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[[File:Fuzzy1.jpg|left|250px]]

[[File:Fuzzy1.jpg|left|250px]]

In this chapter, we will discuss '''fuzzy logic'''. It is called ''fuzzy'' because it is characterized by a gradualness: an object can be attributed a quality that can have ''various degrees of truth''.

In this chapter, we will discuss '''fuzzy logic'''. It is called ''fuzzy'' because it is characterized by a gradualness: an object can be attributed a quality that can have ''various degrees of truth''.

In the first part of this chapter, the meaning of graded truth will be conceptually discussed, while in the second part, we will delve into the mathematical formalism by introducing the membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>: the element that allows us to mathematically synthesize the nuances of this logic of language. The conclusion has a positive aspect. It has been possible to show that with ‘fuzzy’ reasoning, unlike the previous logics of language, the diagnoses show less uncertainty. Despite this, however, the need is still felt to further refine the language method and enrich it with further ‘logics’.{{ArtBy|

In the first part of this chapter, the meaning of graded truth will be conceptually discussed, while in the second part, we will delve into the mathematical formalism by introducing the membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>: the element that allows us to mathematically synthesize the nuances of this logic of language. It has been possible to show that with ‘fuzzy’ reasoning, unlike the previous logics of language, the diagnoses show less uncertainty. Despite this, however, the need is still felt to further refine the language method and enrich it with further ‘logics’.{{ArtBy|

| autore = Gianni Frisardi

| autore = Gianni Frisardi

| autore2 = Riccardo Azzali

| autore2 = Riccardo Azzali

==Introduction==

==Introduction==

We have come this far because we, as colleagues, are very often faced with responsibilities and decisions that are very difficult to take and issues such as conscience, intelligence and humility come into play. n such a situation, however, we are faced with two equally difficult obstacles to manage that of one <math>KB</math> ( Knowledge Basis) as we discussed in the chapter  ‘Logic of probabilistic language’, limited in the time we sign <math>KB_t</math> and one <math>KB</math> limited in the specific context (<math>KB_c</math>). These two parameters of epistemology characterize the scientific age in which we live. Also, both <math>KB_t</math> that the <math>KB_c</math> are dependent variables of our phylogeny, and, in particular of our conceptual plasticity and attitude to change.<ref>{{Cite book  

We have come this far because, as colleagues, are very often faced with responsibilities and decisions that are very difficult to take and issues such as conscience, intelligence and humility come into play. in such a situation, however, we are faced with two equally difficult obstacles to manage that of one <math>KB</math> ( Knowledge Basis), as we discussed in the chapter  ‘Logic of probabilistic language’, limited in the time that we codify in <math>KB_t</math> and one <math>KB</math> limited in the specific context (<math>KB_c</math>). These two parameters of epistemology characterize the scientific age in which we live. Also, both <math>KB_t</math> that the <math>KB_c</math> are dependent variables of our phylogeny, and, in particular of our conceptual plasticity and attitude to change.<ref>{{Cite book  

  | autore = Takeuchi S

  | autore = Takeuchi S

  | autore2 = Okuda S

  | autore2 = Okuda S

#<math>KB_c=</math> 'Temporomandibular disorders AND Orofacial Pain AND Fuzzy logic' 0 articles in the last 10 years <ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Temporomandibular+disorders+AND+Orofacial+Pain+AND+Fuzzy+logic%22&filter=datesearch.y_10 "Temporomandibular disorders AND Orofacial Pain AND Fuzzy logic"] in Pubmed</ref>

#<math>KB_c=</math> 'Temporomandibular disorders AND Orofacial Pain AND Fuzzy logic' 0 articles in the last 10 years <ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Temporomandibular+disorders+AND+Orofacial+Pain+AND+Fuzzy+logic%22&filter=datesearch.y_10 "Temporomandibular disorders AND Orofacial Pain AND Fuzzy logic"] in Pubmed</ref>

The example shown means that the <math>KB_t</math> it is relatively up-to-date individually for the three topics while it decreases dramatically when the topics between contexts are merged and specifically to 9 articles for Point 1) and even to 0 articles for Point 2). So, the <math>KB_t</math> is a time dependent variable while the <math>KB_c</math> is a cognitive variable dependent on our aptitude for the progress of science, as already mentioned—among other things—in the chapter ‘Introduction’.

The example shown means that the <math>KB_t</math> is relatively up-to-date individually for the three topics while it decreases dramatically when the topics between contexts are merged and specifically to 9 articles for Point 1 and even to 0 articles for Point 2. So, the <math>KB_t</math> is a time dependent variable while the <math>KB_c</math> is a cognitive variable dependent on our aptitude for the progress of science, as already mentioned—among other things—in the chapter ‘Introduction’.

{{q2|you almost convinced me|Wait and see}}

{{q2|you almost convinced me|Wait and see}}

{{q2|Probability or Possibility?|}}

{{q2|Probability or Possibility?|}}

==Graduated truth==

==Gradualited truth==

In the ambitious attempt to mathematically translate human rationality, it was thought in the mid-twentieth century to expand the concept of classical logic by formulating fuzzy logic. Fuzzy logic concerns the properties that we could call ‘graduate’, i.e., which can be attributed to an object with different degrees. Examples are the properties ‘being sick’, ‘having pain’, ‘being tall’, ‘being young’, and so on.

In the ambitious attempt to mathematically translate human rationality, it was thought in the mid-twentieth century to expand the concept of classical logic by formulating fuzzy logic. Fuzzy logic concerns the properties that we could call ‘graduality’, i.e., which can be attributed to an object with different degrees. Examples are the properties ‘being sick’, ‘having pain’, ‘being tall’, ‘being young’, and so on.

Mathematically, fuzzy logic allows you to attribute to each proposition a degree of truth between <math>0</math> and <math>1</math>. The most classic example to explain this concept is that of age: we can say that a new-born has a ‘degree of youth’ equal to <math>1</math>, an eighteen-year-old equal to <math>0,8</math>, a sixty-year-old equal to <math>0,4</math>, and so on ...

Mathematically, fuzzy logic allows us to attribute to each proposition a degree of truth between <math>0</math> and <math>1</math>. The most classic example to explain this concept is that of age: we can say that a new-born has a ‘degree of youth’ equal to <math>1</math>, an eighteen-year-old equal to <math>0,8</math>, a sixty-year-old equal to <math>0,4</math>, and so on  

In the context of classical logic, on the other hand, the statements:

In the context of classical logic, on the other hand, the statements:

The importance and the charm of fuzzy logic arise from the fact that it is able to translate the uncertainty inherent in some data of human language into mathematical formalism, coding ‘elastic’ concepts (such as almost high, fairly good, etc.), in order to make them understandable and manageable by computers.

The importance and the charm of fuzzy logic arise from the fact that it is able to translate the uncertainty inherent in some data of human language into mathematical formalism, coding ‘elastic’ concepts (such as almost high, fairly good, etc.), in order to make them understandable and manageable by computers.

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==Set theory==

==*Set theory*==

As mentioned in the previous chapter, the basic concept of fuzzy logic is that of multivalence, i.e., in terms of set theory, of the possibility that an object can belong to a set even partially and, therefore, also to several sets with different degrees. Let us recall from the beginning the basic elements of the theory of ordinary sets. As will be seen, in them appear the formal expressions of the principles of Aristotelian logic, recalled in the previous chapter.

As mentioned in the previous chapter, the basic concept of fuzzy logic is that of multivalence, i.e., in terms of set theory, of the possibility that an object can belong to a set even partially and, therefore, also to several sets with different degrees. Let us recall from the beginning the basic elements of the theory of ordinary sets. As will be seen, in them appear the formal expressions of the principles of Aristotelian logic, recalled in the previous chapter.

=== Quantifiers ===

===Quantifiers===

*Membership: represented by the symbol <math>\in </math> (belongs), - for example the number 13 belongs to the set of odd numbers <math>\in </math> <math>13\in  Odd </math>

*Membership: represented by the symbol <math>\in </math> (belongs), - for example the number 13 belongs to the set of odd numbers <math>\in </math> <math>13\in  Odd </math>

*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)

*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)

=== Set operators ===

===Set operators===

Given the whole universe <math>U</math> we indicate with <math>x</math> its generic element such that<math>x \in U</math>; then we consider two subsets <math>A</math> and <math>B</math>  internal to <math>U</math> such that <math>A \subset U</math> and <math>B \subset U</math>

Given the whole universe <math>U</math> we indicate with <math>x</math> its generic element such that<math>x \in U</math>; then we consider two subsets <math>A</math> and <math>B</math>  internal to <math>U</math> such that <math>A \subset U</math> and <math>B \subset U</math>