Changes

In this chapter we will reconsider the clinical case of the unfortunate Mary Poppins suffering from Orofacial Pain for more than 10 years to which her dentist diagnosed as Temporomandibular Disorders or rather Orofacial Pain from Temporomandibular Disorders (TMDs). To better understand why the exact diagnostic formulation remains complex with a Logic of Classical Language, we should understand the concept on which the philosophy of classical language is based with a brief introduction to the topic.

In this chapter we will reconsider the clinical case of the unfortunate Mary Poppins suffering from Orofacial Pain for more than 10 years to which her dentist diagnosed as Temporomandibular Disorders or rather Orofacial Pain from Temporomandibular Disorders (TMDs). To better understand why the exact diagnostic formulation remains complex with a Logic of Classical Language, we should understand the concept on which the philosophy of classical language is based with a brief introduction to the topic.

'''Propositions'''

=== Propositions ===

Classical logic is based on propositions. It is often said that a proposition is a sentence that asks whether the proposition is true or false. Indeed a proposition in mathematics is usually either true or false, but this is obviously a little too vague to be a definition. It can at best be taken as a warning: if a sentence, expressed in common language, makes no sense to ask whether it is true or false, it will not be a proposition but something else.

Classical logic is based on propositions. It is often said that a proposition is a sentence that asks whether the proposition is true or false. Indeed a proposition in mathematics is usually either true or false, but this is obviously a little too vague to be a definition. It can at best be taken as a warning: if a sentence, expressed in common language, makes no sense to ask whether it is true or false, it will not be a proposition but something else.

#Membership, which is indicated by <math>\in</math> (is an element of) o <math>\not\in</math> (is not an element of):

#Membership, which is indicated by <math>\in</math> (is an element of) o <math>\not\in</math> (is not an element of):

'''Demonstration by absurdity'''

=== Demonstration by absurdity ===

Furthermore, in classical logic there is a principle called the <u>excluded third</u> which declares that a sentence that cannot be false must be taken as true since there is no third possibility.

Furthermore, in classical logic there is a principle called the <u>excluded third</u> which declares that a sentence that cannot be false must be taken as true since there is no third possibility.

Suppose we need to prove that the proposition <math>p</math> is true. The procedure consists in showing that the assumption that <math>p</math> is false leads to a logical contradiction. Thus the proposition <math>p</math> cannot be false, and therefore, according to the law of the excluded third, it must be true. This method of demonstration is called 'demonstration by absurdity'''<ref>Luıs Moniz Pereira and Alexandre Miguel Pinto. [http://www-lia.deis.unibo.it/confs/ArgNMR/proceedings/ArgNMR-proceedings.pdf#page=100 Reductio ad Absurdum Argumentation in Normal Logic Programs]</ref>''

Suppose we need to prove that the proposition <math>p</math> is true. The procedure consists in showing that the assumption that <math>p</math> is false leads to a logical contradiction. Thus the proposition <math>p</math> cannot be false, and therefore, according to the law of the excluded third, it must be true. This method of demonstration is called 'demonstration by absurdity'''<ref>Luıs Moniz Pereira and Alexandre Miguel Pinto. [http://www-lia.deis.unibo.it/confs/ArgNMR/proceedings/ArgNMR-proceedings.pdf#page=100 Reductio ad Absurdum Argumentation in Normal Logic Programs]</ref>''

'''Predicates'''

=== Predicates ===

What we have briefly described so far is the logic of propositions. A proposition asserts something about specific mathematical objects such as: '2 is greater than 1, so 1 is less than 2' or 'a square has no 5 sides then a square is not a pentagon'. Many times, however, the mathematical statements concern not the single object but generic objects of a set such as: '''<math>X</math>'' are taller than 2 meters ' where with ''<math>X</math>'' it denotes a generic group (for example all volleyball players). In this case we speak of predicates.

What we have briefly described so far is the logic of propositions. A proposition asserts something about specific mathematical objects such as: '2 is greater than 1, so 1 is less than 2' or 'a square has no 5 sides then a square is not a pentagon'. Many times, however, the mathematical statements concern not the single object but generic objects of a set such as: '''<math>X</math>'' are taller than 2 meters ' where with ''<math>X</math>'' it denotes a generic group (for example all volleyball players). In this case we speak of predicates.

File:Spasmo emimasticatorio assiografia.jpg|'''Figure 5:''' Axiography of the patient showing a flattening of the chewing pattern on the right condyle

File:Spasmo emimasticatorio assiografia.jpg|'''Figure 5:''' Axiography of the patient showing a flattening of the chewing pattern on the right condyle

File:EMG2.jpg|'''Figure 6:''' EMG Interferential Pattern. Overlapping upper traces corresponding to the right masseter, lower to the left masseter.

File:EMG2.jpg|'''Figure 6:''' EMG Interferential Pattern. Overlapping upper traces corresponding to the right masseter, lower to the left masseter.

</gallery>'''Dental propositions'''

</gallery>

 

=== Dental propositions ===

While seeking to use the mathematical formalism to translate the conclusions reached by the dentist with classical logic language, we consider the following predicates:

While seeking to use the mathematical formalism to translate the conclusions reached by the dentist with classical logic language, we consider the following predicates:

The dentist believes that Mary Poppins' claim that she does not have TMD under these premises is a contradiction so the main claim is true.

The dentist believes that Mary Poppins' claim that she does not have TMD under these premises is a contradiction so the main claim is true.

'''Neurophysiological proposition'''

=== Neurophysiological proposition ===

Let us imagine that the neurologist disagrees with the conclusion <math>(1)</math> and asserts that Mary Poppins is not affected by TMDs or that at least it is not the main cause of Orofacial Pain but that she is affected by a neuropathic Orofacial Pain (_nOP), therefore that she does not belong to the group of 'normal patients' but is to be considered a 'non-specific patient' (uncommon in the specialist context).

Let us imagine that the neurologist disagrees with the conclusion <math>(1)</math> and asserts that Mary Poppins is not affected by TMDs or that at least it is not the main cause of Orofacial Pain but that she is affected by a neuropathic Orofacial Pain (_nOP), therefore that she does not belong to the group of 'normal patients' but is to be considered a 'non-specific patient' (uncommon in the specialist context).