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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 theor*==

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.

===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:Venn0111.svg|sinistra|80px]]

|[[File:Venn0111.svg|sinistra|80px]]

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

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:

[[File:Fuzzy1.jpg|thumb|400x400px|'''Figure 2:''' Representation of the comparison between a classical and fuzzy ensemble.]]

[[File:Fuzzy1.jpg|thumb|400x400px|'''Figure 2:''' Representation of the comparison between a classical and fuzzy ensemble.]]

'''Figure 2:''' Let us imagine the Science Universe <math>U</math> in which there are two parallel worlds or contexts, <math>{A}</math> and <math>\tilde{A}</math>.

'''Figure 2:''' Let us imagine the Science Universe <math>U</math> in which there are two parallel worlds or contexts, <math>{A}</math> and <math>\tilde{A}</math>.

<math>{A}=</math>  In the scientific context, the so-called ‘crisp’, and we have converted into ''the logic'' of ''Classic Language'', in which the physician has an absolute scientific background information <math>KB</math>  with a clear dividing line, we signed <math>KB_c</math>  

<math>{A}=</math>  In the scientific context, the so-called ‘crisp’, and we have converted into ''the logic'' of ''Classic Language'', in which the physician has an absolute scientific background information <math>KB</math>  with a clear dividing line that we have named <math>KB_c</math>.

<math>\tilde{A}=</math> In another scientific context called  ‘fuzzy logic’, and in which there is a union between the subset <math>{A}</math> in <math>\tilde{A}</math> that we can go so far as to say: union between <math>KB_c</math>.

<math>\tilde{A}=</math> In another scientific context called  ‘fuzzy logic’, and in which there is a union between the subset <math>{A}</math> in <math>\tilde{A}</math> that we can go so far as to say: union between <math>KB_c</math>.

==Final considerations==

==Final considerations==

topics that could distract the reader’s attention—was, in fact, essential for demonstrating the message. Normally, in fact, when any more or less brilliant mind allows itself to throw a stone into the pond of Science, a shockwave is generated, typical of the period of Kuhn’s extraordinary science, against which most of the members of the international scientific community row. With good faith, we can say that this phenomenon—as regards the topics we are addressing here—is well represented in the premise at the beginning of the chapter.

Topics that could distract the reader’s attention—was, in fact, essential for demonstrating the message. Normally, in fact, when any more or less brilliant mind allows itself to throw a stone into the pond of Science, a shockwave is generated, typical of the period of Kuhn’s extraordinary science, against which most of the members of the international scientific community row. With good faith, we can say that this phenomenon—as regards the topics we are addressing here—is well represented in the premise at the beginning of the chapter.

 

 

In these chapters, in fact, a fundamental topic for science has been approached: the re-evaluation, the specific weight that has always been given to <math>P-value</math>, awareness of scientific / clinical contexts <math>KB_c</math>, having undertaken a more elastic path of Fuzzy Logic than the Classical one, realizing the extreme importance of <math>KB</math> and ultimately the union of contexts <math>KB_c</math> to increase its diagnostic capacity.

In the next chapter we will be ready to undertake an equally fascinating path that will leads us to the context of a System Language logic and will allow us to deepen our knowledge no longer only in clinical semeiotics but in the understanding of system functions as recently it is approaching in neuromotor disciplines for Parkinson's disease.<ref>Mehrbakhsh Nilashi, Othman Ibrahim, Ali Ahani. Accuracy Improvement for Predicting Parkinson's Disease Progression. Sci Rep. 2016 Sep 30;6:34181.

In the next chapter we will be ready to undertake an equally fascinating path that will leads us to the context of a System Language logic and will allow us to deepen our knowledge no longer only in clinical semeiotics but in the understanding of system functions (in our case it is the Masticatory System.

doi: 10.1038/srep34181.</ref> In Masticationpedia, of course, we will report the topic 'System Inference' in the field of the masticatory  system as we could read in the next chapter entitled 'System logic'.