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Difference between revisions of "Fuzzy language logic"
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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. | 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 [[The logic of classical language| | <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 the classical language|The logic of the classic language]]), | ||
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>'' | 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>'' | ||
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{{q2|<!--46-->Probability or Possibility?|}} | {{q2|<!--46-->Probability or Possibility?|}} | ||
==Fuzzy truth== | ==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. | 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. | ||
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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. | ||
==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. | ||
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*Universal quantifier, which is indicated by the symbol <math>\forall</math> (for each) | *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) | *Demonstration, which is indicated by the symbol <math>\mid</math> (such that) | ||
===Set operators=== | ===Set operators=== | ||
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In other words, if any element does not belong to the whole, it must necessarily belong to its complementary. | 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>== | ==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. | 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. | ||
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[[File:Fuzzy_crisp.svg|alt=|left|thumb|400px|'''Figure 1:''' Types of graphs for the membership function.]] | [[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>; | 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 '''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 | 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> belongs to the whole or not, 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|<!--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: | 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|''' | [[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 | '''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> | <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 | <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>. | ||
We will remarkably notice the following deductions: | We will remarkably notice the following deductions: | ||
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*'''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 | ||
</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|}} | </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== | ==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. | 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. | ||