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Quantifier connectors, such as the universal quantifier (<math>\forall</math>) and the existential quantifier (<math>\exists</math>), allow for extending statements to sets of elements, offering a way to express propositions concerning 'all elements' of a certain set or 'at least one element' of such a set.
Quantifier connectors, such as the universal quantifier (<math>\forall</math>) and the existential quantifier (<math>\exists</math>), allow for extending statements to sets of elements, offering a way to express propositions concerning 'all elements' of a certain set or 'at least one element' of such a set.


By combining these tools, it is possible to construct complex propositions that serve as the foundation for logical arguments and mathematical reasoning, eliminating the ambiguities typical of common language and providing a clear structure for analysis and proof."
By combining these tools, it is possible to construct complex propositions that serve as the foundation for logical arguments and mathematical reasoning, eliminating the ambiguities typical of common language and providing a clear structure for analysis and proof".......................<center><div class="colour-button">[[Special:UserLogin&returnto=Introduction+Page|Read the full chapter]]</div>
===Proof by Contradiction===
 
In classical logic, there exists a principle called "the law of excluded middle", which asserts that a proposition, which cannot be false, must be considered true, since there is no third possibility.
 
Suppose we have to prove that the proposition p is true. The procedure consists of demonstrating that assuming <math>p</math> to be false leads to a logical contradiction. Consequently, the proposition <math>p</math> cannot be false and, therefore, according to the law of excluded middle, must be true. This method of proof is known as proof by contradiction.<ref>{{Cite book
| author = Pereira LM
| author2 = Pinto AM
| title = Reductio ad Absurdum Argumentation in Normal Logic Programs
| url = http://www-lia.deis.unibo.it/confs/ArgNMR/proceedings/ArgNMR-proceedings.pdf#page=100
| volume = Argumentation and Non-Monotonic Reasoning - An LPNMR Workshop
| work =
| year = 2007
| publisher = Arg NMR
| city = Tempe, Arizona - Caparica, Portugal
| ISBN =
| PMID =
| PMCID =
| DOI =
| oaf = <!-- any value -->
| LCCN =
| OCLC =
}}</ref>
===Predicates===
 
What has been briefly described so far represents the logic of propositions, which asserts something about specific mathematical objects. Examples of propositions include: "2 is greater than 1, therefore 1 is less than 2" or "a square does not have 5 sides, therefore it cannot be a pentagon". Often, however, mathematical statements are not limited to individual objects but refer to generic objects within a set, as in the expression "the elements ''<math>X</math>'' are taller than 2 meters", where ''<math>X</math>'' indicates a generic group (for example, all volleyball players). In these cases, we talk about predicates.
 
Intuitively, a predicate is a sentence that concerns a set of elements (which, in our medical context, would be the patients) and makes a statement about them.
{{q2|So, does Mary Poppins suffer from TMD or not?|let's see what classical language logic tells us}}
 
In addition to the confirmations derived from the medical language logic discussed in the previous chapter, the dentist acquires further instrumental data that strengthen his diagnosis. These tests include the analysis of axiographic traces, obtained through the use of a custom-made functional paraocclusal fork. This tool allows for the visualization and quantification of the condylar paths during masticatory functions. As shown in Figure 4, the flattening of the condylar traces on the right side, both in the mediotrusive masticatory kinetics (indicated in green) and in the opening and protrusion cycles (in grey), confirms the anatomical and functional flattening of the right TMJ during the dynamics of mastication.
 
In addition to axiography, surface electromyography on the masseter muscles is performed (see Figure 6), during which the patient is asked to exert maximum muscle force. This type of electromyographic analysis, called 'EMG Interferential Pattern', is characterized by the high-frequency content of the peaks showing phase interference. Indeed, Figure 6 highlights an asymmetry in the recruitment of motor units between the right masseter (upper trace) and the left masseter (lower trace).
<ref>{{cite book
| autore = Castroflorio T
| autore2 = Talpone F
| autore3 = Deregibus A
| autore4 = Piancino MG
| autore5 = Bracco P
| titolo = Effects of a Functional Appliance on Masticatory Muscles of Young Adults Suffering From Muscle-Related Temporomandibular Disorder
| url = https://pubmed.ncbi.nlm.nih.gov/15189308/
| volume =
| opera =  J Oral Rehabil
| anno = 2004
| editore =
| città =
| ISBN =
| PMID = 15189308
| PMCID =
| DOI = 10.1111/j.1365-2842.2004.01274.x
| oaf = <!-- qualsiasi valore -->
}}</ref><ref>{{cite book
| autore = Maeda N
| autore2 = Kodama N
| autore3 = Manda Y
| autore4 = Kawakami S
| autore5 = Oki K
| autore6 = Minagi S
| titolo = Characteristics of Grouped Discharge Waveforms Observed in Long-term Masseter Muscle Electromyographic Recording: A Preliminary Study
| url = http://ousar.lib.okayama-u.ac.jp/files/public/5/56938/20190821181112825794/73_4_357.pdf
| volume =
| opera =  Acta Med Okayama
| anno = 2019
| editore = Okayama University Medical School
| città = Okayama, Japan
| ISBN =
| PMID = 31439959
| PMCID =
| DOI = 10.18926/AMO/56938
| oaf = <!-- qualsiasi valore -->
}}</ref><ref>{{cite book
| autore = Rudy TE
| titolo = Psychophysiological Assessment in Chronic Orofacial Pain
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2190318/pdf/anesthprog00262-0036.pdf
| volume =
| opera = Anesth Prog
| anno = 1990
| editore = American Dental Society of Anesthesiology
| città =
| ISSN = 0003-3006/90
| ISBN =
| PMID = 2085203
| PMCID = PMC2190318
| DOI =
| oaf = <!-- qualsiasi valore -->
}}</ref><ref>{{cite book
| autore = Woźniak K
| autore2 = Piątkowska D
| autore3 = Lipski M
| autore4 = Mehr K
| titolo = Surface electromyography in orthodontics - a literature review
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3673808/pdf/medscimonit-19-416.pdf
| volume =
| opera = Med Sci Monit
| anno = 2013
| editore =
| città =
| ISBN =
| E-ISSN = 1643-3750
| PMID = 23722255
| PMCID =  PMC3673808
| DOI = 10.12659/MSM.883927
| oaf = <!-- qualsiasi valore -->
}}</ref><center>
==2nd Clinical Approach==
(Hover over the images)
<gallery widths="350" heights="282" perrow="2" mode="slideshow">
File:Spasmo emimasticatorio.jpg|Figure 2: Patient reports "orofacial pain" on the right facial hemisphere.
File:Spasmo emimasticatorio ATM.jpg|Figure 3: Stratigraphy of the temporomandibular joint (TMJ) of the patient showing signs of condylar flattening and presence of osteophytes.
File:Atm1 sclerodermia.jpg|Figure 4: Computed tomography of the TMJ
File:Spasmo emimasticatorio assiografia.jpg|Figure 5: Axiography of the patient showing a flattening of the masticatory pattern at the level of the right condyle.
File:EMG2.jpg|Figure 6: Interfering EMG activity. Overlapping upper traces corresponding to the right masseter, below to the left masseter.
</gallery></center>
----
=====Propositions in the Dental Context=====
In an attempt to apply mathematical formalism to interpret the dentist's diagnostic conclusions using classical logic language, we define the following predicates:
 
<math>x \equiv</math> Normal patients (where "normal" refers to patients commonly encountered in a specialist setting)
<math>A(x) \equiv</math> Presence of bone remodeling with detected osteophyte from stratigraphic exams and condylar CT
<math>B(x)\equiv</math> Temporomandibular Disorders (TMD) resulting in orofacial pain (OP)
<math>\mathrm{a}\equiv</math> Specific patient: Mary Poppins
We establish that for every normal patient <math>\mathrm{\mathcal{A}}(\text{x})</math>, if they test positive for the TMJ radiographic examination <math>\mathrm{\mathcal{A}}(\text{x})</math> [see Figures 2 and 3], then they are affected by TMD<math>\rightarrow\mathrm{\mathcal{B}}(\text{x})</math>. Consequently <math>\vdash</math> if Mary Poppins tests positive (and is considered a "normal patient") for the TMJ radiographic exam <math>A(a)</math>, it follows that she too is affected by TMD <math>\rightarrow \mathcal{B}(a)</math>. This can be formally expressed as:
{|
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|<math>{a \in x \mid \forall \text{x} ; A(\text{x}) \rightarrow {B}(\text{x}) \vdash A( a)\rightarrow B(a) }
</math>
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|<math>(1)</math>
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In addition to proving that a certain set of premises implies a particular conclusion, predicate logic is also used to prove the falsity of a statement or the logical compatibility/incompatibility of a certain piece of knowledge with a given piece of evidence.
 
To verify the truthfulness of this proposition, we resort to proof by contradiction. If the negation of the proposition generates a contradiction, we can conclude that the original hypothesis of the dentist is correct:
{|
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|<math>\urcorner{a \in x \mid \forall \text{x} ; A(\text{x}) \rightarrow {B}(\text{x}) \vdash A( a)\rightarrow B(a) }</math>
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|<math>(2)</math>
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Statement (2) asserts that it is not true that patients who test positive for the TMJ CT are affected by TMD, implying that Mary Poppins (a "normal patient" with a positive outcome for the TMJ CT) is not affected by TMD.
 
The dentist believes that this statement, based on the provided premises, constitutes a contradiction, thereby confirming the validity of the main statement.
===Propositions in the Neurological Context===
 
Suppose the neurologist contests conclusion (1), arguing that Mary Poppins does not suffer from TMD or that, at least, TMD is not the primary cause of her Orofacial Pain. Instead, he hypothesizes that Mary suffers from neuromotor type Orofacial Pain (<sub>n</sub>OP), classifying her not as a 'normal patient' but as a 'specific patient' (atypical for the dental specialist).
 
The neurologist's position can be formalized as follows:
{|
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|<math>{a \not\in x \mid \forall \text{x} ; A(\text{x}) \rightarrow {B}(\text{x}) \and A( a)\rightarrow \urcorner B(a) }</math>
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|<math>(3)</math>
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This statement (3) claims that there might exist a patient with a positive outcome for the TMJ CT who does not suffer from TMD. To validate this hypothesis through proof by contradiction, consider its negation:
{|
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|<math>\urcorner{a \not\in x \mid \forall \text{x} ; A(\text{x}) \rightarrow {B}(\text{x}) \and A( a)\rightarrow \urcorner B(a) }</math>
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|<math>(4)</math>
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Analyzing predicate logic, we find no sufficient reasons to believe that negation (4) leads to a contradiction, indicating that the neurologist, unlike the dentist, may not have sufficient logical grounds to confirm his conclusion without further evidence.
 
{{q2|so the dentist triumphs!|don't take it for granted}}
===Compatibility and Incompatibility of Statements===
The complexity arises when the dentist presents a series of statements based on clinical reports, such as stratigraphy and computed tomography (CT) of the temporomandibular joint (TMJ), indicating an anatomical flattening of the joint, axiography of the condylar paths with a reduction of cinematic convexity, and an electromyographic (EMG) interference pattern showing asymmetry on the masseters. These evidences can be considered co-causes of damage to the temporomandibular joint and, consequently, responsible for "Orofacial Pain".
 
Documents, reports, and clinical evidence can be used to make the neurologist's statement incompatible and support the dentist's diagnostic conclusion. To do this, we present some logical rules that describe compatibility or incompatibility according to classical language logic:
 
# A set of sentences <math>\Im</math> and a number <math>n\geq1</math> of other sentences or statements <math>(\delta_1,\delta_2,.....\delta_n \ )</math> are logically compatible if, and only if, their union <math>\Im\cup{\delta_1,\delta_2.....\delta_n}</math> is coherent.
# A set of sentences <math>\Im</math> and a number <math>n\geq1</math> of other sentences or statements <math>(\delta_1,\delta_2,.....\delta_n \ )</math> are logically incompatible if, and only if, their union <math>\Im\cup{\delta_1,\delta_2.....\delta_n}</math> is incoherent.
 
Let's examine this concept with practical examples.
 
The dentist presents the following statement:
 
<math>\Im</math>: Following the personalized techniques suggested by Xin Liang et al.
<ref>{{cite book
| autore = Liang X
| autore2 = Liu S
| autore3 = Qu X
| autore4 = Wang Z
| autore5 = Zheng J
| autore6 = Xie X
| autore7 = Ma G
| autore8 = Zhang Z
| autore9 = Ma X
| titolo = Evaluation of Trabecular Structure Changes in Osteoarthritis of the Temporomandibular Joint With Cone Beam Computed Tomography Imaging
| url = https://pubmed.ncbi.nlm.nih.gov/28732700/
| volume =
| opera =  Oral Surg Oral Med Oral Pathol Oral Radiol
| anno = 2017
| editore =
| città =
| ISBN =
| PMID = 28732700
| PMCID =
| DOI = 10.1016/j.oooo.2017.05.514
| oaf = <!-- qualsiasi valore -->
}}</ref>  that focuses on the quantitative microstructural analysis of the bone value fraction, trabecular number, trabecular thickness, and trabecular separation on each slice of a TMJ CT, it appears that Mary Poppins is affected by Temporomandibular Disorders (TMDs) and the consequence causes orofacial pain.
 
However, to further confirm the diagnosis, the dentist presents a series of additional assertions that should pass the compatibility filter described above, thus establishing a coherent basis for the diagnosis of TMD in Mary Poppins.
 
'''<math>\delta_1=</math> Bone remodeling:''' The flattening of the axiographic traces shown in Figure 5 indicates the joint remodeling of Mary Poppins' right TMJ. This report can be related to a series of research and articles confirming how malocclusion can be associated with morphological changes of the temporomandibular joints, particularly if related to age. Indeed, the presence of chronic malocclusion can aggravate the scenario of bone remodeling.
<ref>{{cite book
| autore = Solberg WK
| autore2 = Bibb CA
| autore3 = Nordström BB
| autore4 = Hansson TL
| titolo = Malocclusion Associated With Temporomandibular Joint Changes in Young Adults at Autopsy
| url = https://pubmed.ncbi.nlm.nih.gov/3457531/
| volume =
| opera = Am J Orthod
| anno = 1986
| editore =
| città =
| ISBN =
| PMID = 3457531
| PMCID =
| DOI = 10.1016/0002-9416(86)90055-2
| oaf = <!-- qualsiasi valore -->
}}</ref> The provided scientific references support the compatibility of the assertion.
 
'''<math>\delta_2=</math> Sensitivity and Specificity of the Axiographic Measurement:'''
 
A study was conducted to evaluate the sensitivity and specificity of the data obtained from a sample of patients with temporomandibular joint disorders, using the ARCUSdigma axiographic system.
<ref>[https://www.kavo.com/de-de/ KaVo Dental GmbH, Biberach / Ris]</ref>
The results showed a sensitivity of 84.21% for the right TMJ and 92.86% for the left TMJ, with a specificity of 93.75% and 95.65%, respectively.
<ref>{{cite book
| autore = Kobs G
| autore2 = Didziulyte A
| autore3 = Kirlys R
| autore4 = Stacevicius M
| titolo = Reliability of ARCUSdigma (KaVo) in Diagnosing Temporomandibular Joint Pathology
| url = https://sbdmj.lsmuni.lt/072/072-03.pdf
| volume =
| opera = Stomatologija
| anno = 2007
| editore =
| città =
| ISBN =
| PMID = 17637527
| PMCID =
| DOI =
| oaf = <!-- qualsiasi valore -->
}}</ref> These scientifically validated data support the compatibility of the statement in the dental field, given the consistency of related studies.
<ref>{{cite book
| autore = Piancino MG
| autore2 = Roberi L
| autore3 = Frongia G
| autore4 = Reverdito M
| autore5 = Slavicek R
| autore6 = Bracco P
| titolo = Computerized axiography in TMD patients before and after therapy with 'function generating bites'
| volume =
| opera = J Oral Rehabil
| anno = 2008
| editore =
| città =
| ISBN =
| PMID = 18197841
| PMCID =
| DOI = 10.1111/j.1365-2842.2007.01815.x
| oaf = <!-- qualsiasi valore -->
}}</ref>
 
'''<math>\delta_3=</math> Alteration in Condylar Paths:'''
 
Urbano Santana-Mora and colleagues
<ref>{{cite book
| autore = López-Cedrún J
| autore2 = Santana-Mora U
| autore3 = Pombo M
| autore4 = Pérez Del Palomar A
| autore5 = Alonso De la Peña V
| autore6 = Mora MJ
| autore7 = Santana U
| titolo = Jaw Biodynamic Data for 24 Patients With Chronic Unilateral Temporomandibular Disorder
| url = https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5674825/pdf/sdata2017168.pdf
| volume =
| opera = Sci Data
| anno = 2017
| editore =
| città =
| ISBN =
| PMID =  29112190
| PMCID = PMC5674825
| DOI = 10.1038/sdata.2017.168
| oaf = YES<!-- qualsiasi valore -->
}}</ref> conducted a study on 24 adult patients suffering from severe chronic unilateral pain, diagnosed with Temporomandibular Disorders (TMD). The research focused on the analysis of various functional and dynamic factors, including masticatory function, the remodeling of the temporomandibular joint (TMJ) or condylar paths (CP), and lateral jaw movement or lateral guidance (LG).
 
The CPs were evaluated using conventional axiography, while the LG was examined through kinesiographic tracing analysis.
<ref>[https://www.myotronics.com/ Myotronics Inc., Kent, WA, US]</ref>
It was found that seventeen patients, corresponding to 71% of the total sample, habitually preferred to chew on one side. The mean and standard deviation of the CP angles were 47.90 (± 9.24) degrees, while the mean of the LG angles was 42.95 (± 11.78) degrees.
 
The study results contributed to the definition of a new paradigm for TMDs, suggesting that the side affected by the disorder might coincide with the habitual chewing side, especially when the lateral mandibular cinematic angle is flatter. This parameter further supports the dental assertion regarding the correlation between chewing habits and the development of TMDs.
 
'''<math>\delta_4=</math> EMG Interference Pattern:'''
 
M.O. Mazzetto and collaborators
<ref>{{cite book
| autore = Oliveira Mazzetto M
| autore2 = Almeida Rodrigues C
| autore3 = Valencise Magri L
| autore4 = Oliveira Melchior M
| autore5 = Paiva G
| titolo = Severity of TMD Related to Age, Sex and Electromyographic Analysis
| url = https://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-64402014000100054&lng=en&nrm=iso&tlng=en
| volume =
| opera = Braz Dent J
| anno = 2014
| editore =
| città =
| ISBN =
| PMID =
| PMCID =
| DOI = 10.1590/0103-6440201302310
| oaf = <!-- qualsiasi valore -->
}}</ref> demonstrated that there is a positive correlation between the electromyographic activity of the anterior temporal muscles and the masseter and the 'Cranio-Mandibular Index' (CMI), with a value of <math>P=0,01</math>. This suggests that the use of CMI to quantify the severity of Temporomandibular Disorders (TMD) and electromyography (EMG) to assess the function of the masticatory muscles can represent a significant diagnostic and therapeutic element. Such scientific references support the compatibility of the assertion.
 
<math>\delta_n=?</math>
 
Given the evidence presented and the statements made, the dentist can legitimately claim that the set of sentences <math>\Im</math>, and a number <math>n\geq1</math> of other positive clinical assertions or data <math>(\delta_1,\delta_2,.....\delta_n \ )</math>​ are logically compatible. This is because their union, <math>\Im\cup{\delta_1,\delta_2.....\delta_n}</math>, turns out to be coherent.
 
{{q2|Following classical language logic, the dentist is right!|It would seem so! But beware, only in their own dental context!}}
 
This statement is so valid that the value of <math>P</math> could be extended indefinitely, even reaching an <math>\alpha=0</math>, which corresponds to infinite significance, as long as it remains within the specific context; however, it may not hold any significance in different contexts, such as the neurological one, for example.
 
==Final Considerations==
Within the scope of this observation, the application of Predicate Logic significantly contributes to strengthening the dentist's deductive process, concurrently solidifying the principle of excluded middle. This principle is emphasized by the coherence of the supplementary statements <math>(\delta_1,\delta_2,...,\delta_n)</math>, providing the dentist with a solid foundation for a coherent diagnosis and to affirm with certainty that 'Poor Mary Poppins is, without a shadow of a doubt, affected by TMD or not.'
 
{{q2|And what if, as research advances, new phenomena emerge supporting the neurologist's theories rather than the dentist's?|}}
----
In essence, considering the compatibility of the assertions <math>(\delta_1,\delta_2,.....\delta_n)</math>, coherently supporting that Orofacial Pain is caused by a Temporomandibular Disorder might become incompatible if another set of equally coherent assertions <math>(\gamma_1,\gamma_2,.....\gamma_n)</math> emerged. This scenario would pave the way for a new interpretation <math>\Im</math>: Mary Poppins could be suffering from Orofacial Pain due to a Neuromotor Disorder (<sub>n</sub>OP), not directly from Temporomandibular Disorders. Within the current medical language, such assertions remain purely theoretical, given that prevailing beliefs and opinions do not facilitate a rapid paradigm shift.
 
Considering also the risk associated with such a change, the analysis of recent studies on the epidemiology of Temporomandibular Disorders
<ref>{{cite book | autore = LeResche L | titolo = Epidemiology of temporomandibular disorders: implications for the investigation of etiologic factors | url = https://pubmed.ncbi.nlm.nih.gov/9260045/ | volume = | opera = Crit Rev Oral Biol Med | anno = 1997 | editore = | città = | ISBN = | PMID = 9260045 | PMCID = | DOI = 10.1177/10454411970080030401 | oaf = <!-- qualsiasi valore --> }}</ref> could prove useful. These studies indicate that, despite methodological and sampling differences, pain in the temporomandibular region is relatively common, affecting about 10% of the population. Therefore, it is reasonable to hypothesize that Mary Poppins could fall within this percentage, classifying her as a patient suffering from Orofacial Pain due to Temporomandibular Disorders (TMD).
 
In conclusion, adopting a classical dichotomous approach in medical language, which tends to categorize conditions in black or white, fails to capture the numerous nuances present in clinical realities. Therefore, exploring a more flexible and suitable linguistic approach is necessary...
 
{{q2|Could we then consider adopting a probabilistic type of linguistic logic?|perhaps}}
 
{{Bib}}
 
{{apm}}
<onlyinclude>
</onlyinclude>
 
 
<center><div class="colour-button">[[Special:UserLogin&returnto=Introduction+Page|Read the full chapter]]</div>
</center><blockquote>
</center><blockquote>
== [[Strategic dental topics for authors to subscribe an article]] ==
== [[Strategic dental topics for authors to subscribe an article]] ==
Classical Logic, Craniofacial Biology, Advanced Technologies, Medical Diagnostics, Epigenetics, Phenomics, Treatment Personalization, Craniofacial Anomalies, Formal Logic, Interdisciplinary Approach, Medical Innovations, Personalized Treatments, Computational Logic, Precision Diagnostics.</blockquote>
Classical Logic, Craniofacial Biology, Advanced Technologies, Medical Diagnostics, Epigenetics, Phenomics, Treatment Personalization, Craniofacial Anomalies, Formal Logic, Interdisciplinary Approach, Medical Innovations, Personalized Treatments, Computational Logic, Precision Diagnostics.</blockquote>
[[Category:Medical Innovations]]
{{Bib}}
[[Category:Craniofacial Biology]]
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