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{{PPX}} [[File:Universe.jpeg|sinistra|250x250px|link=Special:FilePath/Universe.jpeg]] 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.

[[File:Fuzzy1.jpg|left|250x250px]]

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. 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|

*<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>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>

 

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

[[File:Fuzzy crisp.svg|sinistra|miniatura|'''Figure 1:''' Types of graphs for the membership function.]] 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>

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>

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 '<nowiki/>'''possible'''<nowiki/>', while the 'core' represents those deemed more ''''plausible'''<nowiki/>'.

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> [[File:Universe.jpeg|561x561px|'''Figure 2:''' Representation of the comparison between a classical and fuzzy ensemble.|link=Special:FilePath/Universe.jpeg]] 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:

 

'''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>.

*'''Classical Logic''' in the Dental Context <math>{A}</math> in which only a logical process that gives as results <math>\mu_{\displaystyle {{A}}}(x)= 1 </math> it will be possible, or <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> being the range of data <math>D=\{\delta_1,\dots,\delta_4\}</math>reduced to basic knowledge <math>KB</math> in the set <math>{A}</math>. This means that outside the dental world there is a void and that term of set theory, it is written precisely <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> and which is synonymous with a high range of:

*'''Classical Logic''' in the Dental Context <math>{A}</math> in which only a logical process that gives as results <math>\mu_{\displaystyle {{A}}}(x)= 1 </math> it will be possible, or <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> being the range of data <math>D=\{\delta_1,\dots,\delta_4\}</math>reduced to basic knowledge <math>KB</math> in the set <math>{A}</math>. This means that outside the dental world there is a void and that term of set theory, it is written precisely <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> and which is synonymous with a high range of:

{{q2|Differential diagnostic error|}}

 

<br />{{q2|Differential diagnostic error|}}

*'''Fuzzy logic''' in a dental context <math>\tilde{A}</math> in which they are represented beyond the basic knowledge <math>KB</math> of the dental context also those partially acquired from the neurophysiological world <math>0<\mu_ {\tilde {A}}(x) < 1</math> will have the prerogative to return a result <math>\mu_\tilde{A}(x)= 1

*'''Fuzzy logic''' in a dental context <math>\tilde{A}</math> in which they are represented beyond the basic knowledge <math>KB</math> of the dental context also those partially acquired from the neurophysiological world <math>0<\mu_ {\tilde {A}}(x) < 1</math> will have the prerogative to return a result <math>\mu_\tilde{A}(x)= 1