

For Criteria I-III let then

From (7.10) and given subsets from (7.11) we have

Let us introduce the following function (as a part of implication operation)

Then the following is taking place:

From (7.14) and (7.15) we have

(Q.E.D.).

For Criteria IV-2 [19] let then

From (7.16) and given subsets from (7.11) we have

Apparently the following is taking place

therefore

Theorem 3.

If fuzzy sets and are defined as (7.1) and (7.2) respectively and is defined by the following

where

Then Criteria I, II, III and IV-2 [32] are satisfied.

Proof:

Let us introduce the following functions

Therefore from (7.18)-(7.20) for Criteria I-III let then

From (7.20), (7.21) and given subsets from (7.19) we have

Then again the following is taking place:

From (7.23) - (7.25) we have

(Q.E.D.).

For Criteria IV-2 let then

From (7.22), (7.26) and given subsets from (7.19) we have

Apparently the following is taking place

and

therefore

For many real practical applications a decision making apparatus could be based not on a fuzzy conditional proposition of the type: “If x is P then y is Q”, but rather on a rule of the following type

And correspondent fuzzy sets are represented as such that

Given (7.1) and (7.2) a binary relationship for a fuzzy conditional proposition of the Type (7.28) for proposed fuzzy logic is defined as

Theorem 4. If a fuzzy conditional proposition is defined as (7.28), correspondent fuzzy sets of antecedents and consequent are presented as (7.29) and a binary relationship for a fuzzy conditional proposition is from (7.30), and “elementary” binary relationships are defined as following

then the following expression is taking place

and as a result the following is also true.

Proof:

Let ; and let us denote each

whereas

From (7.34)

Since

From (7.35)

Both (7.35) and (7.37) prove (7.33) (Q.E.D.)

Let us consider more complex fuzzy conditional proposition of the type:

For

where are normal fuzzy sets of a type (7.1) and (7.2)

, (7.39)

with unimodal membership functions and are countable, finite universes of discourses, i.e. . Note that a unimodality of (7.39) means that for all singletons the following is taking place

and

Note that each rule of type (7.38) looks like that

Let us point to the fact that (7.40) is a mathematical representation of the following fuzzy conditional rule:

Ant 1:

Ant2:

------------------------------------------------------------- (7.41)

Cons:

In traditional way (7.41) looks like:

But for we have

Let us define

from (7.43) we get

Suppose

and (7.45)

Proposition 4. For discussed fuzzy logic a fuzzy conditional rule of Type (7.41) satisfies the following feature

Let us call the binary relationship matrix from (7.42) by “elementary knowledge” (EK). Each EK is characterized by the following features

Same is also true for. Since both membership functions are unimodal, then

and. (The same is true for).

It is apparent that both. In practical terms it means that if, then either

or

whereas if, then either

or

Definition 7. The EK of the type from (7.42) is called logically contradictive or fruitless if a membership function of a consequent in fuzzy conditional rule (7.41) is non-unimodal or when two different antecedents induct the same consequent.

Given (7.46)-(7.48), based on defined earlier sets and taking into account that suppose that

then

*From now on upper and lower indices of based variables u and v denote not a membership functions, but their correspondent singletons

It is clear, that for from (7.45)

where -are traces of matrixes and

1. Let, then

2. Let, then

3. Let, then either

and, which means that there is a major diagonal of a matrix consists of singles (1), or

which means that there is a peripheral diagonal of a matrix consists of singles. Taking into account (7.49)-(7.54) we put together the following

Definition 8.

If are normal fuzzy sets of a type (7.39) and the sets are defined as the follows , then binary relationship matrix of type (7.40) or its counterpart, have major diagonal, which consists of singles.

Based on this definition and (7.49)-(7.53) let us formulate the following:

Theorem 5.

An EK of the type from (7.42) is logically contradictive or fruitless if the following inequality is taking place

where should be defined based on certain practical considerations, i.e.

Proof:

Let is

then based on (7.42) we have

Let us introduce the following sets, which are

From (7.55) and (7.56) we are getting the following

Taking into account (7.56) we get the following:

therefore we are getting

Note that the set is defined by both and. We are considering the following case first:

1) . Since is normal fuzzy set, then and given (7.49) and (7.50) we get

therefore from (7.59) we get the following (here we consider not a membership functions (7.39), but their correspondent singletons) and therefore

In other words from (7.59) we see that the membership function of a Consequent from fuzzy conditional inference rule (7.41) is poly modal one.

2) . Since is normal fuzzy set, then and given (7.51) and (7.52) we get

therefore when from (7.59) we get the following

Here we also see that the membership function of a Consequent from fuzzy conditional inference rule (7.41) is poly modal one.

3) . Since is normal fuzzy set, then and given (7.53) and (7.54) we get

therefore from (7.59) we get the following

In a meantime one can see that

In other words we see that there are two fuzzy sets, when used as an Antecedents in fuzzy conditional inference rule (7.41), induce the same Consequent, in other words, based on the Definition 7 we have logically contradictive EK from (7.42). (Q.E.D.)

Based on the results of this Theorem we have to present the following

Corollary.

In order to make EK from (7.42) logically non-contradictive, both membership functions of a fuzzy sets from (7.39) be unimodaland, defined in (7.47) and (7.48) correspondingly, would be empty, i.e..

In this paper we proposed new t-norm fuzzy logic in which: 1) Truth values of an implication operator are based on truth values of both antecedent and consequent; 2) Implication operator could be considered as one of the fuzziest implication from the list [34] of known so far; 3) The suggested implication operator is a base for fuzzy conditional inference rules and satisfies the set of important human intuition Criteria; 4) Important features of these rules are investigated.

Tserkovny, A. (2017) A t-Norm Fuzzy Logic for Approximate Reasoning. Journal of Software Engineering and Applications, 10, 639-662. https://doi.org/10.4236/jsea.2017.107035

Criterion I

Ant 1: If x is P then y is Q

Ant 2: x is P

----------------------------------

Cons: y is Q.

Criterion II-1

Ant 1: If x is P then y is Q

Ant 2: x is very P

----------------------------------

Cons: y is very Q.

Criterion II-2

Ant 1: If x is P then y is Q

Ant 2: x is very P

----------------------------------

Cons: y is Q.

Criterion III

Ant 1: If x is P then y is Q

Ant 2: x is more or less P

----------------------------------

Cons: y is more or less Q.

Criterion IV-1

Ant 1: If x is P then y is Q

Ant 2: x is not P

----------------------------------

Cons: y is unknown

Criterion IV-2

Ant 1: If x is P then y is Q

Ant 2: x is not P

----------------------------------

Cons: y is not Q.

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