**3. Fuzzy Logic is not the key of the formalization of Natural Language**

Fuzzy Logic is not the key of a complete formalization of NL. The phenomena of vagueness and the relaxations of classical logical truth are only an aspect of NL that Fuzzy Logic is able to treat. In the essay *Vagueness: An Excercise in Logical Analysis* (1937), the work in which was proposed for first time the idea of vague sets, Black distinguished three kinds of imprecision in NL: the generality, the ambiguity and the vagueness. The generality is the power of a word to refer to a lot of things which can be very different each other. The ambiguity is the possibility that a linguistic expression has many different meanings. The vagueness is the absence of precise confines in the reference of a lot of adjectives and common names of human language, e.g. "table", "house", "tall", "rich", "strong", "young", etc. More precisely, vagueness is an approximate relation between a common name or a *quantitative* adjective2 and the objects of the world which can be referred by this name or predicated of this adjective. Fuzzy Logic has been developed to treat in a formal way the linguistic vagueness.

The successes of Fuzzy Logic in the field of engineering (in the automatic and self-regulating processes of cybernetics) and the birth of fuzzy sets theory from the study of linguistic vagueness (cf. Black 1937, Zadeh 1965) empowered the idea that Fuzzy Logic can give solution to the problems that the bivalent logic leaves unsolved in artificial intelligence (henceforth AI). Kosko (1993) proposes the idea that an artificial system will be a good imitation of a natural system, like a brain, only when the artificial system will be able to learn, to get experience and to change itself without the intervention of a human programmer. I think that this is correct, but I believe that it is not enough to put Fuzzy Logic into a dynamic system to solve the problems of AI. Instead Kosko (cf. 1993: 185-190) hypothesizes that the employment of Fuzzy Logic is the key to give the *common sense* to a system. I think that this is not correct. The common sense is the result of so many experiences and so many complex processes of our knowledge, in social interaction, that it is not enough to substitute bivalent logic with Fuzzy Logic to obtain a system which operates on the basis of common sense. Moreover it is important to remember that classical logic is however the soul of logic, and Zadeh does not think that there is a great difference between classical and Fuzzy Logic.

 2 With the expression "quantitative adjective" I mean an adjective which refers to qualities which have variable intensities, i.e. qualities which can be predicated of the subject more or less.

Fuzzy Logic, Knowledge and Natural Language 13

find the truth-values of the fuzzy system inside the evaluation set of the many-valued calculus, which is the basis of the fuzzy system. This freedom in the choice, of how many truth-values are to employ, makes Fuzzy Logic a very dynamic device to treat the complex phenomena.

*Definitions*. Fuzzy Logic is a logic built over a polyvalent logic. The FL system was proposed by Zadeh in 1975. The basis of FL is the system L1 of Łukasiewicz. L1 is a many-valued logic in which the set of truth values contains all the real numbers of the interval [0,1]. FL admits, as truth values, "linguistic" truth values which belong to a set *T* of infinite cardinality: *T* = {True = {Very True, Not very True, More or less True, Not True, …}, False = {Very False, Not very False, More or less False, not False, ….}, … }. Each linguistic truth-value of FL is a fuzzy subset of the set *T* of the infinite numeric truth values of L1. The employment of linguistic truthvalues permits to formulate vague answers to vague questions. A concrete example permits to understand what is a fuzzy truth. To the question "Is John YOUNG?", I can answer "It is Very True that John is YOUNG". As we see in this example, FL works with vague sentences, i.e. with sentences which contain vague or fuzzy predicates. As in the case of truth values, the fuzzy predicates are fuzzy subsets of the universe of discussion X. The universe of discussion is a classical (non fuzzy) set which contains e.g. ages, temperatures, velocities and all kind of adjectival quantities which can have a numerical translation. In the case of our example the universe of discussion is the "age". The set A ("age") contains finite numeric values [0,120]. Inside the classical set A it is possible to define the linguistic variables (YOUNG, OLD, …) as fuzzy subsets. A = {YOUNG = {Very YOUNG, Not very YOUNG, More or less YOUNG, Not YOUNG,…}, OLD = {Very OLD, Not very OLD, More or less OLD, Not OLD,…}, …}. Given that inside the set A = {0,120}, a subset of values which can be considered internal to the vague concept of "young" is e.g. {0,35}, the employment of modifiers (Very ..., Not very…, More or

*Membership function*. The membership function of the elements of A establish *how much* each element belongs to the fuzzy subset YOUNG3. The membership function YOUNG (*a*n) = [x] of the element *a*n establishes that the age *a*n belongs to the fuzzy set YOUNG at the degree x, which is a real number of the interval [0,1]. The interval [0,1] is called "Evaluation set". Let's make some examples. The membership degree of the age "20" to the fuzzy set YOUNG is 0,80. In symbols this is written as YOUNG (20) = [0,80]. The membership degree of the age "30" to the fuzzy set YOUNG is 0,50. In symbols this is YOUNG (30) = [0,50]. If we write the

3 At the end of this paragraph I will give a mathematical and graphic explanation of the membership

membership degree of each age of the fuzzy set YOUNG we will obtain a bend.

less…, etc.) makes fuzzy the subset YOUNG.

 Fig. 4. Diagram of the fuzzy set YOUNG.

function.

The faith in Fuzzy Logic, with respect to the problems of AI, created the illusion that Fuzzy Logic could be the key of the formalization of NL. But, also admitting that Fuzzy Logic is the best method to formalize all kinds of linguistic vagueness, the vagueness is only one of the many aspects of NL that classical logic cannot treat. A calculus would reproduce a good part of the richness of NL only having, at least, 1) a sufficient meta-linguistic power, 2) the ability to interpret the metaphors and 3) the devices to calculate all the variables of the pragmatic context of the enunciation. The first problem is solvable with a very complicate syntax and the large employment of the set theory (cf. the use of Universal Algebra in Montague, 1974). With regard to the second problem, there are a lot of theories on the treatment of metaphor, but none of them seems to be adequate to reach the objective of a "formal" interpretation. The third problem is instead very far from a solution. The pragmatics studies of physical, cultural and situational context of linguistic expressions show only that the pragmatics is fundamental for semantics: the Wittgenstein's theory of linguistic games (1953) is the best evidence of this fact. With regard to pragmatics, today it is clear that it is impossible that an artificial cognitive system could process semantically sentences, if this system is not also capable of perception and action (cf. Marconi, 1997). The problem of vagueness is important in semantics, but I think that the solutions of the problems 1), 2) and 3) are more important and more structural to reach a good automatic treatment of NL. A good automatic treatment of NL does not necessarily require a rigorous logical formalization of NL. With regard to the formalization of NL, I believe that Tarski (1931) has demonstrated that it is, in principle, an impossible task; formalized languages are always founded on NL and their semantic richness is always a parasitical part of the semantic richness of NL. Thus the objective of the research in NLP (Natural Language Processing) and AI can be only: I) the automatic production of artificial sentences which human speakers can easily understand, and II) a sufficiently correct interpretation of sentences of NL. In a static system, which is not able to program itself, I) and II) will be realized on the basis of a formalized language, NL', which is semantically rich enough to be similar to NL, but the semantic richness of NL' is however founded on NL. In a dynamic system, projected in an advanced technological environment, are conceivable *imitations* of natural phenomena like "the extensibility of the meaning of the words", "the change of meaning of words along time" and other kinds of "rule changing creativity" or of "metaphorical attitude", on the basis of the auto-programming activity of the system. Also in dynamic systems, like the hypothetical neural nets, the substitution of bivalent logic with Fuzzy Logic is a good improvement, but it is not the key of the solution of all problems. The concept of dynamic system, understood as a system which learns and changes on the basis of experience is very important, but we are still very far form a concrete realization of a system like this, and the treatability of semantic vagueness through Fuzzy Logic is only a little solution of this task. Now it is clear how much is far from truth who thinks that Fuzzy Logic is the key of formalization of NL.

#### **4. The position of Fuzzy Logic in the context of the many-valued logics**

Often Kosko affirms, in his book (1993), that "fuzzy logic is a polyvalent logic". This is true, but this proposition hides the fact that fuzzy logic is a special kind of polyvalent logic: fuzzy logic is a polyvalent logic which is always based upon another many-valued calculus. Benzi (1997: 133) proposes a mathematical relationship between Fuzzy Logic and many-valued logics. *A fuzzy logic calculus is a logic in which the truth-values are fuzzy subsets of the set of truthvalues of a non fuzzy many-valued logic*. Thus, a simple many-valued logic has a fix number of truth values (3,4, …, *n*), while a Fuzzy Logic has a free number of truth values: it will be the user who will choose, each time, how many truth values he wants to employ. The user will

The faith in Fuzzy Logic, with respect to the problems of AI, created the illusion that Fuzzy Logic could be the key of the formalization of NL. But, also admitting that Fuzzy Logic is the best method to formalize all kinds of linguistic vagueness, the vagueness is only one of the many aspects of NL that classical logic cannot treat. A calculus would reproduce a good part of the richness of NL only having, at least, 1) a sufficient meta-linguistic power, 2) the ability to interpret the metaphors and 3) the devices to calculate all the variables of the pragmatic context of the enunciation. The first problem is solvable with a very complicate syntax and the large employment of the set theory (cf. the use of Universal Algebra in Montague, 1974). With regard to the second problem, there are a lot of theories on the treatment of metaphor, but none of them seems to be adequate to reach the objective of a "formal" interpretation. The third problem is instead very far from a solution. The pragmatics studies of physical, cultural and situational context of linguistic expressions show only that the pragmatics is fundamental for semantics: the Wittgenstein's theory of linguistic games (1953) is the best evidence of this fact. With regard to pragmatics, today it is clear that it is impossible that an artificial cognitive system could process semantically sentences, if this system is not also capable of perception and action (cf. Marconi, 1997). The problem of vagueness is important in semantics, but I think that the solutions of the problems 1), 2) and 3) are more important and more structural to reach a good automatic treatment of NL. A good automatic treatment of NL does not necessarily require a rigorous logical formalization of NL. With regard to the formalization of NL, I believe that Tarski (1931) has demonstrated that it is, in principle, an impossible task; formalized languages are always founded on NL and their semantic richness is always a parasitical part of the semantic richness of NL. Thus the objective of the research in NLP (Natural Language Processing) and AI can be only: I) the automatic production of artificial sentences which human speakers can easily understand, and II) a sufficiently correct interpretation of sentences of NL. In a static system, which is not able to program itself, I) and II) will be realized on the basis of a formalized language, NL', which is semantically rich enough to be similar to NL, but the semantic richness of NL' is however founded on NL. In a dynamic system, projected in an advanced technological environment, are conceivable *imitations* of natural phenomena like "the extensibility of the meaning of the words", "the change of meaning of words along time" and other kinds of "rule changing creativity" or of "metaphorical attitude", on the basis of the auto-programming activity of the system. Also in dynamic systems, like the hypothetical neural nets, the substitution of bivalent logic with Fuzzy Logic is a good improvement, but it is not the key of the solution of all problems. The concept of dynamic system, understood as a system which learns and changes on the basis of experience is very important, but we are still very far form a concrete realization of a system like this, and the treatability of semantic vagueness through Fuzzy Logic is only a little solution of this task. Now it is clear how much is far from truth who

thinks that Fuzzy Logic is the key of formalization of NL.

**4. The position of Fuzzy Logic in the context of the many-valued logics** 

Often Kosko affirms, in his book (1993), that "fuzzy logic is a polyvalent logic". This is true, but this proposition hides the fact that fuzzy logic is a special kind of polyvalent logic: fuzzy logic is a polyvalent logic which is always based upon another many-valued calculus. Benzi (1997: 133) proposes a mathematical relationship between Fuzzy Logic and many-valued logics. *A fuzzy logic calculus is a logic in which the truth-values are fuzzy subsets of the set of truthvalues of a non fuzzy many-valued logic*. Thus, a simple many-valued logic has a fix number of truth values (3,4, …, *n*), while a Fuzzy Logic has a free number of truth values: it will be the user who will choose, each time, how many truth values he wants to employ. The user will find the truth-values of the fuzzy system inside the evaluation set of the many-valued calculus, which is the basis of the fuzzy system. This freedom in the choice, of how many truth-values are to employ, makes Fuzzy Logic a very dynamic device to treat the complex phenomena.

*Definitions*. Fuzzy Logic is a logic built over a polyvalent logic. The FL system was proposed by Zadeh in 1975. The basis of FL is the system L1 of Łukasiewicz. L1 is a many-valued logic in which the set of truth values contains all the real numbers of the interval [0,1]. FL admits, as truth values, "linguistic" truth values which belong to a set *T* of infinite cardinality: *T* = {True = {Very True, Not very True, More or less True, Not True, …}, False = {Very False, Not very False, More or less False, not False, ….}, … }. Each linguistic truth-value of FL is a fuzzy subset of the set *T* of the infinite numeric truth values of L1. The employment of linguistic truthvalues permits to formulate vague answers to vague questions. A concrete example permits to understand what is a fuzzy truth. To the question "Is John YOUNG?", I can answer "It is Very True that John is YOUNG". As we see in this example, FL works with vague sentences, i.e. with sentences which contain vague or fuzzy predicates. As in the case of truth values, the fuzzy predicates are fuzzy subsets of the universe of discussion X. The universe of discussion is a classical (non fuzzy) set which contains e.g. ages, temperatures, velocities and all kind of adjectival quantities which can have a numerical translation. In the case of our example the universe of discussion is the "age". The set A ("age") contains finite numeric values [0,120]. Inside the classical set A it is possible to define the linguistic variables (YOUNG, OLD, …) as fuzzy subsets. A = {YOUNG = {Very YOUNG, Not very YOUNG, More or less YOUNG, Not YOUNG,…}, OLD = {Very OLD, Not very OLD, More or less OLD, Not OLD,…}, …}. Given that inside the set A = {0,120}, a subset of values which can be considered internal to the vague concept of "young" is e.g. {0,35}, the employment of modifiers (Very ..., Not very…, More or less…, etc.) makes fuzzy the subset YOUNG.

*Membership function*. The membership function of the elements of A establish *how much* each element belongs to the fuzzy subset YOUNG3. The membership function YOUNG (*a*n) = [x] of the element *a*n establishes that the age *a*n belongs to the fuzzy set YOUNG at the degree x, which is a real number of the interval [0,1]. The interval [0,1] is called "Evaluation set". Let's make some examples. The membership degree of the age "20" to the fuzzy set YOUNG is 0,80. In symbols this is written as YOUNG (20) = [0,80]. The membership degree of the age "30" to the fuzzy set YOUNG is 0,50. In symbols this is YOUNG (30) = [0,50]. If we write the membership degree of each age of the fuzzy set YOUNG we will obtain a bend.

Fig. 4. Diagram of the fuzzy set YOUNG.

<sup>3</sup> At the end of this paragraph I will give a mathematical and graphic explanation of the membership function.

Fuzzy Logic, Knowledge and Natural Language 15

values. Moreover, when the phenomenon we want to treat has no a scalar shape, we must give to this phenomenon a scalar shape following its intensity, to put this datum in a fuzzy inference. When, in my works on clinical diagnosis, I put the "pain" or the "liver enlargement" in fuzzy inferences I had to give to these factors a scalar shape, employing percent values. Thus I created tables of correspondence between "linguistic" and "numeric" intensities with the aid of bends (as in Fig. 1): a pain with value "20%" was "Light", a pain with value "50%" was "Mild"; a liver enlargement with value "15%" was "Little", a liver enlargement with value "85%" was "Very big". The employment of these tables in the premises of the diagnostic inferences gave, as result, intensity values for the diseases found in the patient; e.g. I obtained conclusions like "the patient suffers from a 'moderate-severe' Congestive Heart Failure (80%) and from a 'moderate' Congestive Hepatopathy (50%)". The great precision and richness of this new kind of diagnosis is the advantage of the use of Fuzzy Logic in clinical diagnosis. Anyway, It is important to notice here that the correspondence between linguistic and numerical values is what permits the processes of fuzzification and defuzzification. The fuzzification is the transformation of a numerical value in a linguistic value, the defuzzification is the reverse. It is clear that Zadeh's FL system doesn't need diagrams of correspondence, or fuzzification-defuzzification processes, because the linguistic variables (for predicates and for truth-values) represent fuzzy sets through the modifiers. However, also in FL system (as in all fuzzy systems) we find a precise correspondence between numerical and linguistic values. This happens because fuzzy sets can be theorized only as subsets of a classical set X, and the possibility to use linguistic vague predicates is given by the reference to a great number, or an infinite number of values into the classical set X. Thus, the "formal" vagueness of Fuzzy Logic is only the

An "indicator function" or a "characteristic function" is a function defined on a set *X* that indicates the membership of an element *x* to a subset *A* of *X*. The indicator function is a function *A*: *X*→ {0,1}. It is defined as *<sup>A</sup>* (*x*) = 1 if *x A* and *<sup>A</sup>* (*x*) = 0 if *x A* . In the classical set theory the characteristic function of the elements of the subset *A* is assessed in binary terms according to a crisp condition: an element either belongs (1) or does not belong (0) to the subset. By contrast, fuzzy set theory permits a gradual membership of the elements of *X* (universe of discourse) to the subset *B* of *X*. Thus, with a generalization of the indicator

The membership function indicates the degree of membership of an element *x* to the fuzzy subset *B,* its value is from 0 to 1. Let's consider the braces as containing the elements 0 and 1, while the square parentheses as containing the finite/infinite interval from 0 to 1. The generalization of the indicator function corresponds to an extension of the valuation set of *B*: the elements of the valuation set of a classical subset *A* of *X* are 0 and 1, those of the valuation set of a fuzzy subset *B* of *X* are all the real numbers in the interval between 0 and 1. For an element *x*, the value *B* (*x*) is called *"membership degree* of *x* to the fuzzy set *B*",

 *x X*, *X*(*x*) = 1 (11) As the fuzzy set theory needs the classical set theory as its basis, in the same way, the

*<sup>B</sup>* : *X*→ [0,1] (10)

function of classical sets, we obtain the membership function of a fuzzy set:

which is a subset of *X*. The universe *X* is never a fuzzy set, so we can write:

fundament of the logic, also of polyvalent logic, is however the bivalence.

result of a great numerical precision.

The bend gives an idea of the fuzzy membership of the elements to a fuzzy set.

*Modifiers*. The Zadeh's modifiers of FL are arithmetical operators on the value of the membership function of the primary terms. In this way it is possible to obtain secondary terms. If the primary term is "YOUNG", the secondary terms will be "Very YOUNG", "Not very YOUNG", "More or less YOUNG", "Not YOUNG", etc. For the application of the modifiers it is necessary to give an average value to the fuzzy set YOUNG. Let's suppose that the membership value of the average element of YOUNG is YOUNG. The modifier "Not" will define the value "Not YOUNG" in this way:

$$
\mu\_{\text{Not YUMNG}} = 1 - \mu\_{\text{YUMNG}} \tag{7}
$$

The modifier "Very" will define the value "Very YOUNG" in this way:

$$
\mu\_{\text{Very YOUNG}} = (\mu\_{\text{YOUNG}})^2 \tag{8}
$$

The modifier "More or less" will define the value "More or less YOUNG" in this way:

$$\text{\textquotedblleft Morec or less YONING} = \text{\textquotedblright} \text{\textquotedblleft MVMG} \text{\textquotedblright} \tag{9}$$

The Zadeh's modifiers give a concrete idea of what is a fuzzy set, because they are the linguistic translation of the *degree* of membership of the elements. Their mathematical definition is necessary for their employment in the calculus.

In a calculus it is possible to introduce a lot of different universes discourse: "age", "strenght", "temperature", "tallness", etc. A fuzzy system can treat vague sentences like "Maria is rich enough" or "The fever of Bill is very high", or more complicated sentences like "It is not very true that an high fever is dangerous for life". In this way it is possible to build a logical system, which can be employed to translate the vague sentences of natural language in a formal calculus, and it is possible to make formal demonstrations about a scientific phenomenon. It is clear that the fuzziness of the sentence can be transferred from the predicate to the truth value. The sentence "John is Not very YOUNG" is synonym of the sentence "It is Not very True that John is YOUNG". Now it is clear how is possible to have, in a logical system, "linguistic" truth-values as fuzzy sets. These truth-values must be the subsets of a classical set: the set of truth-values of a simple many valued logic.

In (Licata 2007) and (Licata 2010) I demonstrated that it is possible to obtain a great advantage employing Fuzzy Logic in the clinical diagnosis of a concrete clinical case. In those works the linguistic variables were the so called "hedges", i.e. sets of values which are in correspondence to the numerical values of the universe of discourse. So, remaining in the example of the "age", the set A = {0,120} can be split into five subsets: A = {VERY YOUNG, YOUNG, ADULT, OLD, VERY OLD}. The 120 values of A are distributed in this five sets. VERY YOUNG = {0,18}; YOUNG = {19,35}; ADULT = {36,55}; OLD = {56,75}; VERY OLD = {76,120}. These five sets are fuzzy sets because I consider that each value belongs to a subset of A *more or less*, following a fuzzy membership function. I mean that, e.g., the element "70" has a higher degree of membership than the element "60" to the set OLD; or that the element "2" has a higher degree of membership than the element "10" to the set VERY YOUNG. With respect to Zadeh's FL, this is another way to create a correspondence between linguistic variables and fuzzy values. The unchanged matter is that, even if we work with vague predicates, this vagueness has however a precise reference to scalar

*Modifiers*. The Zadeh's modifiers of FL are arithmetical operators on the value of the membership function of the primary terms. In this way it is possible to obtain secondary terms. If the primary term is "YOUNG", the secondary terms will be "Very YOUNG", "Not very YOUNG", "More or less YOUNG", "Not YOUNG", etc. For the application of the modifiers it is necessary to give an average value to the fuzzy set YOUNG. Let's suppose that the membership value of the average element of YOUNG is YOUNG. The modifier "Not"

Not YOUNG = 1 YOUNG (7)

Very YOUNG = (YOUNG)2 (8)

More or less YOUNG = (YOUNG)1/2 (9)

The bend gives an idea of the fuzzy membership of the elements to a fuzzy set.

The modifier "Very" will define the value "Very YOUNG" in this way:

definition is necessary for their employment in the calculus.

The modifier "More or less" will define the value "More or less YOUNG" in this way:

The Zadeh's modifiers give a concrete idea of what is a fuzzy set, because they are the linguistic translation of the *degree* of membership of the elements. Their mathematical

In a calculus it is possible to introduce a lot of different universes discourse: "age", "strenght", "temperature", "tallness", etc. A fuzzy system can treat vague sentences like "Maria is rich enough" or "The fever of Bill is very high", or more complicated sentences like "It is not very true that an high fever is dangerous for life". In this way it is possible to build a logical system, which can be employed to translate the vague sentences of natural language in a formal calculus, and it is possible to make formal demonstrations about a scientific phenomenon. It is clear that the fuzziness of the sentence can be transferred from the predicate to the truth value. The sentence "John is Not very YOUNG" is synonym of the sentence "It is Not very True that John is YOUNG". Now it is clear how is possible to have, in a logical system, "linguistic" truth-values as fuzzy sets. These truth-values must be the

In (Licata 2007) and (Licata 2010) I demonstrated that it is possible to obtain a great advantage employing Fuzzy Logic in the clinical diagnosis of a concrete clinical case. In those works the linguistic variables were the so called "hedges", i.e. sets of values which are in correspondence to the numerical values of the universe of discourse. So, remaining in the example of the "age", the set A = {0,120} can be split into five subsets: A = {VERY YOUNG, YOUNG, ADULT, OLD, VERY OLD}. The 120 values of A are distributed in this five sets. VERY YOUNG = {0,18}; YOUNG = {19,35}; ADULT = {36,55}; OLD = {56,75}; VERY OLD = {76,120}. These five sets are fuzzy sets because I consider that each value belongs to a subset of A *more or less*, following a fuzzy membership function. I mean that, e.g., the element "70" has a higher degree of membership than the element "60" to the set OLD; or that the element "2" has a higher degree of membership than the element "10" to the set VERY YOUNG. With respect to Zadeh's FL, this is another way to create a correspondence between linguistic variables and fuzzy values. The unchanged matter is that, even if we work with vague predicates, this vagueness has however a precise reference to scalar

subsets of a classical set: the set of truth-values of a simple many valued logic.

will define the value "Not YOUNG" in this way:

values. Moreover, when the phenomenon we want to treat has no a scalar shape, we must give to this phenomenon a scalar shape following its intensity, to put this datum in a fuzzy inference. When, in my works on clinical diagnosis, I put the "pain" or the "liver enlargement" in fuzzy inferences I had to give to these factors a scalar shape, employing percent values. Thus I created tables of correspondence between "linguistic" and "numeric" intensities with the aid of bends (as in Fig. 1): a pain with value "20%" was "Light", a pain with value "50%" was "Mild"; a liver enlargement with value "15%" was "Little", a liver enlargement with value "85%" was "Very big". The employment of these tables in the premises of the diagnostic inferences gave, as result, intensity values for the diseases found in the patient; e.g. I obtained conclusions like "the patient suffers from a 'moderate-severe' Congestive Heart Failure (80%) and from a 'moderate' Congestive Hepatopathy (50%)". The great precision and richness of this new kind of diagnosis is the advantage of the use of Fuzzy Logic in clinical diagnosis. Anyway, It is important to notice here that the correspondence between linguistic and numerical values is what permits the processes of fuzzification and defuzzification. The fuzzification is the transformation of a numerical value in a linguistic value, the defuzzification is the reverse. It is clear that Zadeh's FL system doesn't need diagrams of correspondence, or fuzzification-defuzzification processes, because the linguistic variables (for predicates and for truth-values) represent fuzzy sets through the modifiers. However, also in FL system (as in all fuzzy systems) we find a precise correspondence between numerical and linguistic values. This happens because fuzzy sets can be theorized only as subsets of a classical set X, and the possibility to use linguistic vague predicates is given by the reference to a great number, or an infinite number of values into the classical set X. Thus, the "formal" vagueness of Fuzzy Logic is only the result of a great numerical precision.

An "indicator function" or a "characteristic function" is a function defined on a set *X* that indicates the membership of an element *x* to a subset *A* of *X*. The indicator function is a function *A*: *X*→ {0,1}. It is defined as *<sup>A</sup>* (*x*) = 1 if *x A* and *<sup>A</sup>* (*x*) = 0 if *x A* . In the classical set theory the characteristic function of the elements of the subset *A* is assessed in binary terms according to a crisp condition: an element either belongs (1) or does not belong (0) to the subset. By contrast, fuzzy set theory permits a gradual membership of the elements of *X* (universe of discourse) to the subset *B* of *X*. Thus, with a generalization of the indicator function of classical sets, we obtain the membership function of a fuzzy set:

$$
\mu\_{\mathcal{B}} \colon \mathcal{X} \to [0, 1] \tag{10}
$$

The membership function indicates the degree of membership of an element *x* to the fuzzy subset *B,* its value is from 0 to 1. Let's consider the braces as containing the elements 0 and 1, while the square parentheses as containing the finite/infinite interval from 0 to 1. The generalization of the indicator function corresponds to an extension of the valuation set of *B*: the elements of the valuation set of a classical subset *A* of *X* are 0 and 1, those of the valuation set of a fuzzy subset *B* of *X* are all the real numbers in the interval between 0 and 1. For an element *x*, the value *B* (*x*) is called *"membership degree* of *x* to the fuzzy set *B*", which is a subset of *X*. The universe *X* is never a fuzzy set, so we can write:

$$\forall \mathbf{x} \; \mathsf{e} \, X, \, \mu \chi(\mathbf{x}) = 1 \tag{11}$$

As the fuzzy set theory needs the classical set theory as its basis, in the same way, the fundament of the logic, also of polyvalent logic, is however the bivalence.

Fuzzy Logic, Knowledge and Natural Language 17

Schrödinger's equation is very interesting. Indeed it seems to me that Schrödinger's equation regards the quantity and the quantum distribution of matter, and not the probability to find the particle in the region dV. However, in other fields of science it is not useful to try to reduce probabilistic logic to Fuzzy Logic or to treat the problems of probability with Fuzzy Logic. It is also wrong to reduce Fuzzy Logic to probabilistic logic. These two kinds of calculus have different fields of employment, different aims and give different informations about phenomena. An evidence is that probabilistic diagnosis and fuzzy diagnosis give different kinds of information about the health of the patient. In particular: probabilistic diagnosis drives in the choice of the possible diseases which could cause the symptoms, while fuzzy diagnosis gives the exact quantification of the strength of diseases. They are both useful in the study and in the cure of pathology but they do different tasks (cf. Licata, 2007). It is usual in literature to distinguish probabilistic logic from fuzzy logic, telling that the first is a way to formalize the "uncertainty" while the second is a method to treat "vagueness". In technical sense, uncertainty is the incompleteness of information, while vagueness is the absence of precise confines in the reference of *quantitative* adjectives, common names, etc. to objects of world (see §3). Nevertheless, some authors employed Fuzzy Logic to treat uncertainty (in the sense of incompleteness of information) and many theorists of probability think that probabilistic logic is a good way to treat vagueness. In general, it is clear that vagueness and uncertainty (in technical sense) can be theorized as two distinct areas of knowledge, studied by distinct methods. Given that uncertainty is understood as incompleteness of information, while vagueness regards an indefinite relationship between words and objects, it is possible to say that uncertainty and probabilistic logic fall in the area of "subjective knowledge", while vagueness and Fuzzy

I thank Giuseppe Nicolaci and Marco Buzzoni for their irreplaceable help in the

AA.VV., (1988). La Nuova Enciclopedia delle Scienze Garzanti, Garzanti editore, Milano.

Bellman, R.E. & Zadeh, L. (1977). Local and Fuzzy Logics, in: *Modern Uses of Multiple-Valued* 

Birkhoff G. & von Neumann J. (1936), The Logic of Quantum Mechanics, in: *Annals of* 

Black, M. (1937). Vagueness: An Excercise in Logical Analysis, In: *Philosophy of Science*, 4, pp.

Buzzoni, M. (2008). Epistemologia e scienze umane – 1. Il modello nomologico-deduttivo, In:

Benzi, M. (1997). Il ragionamento incerto, Franco Angeli, Milano.

*Logic*, J.M. Dunn & G. Epstein, (ed.), pp. 105-165, Dodrecht.

Capra, F. (1996). The Web of Life, Doubleday-Anchor Book, New York.

ψ|2 in

I think that the fuzzy interpretation, proposed by Kosko, of the wave-function |

**5. Probabilistic and Fuzzy Logic in distinct sides of knowledge** 

Logic fall in the area of "objective knowledge".

*Mathematics*, 37, pp. 823-843.

*Nuova Secondaria*, 8, pp. 50-52.

**6. Acknowledgments** 

**7. References** 

development of my research.

427-455.

Fig. 5. A fuzzy set is always a subset of a classical set.

In conclusion, it seems that the dispute between bivalent and polyvalent logic proposed by Kosko is not a real opposition. The vagueness, the becoming of world and the plenty of points of view request polyvalence, but the Aristotelian bivalence is however fundamental in our knowledge. A lot of circumstances in our life require the bivalence. Often our decisions are choices between two alternatives, and the alternative true/false is one of the most fundamental rule of our language (Wittgenstein, 1953). As Quine (1960) underlines, the learning of a foreign language has at its basis the "yes" and the "no", as answers to sentences. Kosko, instead, affirms that the advent of Fuzzy Logic is a real revolution in science and in philosophy, also from a metaphysical point of view. When he prefers the Buddah principle of "A non A" to the Aristotle principle "A non A", he is meaning that the world can be understood only if we forget the principle of non contradiction, because always the objects of the world have in the same time opposite determinations. On the other hand, Zadeh does not think that Fuzzy Logic is so in contrast with classical logic. Zadeh and Bellman writes (1977: 109): "Although fuzzy logic represents a significant departure from the conventional approaches to the formalization of human reasoning, it constitutes – so far at least – an extension rather than a total abandonment of the currently held views on meaning, truth and inference". Fuzzy logic is just an extension of standard Boolean logic: if we keep the fuzzy values at their extremes of 1 (completely true), and 0 (completely false), the laws of classical logic will be valid. In this sense it is possible to formulate a new principle which considers the importance of bivalence and, in the same time, permits to accept the polyvalence, the Aristotle-and-Buddah principle:

#### (A non A) and (A non A)

In the Aristotle-and-Buddah principle I employed the conjunction "and" of natural language, and not the conjunction of a formalized language, because only the natural language has the power to maintain the conjunction between two principles which express different metaphysical systems. In the spirit of Aristotle-and-Buddah principle, bivalence can be considered a *polarization* of polyvalence, but bivalence is also the fundament of polyvalence: the evidences are that i) bivalence was born as foundation of logic while polyvalence is only a recent result, and that ii) Zadeh (1965) was able to theorize the fuzzy sets only because he considered them as subsets of a classical set. Thus it is clear that Aristotle's thought and bivalence are not outdated themes of philosophy.

In conclusion, it seems that the dispute between bivalent and polyvalent logic proposed by Kosko is not a real opposition. The vagueness, the becoming of world and the plenty of points of view request polyvalence, but the Aristotelian bivalence is however fundamental in our knowledge. A lot of circumstances in our life require the bivalence. Often our decisions are choices between two alternatives, and the alternative true/false is one of the most fundamental rule of our language (Wittgenstein, 1953). As Quine (1960) underlines, the learning of a foreign language has at its basis the "yes" and the "no", as answers to sentences. Kosko, instead, affirms that the advent of Fuzzy Logic is a real revolution in science and in philosophy, also from a metaphysical point of view. When he prefers the Buddah principle of "A non A" to the Aristotle principle "A non A", he is meaning that the world can be understood only if we forget the principle of non contradiction, because always the objects of the world have in the same time opposite determinations. On the other hand, Zadeh does not think that Fuzzy Logic is so in contrast with classical logic. Zadeh and Bellman writes (1977: 109): "Although fuzzy logic represents a significant departure from the conventional approaches to the formalization of human reasoning, it constitutes – so far at least – an extension rather than a total abandonment of the currently held views on meaning, truth and inference". Fuzzy logic is just an extension of standard Boolean logic: if we keep the fuzzy values at their extremes of 1 (completely true), and 0 (completely false), the laws of classical logic will be valid. In this sense it is possible to formulate a new principle which considers the importance of bivalence and, in the same time, permits to

 (A non A) and (A non A) In the Aristotle-and-Buddah principle I employed the conjunction "and" of natural language, and not the conjunction of a formalized language, because only the natural language has the power to maintain the conjunction between two principles which express different metaphysical systems. In the spirit of Aristotle-and-Buddah principle, bivalence can be considered a *polarization* of polyvalence, but bivalence is also the fundament of polyvalence: the evidences are that i) bivalence was born as foundation of logic while polyvalence is only a recent result, and that ii) Zadeh (1965) was able to theorize the fuzzy sets only because he considered them as subsets of a classical set. Thus it is clear that

Fig. 5. A fuzzy set is always a subset of a classical set.

accept the polyvalence, the Aristotle-and-Buddah principle:

Aristotle's thought and bivalence are not outdated themes of philosophy.
