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Despite the advances offered by fuzzy logic, the text acknowledges that there remains a need for continuous improvement and refinement of this logical approach. The chapter suggests that integrating fuzzy logic with other logical frameworks could further enhance its effectiveness and reduce uncertainty in diagnostics.
Despite the advances offered by fuzzy logic, the text acknowledges that there remains a need for continuous improvement and refinement of this logical approach. The chapter suggests that integrating fuzzy logic with other logical frameworks could further enhance its effectiveness and reduce uncertainty in diagnostics.


Overall, the discussion highlights the importance of adopting new logical models like fuzzy logic in improving the precision and reliability of medical diagnostics, especially in complex cases where traditional methods may be inadequate.<blockquote>
Overall, the discussion highlights the importance of adopting new logical models like fuzzy logic in improving the precision and reliability of medical diagnostics, especially in complex cases where traditional methods may be inadequate.
== Keywords ==
'''Fuzzy Logic''' - A form of many-valued logic or probabilistic logic that deals with reasoning that is approximate rather than fixed and exact. It allows for more flexible and nuanced decision-making processes by handling the concept of partial truth.


'''Graded Truth''' - Refers to the concept in fuzzy logic where truths are seen not just in binary (true or false) but in degrees of truth. This allows for a more granular evaluation of statements or conditions.


'''Mathematical Formalism''' - Pertains to the mathematical underpinnings of fuzzy logic, focusing on the formal structures and functions that enable the quantification and analysis of fuzzy states.


'''Membership Function''' - A fundamental concept in fuzzy logic, it defines how each point in the input space is mapped to a degree of membership between 0 and 1. This function is critical for determining the truth value in fuzzy logic.
{{ArtBy|
| autore = Gianni Frisardi
| autore2 = Riccardo Azzali
| autore3 = Flavio Frisardi
}}
 
==Introduction==
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 ‘[[The logic of probabilistic language|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
| autore2 = Okuda S
| titolo = Knowledge base toward understanding actionable alterations and realizing precision oncology
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6373253/
| volume =
| opera =  Int J Clin Oncol
| anno = 2019
| editore =
| città =
| ISBN =
| PMID = 30542800
| PMCID = PMC6373253
| DOI = 10.1007/s10147-018-1378-0
| oaf = yes<!-- qualsiasi valore -->
| LCCN =
| OCLC =
}}</ref>{{q2|I'm not following you|I'll give you a practical example}}


'''Logic of Language''' - Discusses how fuzzy logic can be applied to the nuances and complexities of language, enhancing the precision and adaptability of linguistic models in computational and theoretical frameworks.
*<blockquote><big>How much research has been produced on the topic 'Fuzzy logic'?</big></blockquote>


'''Medical Diagnostics''' - The application of fuzzy logic to medical diagnostics, which improves the accuracy and reliability of diagnosing by incorporating degrees of certainty and uncertainty.
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><ref><!--11-->All statistics collected following visits to the Pubmed site (https://pubmed.ncbi.nlm.nih.gov/). Last checked: December 2020.</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 <!--14-->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 <!--16-->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.


'''Nuanced Interpretations''' - Involves the capability of fuzzy logic to provide detailed and finely differentiated interpretations of data, which is especially useful in fields requiring high precision like medicine.
If, now, we wanted to verify the interconnection between the topics, we will notice that <math>KB_c</math> in the contexts will be the following:
#<math>KB_c=</math> 'Temporomandibular disorders AND Orofacial Pain'<math>\rightarrow</math> 9 articles in the last 10 years<ref>[https://pubmed.ncbi.nlm.nih.gov/?term=%22Temporomandibular+disorders+AND+Orofacial+Pain%22&filter=datesearch.y_10 <!--21-->Temporomandibular disorders AND Orofacial Pain 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 "<!--24-->Temporomandibular disorders AND Orofacial Pain AND Fuzzy logic" in Pubmed]</ref>


'''Degrees of Truth''' - Highlights the flexible approach of fuzzy logic to truth assessment, allowing for more than just absolute truths or falsehoods, thus accommodating the complexities of real-world scenarios.
The example 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|<!--27-->you almost convinced me|<!--28-->Wait and see}}


'''Uncertainty in Diagnostics''' - Focuses on the use of fuzzy logic to manage and mitigate uncertainty in diagnostic processes, particularly in medicine, by applying probabilistic methods.
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.


'''Complex Medical Data Analysis''' - Refers to the use of fuzzy logic techniques in analyzing intricate and voluminous medical data, enabling more effective decision-making based on a spectrum of probabilities rather than clear-cut choices.</blockquote>
<math>\{a \in x \mid \forall \text{x} \; A(\text{x}) \rightarrow {B}(\text{x}) \vdash A( a)\rightarrow B(a) \}</math> (see chapter [[The logic of classical language|Classical Language's Logic]]),


argues that: "every normal patient ''<math>\forall\text{x}</math>'' which is positive on the radiographic examination of the TMJ ''<math>\mathrm{\mathcal{A}}(\text{x})</math>'' has TMDs''<math>\rightarrow\mathrm{\mathcal{B}}(\text{x})</math>'', as a direct consequence ''<math>\vdash</math>'' Mary Poppins being positive (and also being a "normal" patient) on the TMJ x-ray ''<math>A(a)</math>'' then Mary Poppins is also affected by TMDs ''<math>\rightarrow \mathcal{B}(a)</math>''


{{ArtBy|
The limitation of the logical path that has been followed has led us to undertake an alternative path, in which the bivalence or binary nature of classical language logic is avoided and a probabilistic model is followed. The dentist colleague, in fact, changed the vocabulary and preferred a conclusion like:
| autore = Gianni Frisardi
 
| autore2 = Riccardo Azzali
<math>P(D| Deg.TMJ  \cap TMDs)=0.95</math>
| autore3 = Flavio Frisardi
 
}}
and which is, that our Mary Poppins is 95% affected by TMDs since she has a degeneration of the temporomandibular joint supported by the positivity of the data <math>D=\{\delta_1,\dots\delta_4\}</math> in a population sample <math>n</math>. However, we also found that in the process of constructing probabilistic logic (Analysandum <math>  = \{P(D),a\}</math>) which allowed us to formulate the aforementioned differential diagnostic conclusions and choose the most plausible one, there is a crucial element to the whole Analysand'''<math>= \{\pi,a,KB\}</math>''' represented by the term <math>KB</math> which indicates, specifically, a 'Knowledge Base' of the context on which the logic of probabilistic language is built.
 
We therefore concluded that perhaps the dentist colleague should have become aware of his own 'Subjective Uncertainty' (affected by TMDs or <sub>n</sub>OP?) and 'Objective Uncertainty' (probably more affected by TMDs or <sub>n</sub>OP?).
*<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>{{Cite book
| autore = Dubois D
| autore2 = Prade H
| titolo = Fundamentals of Fuzzy Sets
| 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
| volume =
| opera =
| anno = 2000
| editore = Kluwer Academic Publishers
| città = Boston
| ISBN =
| PMID =
| PMCID =
| DOI =
| oaf = <!-- qualsiasi valore -->
| LCCN =
| OCLC =
}}</ref>
 
{{q2|<!--46-->Probability or Possibility?|}}
----
==Fuzzy 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 ‘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 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:
**a ten-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.
 
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==
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===
 
*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>
*Non-membership: represented by the symbol <math>\notin </math> (It does not belong)
*Inclusion: Represented by the symbol <math>\subset</math> (is content), - for example the whole <math>A</math> it is contained within the larger set <math>U</math>, <math>A \subset U</math> (in this case it is said that <math>A</math> is a subset of <math>U</math>)
*Universal quantifier, which is indicated by the symbol <math>\forall</math> (for each)
*Demonstration, which is indicated by the symbol <math>\mid</math> (such that)
----
===Set operators===
 
Given the whole universe <math>U</math> we indicate with <math>x</math> its generic element so that <math>x \in U</math>; then, we consider two subsets <math>A</math> and <math>B</math> internal to <math>U</math> so that <math>A \subset U</math> and <math>B \subset U</math>
{|
|[[File:Venn0111.svg|left|80px]]
|'''Union:''' represented by the symbol <math>\cup</math>, indicates the union of the two sets <math>A</math> and <math>B</math> <math>(A\cup B)</math>. It is defined by all the elements that belong to <math>A</math> and <math>B</math> or both:
 
<math>(A\cup B)=\{\forall x\in U \mid x\in A \land x\in B\}</math>
|-
|[[File:Venn0001.svg|sinistra|80px]]
|'''Intersection:''' represented by the symbol <math>\cap</math>, indicates the elements belonging to both sets:
 
<math>(A\cap B)=\{\forall x\in U \mid x\in A \lor x\in B\}</math>
|-
|[[File:Venn0010.svg|left|80px]]
|'''Difference:''' represented by the symbol <math>-</math>, for example <math>A-B</math> shows all elements of <math>A</math> except those shared with <math>B</math>
|-
|[[File:Venn1000.svg|left|80px]]
|'''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.
----
==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 ''''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>\mu_ {\tilde {A}}(x) = 1\rightarrow </math> if <math>x</math> is totally contained in <math>A</math> (these points are called 'nucleus', they indicate <u>plausible</u> predicate values).
*<math>\mu_ {\tilde {A}}(x) = 0\rightarrow </math> if <math>x</math> is not contained in <math>A</math>
*<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> 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
| autore = Zhang W
| autore2 = Yang J
| autore3 = Fang Y
| autore4 = Chen H
| autore5 = Mao Y
| autore6 = Kumar M
| titolo = Analytical fuzzy approach to biological data analysis
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5372457/
| volume =
| opera = Saudi J Biol Sci
| anno = 2017
| editore =
| città =
| ISBN =
| PMID = 28386181
| PMCID = PMC5372457
| DOI = 10.1016/j.sjbs.2017.01.027
| oaf = <!-- qualsiasi valore -->
| LCCN =
| OCLC =
}}</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
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6371778/
| volume =
| opera = Healthc Technol Lett
| anno = 2016
| editore = The Institution of Engineering and Technology
| città =
| ISBN =
| PMID = 30800318
| PMCID = PMC6371778
| DOI = 10.1049/htl.2016.0022
| oaf = <!-- qualsiasi valore -->
| LCCN =
| OCLC =
}}</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>; <!--131-->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 '''possible''', while the 'core' represents those deemed more '''plausible'''.
 
If <math>{A}</math> <!--134-->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> <!--135-->or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> <!--136-->depending on whether the element <math>x</math> <!--137-->belongs to the whole or not, as considered. <!--138-->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|<!--139-->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|'''<!--141-->Figure 2:''' <!--142-->Representation of the comparison between a classical and fuzzy ensemble.]]
'''Figure 2:''' Let us imagine the Science Universe <math>U</math> <!--145-->in which there are two parallel worlds or contexts, <math>{A}</math> <!--146-->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>  <!--148-->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> <!--150-->in <math>\tilde{A}</math> <!--151-->that we can go so far as to say: union between <math>KB_c</math>.
 
We will remarkably notice the following deductions:
 
*'''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> 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 is written precisely <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> and which is synonymous with a high range of:


{{:Store:FLen01}}
{{q2|Differential diagnostic error|}}


{{:Store:FLen02}}
*'''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
</math> and a result <math>0<\mu_ {\tilde {A}}(x) < 1</math> because of basic knowledge <math>KB</math> which at this point is represented by the union of <math>KB_c</math> dental and neurological contexts. The result of this scientific-clinical implementation of dentistry would allow a {{q2|Reduction of differential diagnostic error|}}
----
==Consideraciones finales==
Los temas que podían distraer la atención del lector eran, de hecho, esenciales para demostrar el mensaje. Normalmente, en efecto, cuando cualquier mente más o menos brillante se permite arrojar una piedra al estanque de la Ciencia, se genera una onda expansiva, propia del período de la ciencia extraordinaria de Kuhn, contra la que se pelean la mayoría de los miembros de la comunidad científica internacional. De buena fe podemos decir que este fenómeno —en lo que se refiere a los temas que aquí abordamos— está bien representado en la premisa al inicio del capítulo.


{{:Store:FLen03}}
En estos capítulos, en realidad, se ha abordado un tema fundamental para la ciencia: la reevaluación, el peso específico que siempre se le ha dado al {{PV}}, la toma de conciencia de los contextos científico/clínicos <math>KB_c</math>, habiendo emprendido un camino más elástico de la Lógica Difusa que el Clásica, dándose cuenta de la extrema importancia del <math>KB</math> y en definitiva de la unión de los contextos <math>KB_c</math> para aumentar su capacidad diagnóstica.<ref>Mehrdad Farzandipour, Ehsan Nabovati, Soheila Saeedi, Esmaeil Fakharian. [https://pubmed.ncbi.nlm.nih.gov/30119845/ Fuzzy decision support systems to diagnose musculoskeletal disorders: A systematic literature review] . Comput Methods Programs Biomed. 2018 Sep;163:101-109. doi: 10.1016/j.cmpb.2018.06.002. Epub 2018 Jun 6.</ref><ref>Long Huang, Shaohua Xu, Kun Liu, Ruiping Yang, Lu Wu. [https://pubmed.ncbi.nlm.nih.gov/34257635/ A Fuzzy Radial Basis Adaptive Inference Network and Its Application to Time-Varying Signal Classification] . Comput Intell Neurosci, 2021 Jun 23;2021:5528291.<br>doi: 10.1155/2021/5528291.eCollection 2021.</ref>


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En el próximo capítulo estaremos listos para emprender un camino igualmente fascinante: nos conducirá al contexto de una lógica de Lenguaje de Sistemas, y nos permitirá profundizar nuestro conocimiento, ya no solo en la semiótica clínica, sino en la comprensión de la lógica de sistemas. funciones (recientemente se está evaluando en disciplinas neuromotoras para la enfermedad de Parkinson).<ref>Mehrbakhsh Nilashi, Othman Ibrahim, Ali Ahani. [https://pubmed.ncbi.nlm.nih.gov/27686748/ Accuracy Improvement for Predicting Parkinson's Disease Progression.] Sci Rep. 2016 Sep 30;6:34181.


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doi: 10.1038/srep34181.</ref>


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En Masticationpedia, por supuesto, informaremos sobre el tema 'Inferencia del sistema' en el campo del sistema masticatorio, como podemos leer en el próximo capítulo titulado 'Lógica del sistema'.


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