Changes

| autore3 = Flavio Frisardi

| autore3 = Flavio Frisardi

}}

}}

==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>Shiho Takeuchi, Shujiro Okuda . [https://pubmed.ncbi.nlm.nih.gov/30542800/ Knowledge base toward understanding actionable alterations and realizing precision oncology.] Int J Clin Oncol. 2019 Feb;24(2):123-130.doi: 10.1007/ s10147-018-1378-0.Epub 2018 Dec 12.</ref>{{q2|I'm not following you|I'll give you a practical example}}

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

| autore = Takeuchi S

| autore2 = Okuda S

| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6373253/

| opera =  Int J Clin Oncol

  | anno = 2019

| DOI = 10.1007/s10147-018-1378-0  

}}</ref>{{q2|I'm not following you|I'll give you a practical example}}

*<blockquote><big>How much research has been produced on the topic 'Fuzzy logic'?</big></blockquote>

*<blockquote><big>How much research has been produced on the topic 'Fuzzy logic'?</big></blockquote>

Pubmed responds with 2862 articles in the last 10 years<ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Fuzzy+logic%22&filter=datesearch.y_10 Fuzzy logic su Pubmed]</ref>, so that we can say that ours is current  and is sufficiently updated. However, if we wanted to focus attention on a specific topic like ‘Temporomandibular Disorders’, the database will respond with as many as 2,235 articles. <ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Temporomandibular+disorders%22&filter=datesearch.y_10 Temporomandibular Disorders in Pubmed]</ref>  Hence, if we wanted to check another topic like ‘Orofacial Pain’, Pubmed gives us 1,986 articles.<ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22orofacial+Pain%22&filter=datesearch.y_10 Orofacial Pain in Pubmed]</ref> This means that the <math>KB_t</math> for these three topics in the last 10 years it has been sufficiently updated.

Pubmed responds with 2862 articles in the last 10 years<ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Fuzzy+logic%22&filter=datesearch.y_10 Fuzzy logic on Pubmed]</ref>, so that we can say that ours is current  and is sufficiently updated. However, if we wanted to focus attention on a specific topic like ‘Temporomandibular Disorders’, the database will respond with as many as 2,235 articles. <ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Temporomandibular+disorders%22&filter=datesearch.y_10 Temporomandibular Disorders in Pubmed]</ref>  Hence, if we wanted to check another topic like ‘Orofacial Pain’, Pubmed gives us 1,986 articles.<ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22orofacial+Pain%22&filter=datesearch.y_10 Orofacial Pain in Pubmed]</ref> This means that the <math>KB_t</math> for these three topics in the last 10 years it has been sufficiently updated.

If, now, we wanted to verify the interconnection between the topics we will notice that <math>KB_c</math> in the contexts it will be the following:

If, now, we wanted to verify the interconnection between the topics we will notice that <math>KB_c</math> in the contexts it will be the following:

#<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’.{{q2|you almost convinced me|Wait and see}}We ended the previous chapter by asserting that the logic of a classical language and subsequently probabilistic logic have helped us a lot in the progress of medical science and diagnostics but implicitly carry within themselves the limits of their own logic of language, which limits the vision of the biological universe. We also verified that with the logic of a classical language—so to speak, Aristotelian—the logical syntax that is derived from it in the diagnostics of our Mary Poppins limits, in fact, the clinical conclusion.

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’.

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

 

We ended the previous chapter by asserting that the logic of a classical language and subsequently probabilistic logic have helped us a lot in the progress of medical science and diagnostics but implicitly carry within themselves the limits of their own logic of language, which limits the vision of the biological universe. We also verified that with the logic of a classical language—so to speak, Aristotelian—the logical syntax that is derived from it in the diagnostics of our Mary Poppins limits, in fact, the clinical conclusion.

<math>\{a \in x \mid \forall \text{x} \; A(\text{x}) \rightarrow {B}(\text{x}) \vdash A( a)\rightarrow B(a) \}</math> ( see chapter [[Logica di Linguaggio Classica/en|Classical Language's Logic]]),

<math>\{a \in x \mid \forall \text{x} \; A(\text{x}) \rightarrow {B}(\text{x}) \vdash A( a)\rightarrow B(a) \}</math> ( see chapter [[Logica di Linguaggio Classica/en|Classical Language's Logic]]),

*<blockquote><big>Why have we come to these critical conclusions?</big></blockquote>

*<blockquote><big>Why have we come to these critical conclusions?</big></blockquote>

For a widely shared form of the representation of reality, supported by the testimony of authoritative figures who confirm its criticality.  This has given rise to a vision of reality which, at first glance, would seem unsuitable for medical language; in fact, expressions such as ‘about 2’ or ‘moderately’ can arouse legitimate perplexity and seem an anachronistic return to pre-scientific concepts. On the contrary, however, the use of fuzzy numbers or assertions allows scientific data to be treated in contexts in which one cannot speak of ‘'''probability'''’ but only of ‘'''possibility’.'''<ref>DUBOIS D., PRADE H. 2000, [https://www.springer.com/gp/book/9780792377320 Fundamentals of Fuzzy Sets], Boston: Kluwer Academic Publishers.</ref>{{q2|Probability or Possibility?|}}

For a widely shared form of the representation of reality, supported by the testimony of authoritative figures who confirm its criticality.  This has given rise to a vision of reality which, at first glance, would seem unsuitable for medical language; in fact, expressions such as ‘about 2’ or ‘moderately’ can arouse legitimate perplexity and seem an anachronistic return to pre-scientific concepts. On the contrary, however, the use of fuzzy numbers or assertions allows scientific data to be treated in contexts in which one cannot speak of ‘'''probability'''’ but only of ‘'''possibility’.'''<ref>{{Cite book

| autore = Dubois D

| autore2 = Prade H

| url = https://books.google.it/books?id=OCznBwAAQBAJ&lpg=PR15&ots=TXlc29Hczd&dq=Fundamentals%20of%20Fuzzy%20Sets%20Editors%3A%20Dubois%2C%20Didier%2C%20Prade%2C%20Henri&lr&hl=it&pg=PR15#v=onepage&q=Fundamentals%20of%20Fuzzy%20Sets%20Editors:%20Dubois,%20Didier,%20Prade,%20Henri&f=false

| editore = Kluwer Academic Publishers

}}</ref>

 

{{q2|Probability or Possibility?|}}

 

==Graduated truth==

==Graduated 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 ‘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.

**a ten-year-old is young

**a ten-year-old is young

**a thirty-year-old is young

**a thirty-year-old is young

are both true. However, in the case of classical logic (which allows only the two true or false data), this would mean that the infant and the thirty-year-old are equally young. Which is obviously wrong.

are both true. However, in the case of classical logic (which allows only the two true or false data), this would mean that the infant and the thirty-year-old are equally young. Which is obviously wrong.

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.

.

==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>

|[[File:Venn1000.svg|sinistra|80x80px]]

|[[File:Venn1000.svg|sinistra|80x80px]]

|'''Complementary:''' represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> the complementary of <math>A</math>, that is, the set of elements that belong to the whole universe except those of <math>A</math>, in formulas: <math>\bar{A}=U-A</math><br />

|'''Complementary:''' represented by a bar above the name of the collection, it indicates by <math>\bar{A}</math> the complementary of <math>A</math>, that is, the set of elements that belong to the whole universe except those of <math>A</math>, in formulas: <math>\bar{A}=U-A</math><br />

|}The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. Remember that in classical logic, given the set <math>A</math> and its complementary <math>\bar{A}</math>, the principle of non-contradiction states that if an element belongs to the whole <math>A</math> it cannot at the same time also belong to its complementary <math>\bar{A}</math>; according to the principle of the excluded third, however, the union of a whole <math>A</math> and its complementary <math>\bar{A}</math> constitutes the complete universe <math>U</math>. In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.

|}The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid. Remember that in classical logic, given the set <math>A</math> and its complementary <math>\bar{A}</math>, the principle of non-contradiction states that if an element belongs to the whole <math>A</math> it cannot at the same time also belong to its complementary <math>\bar{A}</math>; according to the principle of the excluded third, however, the union of a whole <math>A</math> and its complementary <math>\bar{A}</math> constitutes the complete universe <math>U</math>.  

==Fuzzy set <math>\tilde{A}</math> and membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==

 

We choose -as a formalism- to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.

In other words, if any element does not belong to the whole, it must necessarily belong to its complementary.

 

==Fuzzy set <math>\tilde{A}</math> and membership function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==

We choose - as a formalism - to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.

To mathematically represent this degree of belonging is the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> called '<nowiki/>'''Membership Function''''. The function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> is a continuous function defined in the interval <math>[0;1]</math>where it is:

To mathematically represent this degree of belonging is the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> called '<nowiki/>'''Membership Function''''. The function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> is a continuous function defined in the interval <math>[0;1]</math>where it is:

*<math>0<\mu_ {\tilde {A}}(x) < 1 \;\rightarrow </math> if <math>x</math> is partially contained in <math>A</math> (these points are called 'support', they indicate the <u>possible</u> predicate values).

*<math>0<\mu_ {\tilde {A}}(x) < 1 \;\rightarrow </math> if <math>x</math> is partially contained in <math>A</math> (these points are called 'support', they indicate the <u>possible</u> predicate values).

The graphical representation of the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> it can be varied; from those with linear lines (triangular, trapezoidal) to those in the shape of bells or 'S' (sigmoidal) as depicted in Figure 1, which contains the whole graphic concept of the function of belonging.<ref>Weiping Zhang, Jingzhi Yang, Yanling Fang, Huanyu Chen, Yihua Mao, Mohit Kumar. [https://pubmed.ncbi.nlm.nih.gov/28386181/ Analytical fuzzy approach to biological data analysis] Saudi J Biol Sci. 2017 Mar;24(3):563-573. doi: 10.1016/j.sjbs.2017.01.027. Epub 2017 Jan 25.</ref><ref>Prinza Lazar, Rajeesh Jayapathy, Jordina Torrents-Barrena,Beena Mol, Mohanalin, Domenec Puig. [https://pubmed.ncbi.nlm.nih.gov/30800318/ Fuzzy-entropy threshold based on a complex wavelet denoising technique to diagnose Alzheimer disease] Healthc Technol Lett.. 2016 Jul 1;3(3):230-238. doi: 10.1049/htl.2016.0022.eCollection 2016 Sep.</ref>

The graphical representation of the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> it can be varied; from those with linear lines (triangular, trapezoidal) to those in the shape of bells or 'S' (sigmoidal) as depicted in Figure 1, which contains the whole graphic concept of the function of belonging.<ref>{{Cite book

[[File:Fuzzy_crisp.svg|alt=|left|thumb|400x400px|'''Figure 1:''' Types of graphs for the membership function.]]

| autore = Zhang W

The '<nowiki/>'''support set'<nowiki/>''' of a fuzzy set is defined as the zone in which the degree of membership results <math>0<\mu_ {\tilde {A}}(x) < 1</math>; on the other hand, the '<nowiki/>'''core'''' is defined as the area in which the degree of belonging assumes value <math>\mu_ {\tilde {A}}(x) = 1</math>

| autore2 = Yang J

| autore3 = Fang Y

| autore4 = Chen H

| autore5 = Mao Y

| autore6 = Kumar M

| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5372457/

| opera = Saudi J Biol Sci

| anno = 2017

| DOI = 10.1016/j.sjbs.2017.01.027

}}</ref><ref>{{Cite book

| autore = Lazar P

| autore2 = Jayapathy R

| autore3 = Torrents-Barrena J

| autore4 = Mol B

| autore5 = Mohanalin

| autore6 = Puig D

| titolo = Fuzzy-entropy threshold based on a complex wavelet denoising technique to diagnose Alzheimer disease

| opera = Healthc Technol Lett

| anno = 2016

| DOI = 10.1049/htl.2016.0022

}}</ref>

[[File:Fuzzy_crisp.svg|alt=|left|thumb|400px|'''Figure 1:''' Types of graphs for the membership function.]]

 

The '''support set''' of a fuzzy set is defined as the zone in which the degree of membership results <math>0<\mu_ {\tilde {A}}(x) < 1</math>; on the other hand, the '''core''' is defined as the area in which the degree of belonging assumes value <math>\mu_ {\tilde {A}}(x) = 1</math>

The 'Support set' represents the values of the predicate deemed '<nowiki/>'''possible'''<nowiki/>', while the 'core' represents those deemed more ''''plausible'''<nowiki/>'.

The 'Support set' represents the values of the predicate deemed '''possible''', while the 'core' represents those deemed more '''plausible'''.

If <math>{A}</math> represented a set in the ordinary sense of the term or classical language logic previously described, its membership function could assume only the values <math>1</math> or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> depending on whether the element <math>x</math> whether or not it belongs to the whole, as considered. Figure 2 shows a graphic representation of the crisp (rigidly defined) or fuzzy concept of membership, which clearly recalls Smuts's considerations.<ref name=":0">•SMUTS J.C. 1926, [[wikipedia:Holism_and_Evolution|Holism and Evolution]], London: Macmillan.</ref> Let us go back to the specific case of our Mary Poppins, in which we see a discrepancy between the assertions of the dentist and the neurologist and we look for a comparison between classical logic and fuzzy logic:

If <math>{A}</math> represented a set in the ordinary sense of the term or classical language logic previously described, its membership function could assume only the values <math>1</math> or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> depending on whether the element <math>x</math> whether or not it belongs to the whole, as considered. Figure 2 shows a graphic representation of the crisp (rigidly defined) or fuzzy concept of membership, which clearly recalls Smuts's considerations.<ref name=":0">•SMUTS J.C. 1926, [[wikipedia:Holism_and_Evolution|Holism and Evolution]], London: Macmillan.</ref> Let us go back to the specific case of our Mary Poppins, in which we see a discrepancy between the assertions of the dentist and the neurologist and we look for a comparison between classical logic and fuzzy logic: