RESOURCE DESCRIPTION FRAMEWORK
FOR MUCOSAL SURFACE PATHOLOGY.
DRAFT COPY ONLY.
3/5/2008.

G. William Moore, MD, PhD.
Grace F. Kao, MD.
Lawrence A. Brown, MD.
http://www.netautopsy.org/mucordfh.htm
http://www.netautopsy.org/mucordfh.ppt


Dr. Moore     Dr. Kao     Dr. Brown

Send comments and correspondence to: George.Moore4@va.gov


1. DISCLAIMER.

United States Government Work, uncopyrighted, public-domain, DRAFT COPY ONLY. This document does not necessarily represent the views or policies of any United States Government agency. This document is provided "as is", without warranty of any kind, express or implied, including but not limited to the warranties of merchantability, fitness for a particular purpose and non-infringement. In no event shall the authors be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, arising from, out of, or in connection with the document or the use or other dealings made with the document.

2. TABLE OF CONTENTS.

1. Disclaimer.
2. Table of Contents.
3. Abstract.
4. Introduction.
5. Namespace.
6. Relationspace.
7. Fuzzyspace.
8. Propositional Logic, Set Theory.
9. Mathematical Theorems.
10. Live Computer Demonstration.
11. Model for Ethical Data Collection.
12. Examples.
13. Dermatopathogenesis.
14. Discussion.
19. References.
20. Appendix A. Resource Description Framework (RDF). Example.
21. Appendix B. Anatomic Names.
22. Appendix C. Skin Alterations.
23. Appendix D. Dermatopathologic Clues.
24. Appendix E. Large Specimen Checklists.
25. Appendix F. Specimen Accessioning.
26. Appendix G. Quality Assurance. Followup.
27. Appendix H. Symbolic Logic Theorems.
28. Appendix I. Embryogenesis.
29. Appendix J. Dermatopathology Diagnosis.


3. ABSTRACT: APIII #324.

 Resource Description Framework for Mucosal Surface Pathology.
 G. William Moore, MD, PhD (George.Moore4@va.gov) [1,2,3];
 Lawrence A. Brown, MD [1,2]; Grace F. Kao, MD [1,4].
 Pathology and Laboratory Medicine Service, Veterans Affairs
 Maryland Health Care System, Baltimore, MD [1]; Department of Pathology,
 University of Maryland Medical System, Baltimore, MD [2]; Department
 of Pathology, The Johns Hopkins Medical Institutions, Baltimore, MD [3];
 and Department of Dermatology, George Washington University School
 of Medicine, Washington, DC [4].
                   http://www.netautopsy.org/mucordfh.htm 
                                       
 Content: Tumors of mucosal surfaces are among the most common human
 malignancies.  The pathogenesis of these tumors is well-studied, but
 scattered in articles and textbooks. Resource Description Framework (RDF)
 is a general syntax for writing computer-parsable ordered triples, that
 export meaning among databases on the semantic worldwide web, by binding
 a described datum to a specified subject. Internet web-crawler programs
 can interrogate multiple RDF documents, and draw inferences from these
 ordered triples.
                                
 Technology: Perl programming language, classical propositional logic,
 non-monotonic logic.
                                      
 Design: We propose a hierarchical classification for human mucosal surface
 tumors, and present a Perl computer script for translating this hierarchy
 into RDF code, in the style of the Laboratory Data Imaging Project.
 This hierarchical classification employs classical logic, with additional
 features to handle non-monotonic logic ("Sutton's Law") and ethical
 constraints ("first do no harm").
                    
 Results: This human mucosal surface tumor RDF class hierarchy is
 mathematically consistent. Over 200 theorems of classical and modal logic
 are proved in the system. An Intercalation Theorem (for inserting
 new concepts) and a Retirement Theorem (for removing obsolete concepts)
 are stated and proved.
                                               
 Conclusion: This RDF hierarchy serves to organize the vast knowledge of
 mucosal surface pathology in the format of the semantic worldwide web,
 in a manner that incorporates both clinical and pathologic findings.

OUTLINE OF EPOSTER.



Tumors of mucosal surfaces.


1. Most common human malignancies.
2. Pathogenesis well-studied.
3. Details scattered in articles, textbooks.

Resource Description Framework (RDF).


1. General syntax for writing ordered triples.
2. Computer-parsable.
3. Export meaning among databases on semantic worldwide web.
4. Binding a described datum to a specified subject.
5. Internet web-crawler programs draw inferences from ordered triples.

Technology.


1. Perl programming language.
2. Classical propositional logic.
3. Non-monotonic logic.

Methods.


1. Hierarchical classification for mucosal surface tumors.
2. Perl script for translating into RDF code.
3. Style of Laboratory Data Imaging Project (LDIP).

Logic model.


1. Classical propositional logic.
2. Non-monotonic logic ("Sutton's Law").
3. Ethical constraints ("first do no harm").

Namespace.


1. Strict hierarchy of all concepts in the ontology.
2. Each name appears exactly once.
3. Each name has exactly one parent, except...
4. Ultimate name has no parent.
5. Negations not allowed.

Resource Description Framework (RDF).


1. Mathematical consistency:
2. No statement and its negation may be deduced.
3. Negations and multiple parents not allowed in RDF.
4. Namespace formalism is consistent by construction. (Theorem§9.1).

Laboratory Digital Imaging Project.


1. RDF specification for anatomic pathology.
2. Ultimate_Class.
3. Seven subClasses:
4. Person, Event, Data_object, Specimen, Reagent, Instrument, Terminology.

Event subclass.


1. Temporal, Spatial, Mass, Temperature, Homunculus.
2. Homunculus: Cardiovascular_system, Respiratory_system, Gastrointestinal_system, Genitourinary_system, Endocrine_system, Musculoskeletal_system, Lymphoreticular_system, Nervous_system, Integumentary_system.
3. Some lumping or spitting is possible
4. Monoparental hierarchy is required.
5. Namespace designer must decide:
     5a. Fundamental organizing principle of the hierarchy: ontologic commitment (Quine).

Example: LDIP for Class Patient.

 Identifier:ldip:Patient
 Class Label:Patient
 versionInfo (required): 0.1
 Registration Authority (required): Association for Pathology Informatics
 Language:en
 Obligation:optional
 Maximum occurrence:Unlimited
 Cardinality (required):/[0-9]+/
 Datatype: Literal
 comment: The patient, unambiguously denoted by the required
 ordered quadruple: patient_name (=patient_surname, patient_givenname,
 patient_honorific), patient_social_security_number, patient_date_of_birth,
 and patient_gender. Includes: patient_insurance.
 subClassOf:Person
 Contributor:Bill Moore
 Date_of_contribution:11-13-2006

Monoparentality assumption.


1. Limited mathematical power.
2. Limited opportunities for mathematical mischief:
      2a. Inconsistency.
      2b. Incomputability.

Biomedical Semantic Content (Meaning).


1. Pair of (metadata, data), bound to unique subject.
2. Ordered triple: <subject,metadatum,datum>.
3. Example: <bill_moore,actinic_keratosis,yes>.
4. Mathematical notation: <argument,function,value> or <x,f,v>, where:
5. function(argument)=value, or f(x)=y.

Sample Ordered Triples.


1. <+g_moore,+left_axillary_lymphadenopathy,+no>.
2. <+g_moore,+right_eye_lens_prosthesis,+yes>.
3. <+superman,+xray_vision,+yes>.
4. <+lou_gehrig,+amyotrophic_lateral_sclerosis,+yes>.

Spelling, Punctuation Conventions.


1. American-English spelling, American-Roman alphabet, all lower case.
2. Blankspace, hyphen, and other punctuation replaced by underline (_).
3. Metadata inheritance: carriagereturn, indentation: ⇒.
4. Metadata sibship: carriagereturn, no indentation: |.
5. Existing standards used, where applicable, freely available.
6. Date/time convention: International Standards Organization, ISO 8601.
7. All stopwords, barrierwords, or low-information words removed.

Online Embryology RDF Resources.


1. Nomina Embryologica Veterinaria:
      http://www.wava-amav.org/Downloads/nev_2006.pdf
2. Univ Ca Berkeley: Human Developmental Anatomy, Staged:
      http://www.berkeleybop.org/ontologies/obo-all/human-dev-anat-staged/human-dev-anat-staged.obo_xml

Online Anatomy RDF Resources.


1. Nomina Anatomica Veterinaria:

Relationspace.


1. List of all relationships among concepts.
2. Includes concepts with negations and multiple parents.

Fuzzyspace.


1. Levels of certainty within the same concept.
2. Based upon fuzzy set theory.
3. May be necessary to take actions based upon incomplete information.
4. Including: therapy based upon a presumptive diagnosis; decision to collect additional information, such as a skin biopsy.
5. Might inconvenience, or even injure, the patient.

Sutton's Law.


1. Medical slang term.
2. In the face of diagnostic uncertainty, use the most likely diagnosis.
3. Named after notorious bank robber, Willie Sutton.
4. "Go where the money is."
5. Fevers of unknown origin (Petersdorf and Beeson, 1961):
6. Treat with the most likely effective antibiotic.

Zebra Rule.


1. Another version of Sutton's Law: "If you hear hoofbeats in the street, think of horses not zebras."
2. Tertiary-care medical institution sometimes called a zebra farm.
3. Computer science: logic of jumping to conclusions (Brewka et al, 1997).
4. When the most likely outcome is FALSE, then in classical logic, one obtains an inconsistency.

Zebra Rule: Modal Logic.


1. "If you hear hoofbeats in the street, think of horses not zebras." 

 +∀
    ...
       +□∞+hoofbeats_in_street
          +□5+horses
          +□1+zebras
where □p denotes necessarily p; and ◇p denotes possibly p.
2. Paraphrased: "If you see a solitary, non-hemorrhagic pigmented lesion unchanged for 20 years, think of seborrheic keratoses not melanomas." 

 +∀
    ...
       +□∞+non_hemorrhagic_pigmented_long_duration
          +□5+seborrheic_keratosis
          +□1+melanoma

Classical/Crisp Set Theory.


1. Mathematical theory: collections of abstract objects, or sets.
2. Two PRIMARY OBJECTS of classical set theory:
      2a. SET MEMBERSHIP: ∈;
      2b. EMPTY SET or NULL SET: {} or Ø.

Set Theory Operations.


1. Not:   ~.
2. Membership, ∈:   x ∈ X: x belongs to X.
3. Union, ∪:   X = (Y ∪ Z): set of all members of Y or Z or both.
4. Intersection, ∩:   X = (Y ∩ Z): set of all members of both Y and Z.
5. Subset, ⊆:   X ⊆ Y if and only if every member of X is also a member of Y.
6. Superset, ⊇:   X ⊇ Y if and only if every member of Y is also a member of X.
7. Set_subtraction, -:   X = (Y - Z): set of all members of Y but not Z.

Fuzzy Set Theory.


1. Represents different levels of certainty for the same concept.
2. Element p has partial membership in set P: pμvP.
3. v: assumes any value along closed interval, [0,1].
4. Fuzzy is NOT probability.
5. Despite its quirky name, fuzzy is serious mathematics.
6. Ordinal property: If pμvP, and v>w, then pμwP.
7. Classical set theory: special case of fuzzy set theory: either v=0 or v=1.

Fuzzy Set Theory: Example.


1.

Propositional Logic.


1. Represents declarative sentences in algebraic form.
2. Logical operators: NOT, AND, INCLUSIVE_OR, EXCLUSIVE_OR, IMPLIES,....
3. Aristotle (384-322 BC). Modernized by George Boole (1815-1864).
4. Proposition: statement that may be evaluated as true or false, not both, not neither.
5. Every statement always HAS a true_false value.
6. Statement both true and false: INCONSISTENCY.
7. Syntactically correct namespace is always consistent.

Rules of Propositional Logic.


1. Double Negative Rule: --p = +p.
2. Demorgan's Rules: -(+p|+q) = (-p&-q); and -(+p&+q) = (-p|-q)
3. Distributive Rules: ((+p|+q)&+r) = ((+p&+r)|(+q&+r)); and ((+p&+q)|+r) = ((+p|+r)&(+q|+r)).
4. Whitehead-Russell Transformation: (+p⇒+q) = (-p|+q)

Nand.


1. NAND ("not_and"): most fundamental operation of classical logic.
2. All other logic operations: constructible from NAND.
3. NANDSET: Set of propositions that are nanded to one another.
4. Invented by American philosopher Charles S. Peirce (1839-1914).
5. Known as Scheffer's stroke or Scheffer's dee, δ, by logicians.

Nandset.


1. Nandset that contains an element and its exact negation is vacuous.
2. Empty nandset is inconsistent.
3. If nandset X is a subset of set Y, then set Y is also a nandset.
4. These properties of nandsets: basis for mathematical proofs.

Possible Worlds / Possible Patients Model.


1. Possible patient description: is a set that contains anypatient (∀), and:
2. Exactly one true_false value for each proposition.
3. Example: four possible patient descriptions for two propositions skin_biopsy and basal_cell_carcinoma:
{+∀, +skin_biopsy, +basal_cell_carcinoma}
{+∀, +skin_biopsy, -basal_cell_carcinoma},
{+∀, -skin_biopsy, +basal_cell_carcinoma},
{+∀, -skin_biopsy, -basal_cell_carcinoma}.

4. Generally: 2n possible patient descriptions for n propositions.
5. Set of all possible patient descriptions: truth table.

Mathematical Theorem.


1. Mathematically precise statement, provable by deductive, step-by-step argument.
2. Similar to arguments in Euclid's (330-275 BC) Elements.
3. Not all true statements are provable (Gödel, 1931).
4. All provable statements are true.

Mathematical Theorems in Relationspace.


Theorem §9.1. Consistency of Namespace.
Theorem §9.2. Identity.
Theorem §9.3. Or-expansion.
Theorem §9.4. Telescoping.
Theorem §9.5. Contextualization.
Theorem §9.6. Intercalation.
Theorem §9.7. Retirement.

Theorem §9.1. Consistency of Namespace. 


 Theorem §9.1.
 +∀
    +p
       +q
          ...
             +r
                +t
                +u
             +s
                   ...
is consistent.

Theorem §9.2. Identity. 


 Theorem §9.2.
 +p
    +p

Theorem §9.3. Or-expansion.


 Theorem §9.3.
 +p                      +p
    +q         ⇒             +q
                            +q
                            +q
                            ...

Theorem §9.4. Telescoping.

Theorem §9.4. ((+p ⇒ +q) & ((+p & +q) ⇒ +r) & ((+p & +q & +r) ⇒ +s)) ⇒ (+p ⇒ +s).

Theorem §9.5. Contextualization.


 Theorem §9.5.
 +p                      +p
    +q         ⇔            +p
       +r                      +q
       +s                   +p
                               +r
                               +s

Theorem §9.6. Intercalation.


1. Procedure for inserting (intercalating) a new subhierarchy into the hierarchy.
      1a. Not disturbing the remaining hierarchy. 

 +p                      +p
    +p                      +q
       +q         ⇒         +r
       +r
    +p
       +s
       +t

Theorem §9.7. Retirement.


1. Procedure for removing a subhierarchy (obsolete concept). without disturbing the remainder of the hierarchy.
      1a. Not disturbing the remaining hierarchy. 

 +p                      +p
    +p            ⇒         +q
       +q
       +r
    +p
       -r

Mathematical Theorems: Proof Strategies.


1. Translate the statement into nandsets.
2. Verify that all nandsets are negative.
3. Example: +p ⇒ (+q ⇒ +p).
4. Whitehead/Russell Transformation: +p|-q|+p.
5. Nandset: {+p,+q,-p}.
6. Since ±p∈{+p,+q,-p}, the nandset is vacuous, and the theorem is proved.

Theorems of Classical/Modal Logic.


1.1. CCpCqrCCpqCpr. Restated: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r)).
1.2. CpCqp. Restated: +p ⇒ (+q ⇒ +p).
1.3. CCpqCpp. Restated: (+p ⇒ +q) ⇒ (+p ⇒ +p).
1.4. Cpp. Restated: (+p ⇒ +p).
1.5. CCpqCCqrCpr. Restated: (+p ⇒ +q) ⇒ ((+q ⇒ +r) ⇒ (+p ⇒ +r)).
1.6. CCCCqrCprsCCpqs. Restated: ((+q ⇒ +r) ⇒ ((+p ⇒ +r) ⇒ +s)) ⇒ ((+p ⇒ +q) ⇒ +s).

Live Computer Demonstration.



Public-domain Perl Source Code.



Model for Ethical Data Collection.


Rule 1. Complementizer Absorbs Negation.
Rule 2. Fuzzy Certainty.
Rule 3. Data are Crisp.
Rule 4. Hippocratic.
Rule 5. Converse Hippocratic.
Rule 6. Vexative.
Rule 7. Ontologic.
Rule 8. Ethical Data Registration.
Rule 9. Schrödinger's Opening.

Complementizers for Ethical Data Collection.

1. $zp: It is certain at level z whether p (doxastic modal logic, Greek: δοξα = doxa = belief);

2. #: It is demanded to know whether d; (deontic modal logic, Greek: δεον = deon = obligation, command);

3. !: It is paid to know whether d. (telontic modal logic, Greek: τελων = telón = payment, taxation).

Model for Ethical Data Collection.


Rule 1. Complementizer Absorbs Negation:
      +$zp = +$z+p = +$z-p.
      +#d=+#+d=+#-d.
      +!d=+!+d=+!-d.
Rule 2. Fuzzy Certainty. For proposition p and positive integers v>w, $vp ⇒ $wp.
Rule 3. Data are Crisp. For any datum, d, $d ⇒ $∞d.
Rule 4. Hippocratic. Datum d is Hippocratic if and only if !d ⇒ #d.
Rule 5. Converse Hippocratic. Datum d is converse Hippocratic if and only if (-$d & #d) ⇒ !d.
Rule 6. Vexative. (+□k+e & +□Δ & -□+d) ⇒ (+#+d | +□k+1-e).
Rule 7. Ontologic. For datum, d and entity, e: +□Δ ⇒ (+□ke | +□k+1-e), where Δ ≠ Ø and there exists no Δ' ⊆ Δ such that +□Δ' ⇒ (+□k-e |+□k+1+e).
Rule 8. Ethical Data Registration.
Rule 9. Schrödinger's Opening.

Summary of Results.


1. Mathematical consistency.
2. 200 theorems of classical and modal logic.
3. Intercalation Theorem (inserting new concepts).
4. Retirement Theorem (removing obsolete concepts).

Conclusions.


1. Organize knowledge of mucosal surface pathology.
2. Format of the semantic worldwide web.
3. Incorporates clinical and pathologic findings.

4. INTRODUCTION.

Dermatopathology is one of the most complex subspecialties of anatomic pathology. Skin is the largest organ in the human body, with direct exposure to environmental insults, as well as accessibility for observation and biopsy, so that there is an enormous variety and number of skin diseases with described pathologic lesions. Several leading textbooks of dermatopathology are over nine hundred pages long (McKee et al, 2005, Barnhill, 2004, Weedon, 2002, Farmer, 1999). Weedon (2002) has 21,998 literature references in 41 chapters. Clinical history is often an essential part of a dermatopathology diagnosis, and contributes to prognosis and therapy. The general pathologist is easily overwhelmed by all this detail and complexity.

An ontology is the core knowledge base and fundamental assumptions for a field of study. The ontology for dermatopathology may be broadly classified into image recognition and medical reasoning components. This report proposes an ontology for the medical reasoning concepts of dermatopathology, based upon the general principle that even in the large realm of possibilities, some choices can be eliminated based upon a distinctive clinical setting or pathologic features; and other choices can be eliminated, at least provisionally, based upon their unlikeliness ("Sutton's Law").

The ontology model employs a hierarchical namespace of unique names; a relationspace, listing all relationships among concepts; and a fuzzyspace, based upon fuzzy set theory, for levels of certainty within the same concept. This ontology includes tests of mathematical consistency and completeness; techniques for managing large lists of concepts; and a formal method for introducing new ideas and retiring obsolete ideas from the hierarchy. The mathematical model is supported by mathematical definitions, theorems, and proofs, and examples from dermatopathology diagnostic principles. This mathematical model has promise for automated review of the emerging electronic medical record, and detecting possible quality assurance anomalies in a large medical database.

5. NAMESPACE.

A NAMESPACE is a strict hierarchy of all concepts in the ontology, where each name appears exactly once. Each name, except for the ultimate name, has exactly one parent, and negations are not allowed. Mathematical consistency is the property of a system that no statement and its negation may be deduced from the system. Since negations and multiple parents are not allowed in a namespace, the namespace formalism is inherently consistent (Theorem §9.1). This formal structure is equivalent to Classes in the Resource Description Framework (RDF), a hierarchical ontology specification language of the semantic worldwide web. The Laboratory Digital Imaging Project (LDIP) of the Association for Pathology Informatics (API) has proposed an RDF specification for a large area of anatomic pathology, in which there is an Ultimate_Class for anatomic pathology concepts, consisting of seven subClasses: Person, Event, Data_object, Specimen, Reagent, Instrument, and Terminology. This ultimate_class serves as the origin, named anypatient and denoted ∀, for all elements in the hierarchy. Present is denoted +; and absent is denoted -. In the namespace, all names are present.

The Event Class contains temporal, spatial, mass, temperature, homunculus, and morbulus. The homunculus Class is a reference model of an idealized human body, that may be divided broadly into nine systems: cardiovascular_system, respiratory_system, gastrointestinal_system, genitourinary_system, endocrine_system, musculoskeletal_system, lymphoreticular_system, nervous_system, and integumentary_system: 

 +ultimate_class=+∀
    +person
    +event
       +temporal
       +spatial
       +mass
       +temperature
       +homunculus
          +cardiovascular_system
          +respiratory_system
          +gastrointestinal_system
          +genitourinary_system
          +endocrine_system
          +musculoskeletal_system
          +lymphoreticular_system
          +nervous_system
          +integumentary_system
       +morbulus
    +data_object
    +specimen
    +reagent
    +instrument
    +terminology
One may focus on the integumentary_system, under which there are epidermis, dermis, skin_appendage, and subcutaneous_tissue, summarized as follows: 

 +ultimate_class=+∀
    ...
    +event
       ...
       +homunculus
          ...
          +integumentary_system
             +epidermis
                +stratum_corneum
                +stratum_granulosum
                +stratum_spinosum
                +stratum_basale
             +dermis
                +papillary_dermis
                +reticular_dermis
             +skin_appendage
                +hair
                +nail
                +apocrine_gland
                +eccrine_gland
                +sebaceous_gland
             +subcutaneous_tissue
                +adipose
                +connective_tissue
                +vascular
                +nerve
Analogously, he Morbulus Class is a reference model of human diseases, that may be divided broadly into seven systems: 

 +Ultimate_class=+∀
    ...
    +Event
       ...
       +Morbulus
          +normal_variant
          +congenital
          +inflammatory
             +non_infectious
             +infectious
                +bacterial
                +fungal
                +mycobacterial
                +viral
                +rickettsial
                +treponemal
          +vascular
             +traumatic
             +ischemic
             +vasculopathy
          +neoplastic
             +benign
             +dysplastic
             +neoplasm_primary
             +neoplasm_metastatic
          +metabolic
          +systemic
The homunculus and morbulus namespace subhierarchies may be constructed to an arbitrary level of refinement. For anatomic structures, the Nomina Anatomica provides a detailed model for such an RDF hierarchy. For diseases, one may use text tables in standard pathology textbooks. Other event classes might include: embryonula, physiologula, genomula, traumula, etc.

The main problem with the namespace/RDF formalism is that each name can only appear ONCE in the hierarchy. This constraint makes for a tidy classification, but it also means that the namespace designer is forced into making an executive decision (sometimes arbitrary, often controversial) on what represents the most fundamental organizing principle of the hierarchy. My preference is embryologic or even evolutionary (comparative anatomic) origin of the structure. This unique-name constraint has limited mathematical power, but also limited opportunities for getting into mathematical mischief, such as inconsistency, incomputability, etc.

In biomedical informatics, assertions have semantic content, or meaning, whenever a pair of metadata and data (the descriptor for the datum and the datum itself) is assigned, or bound, to a unique subject. An ordered triple consists of: <subject,metadatum,datum>, where each metadatum appears in the ontology namespace; a subject is a patient; and a datum is the value of that metadatum for that patient. In ordinary mathematical notation, this structure corresponds to <argument,function,value>, where function(argument)=value, or f(x)=y.

Some ordered triples that might be found in a medical dataset include:
<+g_moore,+left_axillary_lymphadenopathy,+no>
<+g_moore,+right_eye_lens_prosthesis,+yes>
<+superman,+xray_vision,+yes>
<+lou_gehrig,+amyotrophic_lateral_sclerosis,+yes>
In the first two ordered triples, the subject is ambiguous. For example, in the 2007 Baltimore, MD, telephone directory alone, there are xxx g_moores; however, there is only one g_moore who serves as a staff pathologist at the Baltimore VA Medical center. The last two ordered triples have uniquely identified subjects.

We employ the following spelling conventions, for simplicity and consistency:
1. American-English spelling, American-Roman alphabet, all lower case. Examples: hemochromatosis NOT haemochromatosis; esophagus NOT oesophagus; behcet NOT Behçet.

2. Blankspace, hyphen, and other punctuation replaced by underline (_). No apostrophe or apostrophe_s. All nouns singular. Examples: hashimoto_thyroiditis, wilms_tumor, graves_disease, hodgkin_disease.

3. Metadata inheritance denoted by carriagereturn and indentation, or ⇒.

4. Metadata sibship denoted by carriagereturn and no indentation, or |.

5. Existing standards used, where applicable and freely available.

6. Date/time convention: International Standards Organization, ISO 8601.

7. All stopwords, barrierwords, or low-information words (prepositions, conjunctions, articles, pronouns, auxiliary verbs, and disease, syndrome, condition, method, modified, solution, and technique) are removed or placed at the end of the phrase. Example: zuckerkandl_organ NOT Organ of Zuckerkandl.
Ordered triples can export their meaning between different databases on the worldwide web, because they bind a described datum to a specified subject. This feature of ordered triples supports data integration of heterogeneous data; and facilitates the design of internet web-crawler programs, called software agents, that can interrogate multiple RDF documents on the worldwide web, and initiate their own actions, based on inferences yielded from retrieved ordered triples. Resource Description Framework (RDF) is a general syntax for writing computer-parsable ordered triples. Detailed instructions and computer software for preparing web-ready RDF files are available in the public domain (Berman, 2007, Berman and Moore, 2007). Berman (2004) has constructed a strict hierarchy for over 100,000 human neoplasms, organized by embryologic origin, that is computer-parsable and RDF-translatable.

The greatest problem with a namespace for an emerging field of study such as anatomic pathology informatics, is that standardization committees often cannot settle upon a unique organizing principle. In the short term, it may make more sense to allow several organizing principles to exist side-by-side, including negations and multiple parents for some elements; and to have a mechanism for introducing new concept subhierarchies and retiring obsolete concept subhierarchies. At a later time, redundant RDF classes may be recast as RDF properties, which permit multiple parents.

6. RELATIONSPACE.

The RELATIONSPACE is the general hierarchy of relationships among concepts appearing in the namespace. In the relationspace, unlike the namespace, a name may occur in many, separate contexts, as for example, anatomic, pathophysiologic, embryonic, environmental susceptibility, tumor susceptibility, etc. A mathematical relation is a correspondence between two objects in a hierarchy, where multiple occurrences of the same element are allowed; multiple parents for the same element are allowed; and the negation of an element is allowed. We use +p to denote it is true that p; and -p to denote it is false that p.

7. FUZZYSPACE.

In many areas of clinical medicine, it may be necessary to take actions based upon incomplete information about a particular patient. These actions might include therapy based upon a presumptive diagnosis; or even the decision to collect additional information, such as a skin biopsy, that might inconvenience, or even injure, the patient. Sutton's law is a medical slang term asserting that, in the face of diagnostic uncertainty, one should take action based upon the most likely diagnosis. Sutton's Law is named after the notorious bank robber, Willie Sutton, who preferred to rob banks because they were the most likely places to find money. The idea first entered the medical literature in 1961, in the context of fevers of unknown origin (Petersdorf and Beeson, 1961), i.e., in the absence of immediate culture results, treat with the most likely effective antibiotic; and has wide applicability in clinical medicine. Another version of Sutton's Law is the Zebra Rule: if you hear hoofbeats in the street, think of horses not zebras. A tertiary-care medical institution that cares for patients with unusual diseases is sometimes called a zebra farm. Computer scientists call this logic the logic of jumping to conclusions (Brewka et al, 1997).

A FUZZYSPACE, based upon fuzzy set theory, is the system of levels of certainty for concepts in the relationspace. Classical set theory, or crisp set theory, is the mathematical theory of collections of abstract objects, or sets. There are two primary objects of classical set theory, i.e., set-membership, denoted ∈; and the empty set or null set, denoted {} or Ø, the set that contains no members. A set is defined exactly by its members. That is, two sets are considered equal if and only if they contain the same members. In classical set theory, we say that a proposition, p, is either a member of set P, denoted p∈P; or else p is not a member of set P, denoted p~∈P.

In the (presumably rare) instances in which the most likely outcome is not true, then in classical logic, one obtains an inconsistency. For example, suppose that a dermatologist biopsies a lesion that is worrisome for malignant melanoma. The biopsy results are benign, and the patient is given a return appointment at the next routine followup interval. However, to justify the biopsy in the first place, there was a presumptive diagnosis of malignant melanoma. Thus in classical logic, the patient has both +malignant melanoma and -malignant melanoma, an inconsistency. There are various devices for avoiding this potential inconsistency, including non-monotonic logic, modal logic, deviant logic, circumscription logic, and fuzzy logic.

Fuzzy set theory is a generalization of classical set theory, in which proposition p can be a partial member of set P, on a sliding scale of fuzzy values, from v=0 to v=1, inclusive. For fuzzy value v, we write pμvP. For fuzzy value v=1, pμ1P corresponds to full membership in classical set theory, or p∈P; for v=0, pμ0P corresponds to full non-membership in classical set theory, or p~∈P; for v=½, pμ½P corresponds to half-membership, etc. Fuzzy set theory has an ordinal property, i.e., if pμvP and v>w, then pμwP. Certainty levels, numbered z=0, 1, 2,..., correspond to fuzzy values, v, by the formula v=(1-2-z). Certainty level z=∞ corresponds to v=1, or full membership in classical set theory, p∈P. Certainty level z=0 corresponds to v=0, or full non-membership in classical set theory, p~∈P. Certainty level z=1 corresponds to v=½; certainty level z=2 corresponds to v=¾; certainty level z=3 corresponds to v=⅞, etc. Many concepts in medicine are characterized by their relative certainty in a given clinical setting, as rare, common, very frequent, etc., without assigning exact numeric values, so-called computing with words (Zadeh, 2001, 2006). 

 ∞ : absolutely_certain.
 .....
 6 : very_frequent
 5 : frequent
 4 : common
 3 : uncommon
 2 : rare
 1 : very_rare
 0 : absolutely_uncertain.
In memory of Willie Sutton's proclivity for dollars ($) (Moore et al, 1978), we write: 

 $∞ : absolutely_certain.
 .....
 $6 : very_frequent
 $5 : frequent
 $4 : common
 $3 : uncommon
 $2 : rare
 $1 : very_rare
 $0 : absolutely_uncertain.
The Zebra Rule might be formalized as: 

 +∀
    ...
       +hoofbeats_in_street
          +horses
             +$5horses
          +zebras
             +$1zebras
For example, +hoofbeats_in_street implies +horses or +zebras. If there are +hoofbeats_in_street, then it is frequent that there are +horses, but very_rare that there are +zebras.

Suppose we hear +hoofbeats_in_street, and we need to make an immediate decision, based upon whether there are +horses or +zebras in the street. By application of Sutton's Law, we can assume (temporarily) that there are no very rare events. From this, we may conclude that 

 +∀
    ...
       +hoofbeats_in_street
          +horses
             +$5horses
(Details of calculation are given below).

As a notational convenience, we may define necessarily p, denoted □p as □p = ($p & p); and possibly p, denoted ◇p as ◇p = (-$p | p). Then: 

 +∀
    ...
       +□∞+hoofbeats_in_street
          +□5+horses
          +□1+zebras


Now consider a raised, slightly irregular, pigmented skin lesion of uncertain duration on the upper back of a fair-skinned middle-aged patient, which has recently started to bleed. The patient gives an uncertain history of trauma at this site. The differential diagnosis for this lesion might include pigmented seborrheic keratosis, intradermal nevus, atypical compound nevus, and malignant melanoma: 

 +∀
    ...
       +bleeding_pigmented_skin_lesion
          +pigmented_seborrheic_keratosis
          +intradermal_nevus
          +atypical_compound_nevus
          +malignant_melanoma
The courses of action include: 

 +∀
    ...
       +bleeding_pigmented_skin_lesion
          +routine_followup
          +early_followup
          +immediate_skin_biopsy
Then: 

 +∀
    ...
       +bleeding_pigmented_skin_lesion_long_duration_questionable_history_trauma
          +$4malignant_melanoma
             +malignant_melanoma
Because of the possibility of a malignant_melanoma (+$4malignant_melanoma), and the seriousness of this diagnosis if present, the dermatologist has an indication for biopsying the lesion: 

 +∀
    ...
       +malignant_melanoma
          +$4malignant_melanoma
             +immediate_skin_biopsy
          +$1malignant_melanoma
             +early_followup
On the other hand, if the patient is certain that the lesion has been present since childhood, and was recently traumatized, then: 

 +∀
    ...
       +bleeding_pigmented_skin_lesion_short_duration_trauma
          +$1malignant_melanoma
             +malignant_melanoma
In this case, since the possibility of a malignant_melanoma is very_rare (+$1malignant_melanoma), there is an indication for followup only. These two scenarios have the same differential diagnosis but different certainty levels, resulting potentially in different clinical actions.

Certainty levels are akin to the necessarily and possibly operators of modal logic. The logic of certainty or belief is called doxastic modal logic (Greek: δοξα = doxa = belief); the logic of medical obligation or indication is called deontic modal logic (Greek: δεον = deon = obligation, command); and the logic of payment or injury is called telontic modal logic (Greek: τελων = telón = payment, taxation).

8. PROPOSITIONAL LOGIC, SET THEORY.

Propositional logic is a method for representing declarative sentences in algebraic form, and making deductions based upon logical operators, such as not, and, inclusive_or, implies, .... Formal logic dates back to Aristotle (384-322 BC), and was modernized by George Boole (1815-1864). A proposition is a statement that may be evaluated as true or false, not both and not neither. Although the true_false value for a particular statement may be unknown in a particular setting, in principle, the statement always HAS a true_false value. If it is possible to deduce mathematically that particular statement is both true and false, then the logical system is inconsistent. It is shown below (Theorem §9.1) that a syntactically correct namespace is always consistent.

NAND ("not_and") is the most fundamental operation of classical symbolic logic, because all other logic operations can be constructed from nand. The device for keeping track of logical relationships is the nandset, or set of propositions that are nanded to one another, i.e., cannot all be true at once. For example, {+g_moore, +serum_potassium_9.2_ng/dL} is a nandset. The nand-operator was invented by American philosopher and logician, Charles S. Peirce (1839-1914) (1880, unpublished). This fundamental property of the nand-operator was first published by Henry M. Scheffer in 1913, and has become known as Scheffer's stroke or Scheffer's dee, δ, among logicians (Haack, 1996). The nand-operator was further developed by the Lvov-Warsaw school of exact logic (Łukasiewicz (1878-1956) and others) in the early twentieth century, so-called Polish logic. Nandsets have the properties that: a nandset that contains an element and its exact negation is vacuous; and if nandset X is a subset of set Y, then set Y is also a nandset. These properties of nandsets serve as the linchpins for many of the mathematical proofs in this report.

The sentence, (+p nand +q), means that not both +p and +q can be true. The sentence, (+p nand +q nand +r nand ...), means that not all of +p, +q, +r,..., can be true. Nand is the mathematical analogue of the transistor (formerly, mechanical relay, then vacuum tube), the basic building-block of the digital computer. A nandset is any set of propositions that cannot all be true. Commonly-used nand-transformations include: 

 -p             +p nand +p
 +p ⇒ +q       +p nand -q
 +p & +q        -p nand -p, -q nand -q
 +p | +q        -p nand -q
 +p ⇔ +q       +p nand -q, -p nand +q


A relationspace is a formalism for organizing propositions in a hierarchy, in the general format: 

 +p
    +q
       +s
       +t
       ...
    +r
       +u
       +v
       ...
    ...
For example: 

 +∀
    ...
       +anatomy
          +skin
              +epidermis
                 +stratum_corneum
                 +stratum_granulosum
                 ...
              +dermis
                 +papillary_dermis
                 +reticular_dermis
                 ...
              +skin_appendage
                 +hair
                 +nail
                 +eccrine_gland
                 +apocrine_gland
                 ...
              ...
For each element in a relationspace hierarchy that has children, the following propositional logic sentence obtains: (the_element & its_parent & its_grandparent &...) ⇒ (its_child | its_child | its_child | ...), where ⇒ means "implies" or "if...then"; & means "and"; and | means "inclusive_or". In the above example, +skin⇒(+epidermis|+dermis|+skin_appendage|...); (+skin&+epidermis)⇒(+stratum_corneum|stratum_granulosum|...), (+skin&+dermis)⇒(+papillary_dermis|+reticular_dermis|...), etc.

A namespace is a special case of a relationspace in which every name appears exactly once, and all names are positive. It is shown in Theorem §9.1 that a syntactically correct namespace is always consistent.

Generally applicable rules of propositional logic are as follows:
Double Negative Rule: --p = +p.
Demorgan's Rules: -(+p|+q) = (-p&-q); and -(+p&+q) = (-p|-q)
Distributive Rules: ((+p|+q)&+r) = ((+p&+r)|(+q&+r)); and ((+p&+q)|+r) = ((+p|+r)&(+q|+r)).
Whitehead-Russell Transform: (+p⇒+q) = (-p|+q)
That is, a double negative is a positive; not (+p or +q) equals (-p and -q); not (+p and +q) equals (-p or -q), etc. These rules are used to solve expressions in a manner similar to high-school algebra.

SET THEORY is the mathematical theory of collections of abstract objects, or sets. There are two primary objects of classical set theory, namely, set-membership, denoted ∈; and the empty set or null set, denoted {} or Ø). We denote a set as the list of elements that belong to, or are members of, that set, enclosed in curly brackets, {,,,,}, separated by commas. The order of elements is irrelevant, and repeated elements are redundant. For example, sets {a,b,c,d,e} and {e,d,c,b,a,a,a} are the same set. The null set is the set that contains no members. A set is defined exactly by its members. That is, two sets are considered equal if and only if they contain the same members. Operations commonly used in elementary set theory include:
Not:   ~.
Membership, ∈:   x ∈ X denotes: x belongs to (or is a member of) X.
Union, ∪:   X = (Y ∪ Z) is the set of all members of Y or Z or both.
Intersection, ∩:   X = (Y ∩ Z) is the set of all members of both Y and Z.
Subset, ⊆:   X ⊆ Y if and only if every member of X is also a member of Y.
Superset, ⊇:   X ⊇ Y if and only if every member of Y is also a member of X.
Set_subtraction, -:   X = (Y - Z) is the set of all members of Y but not Z.
A possible patient description is a set that contains anypatient, denoted ∀, and exactly one true_false value for each proposition. For example, the four possible patient descriptions for a system containing propositions skin_biopsy and basal_cell_carcinoma are:
{+∀, +skin_biopsy, +basal_cell_carcinoma}
{+∀, +skin_biopsy, -basal_cell_carcinoma},
{+∀, -skin_biopsy, +basal_cell_carcinoma},
{+∀, -skin_biopsy, -basal_cell_carcinoma}.
The collection of all possible patient descriptions is a possible patient description table or a truth table. This concept is developed in the philosophy literature as possible worlds.

A logic sentence is a set of propositions in logical relation to one another, that makes an assertion about members of the possible patient description table. For example, if we assert that a patient has a skin_biopsy with a basal_cell_carcinoma, then we eliminate those possible patient descriptions in which -skin_biopsy or -basal_cell_carcinoma are present, as follows:
{+∀, +skin_biopsy, +basal_cell_carcinoma}
{+∀, +skin_biopsy, -basal_cell_carcinoma},
{+∀, -skin_biopsy, +basal_cell_carcinoma},
{+∀, -skin_biopsy, -basal_cell_carcinoma}.


A theorem is an assertion of the form, if H, then C is true, where H is the hypothesis and C is the conclusion. A proof for the theorem is a stepwise demonstration, starting with H and concluding with C, in which each step follows logically from the previous step. In classical symbolic logic, is_true corresponds to a vacuous nandset, i.e., a nandset that removes no possible_patient_descriptions. Thus, the proof of a theorem may be executed by converting the theorem into nandsets, and then showing that all such nandsets are vacuous. For example, the theorem that (+p⇒+p) may be proved first by showing that it equals (-p|+p) (Whitehead-Russell transformation); and then that (-p|+p) equals the nandset, {+p,-p} (DeMorgan's Laws). Then the nandset {+p,-p} is vacuous, because it is not a subset of any possible_patient_description (which cannot contain both an element and its negation). Over 200 theorems are proven by this method in Chapter 27. Appendix H. Theorems, a leading textbook of symbolic logic.

9. MATHEMATICAL THEOREMS.



A theorem is a mathematically precise statement that can be proven true by a deductive, step-by-step argument, similar to those arguments made in Euclid's (330-275 BC) Elements. Not all true statements are provable (Gödel, 1931), but all provable statements are true.

There are numerous, established mathematical theorems for this name/relation/fuzzy model (Appendix H). Some mathematical properties are particularly useful in medical reasoning applications:
Theorem §9.1. Consistency of Namespace.
Theorem §9.2. Identity.
Theorem §9.3. Or-expansion:
Theorem §9.4. Telescoping.
Theorem §9.5. Contextualization:
Theorem §9.6. Intercalation.
Definitions.
Definition §9.1. Consistency is the property of a system that no statement and its negation may be deduced from the system. That is, {+∀} is not a nandset for the system.
Proofs.
Theorem §9.1. Consistency of Namespace. 

 +∀
    +p
       +q
          ...
             +r
                +t
                +u
             +s
                   ...
is consistent.
Proof. We show that {+∀} is not a nandset for the namespace. For names p,q,r,..., construct possible patient descriptor nandset, {+∀, +p, +q, ..., +r, +s, ..., +t, +u, ...}. Then each name, r, occurs uniquely as a child in nandset, {+∀, +p, +q, ..., -r, -s, ...}, and as a parent in nansets such as {+∀, +p, +q, ..., +r, -t, -u, ...}. None of these nandsets are a subset of the constructed nandset, {+∀, +p, +q, ..., +r, +s, ..., +t, +u, ...}. Therefore, {+∀} is not a nandset for the namespace.

Theorem §9.2. Identity. A statement implies itself. 

 +p
    +p

Proof. +p ⇒ +p. Nandset {+p, -p} is vacuous.

Theorem §9.3. Or-expansion: If +p implies +q, then +p implies +q or +q or +q or.... 

 +p                      +p
    +q         ⇒             +q
                            +q
                            +q
                            ...

Proof. IF. (+p ⇒ +q) ⇒ (+p ⇒ (+q | +q | +q |...)). Nandsets {-p, +p, -q, -q, -q,...} and {+q, +p, -q, -q, -q,...} are vacuous.
Proof. ONLY IF. (+p ⇒ (+q | +q | +q |...)) ⇒ (+p ⇒ +q). Nandsets {-p, +p, -q}, {+q, +p, -q}, {+q, +p, -q}, {+q, +p, -q}, ... are vacuous.

Theorem §9.4. Telescoping.
Proof. ((+p ⇒ +q) & ((+p & +q) ⇒ +r) & ((+p & +q & +r) ⇒ +s)) ⇒ (+p ⇒ +s). Nandsets {-p, -p, -p, +p, -s}, {-p, -p, -q, +p, -s}, {-p, -p, -r, +p, -s}, {-p, -p, +s, +p, -s}, {-p, -q, -p, +p, -s}, {-p, -q, -q, +p, -s}, {-p, -q, -r, +p, -s}, {-p, -q, +s, +p, -s}, {-p, +r, -p, +p, -s}, {-p, +r, -q, +p, -s}, {-p, +r, -r, +p, -s}, {-p, +r, +s, +p, -s}, {+q, -p, -p, +p, -s}, {+q, -p, -q, +p, -s}, {+q, -p, -r, +p, -s}, {+q, -p, +s, +p, -s}, {+q, -q, -p, +p, -s}, {+q, -q, -q, +p, -s}, {+q, -q, -r, +p, -s}, {+q, -q, +s, +p, -s}, {+q, +r, -p, +p, -s}, {+q, +r, -q, +p, -s}, {+q, +r, -r, +p, -s}, and {-q, +r, +s, +p, -s} are vacuous.

Theorem §9.5. Contextualization: 

 +p                      +p
    +q         ⇔            +p
       +r                      +q
       +s                   +p
                               +r
                               +s

Proof. ((+p ⇒ +q) & (+p ⇒ (+r | +s))) ⇒ (+p ⇒ +q) & ((+p & +q) ⇒ (+r | +s)). Nandsets {-p, +p, +q, -r, -s}, {+r, +p, +q, -r, -s}, and {+s, +p, +q, -r, -s} are vacuous.

Theorem §9.6. Intercalation. Procedure for inserting (intercalating) a new subhierarchy into the hierarchy, without disturbing the remainder of the hierarchy. 

 +p                      +p
    +p                      +q
       +q         ⇒         +r
       +r
    +p
       +s
       +t

Proof. (+p ⇒ (+q | +r)) & (+p ⇒ (+s | +t)) ⇒ (+p ⇒ (+q | +r)). {-p, -p, +p, -q, -r}, {-p, +s, +p, -q, -r}, {-p, +t, +p, -q, -r}, {+q, -p, +p, -q, -r}, {+q, +s, +p, -q, -r}, {+q, +t, +p, -q, -r}, {+r, -p, +p, -q, -r}, {+r, +s, +p, -q, -r}, and

Theorem §9.7. Retirement. Procedure for removing a subhierarchy (obsolete concept) without disturbing the remainder of the hierarchy. 

 +p                      +p
    +p            ⇒         +q
       +q
       +r
    +p
       -r

Proof. (+p ⇒ (+q | +r)) ⇒ (+p ⇒ +q). Nandsets {-p, -p, +p, -q}, {-p, -r, +p, -q}, {+q, -p, +p, -q}, {+q, -r, +p, -q}, {+r, -p, +p, -q}, and {+r, -r, +p, -q} are vacuous.

INTERCALATION.

INTERCALATION is a procedure for inserting (intercalating) a new subhierarchy into the hierarchy, without disturbing the remainder of the hierarchy.

RETIREMENT.



RETIREMENT is a procedure for removing a subhierarchy (obsolete concept) without disturbing the remainder of the hierarchy.

10. LIVE COMPUTER DEMONSTRATION.



As part of the mathematical model, we have included a simple, public-domain theorem-verification program, with Perl source code, available to the public for small demonstration datasets. For economic reasons, the input datasets submitted to this software on the author's private internet site are limited to inputs up to 5 KB, 1000 lines, and 200 variables. The input should consist of one or more hierarchies, using the style guidelines provided above. The ultimate_class is denoted +a, and indented one space from the left margin. Each child_class is indented at multiples of three spaces right of the ultimate_class, as appropriate. The left margin may be bounded by : and the right margin by ;, in order to include brief comments or line numbers. The program is very sensitive to syntax errors, and does not provide detailed error messages. Some proofs may fail after too many computing cycles. A demonstration (Theorems §9.2 - §9.7) is provided.

To use the theorem prover, SELECT and COPY a cascade hierarchy text-image from a NOTEPAD® or other text file; PASTE the text-image into the text box below; and click on the SUBMIT button.



The sentence parser is very persnickety and error-intolerant. The upper-left corner MUST contain +0, one space right of the left-margin. Each subsequent row must contain exactly one variable name, preceded by + or -, up to the final row. Indentation must be EXACTLY THREE SPACES.

11. MODEL FOR ETHICAL DATA COLLECTION.



The universe of discourse, W (German: Welt = universe), consists of two, non-overlapping sets: data, D, and medical entities, E, where D ∪ E = W and D ∩ E = Ø. Every proposition, +w ∈ W, has an exact negation, -w ∈ W, where it is understood that +w = --w, i.e., a double-negative is positive. Furthermore, for every +d ∈ D, -d ∈ D; and for every +e ∈ E, -e ∈ E.

For the individual patient, data, D, are collected in order to establish the presence or absence of medical entities, E.

Each proposition, +w ∈ W, has a fuzzy certainty level, ranging from $0w (completely uncertain) to $∞w (completely certain), where $zp corresponds to fuzzy membership value, v = (1-2-z), in the fuzzy membership expression, wμvW.

Each datum, d, also has a demand/value status (#d); and an effort/cost status (!d).

Data collection may include medical history, physical findings, laboratory tests, radiology, and any special tests indicated by medical findings. Every datum collected, d ∈ D, requires some effort/cost, !d, ranging from risk to confidentiality for historical information, through procedures with risk of morbidity or death. Every datum collected must have a medical mandate or indication, #d, that includes consent from the patient or patient_advocate. If the patient schedules an appointment to see a physician, this action in itself is an implied consent for simple medical history taking; major procedures must be justified by significant medical concern.

The relationships between data collection and medical mandates are summarized in nine rules:
Rule 1. Complementizer Absorbs Negation.

Rule 2. Fuzzy Certainty.

Rule 3. Data are Crisp.

Rule 4. Hippocratic.

Rule 5. Conative.

Rule 6. Vexative.

Rule 7. Ontologic.

Rule 8. Data Registration.

Rule 9. Schrödinger's Opening.
Rule 1. Complementizer Absorbs Negation. In linguistics, within a statement of the form, it is said that Homer was blind, the word that is a complementizer (Latin: complére: to fill up, complete), which connects the principal clause, namely it is said, with the dependent clause, namely, Homer was blind. Generalizing from this concept, we may regard the particles, $, #, !, as complementizers for datum d, or for entity e:
$ze: It is certain at level z whether e;
$zd: It is certain at level z whether d;
#: It is demanded whether d;
!: It costs whether d.
These complementizers are negation neutral, i.e., if you are certain whether +d at level z, then you are certain that +d at level z, as well as that -d at level z. That is, $zd = $z+d = $z-d; likewise for #d, !d, and $ze.

Rule 2. Fuzzy Certainty. Fuzzy certainty is the assertion that for any atom a and positive integers v>w, $va ⇒ $wa. This assertion derives from the ordinal property of fuzzy set theory, i.e., if pμvP and v>w, then pμwP, where certainty levels, numbered z=0, 1, 2,..., correspond to fuzzy values, v, by the formula v=(1-2-z).

Rule 3. Data are Crisp. Data are Crisp is the assertion that for any datum, d, $d ⇒ $∞d. That is, every datum is known with complete certainty, although entities are known only on a sliding scale. For this purpose, a datum is understood in its most atomized form, namely, a particular measurement taken at a particular moment in time, with a date/time stamp, as well as other features as required by the LDIP protocol (instrumentation, reagents, method, responsible pathologist, etc). A logic statement is required to link, say, "hyperkalemia on 1/30/2007" (entity) to the datum, "serum potassium 7.2, 1/30/2007 at 8:00 AM" (datum).

Rule 4. Hippocratic. This rule formalizes the famous dictum of Hippocrates (460-370 BC): first do no harm. Datum d is Hippocratic if and only if !d ⇒ #d. That is, each cost/effort, (!d), must be justified/indicated by a medical mandate, #d.

Rule 5. Converse Hippocratic. Try if you must. This rule formalizes the obligation to collect information if it is medically indicated. Datum d is converse Hippocratic if and only if (-$d & #d) ⇒ !d. That is, each datum, d, which is uncertain (-$d) but has a medical mandate (+#d), should be sought (!d).

Rule 6. Vexative. If you know certain entities and data, then this generates a need for an additional datum. That is, you become vexed by your ignorance of that additional datum. For example, if you know that an elderly male patient has not had a serum-prostatic-specific-antigen in the past five years, you become vexed regarding that missing-datum.

(+□k+e & +□Δ & -□+d) ⇒ (+#+d | +□k+1-e).

Nandset definition: {+$ke,e,+$kδ,δ,..,-$d,-#d, -$k+1e} ∈ S0, for 1 < k < M-2, δ∈ D, and e ∈ E.

Rule 7. Ontologic. If you know certain entities and data, then this generates the knowledge/certainty of an additional entity. For example, if this patient has an elevated serum-prostatic-specific-antigen, then you become more certain that the patient has prostate cancer.

+□Δ ⇒ (+□ke | +□k+1-e), where Δ ≠ Ø and there exists no Δ' ⊆ Δ such that +□Δ' ⇒ (+□k-e |+□k+1+e).

Nandset definition: {+$kδ,δ,..,-e,-$k+1e} ∈ S0 and {+$kδ,δ,..,-$ke} ∈ S0, for 1 < k < M-2, d ∈ D, δ ⊆ (D - {+d,-d}).

Rule 8. Ethical Data Collection. For each datum, there is a data-collection step, J, at which the datum is collected and is true; or the datum is collected and is false; or the datum collection attempt fails and the datum is unknown. Otherwise, the datum is never attempted and never collected.

Rule 9. Schrödinger's Opening. Schrödinger's cat is hypothetical scenario used to illutrated principles of quantum mechanics. In this scenario, there is a cat in a soundproof box, which has a probability of being alive. As long as the cat is unobserved, the cat has only a probability of being alive. However, as soon as the box is opened, the cat is either fully alive or fully dead.

In a Schrödinger's opening, the cat has a known alive/dead status each time a new medical datum is collected; but the cat reverts to a probabilistic state before a new datum is mandated and collected. This device is used to sidestep the so-called black crow paradox. According to this paradox: all crows are black; Charley is a crow; but oops!, Charley is an albino. Likewise, all swans are white; Charley is a swan; but oops!, Charley is black. (See: Bernstein PL. Against the Gods. The Remarkable Story of Risk. and Taleb NN. The Black Swan: The Impact of the Highly Improbable. This is analogous to assuming that hoofbeats in the street are USUALLY horses; but oops!, this time they are zebras. In classical logic, this event produces an inconsistency.

12. EXAMPLES.




14. DERMATOPATHOGENESIS.



The entirety of human dermatopathologic disease may be sumsumed under the general heading anypatient, denoted ∀; with two major subheadings: homunculus (Latin: small-human); morbulus (Latin: small-disease). 

 +∀
    +homunculus
    +morbulus
Homunculus is the image of normal human anatomy, which serves as the reference point for all processes, normal and abnormal, within human medicine. 

 +∀
    -homunculus
    +homunculus
       +gender
          +female
          +male
          +gender_variant
             +rare
       +position
          +left|+right|+bilateral|+midline|+laterality_not_specified.
          +superior|+inferior|+anterior|+posterior|
          +superficial|+deep|+lateral|+medial
       +organ_system
          +cardiovascular_system
          +respiratory_system
          +gastrointestinal_system
          +genitourinary_system
          +endocrine_system
          +musculoskeletal_system
          +lymphoreticular_system
          +nervous_system
          +integumentary_system
             +epidermis
                +stratum_corneum
                +stratum_granulosum
                +stratum_spinosum
                +stratum_basale
             +dermis
             +skin_appendage
             +subcutaneous_tissue
       +morbulus
Morbulus comprises the general disease categories, and serves as a reference point for all diseases within human medicine. 

 +∀
    -morbulus
    +morbulus
       +skin
          +congenital
             +genodermatosis
                +ichthyosis
                   +ichthyosis_vulgaris
                   +ichthyosis_x_linked
                   +ichthyosis_epidermolytic_hyperkeratosis
                   +ichthyosis_autosomal_recessive
                   +ichthyosis_erythroderma_variabilis
                   +ichthyosis_linearis_circumflexa
                +acrokeratoelastoidosis
                +dyskeratosis_congenita
                +porokeratosis
                +xeroderma_pigmentosum
                +ectodermal_dysplasia
                +epidermolysis_bullosa
                +focal_dermal_hypoplasia_syndrom
                +aplasia_cutis_congenita
                +poikiloderma_congenitale
                +bloom_syndrome
                +ataxia_telangiectasia
                +werner_syndrome
                +epidermolysis_bullosa
                +epidermolysis_bullosa_acquisita
                +keratosis_follicularis_darier
                +familial_benign_pemphigus_hailey_hailey
                +acrodermatitis_verruciformis_hopf
                +pseudoxanthoma_elasticum
                +connective_tissue_nevus
                +linear_melorheostotic_scleroderma
                +winchester_syndome
                +ehler_danlos_syndrome
                +cutis_laxa
                +pachydermoperiostosis
                +urticaria_pigmentosa
                +incontinentia_pigmenti
                +hypomelanosis_ito
          +inflammatory
             +non_infectious
                +autoimmune
                   +
                +papulosquamous
                   +lichen_planus
                   +benign_lichenoid_keratosis
                   +keratosis_lichenoides_chronica
                   +lichen_nitidus
                   +lichen_striatus
                   +pityriasis_rubra_pilaris
                   +pityriasis_lichenoides
                   +lymphomatoid_papulosis
                +vesiculobullous
                   +miliaria
                   +erythema_toxicum_neonatorum
                   +acropustulosis_infancy
                   +pemphigus
                      +pemphigus_vulgaris
                      +pemphigus_vegetans
                      +pemphigus_foliaceus
                      +pemphigus_erythematosus
                   +bullous_pemphigoid
                   +cicatricial_pemphigoid
                   +herpes_gestationis
                   +dermatitis_herpatiformis
                   +erythema_multiforme
                   +graft_vs_host_disease
                   +subcorneal_pustular_dermatosis
                   +transient_acantholytic_dermatosis
                   +friction_blister
                   +burn
                      +burn_electric
                      +burn_thermal
                +granulomatous
                   +sarcoidosis
                   +cheilitis_granulomatosa
                      +=+mischer_melkersson_rosenthal_syndrome
                   +cheilitis_glandularis
                   +granuloma_annulare
                   +necrobiosis_lipoidica
                   +rheumatoid_nodule
                   +annular_elastolytic_granuloma
                   +granuloma_gluteal_infantum
                +neutrophilic
                   +
                +eosinophilic
                   +
             +infectious
                +bacterial
                   +impetigo
                      +bullous_impetigo
                      +staphylococcal_scalded_skin_syndrome
                      +ecthyma
                   +erysipelas
                      +necrotizing_fasciitis
                   +acute_superficial_folliculitis
                   +pseudomonas_folliculitis
                   +acute_deep_folliculitis
                   +chronic_superficial_folliculitis
                   +pseudofolliculitis_beard
                   +follicular_occlusion_triad
                   +hidradenitis_suppurativa
                   +acne_conglobata
                   +perifolliculitis_capitis_abscedens_suffodiens
                   +blastomycosis_like_pyuoderma_vegetans
                   +toxic_shock_syndrome
                   +acute_septicemia
                      +acute_menigococcemia
                      +pseudomonas_septicemia
                      +vibrio_vulnificus_septicemia
                   +chronic_septicemia
                      +chronic_menigococcemia
                      +chronic_gonococcemia
                   +malakoplakia
                   +tuberculosis
                +fungal
                   +dermatophytosis
                      +erythrasma
                   +candidasis
                      +acute_mucocutaneous_candidasis
                      +chronic_mucocutaneous_candidasis
                      +disseminated_candidasis
                   +aspergillosis
                   +phycomycosis_mucormycosis
                   +cutaneous_alternariosis
                   +cutaneous_protothecosis
                   +north_american_blastomycosis
                   +paracoccidioidomycosis
                   +lobomycosis
                   +chromomycosis
                   +coccidioidomycosis
                   +cryptococcosis
                   +histoplasmosis
                   +african_histoplasmosis
                   +sporotrichosis
                   +actinomycosis
                   +nocardiosis
                   +mycetoma
                   +botryomycosis
                +mycobacterial
                   +tuberculosis_primary
                      +tuberculosis_miliary
                      +lupus_vulgaris
                      +tuberculosis_verrucosa_cutis
                      +scrofuloderma
                      +tuberculosis_cutis_orificialis
                   +tuberculid
                      +papulonecrotic_tuberculid
                      +lichenoid_scrofulosorum
                +viral
                   +herpes_simplex
                   +varicella_herpes_zoster
                   +variola
                   +human_cowpox
                   +eczema_herpeticum
                   +eczema_vaccinatum
                   +cytomegalic_inclusion_disease
                   +parapox_infection
                   +molluscum_contagiosum
                   +verruca
                      +verruca_vulgaris
                      +verruca_deep_palmoplantar
                      +verruca_plana
                      +epidermodysplasia_verricuformis
                      +condyloma_acuminatum
                   +bowenoid_papulosis_genitalia
                   +hand_foot_mouth_disease
                   +acquired_immunodeficiency_syndrome_aids
                +trepomemal
                   +syphilis
                   +yaws
                   +pina
                   +lyme_borreliosis
                +rickettsial
                +protozoal
                   +leishmaniasis_oriental
                   +leishmaniasis_american
                   +leishmaniasis_post_kala_azar_dermal
             +vascular
                +traumatic
                +ischemic
                +noninflammatory_purpura
                   +senile_purpura
                   +scurvy
                +idiopathic_thrombocytopenic_purpura
                   +autoerythrocyte_sensitization_syndrome
                   +coumadin_necrosis
                   +purpura_fulminans
                   +thrombotic_thrombocytopenic_purpura
                +inflammatory_purpura
                   +leukocytoclastic_vasculutis
                   +cryoglobulinemia
                   +pustulosis_acuta_generalisata
                   +purpura_pigmentosa_chronica
                      +=+majocchi_schamberg_disease
                +granuloma_faciale
                +erythema_elevatum_diutinum
                +acute_febrile_neutrophilic_dermatosis_sweet
                +polyarteritis_nodosa
                +vasculitis_granulomatosis
                   +allergic_granulomatosis
                   +wegener_granulomatosis
                   +lymphomatoid_granulomatosis
                +midline_granuloma_face
                +temporal_giant_cell_arteritis
                +malignant_atrophic_papulosis_degos
                +atrophie_blanche
                +cutaneous_cholesterol_embolism
                +livedo_reticularis
                +sclerosing_lymphangitis_penis
          +neoplastic
             +benign
             +dysplastic
             +neoplasm_primary
             +neoplasm_metastatic
          +neoplastic
             +epidermis
                +cyst
                   +epidermal_inclusion_cyst
                   +milium_cyst
                   +trichilemmal_cyst
                   +steatocystoma_multiplex_cyst
                   +pigmented_follicular_cyst
                   +dermoid_cyst
                   +bronchogenic_cyst
                   +thyroglossal_duct_cyst
                   +cutaneous_ciliated_cyst
                   +median_raphe_penis_cyst
                   +eruptive_vellus_hair_cyst
                +noncyst
                   +linear_epidermal_nevus
                   +nevus_comedonicus
                   +epidermolytic_acanthoma
                   +epidermolytic_acanthoma_isolated
                   +epidermolytic_acanthoma_disseminated
                   +oral_white_sponge_nevus
                   +seborrheic_keratosis
                   +large_cell_acanthoma
                   +clear_cell_acanthoma
             +melanocytic
             +appendage_hair
                +hair_follicle_nevus
                +trichofolliculoma
                +dilated_pore
                +pilar_sheath_acanthoma
                +fibrofolliculoma_multiple
                +trichodiscoma_multiple
                +trichoepithelioma
                +hair_follicle_hamartoma
                   +hair_follicle_hamartoma_generalized
                   +hair_follicle_hamartoma_localized
                +pilomatricoma
             +appendage_sebaceous
             +appendage_apocrine
             +appendage_eccrine
             +fibrous_tissue
             +fat
             +muscle
             +cartilage
             +bone
             +neural
             +neuroendocrine
             +vascular
             +metastasis
             +infiltrate_non_lymphoid
             +infiltrate_lymphoid
             +infiltrate_leukemic
          +metabolic
             +lipidosis
                +hyperlipoproteinemia
                +tangier_disease
                +niemann_pick_disease
                +gaucher_disease
                +angiokeratoma_corporis_diffusum_fabry
                +lipogranulomatosis_farber
                +histiocytosis_x
                +congenital_self_healing_reticulohistiocytosis
                +indeterminate_cell_proliferative_disorder
                +xanthoma_disseminatum
                +diffuse_normolipemic_plane_xanthoma
                +verriform_xanthoma
                +juvenile_xanthogranuloma_paraproteinemia
                +reticulohistiocytosis
                +progressive_nodular_histiocytoma
                +hereditary_progressive_mucinous_histiocytosis
                +generalized_eruptive_histiocytoma
                +benign_cephalic_histiocytosis
             +amyloidosis
                +primary_systemic_amyloidosis
                +secondary_systemic_amyloidosis
                +lichenoid_macular_amyloidosis
                +nodular_amyloidosis
             +colloid_milium_degeneration
             +nodular_colloid_degeneration
             +hyalinosis_cutis_mucosae
             +porphyria
                +pseudoporphyria_cutanea_tarda
             +calcinosis_cutis
                +metastatic_calcinosis_cutis
                +dystrophic_calcinosis_cutis
                +idiopathic_calcinosis_cutis
                +idiopathic_calcinosis_scrotum
                +subepidermal_calcified_nodule
             +gout
             +ochronosis
             +mucinosis
                +
             +mucopolysaccharidosis
             +acanthosis_nigricans
             +idiopathic_hemochromatosis
             +phrynoderma_vitamin_a_deficiency
             +pellagra_vitamin_b_deficiency
                +hartnup_disease
             +oculocutaneous_tyrosinosis
          +systemic
             +lupus_erythematosus
                +lupus_erythematosus_systemic
                +lupus_erythematosus_discoid
                +lupus_erythematosus_subacute

15. DISCUSSION.



... The time is now for building a dermatopathology ontology:
instant internet communication, publication
textbooks, indexes, tables of contents.
MS® Powerpoint® presentation subhierarchies.


... The U. S. Veterans Hospital Administration (USVHA), as well as some 5% of non-VHA hospitals nationwide, now uses an entirely paperless electronic medical record (EMR) for all patient care activities/processes. Except for quantitative results from the clinical laboratory, many of the traditionally non-quantitative records represent little more than typewritten versions of paper records, replete with spelling and grammatical errors, as well as a dizzying number of barely comprehensible medical acronyms and abbreviations (Berman, 2007). Some of these ambiguous abbreviations are so dangerous that they have been banned by the Joint Commission for the Accredication of Healthcare Organizations (JCAHO). While there have been proposals for standardizing the EMR, there has been a preference in actual practice to favor free/open expression, so as not to inhibit the communicative power of the reports. For example, the CAP/ACS regulations for pathology reports of large-specimen cancer resections permit free/open expression of the required scientifically valid data elements (SVDEs). This means that a physician reading the report should be able to understand the report and find the SVDEs, but not necessarily computer software.

Hospital inspectors have the right to examine ANY medical record on ANY patient seen at the institution. Wouldn't it be better if one faced an inspection confident that ALL records met the hospital policy requirements? If the EMR were in standard form, then the software could periodically scan ALL records for compliance. One part of this program would be to have an electronic ontology, and a detector for non-compliant ontologic events.


  • Free/open source, available for public comment.
  • Hierarchical data structure, easy to explore.
  • Easy to include a new subhierarchy, backtracking.
  • Easy to update: intercalation, retirement.
  • Points to the semantic web, LDIP/RDH.
  • Covers all major areas of dermatopatholy.
  • Quality assurance in EMRs.
  • Target language for automated translation (see (Berman, 2007).
    1. Dermatopathology
    a. Complex subspecialty of anatomic pathology
    b. Large variety and number of skin diseases.
    c. Described pathologic lesions.
    d. Largest organ in the human body.
    e. Direct exposure to environmental insults.
    f. Accessibility for observation and biopsy.
    g. Complex classification.
    h. Clinicopathologic correlation important.
    i. Published hierarchies.

    2. Emergence of computers in anatomic pathology:
    a. diagnostic reports.
    b. quality assurance: turnaround time, outliers.
    c. large specimen protocols for cancer therapy.

    3. Standardized common data elements for:
    a. sharing reports between institutions.
    b. health care billing.
    c. national health care policy.

    4. Resource description framework hierarchy.
    a. developed by Association for Pathology Informatics (LDIP project).
    b. strict hierarchy of classes.
    c. non-strict hierarchy of properties.

    5. Possible patient descriptors model:
    a. classical symbolic logic.
    b. set theory
    c. intercalation, retirement, theorems.
    d. fuzzy set theory for major classes.

    6. Discussion.
    a. emerging electronic medical records; only 3% of U. S. hospitals.
    b. large specimen protocols for cancer therapy, CAP, ACS; no std syntax.
    c. common data elements for sharing reports, billing.
    d. dermatopathology: complexity, interdisciplinary; needs standards.
    e. open source; collegial discussion; improve patient care.
    f. tracking cases electronically.
    g. canary in the mine.
    h. stratified sets, layered sets.

    NOTES




    RELATIONSPACE.



    The RELATIONSPACE is the general hierarchy of relationships among concepts appearing in the namespace. In the relationspace, unlike the namespace, a name may occur in many, separate contexts, as for example, anatomic, pathophysiologic, embryonic, environmental susceptibility, tumor susceptibility, etc. A mathematical relation is a correspondence between two objects in a hierarchy, where multiple occurrences of the same element are allowed; multiple parents for the same element are allowed; and the negation of an element is allowed. We use +x to denote it is true that x; and -x to denote it is false that x. For example, the principal layers of the skin, namely, epidermis, dermis, appendages, subcutaneous_tissue, might be named by their embryologic origins: 
     ...
    +homunculus
       +skin
           +ectoderm
              +ordinary_ectoderm
                 +epidermis_surface
                    +stratum_corneum
                    +stratum_granulosum
                    +stratum_spinosum
                    +stratum_basale
                 +hair
                 +nail
                 +eccrine_sweat_gland
                 +apocrine_sweat_gland
                 +sebaceous_gland
              +neurectoderm
                 +dendritic_cell
                 +neural_crest_cell
           +mesoderm
              +dermis
                 +corium
                 +subcorium
              +subcutaneous_fat
              +subcutaneous_connective_tissue
              +blood_element
              +lymphatic_element
              +arrector_pili_muscle
           +endoderm
              +blood_vessel_endothelium
    Theorem §9.1 However, for some purposes, it might be advantageous to classify the skin_specimen by surface anatomy: 
     +skin_specimen
    +surface anatomy
       +face
          +face_frontal_eminence 
          +face_glabella 
          +face_zygomatic_arch 
          +face_mental_protuberance 
          +face_mandibular_angle 
          +face_mandibular_inferior_border 
          +face_mastoid_process 
       +eye
          +eye_iris
             -skin_specimen
          +eye_pupil
             -skin_specimen
          +eye_palpebral_fissure
             +eye_palpebral_fissure_superior 
             +eye_palpebral_fissure_inferior 
          +eye_semilunar_fold 
          +eye_conjunctiva 
          +eye_lacrimal_caruncle 
          +eye_medial_angle 
          +eye_lateral_angle 
       +ear
          +ear_tragus 
          +ear_antitragus 
          +ear_intertragic_incisure 
          +ear_lobule 
          +ear_acoustic_meatus_external 
          +ear_helix 
          +ear_antihelix 
       +oral_cavity
          +oral_cavity_uvula 
          +oral_cavity_palatopharyngeal_notch 
          +oral_cavity_palatine_tonsil 
          +oral_cavity_palatoglossal_arch 
          +oral_cavity_tongue_vallate_papilla 
          +oral_cavity_tongue_fungiform_papilla 
       +neck_anterior
          +neck_anterior_hyoid_bone 
          +neck_anterior_thyroid_cartilage 
          +neck_anterior_cricoid_cartilage 
          +neck_anterior_thyroid_gland 
          +neck_anterior_carotid_triangle 
          +neck_anterior_submental_triangle 
          +neck_anterior_submandibular_triangle 
          +neck_anterior_anterior_triangle 
          +neck_anterior_posterior_triangle 
       +neck_posterior
       +lymph_node_cervical
          +lymph_node_cervical_preauricular 
          +lymph_node_cervical_submental 
          +lymph_node_cervical_anterior 
          +lymph_node_cervical_posterior 
          +lymph_node_cervical_supraclavicular 
       +chest
          +chest_jugular_notch 
          +chest_clavicle 
          +chest_sternal_angle 
          +chest_sternal_manubrium 
          +chest_costal_margin 
          +chest_xiphoid_process 
          +lymph_node_axillary
          +lymph_node_axillary_lateral 
          +lymph_node_axillary_central 
          +lymph_node_axillary_apical 
          +lymph_node_axillary_anterior 
          +lymph_node_axillary_posterior 
       +abdomen_anterior
          +abdomen_anterior_hypochondriac_left 
          +abdomen_anterior_epigastric 
          +abdomen_anterior_hypochondriac_right 
          +abdomen_anterior_lumbar_left 
          +abdomen_anterior_periumbilical 
          +abdomen_anterior_lumbar_right 
          +abdomen_anterior_iliac_left 
          +abdomen_anterior_hypogastric 
          +abdomen_anterior_iliac_right 
          +abdomen_anterior_linea_alba 
          +abdomen_anterior_mcburney_line 
          +abdomen_anterior_arcuate_line 
          +abdomen_anterior_inguinal_ligament 
          +abdomen_anterior_superior_iliac_spine 
          +abdomen_anterior_pubic_tubercle 
          +abdomen_posterior_iliac_crest 
       +back
          +back_external_occipital_protuberance 
          +back_mastoid_process 
          +back_scapular_acromion 
          +back_scapular_spine 
          +back_spina_prominens 
          +back_vertebral_spinous_process 
          +back_sacral_dorsum 
          +back_iliac_crest 
       +upper_extremity
          +upper_extremity_dorsal
          +upper_extremity_ventral
          +upper_extremity_proximal
          +upper_extremity_distal
       +lower_extremity
          +lower_extremity_dorsal
          +lower_extremity_ventral
          +lower_extremity_proximal
          +lower_extremity_distal
    Exhaustive logical-set operations consist of:
    Subset operation: if X is a valid nandset and X ⊆ Y, then Y is a valid nandset.
    Theorem. (+p|+q) ⇒ (+p|+q|+r|...).
    Proof. Nandsets {+p,-p,-q,-r,...} and {+q,-p,-q,-r,...} are both vacuous.

    Set multiplication: if X,Y are valid nandsets; there exists an +x ∈ X such that -x ∈ Y; and X×Y = ((X ∪ Y) - {+x,-x}); then X×Y is a valid nandset.

    Theorem. Modus Ponens. If ((+p ⇒ +q) and (+q ⇒ +r), then ((+p ⇒ +r).
    Proof. Nandsets {-p,-q,+p,-r}, {-p,+r,+p,-r}, {+q,-q,+p,-r}, and {+q,+r,+p,-r} are all vacuous.

    Theorem. Multiplication. If ((+p|+q) ⇒ +r) and (+r ⇒ (+s|+t)), then ((+p|+q) ⇒ (+s|+t)).
    Proof. Nandsets {+p,-q,-r,+s,+t,+r}, {+p,-q,-r,+s,+t,-s}, and {+p,-q,-r,+s,+t,-t}, are all vacuous.
    Proof that exhaustive logical-set operations suffice to determine all and only the valid nandsets is given in Chapter 17. Appendix H. Theorems.

    Theorem §9.5. BACKTRACKING.



    BACKTRACKING is the process of isolating a subhierarchy within the main hierarchy, and repeating it near the origin, for easier readability. That is: 
     +∀
    +p
        .
        .
        .
       +q
          +r
          +s
          ...
    is implied by: 
     +∀
    +q
       +r
       +s
       ...
    BACKTRACK THEOREM. (+∀ ⇒ (+q | -q)) and (+∀ & +q) ⇒ (+r | +s | ...) implies (+∀ & +p & ... & +q) ⇒ (+r | +s | ...).
    Proof. The corresponding nandsets, {+∀,+q,-r,-s,...) ⊆ {+∀,+p, ..., +q,-r,-s,...}.

    Theorem §9.6. INTERCALATION.

    INTERCALATION is a procedure for inserting (intercalating) a new subhierarchy into the hierarchy, without disturbing the remainder of the hierarchy. 
     +∀
    +p
       +q
          +s
          +t
          ...
       +r
          +u
          +v
          ...
    implies: 
     +∀
    +p
       +q
          +s
          +t
          ...
       +r


    INTERCALATION THEOREM. (+∀ ⇒ +p), (+∀ & +p) ⇒ (+q | +r), (+∀ & +p & +q) ⇒ (+s | +t |...), and (+∀ & +p & +r) ⇒ (+u | +v |...) implies (+∀ ⇒ +p), (+∀ & +p) ⇒ (+q | +r), and (+∀ & +p & +q) ⇒ (+s | +t |...).
    Proof. The collection of sets {+∀,-p}, {+∀,+p,-q,-r}, and {+∀,+p,+q,-s,-t} is a subset of the collection of sets {+∀,-p}, {+∀,+p,-q,-r}, {+∀,+p,+q,-s,-t}, and {+∀,+p,+r,-u,-v}.

    Theorem §9.7. RETIREMENT.



    RETIREMENT is a procedure for removing a subhierarchy (obsolete concept) without disturbing the remainder of the hierarchy. 
     +∀
    +p
       +r
          +u
          +v
          ...
    is equivalent to: 
     +∀
    +p
       -q
       +r
    +p
       +q
          +s
          +t
          +∀
          ...
       +r
          +u
          +v
          ...


    Theorem §9.7. RETIREMENT THEOREM. (+∀ ⇒ +p), (+∀ & +p) ⇒ +r), and (+∀ & +p & +r) ⇒ (+u | +v |...) is equivalent to: (+∀ ⇒ (+p | +p)), (+∀ & +p) ⇒ (-q | +r)), (+∀ & +p) ⇒ (+q | +r)), (+∀ & +p & +q) ⇒ (+s | +t | ... | +∀), and (+∀ & +p & +r) ⇒ (+u | +v |...)
    Proof. The collection of sets {+∀,-p}, {+∀,+p,-r}, and {+∀,+p,+r,-u,-v} is equivalent to the collection of sets: {+∀,-p}, {+∀,+p,+q,-r}, {+∀,+p,-q,-r}, {+∀,+p,+q,-s,-t,-∀} (vacuous), and {+∀,+p,+r,-u,-v}, since {+∀,+p,+q,-r} × {+∀,+p,-q,-r} = {+∀,+p,-r}.

    For example, a small round blue cell tumor (SRBCT) in a skin biopsy has a lengthy differential diagnosis, including primary Merkel_cell_tumor, and metastases from numerous other primary tumors, namely, small_cell_carcinoma_lung, esthesioneuroblastoma, Ewing_sarcoma, retinoblastoma, nephroblastoma, neuroblastoma, lymphoma, peripheral_neurendocrine_tumor, pinealoblastoma, rhabdomyosarcoma, and medulloblastoma. However, simply the age of the patient or other readily available clinical data serve to rule out many highly unlikely possibilities. These likelihoods may be stated on a variable scale ((Zadeh, 2006)), as for example: 
     z: word_description
 ___________________
 6: nearly_certain
 5: frequent
 4: common
 3: uncommon
 2: rare
 1: very_rare
    where z represents certainty level. The partial membership value, v, is defined as: v = (1 - 2-z). Then as z ⇒ ∞, one is completely certain; whereas when z = 0, one is completely uncertain.

    For discussion, we simplify the differential diagnosis to Merkel_cell_tumor, small_cell_carcinoma_lung, and retinoblastoma. Then the simple differential diagnosis: 
     +skin_specimen
   +small_round_blue_cell_tumor
      +Merkel cell tumor
      +small_cell_carcinoma_lung
      +retinoblastoma
    may be expanded into graded differential diagnoses, as follows: 
     +skin_specimen
   +small_round_blue_cell_tumor
      +adult
         +heavy_smoker
            +small_cell_carcinoma_lung
               +common
            +Merkel_cell_tumor
               +rare
            +retinoblastoma
               +very_rare
         +non_smoker
            +Merkel_cell_tumor
               +common
            +small_cell_carcinoma_lung
               +uncommon
            +retinoblastoma
               +very_rare
      +child
         +retinoblastoma
            +common
         +small_cell_carcinoma_lung
            +very_rare
         +Merkel_cell_tumor
            +very_rare
    We then invoke SUTTON'S LAW to narrow the possibilities, by abandoning the rare and very_rare possibilities for the patient, +p, i.e., (+p ⇒ -rare) and (+p ⇒ -very_rare) i.e., by NOT robbing non-banks, where the money is unlikely to be. If the patient is an adult heavy_smoker, then we may deduce that (+p ⇒ +small_cell_carcinoma_lung)

    If the patient is an adult non_smoker, then we initially deduce that (+p ⇒ +Merkel_cell_tumor | +small_cell_carcinoma_lung). If we now abandon the uncommon possibility, then we may finally deduce that (+p ⇒ +Merkel_cell_tumor).

    The FUZZYSPACE, based upon fuzzy set theory (Zadeh, 1965) is a formalism for representing different levels of certainty for the same concept. For example, a small round blue cell tumor (SRBCT) in a skin biopsy has a lengthy differential diagnosis; however, most possibilities may be ruled out as unlikely based upon the clinical setting:

    Small_round_blue_cell_tumor.

    Number Tumor
    Name    		Primary
    Site    		Age Clinical
    Setting
    1 Merkel Cell TumorSkinOlder
    Adult    		 Isolated
    Lesion
    2Small Cell
    Carcinoma    		Lung AdultLung Mass,
    Widened
    Mediastinum
    3 EsthesioneuroblastomaNasopharynxAdult Nasopharynx
    Mass
    4Ewing sarcoma Bone,
    Soft tissue    		Adolescent,
    Adult    		 Bone, Soft tissue
    Mass
    5 RetinoblastomaEyeYoung
    Child    		 Eye
    Mass
    6NephroblastomaKidney Young
    Child    		Renal
    Mass
    7NeuroblastomaAdrenal Young
    Child    		Adrenal
    Mass
    8MedulloblastomaBrainstem Young