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: pμ_vP.
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 pμ_vP, and v>w, then pμ_wP.
7. Classical set theory: special case of fuzzy set theory: either v=0 or v=1.

Fuzzy Set Theory: Example.

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.

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 pμ_vP. 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, pμ_½P corresponds to half-membership, etc. Fuzzy set theory has an ordinal property, i.e., if pμ_vP and v>w, then pμ_wP. 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, wμ_vW.

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 pμ_vP and v>w, then pμ_wP, 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 Tumor	Skin	Older Adult	Isolated Lesion
2	Small Cell Carcinoma	Lung	Adult	Lung Mass, Widened Mediastinum
3	Esthesioneuroblastoma	Nasopharynx	Adult	Nasopharynx Mass
4	Ewing sarcoma	Bone, Soft tissue	Adolescent, Adult	Bone, Soft tissue Mass
5	Retinoblastoma	Eye	Young Child	Eye Mass
6	Nephroblastoma	Kidney	Young Child	Renal Mass
7	Neuroblastoma	Adrenal	Young Child	Adrenal Mass
8	Medulloblastoma	Brainstem	Young Child	Brainstem Mass
9	Lymphoma	.	.	.
10	Peripheral Neurendocrine Tumor	.	.	.
11	Pinealoblastoma	.	.	.
12	Rhabdomyosarcoma	.	.	.

The FUZZYSPACE is a system that assigns levels of certainty to a particular dermatopathology concept. For example, let a small_round_blue_cell_tumor in a skin biopsy be denoted p, with a differential diagnosis that includes: q=Merkel_cell_tumor, r=metastatic_retinoblastoma, and s=metastatic_small_cell_carcinoma_lung. If the patient is an elderly non-smoker, then the likely order of diagnoses is: p ⇒ (q³ | s² | r¹). That is, p implies q at certainty 3; or s at certainty 2; or r at certainty 1. Alternatively, if the patient is a middle-aged heavy smoker, then the likely order of diagnoses is: p ⇒ (s³ | q² | r¹). That is, p implies s at certainty 3; or q at certainty 2; or r at certainty 1. If the patient is a two-year-old with an ocular enucleation for cancer, then the likely order of diagnoses is: p ⇒ (r³ | s² | q¹). That is, p implies r at certainty 3; or s at certainty 2; or q at certainty 1. The named contents of this differential diagnosis list is exactly the same for all three patient profiles; but in each case, one diagnosis is far more likely than the other two.

One can characterize this priority-order with so-called FUZZY SET THEORY (Zadeh, 1965), in which an element, x, is a member of set X, denoted x μ_v X, on a sliding scale from v=0 (non-member) to v=1 (full-member). In classical set theory (CST), an element, x, is either a member of a set, X, denoted x ∈ X; or else x is not a member of a set X, denoted x ~∈ X. In fuzzy set theory, an element x may have partial membership in set x, denoted x μ_v X; where v may assume any value along the closed interval [0,1], i.e., 0 < v < 1. Thus, classical set theory is a special case of fuzzy set theory, where v may assume only two values, i.e., v=1 corresponding to x μ_v X or x ∈ X; and v=0 corresponding to x μ₀ X or x ~∈ X.

Let M > 2 denote a MAXIMUM, where M is greater than the longest differential diagnosis list. For the sample differential diagnosis with three elements apiece, let M=4. For x a partial member of X, denoted x μ_v X, let v = (1 - 2^M-i) for likelihood number, i. Then for the elderly non-smoker, p ⇒ (q³ | s² | r¹) has fuzzy-memberships q μ_7/8 Q, or s μ_3/4 S, or r μ_1/2 R. That is, the elderly non-smoker patient has 7/8 fuzzy-membership in set Q (i.e., the set of patients with primary Merkel cell tumor); or 3/4 fuzzy-membership in set S (i.e., the set of patients with metastatic small cell carcinoma lung); or 1/2 fuzzy-membership in set R (i.e., the set of patients with metastatic retinoblastoma). These fuzzy-membership levels are NOT probabilities: not because it wouldn't be desirable to know the exact probabilities for medical decisions; but because determining such exact probabilities is a hopeless task in practice. By contrast, ordering relations, e.g., q is more likely than p, are often well-known. In cases where credible probabilistic information does exist, it may be used to verify the order relations.

A calculus for these ordered fuzzy-memberships is provided below.

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. The idea first entered the medical literature in 1960, in the context of fevers of unknown origin (Petersdorf and Beeson, 1960), and has wide applicability in clinical medicine.

Another version of this idea is the saying: if you hear hoofbeats in the street, think of horses not zebras. In other words, in the face of uncertainty, one should take action based upon the more likely event, namely, horses rather than zebras. An academic tertiary-care hospital, which treats patients with unusual diseases, is sometimes called a zebra farm.

SUTTON'S CORRAL (our definition) is the collection/set of abnormal and relatively unlikely conditions. For a small_round_blue_cell_tumor in an adult, this collection would include: retinoblastoma, medulloblastoma, neuroblastoma, nephroblastoma,...., all of which are typically found in children. In other terms, Sutton's corral is an enclosure full of zebras. SUTTON'S LASSO is a priority-based calculation method described below, used to methodically eliminate unlikely conditions from a differential diagnosis of all possibilities, i.e., the contents of Sutton's corral.

MATHEMATICAL MODEL.

The mathematical model employs the idea of a set of all possible patient descriptions. For example, let us consider a simplified namespace with true-false (=binary) elements, as follows:

p=small_round_blue_cell_tumor
q=Merkel_cell_tumor
r=Retinoblastoma
s=Small_cell_carcinoma_lung
t=heavy_smoker

Then there are 16 possible patient descriptions with positive small_round_blue_cell_tumor in this simplified namespace, as follows:

Possible Patient Description Table /
Small_round_blue_cell_tumor.

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	+	-	-	-	+
16	+	-	-	-	-

If we assert that EVERY Small_round_blue_cell_tumor must be either Merkel_cell_tumor, Retinoblastoma, or Small_cell_carcinoma_lung, then THERE CANNOT EXIST a Small_round_blue_cell_tumor in which ALL of Merkel_cell_tumor, Retinoblastoma, and Small_cell_carcinoma_lung are FALSE. That is, the set {+Small_round_blue_cell_tumor, -Merkel_cell_tumor, -Retinoblastoma, -Small_cell_carcinoma_lung} is false. This set of things that cannot all coexist is called a NANDSET. NAND is the most fundamental operation of classical symbolic logic, because all other logic operations (not, and, implies, inclusive_or, etc) can be constructed from nand. Nand is the mathematical analogue of the TRANSISTOR, the basic building-block of the digital computer.

We can represent this assertion by SHADING OUT (▒) Descriptions #15,16, as follows:

Possible Patient Description Table /
SHADED Small_round_blue_cell_tumor.

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	▒	▒	▒	▒	▒
16	▒	▒	▒	▒	▒

We may further narrow the search by observing that collision tumors are unlikely. In this example, the likelihood of a collision tumor of all three malignancies (Descriptions #1,2) is vanishingly small; and the likelihood of a collision tumor of two malignancies (Descriptions #3,4,5,6,9,10) is also very small. That is, if we know that we have tumor q then we may imply that tumor r is absent at priority 1. That is, +q^∞ ⇒ -r¹, corresponding to nandset {+q^∞,+r¹}. All combinations of collision tumors among these three tumors are assigned low priorities with these six nandsets:

{+q^∞,+r¹}
{+q^∞,+s¹}
{+r^∞,+q¹}
{+r^∞,+s¹}
{+s^∞,+q¹}
{+s^∞,+r¹}

Finally, the likelihoods of Descriptions #12, 13 are high. ........................

PROPOSITIONAL LOGIC.

There is a tradition of PROPOSITIONAL LOGIC dating back to Aristotle (384-322 BC), and modernized by George Boole (1815-1864). A PROPOSITION is a statement that may be evaluated as TRUE OR FALSE, not both and not neither, as for example, "the patient has a focal reddened area of the of the right maxillary buccal mucosa"; or "the patient has lichen planus of the right maxillary buccal mucosa". In principle, every proposition HAS a true-false value, even though this value might not be known to a particular observer (patient or healthcare provider) or even be knowable based upon a given data set.

The classical operators for propositional logic include: NOT, AND, INCLUSIVE_OR, EXCLUSIVE_OR, IMPLIES, and EQUALS. That is, proposition NOT_P, denoted -P, is true if and only if proposition +P is false; proposition P_AND_Q denoted P&Q is true if and only if propositions +P and +Q are both true, etc. Of particular interest is the NAND ("not_and") operator, where P_NAND_Q is true if and only if propositions +P and +Q are both false. NAND is the propositional logic equivalent of the transistor, the fundamental component of modern digital computers.

There are thousands of theorems already proven in classical propositional logic. See Appendix H.

PROPOSITIONAL LOGIC THEOREMS FOR MEDICAL HEIRARCHIES.

Theorems:

Theorem: p ⇒ p. Proof. Nandset {+p,-p} is vacuous.
Theorem: p ⇒ (p | p). Proof. Nandset {+p,-p,-p} is vacuous.
Theorem: p ⇒ (p | q). Proof. Nandset {+p,-p,-q} is vacuous.
Theorem: p ⇒ (p | q). Proof. Nandset {+p,-p,-q} is vacuous.
Theorem: p ⇒ (p | q | r | ...). Proof. Nandset {+p,-p,-q,-r,...} is vacuous.
Theorem: p ⇒ (q | -q). Proof. Nandset {+p,-q,+q} is vacuous.
Theorem: (p & q) ⇒ (r | -r). Proof. Nandset {+p,+q,-r,+r} is vacuous.
Theorem: (p & q & r & ...) ⇒ (z | -z). Proof. Nandset {+p,+q,+r,...,-z,+z} is vacuous.
Theorem: +p ⇒ (+q | +q | +r) is equivalent to +p ⇒ (+q | +r) Proof. Nandset .....
Theorem: (+p ⇒ -p) implies that -p is true. Proof. Nandset .....

Corollaries:

Corollary: ∀ ⇒ ∀. Proof. Nandset {+∀,-∀} is vacuous.
Corollary: ∀ ⇒ (p | -p). Proof. Nandset {+∀,-p,+p} is vacuous.
Corollary: ∀ ⇒ (∀ | ∀). Proof. Nandset {+∀,-∀} is vacuous.
Corollary: ∀ ⇒ (∀ | ∀ | ...). Proof. Nandset {+∀,-∀,-∀,...} is vacuous.
Corollary: ∀ ⇒ (∀ (⇒ (q | -q| ...)) | (⇒ (r | -r| ...)). Proof. Nandsets {+∀,-∀,-∀,...} {+∀,-∀}, {+∀,+∀-q,+q}, and {+∀,+∀-r,+r} are vacuous.

In classical logic, a TRUTH TABLE ELEMENT is a list of all propositions, each designated true or false, with no propositions missing and no propositions both true and false. A TRUTH TABLE is the collection of all possible truth table elements. In this report, we consider a truth table as the list of all POSSIBLE PATIENT DESCRIPTIONS. For example, if a patient is completely described by (1) sex; (2) age at least 18 years; and (3) fair-skinned, then the list of eight (=2³) possible patient descriptions is:

Truth Table /
Possible Patient Descriptions.

Number	Female	>18 years	Fair skin
1	+	+	+
2	+	+	-
3	+	-	+
4	+	-	-
5	-	+	+
6	-	+	-
7	-	-	+
8	-	-	-

In general, for n propositions, there are 2ⁿ possible patient descriptions.

Now suppose that we add "(4) has a uterus" to the list of propositions. Then the possible patient description table has 16=2⁴ entries, as follows:

Possible Patient Description Table.

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	-	+	+	+
10	-	+	+	-
11	-	+	-	+
12	-	+	-	-
13	-	-	+	+
14	-	-	+	-
15	-	-	-	+
16	-	-	-	-

While there are some females without a uterus (through traumatic or elective hysterectomy), there are no males with a uterus. That is, it is true that MALE IMPLIES NO UTERUS or MALE NAND UTERUS. This anatomic fact may be represented on the Possible Patient Description Table by shading (▒) the anatomically impossible rows on the table, as follows:

Possible Patient Description Table /
Anatomically Possible.

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	▒	▒	▒	▒
10	-	+	+	-
11	▒	▒	▒	▒
12	-	+	-	-
13	▒	▒	▒	▒
14	-	-	+	-
15	▒	▒	▒	▒
16	-	-	-	-

SUTTON'S LAW / ZEBRAS.

Clinical medicine is an action-oriented discipline, beset by uncertainty. Physicians are ethically obliged to begin treatment for presumptive diseases, before the diseases are fully diagnosed or their course fully understood. SUTTON'S LAW is the assertion that, in the face of diagnostic uncertainty, one should take action based upon the most likely diagnosis.

This logical paradox is variously ........... SUTTON'S LAW is a medical slang term, named after the notorious bank robber, Willie Sutton. When asked why he always robbed banks, Willie Sutton reportedly said, "Because that's where the money is". ......................

SUTTON'S SIEVE is a collection of abnormal and relatively unlikely conditions, say, {+q, +r, +s, ...}, where q=small_cell_carcinoma_lung, r=Merkel_cell_tumor, s=retinoblastoma. NOT included in Sutton's sieve are common conditions, such as hypertension; normal variants, for example, male/female, four_decades_old, etc. Also not included are the negations of Sutton_sieve propositions. Therefore, for every member, +s, of Sutton's sieve, S, -s is NOT a member of Sutton's sieve.

SUTTON'S CORRAL is the collection/set of abnormal and relatively unlikely conditions. In other terms, Sutton's corral is an enclosure full of zebras. SUTTON'S LASSO is a priority-based method for removing the most unlikely members of the corral.

p μ₁ P same as p ∈ P.
p μ₀ P same as p ~∈ P.
p μ_v P: Crisp/classical set theory: v ∈ {0,1}.
p μ_v P: Fuzzy set theory: v ∈ [0,1].
Fuzzy Order Axiom: If 0 < v < w < 1 and p μ_w P, then p μ_v P.

DEFINITION:

pⁿ means that pⁿ μ_v P. where v = (1 - 2^-n).

Simplified model:

p = small round blue cell tumor.
q = Merkel cell tumor.
r = retinoblastoma.
s = small cell carcinoma.
t = old non-smoker.
u = middle-aged heavy smoker.
M = Max, say 4.

Then for the old non-smoker:

p -> (q|r|s).
(p&t) -> (q^M-1 | r^M-3 | s^M-2)

For the middle-aged heavy-smoker:

p -> (q|r|s).
(p&t) -> ( q^M-2 | r^M-3 | s^M-1)

A system is M-FUZZY if and only if for every 1 < i < M, p^-i ⇒ p^-i-1. We define PARTIAL MEMBERSHIP for p^-i as p^-i μ_v p where v = (1-2^-i).

We perform a i-Sutton filtration by inserting the nandsets: {+q^-i}, {+r^-i}, {+s^-i}, ... Then, for example, if p^-0 ⇒ (q^-0 | r^-1), then under Sutton filtration, p^-0 ⇒ q^-0. ......................

FUZZY THEORY.

In FUZZYSPACE, a FUZZY-ELEMENT is an element that can be known at different levels of certainty. For example, a so-called small blue cell tumor occurring in the skin might be a (primary) Merkel cell tumor of the skin, or else a metastasis from carcinomas of the lung (small cell carcinoma), nose (esthesioneuroblastoma), eye (retinoblastoma), brainstem (medulloblastoma), bone (Ewing sarcoma), lung (peripheral neurendocrine tumor), rectum (carcinoid), muscle (rhabdomyosarcoma), kidney (nephroblastoma, blastema type), adrenal (neuroblastoma), etc. With no additional information given about the patient, these possibilities have no preferential likelihood. However, if this is a middle-aged, 40-pack-year smoker with a lung mass ............... Alternatively, if this is a elderly non-smoker with no other known primary tumor, ............... If this is a man in the fourth decade with a large nasal mass ............... If this is a teenager with a long bone mass ............... If this is a two year old child with an ocular mass ............... If this is a two year old child with a brainstem mass ............... If this is a six month old child with a renal mass ............... If this is a six month old child with an adrenal mass ............... The traditional definition of a FUZZY ELEMENT in FUZZY SET THEORY (FST) is given by Zadeh (1968). In classical set theory (CST), an element, x, is either a member of a set, X, denoted x ∈ X; or else x is not a member of a set X, denoted x ~∈ X. In fuzzy set theory, an element x may have PARTIAL MEMBERSHIP in set X, denoted x μ_v X; where v may assume any value along the closed interval [0,1], i.e., 0 < v < 1. Thus, classical set theory is a special case of fuzzy set theory, where v may assume only two values, i.e., v=1 corresponding to x μ_v X or x ∈ X; and v=0 corresponding to x μ₀ X or x ~∈ X.

Fuzzy Membership.

v	(1 - 2^-v)	μ
7	127/128	0.9921875
6	63/64	0.984375
5	31/32	0.96875
4	15/16	0.9375
3	7/8	0.875
2	3/4	0.75
1	1/2	0.5

MATERIALS AND METHODS.

The NAMESPACE is a strict hierarchy, in which each name appears exactly once, and has exactly one parent in the hierarchy..........

For example, if we decide to classify human tissues by their embryologic origin, then all tissues arise from ectoderm, mesoderm, or ectoderm. for completeness, let ∀ denote anypatient, and let anypatient have human tissues. Then:

CLASSIFICATION 1. ∀ human_tissue

which may be expanded into:

CLASSIFICATION 1. ∀ human_tissue ectoderm mesoderm endoderm

This classification may further be expanded as:

CLASSIFICATION 1. ∀ human_tissue ectoderm epidermis neural_crest sensory_epithelium contractile_epithelium glandular_epithelium stomatodeal_epithelium proctodeal_epithelium mesoderm endothelium_mesodermal mesothelium mesenchyme endoderm endothelium_endodermal

and so forth. This hierarchical classification of names can be extended indefinitely, subject to the constraint that every name OCCURS EXACTLY ONCE; and every name except anypatient HAS EXACTLY ONE PARENT. For example, we might construct a classification for the same in ALPHABETICAL ORDER, but this second classification would be incompatible with the first classification:

CLASSIFICATION 2. ∀ human_tissue human_tissue_c contractile_epithelium human_tissue_e ectoderm endoderm endothelium_endodermal endothelium_mesodermal epidermis human_tissue_g glandular_epithelium human_tissue_m mesenchyme mesoderm mesothelium human_tissue_n neural_crest human_tissue_p proctodeal_epithelium human_tissue_s sensory_epithelium stomatodeal_epithelium

However, we may combine these two classifications as a MATHEMATICAL RELATION, in which a single name may have several parents

In the hierarchy of concept-names, each name appears only once...... For example, the embryologic appearance of tumors of the skin. For example, SMALL BLUE CELL TUMOR. ......

A MATHEMATICAL RELATION is a multiple hierarchy of concepts that appear in the name hierarchy. For example, small blue cell tumors may appear as hierarchies of embryologic origins; primary anatomic sites; tumor chemical products; pattern of metastases; tumor aggressiveness, etc. 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

Finally, FUZZY STRATA represent the certainty that we are dealing with a particular entity. ................

These concepts can be assembled as a hierarchy of LISTS, and arbitarily many sublevels of SUBLISTS. For example:

skin anatomy epidermis stratum_corneum stratum_granulosum stratum_spinosum stratum_basale dermis papillary_dermis reticular_dermis appendage hair hair_shaft hair_ostium hair_infundibulum hair_arrector_pili hair_outer_root_sheath hair_inner_root_sheath hair_papilla hair_matrix_cell nail apocrine_gland eccrine_gland subcutaneous_tissue adipose_tissue vascular_tissue muscular_tissue physiology heat_regulation, water_regulation, ... pharmacology therapeutic_medications, medication side effects, ...

A SPECIFICATION is a sufficiently detailed description of an object that a domain-expert can understand that object as a unique entity. For example, Bill_Moore (the first author of this report) is an object in the class of humans. Since there are many humans named Bill_Moore, this designation alone is not a unique identifier. However, Bill_Moore, staff pathologist at the Baltimore VA Medical Center, Baltimore, Maryland, USA, in year 2006, is uniquely specified.

A specification is not as demanding a requirement as a STANDARD, which may involve highly stylized rules of syntax, a narrow range of validity, etc. The only real constraints on a specification are: one name for each concept; a single position on the hierarchy for each concept; understandability to domain_experts; and consistency.

In this report, we propose that RDF metadata comprise a suitable structure for the major components of dermatopathology; and we suggest that RDF metadata are indefinitely extensible to finer concepts. Furthermore, the RDF structure accommodates the growth of new concepts, and the retirement of obsolete concepts, as described below.

Using a SET THEORY MODEL, ............ CONSISTENCY. ............

RESOURCE DESCRIPTION FRAMEWORK (RDF) is a hierarchical ontology specification language, that has gained wide acceptance in the biomedical community, because of its scope, precision, and simplicity. RDF has been applied to a great variety of biological processes. RDF is used to designate MEANING in the semantic worldwide web of the internet (Berman, 2007). RDF has recently been employed as the specification language for the Laboratory Digital Imaging Project (LDIP) of the Association for Pathology Informatics (API).

The RDF format is precise enough to be interpreted / parsed by widely available, cost-free computer software; but it has a simple structure consisting solely of DATA TRIPLES: (UNIQUE IDENTIFIER, METADATA, DATA). For example, the assertion that Bill Moore has a hypertrophic actinic keratosis on his right dorsal hand , would have UNIQUE_IDENTIFIER=Bill_Moore, METADATA=hypertrophic_actinic_keratosis_right_dorsal_hand, and DATA=present.

RDF metadata represent the collected knowledge of a particular discipline, such as dermatopathology. RDF Metadata are assigned into two categories: CLASS and PROPERTY. Every RDF CLASS except for the ULTIMATE RDF CLASS, has exactly one PARENT, and zero or more CHILDREN. The ultimate RDF class has no parents.

In this report,..............

LOGICAL STRUCTURE OF DERMATOPATHOLOGY.

In a simple RDF model, we might state that every skin_specimen is taken from the head_skin, chest_skin, abdomen_skin, back_skin, arm_skin, or leg_skin, diagrammed as follows:

skin_specimen head_skin chest_skin abdomen_skin back_skin arm_skin leg_skin

We may interpret the above hierarchy as: +skin_specimen implies +head_skin, +chest_skin, +abdomen_skin, +back_skin, +arm_skin, or +leg_skin, where + denotes POSITIVE and - denotes NEGATIVE.

We postulate a logical, consistent PATIENT_UNIVERSE, denoted +∀ ("anypatient"). "Anypatient" may have a skin biopsy or not:

+∀ +skin_specimen -skin_specimen

Then:

+∀ +skin_specimen +head_skin +chest_skin +abdomen_skin +back_skin +arm_skin +leg_skin -skin_specimen

In general, we interpret

+∀ +A -A

as +∀ implies (+A or -A).

We interpret

+∀ +A +B +C -A +D +E

as +∀ implies (+A or -A); (+∀ and +A) implies (+B or +C); (+∀ and -A) implies (+D or +E), etc.

We can use this logical structure to construct dermatopathologic hierarchies of arbitrary size and complexity, that may be manipulated with little more sophistication than what is required from high-school algebra. Let us begin with a simple model in which skin biopsies may be interpreted under one (or more) of the following six subcategories:

+∀ +skin_specimen +inflammation +major_tissue_reaction_noninflammatory +minor_tissue_reaction_noninflammatory +dysplastic_neoplastic +unremarkable +unsatisfactory -skin_specimen

That is, every skin_specimen shows either inflammation, major_tissue_reaction, minor_tissue_reaction, dysplasia_neoplasia; is unremarkable (normal); or is technically unsatisfactory.

According to Weedon (2002), there are six non-inflammatory major_tissue_reactions, as follows:

+∀ +skin_specimen +major_tissue_reaction_noninflammatory +lichenoid_reaction +psoriasiform_reaction +spongiotic_reaction +vesiculobullous_reaction +granulomatous_reaction +vasculopathic_reaction -major_tissue_reaction_noninflammatory -skin_specimen

Among lichenoid reactions, there are 21 major headings shown here; among lichen_planus_variants, there are eleven possibilities:

+∀ +skin_specimen +major_tissue_reaction_noninflammatory +lichenoid_reaction +lichen_planus +lichen_planus_variant +lichen_planus_atrophic +lichen_planus_hypertrophic +lichen_planus_linear +lichen_planus_ulcerative +lichen_planus_erythematosus +lichen_planus_erythema_dyschromicum_perstans +lichen_planus_actinicus +lichen_planus_planopilaris +lichen_planus_pemphigoides +lichen_planus_keratosis_chronica +lichen_planus_lupus_erythematosus_overlap +lichen_nitidus +lichen_striatus +drug_eruption_lichenoid +lichen_planus_like_keratosis +fixed_drug_eruption_lichenoid +erythema_multiforme +graft_vs_host_disease +aids_interface_dermatitis +eruption_lymphocyte_recovery +lupus_erythematosus +dermatomyositis +poikiloderma +lichen_sclerosis_et_atrophicus +pityriasis_lichenoides +persistent_viral_reaction +perniosis +paraneoplastic_pemphigus +lichenoid_purpura +lichenoid_contact_dermatitis -lichenoid_reaction -major_tissue_reaction_noninflammatory -skin_specimen

This logical model supports ordinary / classical concepts of CONSISTENCY, LOGICAL VACUITY, REDUNDANCY, and INFERENCE; and two additional properties of INTERCALATION and RETIREMENT (see below).

CONSISTENCY is the property that no statement can be both true and false. For example, the assertion that +anypatient implies -anypatient is inconsistent.

+∀ -∀

VACUITY is the property that a of a particular statement adds nothing to the logical structure, i.e., the statement can be removed without altering the logical structure. For example, the assertion that +anypatient implies +anypatient is vacuous, because a statement always implies itself.

+∀ +∀

REDUNDANCY is the property that one statement adds nothing to the logical structure created by another statement. For example: if +A implies +B, then it is redundant to assert that: +A implies (+B or +C). Thus:

+∀ +A +B +A +B +C -A

is a redundant expression for:

+∀ +A +B -A

An INFERENCE is one logical expression implied by another. For example:

+∀ +A +B +C +A -B +C -A

implies:

+∀ +A +C -A

INTERCALATION is an operation in which a new theory might be placed side-by-side (i.e., intercalated) into an existing theory, without disturbing the structure of the existing theory. For example, consider Hippocrates humoral theory of human_disease_pathogenesis:

+∀ +human_disease_pathogenesis +phlegm +blood +urine +bile -human_disease_pathogenesis

We may INTERCALATE the modern pathoanatomic model of human disease pathogenesis, without disturbing the existing logical structure:

+∀ +human_disease_pathogenesis +humoral_human_disease_pathogenesis +phlegm +blood +urine +bile +pathoanatomic_human_disease_pathogenesis +anatomy +physiology +pathologic_anatomy -human_disease_pathogenesis

RETIREMENT is the process of gracefully removing a defunct theory, that has proved less useful than the new theory. In the above example, placing +∀ into the humoral human disease pathogenesis children, renders it vacuous:

+∀ +human_disease_pathogenesis -humoral_human_disease_pathogenesis +human_disease_pathogenesis +humoral_human_disease_pathogenesis +phlegm +blood +urine +bile +∀ +pathoanatomic_human_disease_pathogenesis +anatomy +physiology +pathologic_anatomy -human_disease_pathogenesis

FALSE PROOF!

As a SINGLE LINE SHORTHAND, we let : denote carriage_return_indent (i.e., parent); , denote carriage_return_no_indent (i.e., sibling); and & denote logical_and. Thus the single line shorthand for

+∀ +A -A

is +∀:+A,-A.

The single line shorthand for

+∀ +A +B +C -A +D +E

is +∀:+A(:+B,+C),-A(:+D,+E).

The RDF hierarchy provides a structure in which to organize different concepts, as for example, +skin:+epidermis,+dermis,+subcutaneous_tissue. Within each single concept, there are links/preferences/progressions/layers of certainty. For example, a small_blue_cell_tumor of skin in a 60 year old, 40-pack-year smoker is: +∀:+small_blue_cell_tumor_skin: +metastatic_small_cell_carcinoma, +merkel_cell_tumor, +neuroblastoma, +esthesioblastoma, +retinoblastoma, +medulloblastoma,..., where the first choice is most likely; the second choice is less likely; and the subsequent choices are vanishingly unlikely. Thus, it makes the most sense to work up the patient for the first two items. These diminishing likelihoods might be denoted as: +∀:+small_blue_cell_tumor_skin: +metastatic_small_cell_carcinoma⁰, +merkel_cell_tumor^-1, +neuroblastoma^-2, +esthesioblastoma^-2, +retinoblastoma^-2, +medulloblastoma^-2, ..., as discussed below.

However, the identical morphological finding of a small_blue_cell_tumor of skin in the clinical setting of a a 80 year old, non-smoker is: +∀:+small_blue_cell_tumor_skin: +merkel_cell_tumor, +metastatic_small_cell_carcinoma, +neuroblastoma, +esthesioblastoma, +retinoblastoma, +medulloblastoma,..., where +∀:+small_blue_cell_tumor_skin: +merkel_cell_tumor⁰, +metastatic_small_cell_carcinoma^-1, +neuroblastoma^-2, +esthesioblastoma^-2, +retinoblastoma^-2, +medulloblastoma^-2, ..., as discussed below.

And the morphological finding of a small_blue_cell_tumor of skin in a 2 year old with an abdominal mass is: +∀:+small_blue_cell_tumor_skin: +neuroblastoma, +retinoblastoma, +merkel_cell_tumor, +metastatic_small_cell_carcinoma, +esthesioblastoma, +medulloblastoma,..., where....

The required simplicity of RDF prevents its comprehensive / immediate use in logical deductions. Mainly, an RDF class can have only a single parent, and can occur only once in the entire hierarchy. For logical operations, it is necessary to be able to use the NEGATION of a class, and for the class to occur in several places in the hierarchy. The CANONICAL POSITION may be used for constructing the RDF.

The SKIN is the largest organ in the human body. Because of its size, direct/immediate exposure to environmental insults, and accessibility for biopsy, there is a huge/bewildering number of skin diseases with described/known pathologic correlations/lesions. Several leading textbooks of dermatopathology that describe these lesions, are over nine hundred pages long (Barnhill, 2004, Farmer, 1999, Weedon, 2002, McKee et al, 2005).

On the other hand, the skin has a relatively simple structure anatomically (epidermis, dermis, subcutanous tissue, and appendages); and a limited range and gross and microscopic alterations (inflammation, necrosis, epidermal/dermal interface changes, dysplasia/neoplasia). As a result, many different diseases have overlapping/similar manifestations in the skin. For the general pathologist facing an unusual skin biopsy, the twenty-page, fine-print index of confusing Latin names in the dermatopathology textbook is of little assistance. One colleague told me that his job was to distinguish "easy" from "hard" skin biopsies, and to forward the "hard cases" to the appropriate consultant.

The past two decades have witnessed a significant growth in the use of computer systems in anatomic pathology, primarily as a means/aid for issuing accurate, timely pathology reports. With the increased regulation of hospital processes by external inspection agencies (College of American Pathologists, American College of Surgeons, etc.) has come a requirement for standardized reporting protocols as a supplement to the traditional, free-text report on large, complex tissue excisions.

The growth of the internet for sharing information has stimulated interest in standardized, object-oriented computer languages, and in the development of consistent resource descriptions for biomedical applications. Just as each internet site can be uniquely located by a UNIVERSAL RESOURCE LOCATOR (URL), each physical or biological object may be uniquely identified UNIVERSAL RESOURCE IDENTIFIER (URI), and changes in these objects may be tracked in time and space.

RESOURCE DESCRIPTION FRAMEWORK (RDF) is a knowledge representation tool that has gained wide acceptance in the biomedical sciences [(Wang, 2005),,,], and is used by the Association for Pathology Informatics (API) in their Laboratory Digital Imaging Project (LDIP).

Finally, there is growing use of the hospitalwide Electronic Medical Record (EMR) as a means for preserving records and making them simultaneously available to all members of the healthcare team. These records could eventually be data-mined against public-domain, open-source, openly discussed standards and policies in medical care. While the unverified/unchecked/free syntax of anatomic pathology reports currently does not support such data-mining with acceptable error rates, one can foresee the day when payment for anatomic pathology reports might be withheld until they are data-minable by suitably regulatory agencies.

Many biomedical processes are readily organized as HIERARCHIES. These include: anatomy (Basel Nomina Anatomica); major pathologic processes (inflammation, dysplasia, neoplasia,...); differential diagnoses; large specimen checklists; tissue-processing; patient scheduling; specimen-accessioning; and billing. RDF is a strict hierarchy of CLASSES, with a non-strict subhierarchy of PROPERTIES. In RDF, a CLASS is.....; a PROPERTY is.....

The Laboratory Digital Imaging Project (LDIP) of the Association for Pathology Informatics (API) has seven major classes, that cover all aspects of specimen collection, reporting, billing, and followup in the pathology laboratory, as follows: Person: (patient, clinical_provider, pathologist, credentialing_office); Event:(sequence, temporal, spatial, mass, homunculus); Data_object: (report, method, image, electronic_medical_record); Specimen; Reagent:(hematoxylin, eosin_yellowish, ethanol,...); Instrument: (microscope, camera...); Terminology: (English, SNOMED, UMLS, MeSH,...). Each terminal object described in the LDIP has a UNIVERSAL RESOURCE IDENTIFIER (URI), analogous to each internet site being uniquely located by a UNIVERSAL RESOURCE LOCATOR (URL). For example, the URL for this document is:

http://www.gwmoore.org/dermpath/dermrdfh.htm

The computer workstation on which this document is being prepared on 11/24/2006, is located in Room 4D-137, Baltimore VA Maryland Health Care System, 10 North Greene Street, Baltimore, MD 21201-1524. Since there is only one computer workstation in this room, the document preparation device has been uniquely identified. In the event of instrument malfunction, this device could be located and examined. This occurrence typically isn't a big problem with computer workstations, but might become an issue, say, with microtomes, cryostats, automated staining devices, etc. Thus in principle, the quality of any process could be backtracked through a unique, defined pathway.

While some of the precision and refinement in this classification may seem persnickety, it is done for the purpose of resolving potential ambiguities. It is not expected that every skin_specimen_report contains all this information every time; although much of it could be reconstructed if the context is known. For example, a biopsy that is taken at 10:00 PM, 11/24/2006, at the Baltimore VA Medical Center corresponds to 2:00 PM, 11/25/2006, on the Universal Time Clock (UTC)! One could imagine a situation in which an urgent laboratory test is taken from the in East St. Louis, IL (Central Time Zone); delivered to a major laboratory in St. Louis, MO (Mountain Time Zone); and read out in less than an hour. If only local times were used, it would appear that the turnaround time was a negative number. For the individual patient, this paradox would not be confusing; for national healthcare data analysis, the paradox could result in drawing false statistical conclusions, and formulating ill-advised healthcare policies.

In this LDIP classification, the Class:Person includes the patient, the clinical_provider who examines the patient and takes the biopsy, the pathologist who issues the report, and the credentialing_office, namely, that group of persons who permit the clinical_provider and the pathologist to practice medicine in their institution. In the LDIP system, Class:Patient is a subset of Class:Person in the sense that every patient is_a person. One does NOT understand, say, that a person's liver is a subset of Class:Person. An individual member of Class:Person is uniquely identified by four properties: person_name (consisting of person_surname, person_givenname, person_honorific); person_date_of_birth; person_social_security_number; and person_gender. A person may, but does not necessarily, have person_insurance and a person_credential.

Class:Event comprises the mathematical, physical, and biological measures required for the case, including sequence (i.e., the whole numbers from zero to infinity, from which the entire number system can be formulated, if necessary); temporal (date_time, as ISO 8601 standard, or as seconds since birth); spatial (length, area, volume, in cm, cm², cm³; or longitude_latitude for location where the biopsy was taken; or in a broad sense to include wavelength, for color); mass (as grams, kilograms, nanograms, etc.); and homunculus.

Homunculus is the biological map of a standard human body, as named by the Nomina Anatomica Basel. This measure is necessary to locate a skin_specimen on the surface of the body, where it is more useful to provide the skin_specimen_site as, say, left ischial tuberosity, than a distance from the center-of-gravity of the patient.

Class:Data_object is an ensemble of data collected for a particular purpose, including report, method, image, electronic_medical_record, etc. The pathology report has properties including identities of the patient, clinical_provider, pathologist, etc., relevant clinical history, gross description, microscopic description, etc.

Class:Specimen comprises specimen_container and all its appropriate labels, including patient_name, collection_date_time, delivery_date_time, body_site, etc.

Class:Reagent comprises all the reagents used in preparation of the specimen, such as ethanol, 10%_formalin, hematoxylin, lithium_carbonate_solution, eosin_yellowish, xylene, etc., as well as manufacturer, purchase_date, expiration_date, quality_control_documentation, etc.

Class:Instrument comprises all instrumentation, such as microscope, camera, microtome, cryostat, etc., including location, maintenance records, etc.

Class:Terminology comprises all languages in which the report is written, including English, SNOMED, UMLS, MeSH, etc.

Unlike physical events such as time, space, or mass, there is another aspect of biomedical classifications that falls under the Class:Event that depends upon our understanding of human anatomy, physiology, and pathology. I call this the HOMUNCULUS (Latin: small human). It is a model of a standardized human body that names and locates all its parts and their possible alterations.

Normal Human anatomy may be conveniently divided into eleven major categories, and further divided according to anatomy textbooks, as follows:

1. Surface_Anatomy.
2. Cardiovascular_System.
3. Respiratory_System.
4. Tubular_gastrointestinal system.
5. Hepatobiliary_System.
6. Genitourinary_System.
7. Integumentary_System.
8. Musculoskeletal_System.
9. Endocrine_System.
10. Lymphoreticular_System.
11. Central_Nervous_System.

The classes for surface_anatomy and integumentary_system have the greatest relevance for dermatopathology.

By formulating all skin biopsy interpretation as a hierarchy of concepts, we may tap into powerful, existing mathematical models of this structure. They may be manipulated with little more sophistication than what is required from high-school algebra. Let us begin with a simple model in which skin biopsies may be interpreted under one (or more) of the following six subcategories:

+∀ +skin_specimen +inflammation +major_tissue_reaction_noninflammatory +minor_tissue_reaction_noninflammatory +dysplastic_neoplastic +unremarkable +unsatisfactory -skin_specimen

Read +∀ as "anypatient"; and we assert from the first-right-indentation of the above hierarchy that "anypatient" either IS a skin_specimen (i.e., +skin_specimen) or IS NOT a skin_specimen (i.e., -skin_specimen). That is: +∀ IMPLIES (+skin_specimen OR -skin_specimen). This isn't a very interesting conclusion. As we shall see below, this expression is MATHEMATICALLY VACUOUS.

Under +skin_specimen in the second-right-indentation of the above hierarchy: ("anypatient" AND +skin_specimen) IMPLIES (+inflammation OR +major_tissue_reaction_noninflammatory OR +minor_tissue_reaction_noninflammatory OR +dysplastic_neoplastic OR +unremarkable OR +unsatisfactory).

The OR given in this expression is INCLUSIVE_OR, i.e., one or more features may be present at once. For example, a particular skin_specimen may exhibit both inflammation and a major_tissue_reaction; a major_tissue_reaction and a minor_tissue_reaction; or inflammation and neoplasia. Some entirely normal/unremarkable skin biopsies will be encountered, and in some cases the specimen will not be satisfactory for diagnostic evaluation. However, AT LEAST ONE FEATURE MUST BE PRESENT in every skin biopsy. The absence of ALL these features results in an MATHEMATICAL INCONSISTENCY, which can be calculated by purely mathematical methods. Therefore, anyone who formulates such a hierarchy must consider ALL the possibilities.

The other significant feature of a hierarchy classification is a mathematical model which includes the properties of INTERCALATION and RETIREMENT, which may be demonstrated mathematically. INTERCALATION is....... RETIREMENT is.........

The universal language of biomedical discourse is declarative sentences. These sentences are written the vocabulary and syntax of an advanced medical culture, such as English, German, French, or Japanese. Such sentences, when controlled for syntax, synonymy, ambiguity, and consistency, have the apparent strength of classical predicate logic. Furthermore, these sentences exist in electronic form in textbooks of dermatopathology, and could possibly be parsed for their scientific content. Quite aside from questions of copyright and professional disagreements for these source documents, there is no accepted concept-structure in which to organize the concepts, with a proven track record in organizing biomedical knowledge.

RESOURCE DESCRIPTION FRAMEWORK (RDF) is a knowledge representation tool that has gained wide acceptance in the biomedical sciences [,,,]. RDF has been used to organize the fundamental ideas of ..., genomics, proteomics, and metabolomics. The LABORATORY DIGITAL IMAGING PROJECT is a project, sponsored by the Association for Pathology Informatics (API), that seeks to develop and promote the use of pathology images to improve patient care and improve the quality of research in pathology, by the publication of open-access standards relating to the interchange of various types of images.

RDF has a simple, hierarchical structure that lends itself well to biological processes/reasoning. RDF structures are well-handled by the emerging, object-oriented language, RUBY. An RDF structure consists of two categories of OBJECT: CLASS and PROPERTY. In general terms, a class is a collection of things, such as a collection of numbers, patients, tissues, cells, etc.; whereas a property is a feature of one member of a class. The single, ULTIMATE CLASS has no PARENT-CLASS, and any number of CHILD-CLASSES. Every other class has exactly one parent-class, and any number of child-classes, including zero child-classes. A class with no children is a TERMINAL CLASS. A PROPERTY is an RDF-object with at least one class-parent or one-or-more property-parents; and any number, including zero, of property-children.

In mathematics, the RDF class-structure forms a DIRECTED GRAPH, where each class-object is a VERTEX and the parent-to-child connection is an EDGE. As we shall show herein, there are many desirable features of this structure for describing dermatopathology concepts.

We propose herewith a structure suitable for the complete logical description of the field of dermatopathology. The structure has formal properties for dealing with redundancy, inconsistency, synonymy, and hyponymy. The structure has potential applicability to quality assurance of electronic medical records (EMRs).

The MATHEMATICAL MODEL underlying this formalism is the set of all POSSIBLE_PATIENT_DESCRIPTIONS, or TRUTH_TABLE in classical logic. For n true_false features, there are 2ⁿ descriptions in the table. For simplicity, let the possible outcome of a skin_specimen=s (feature #1) be reduced to either inflammation=i (feature #2) or neoplasia=n (feature #3). Then considering anypatient, ∀ who might undergo a skin_specimen, yields the following eight possible_patient_descriptions:

Skin Biopsy
Possible Patient
Descriptions Model.

Row:	Anypatient ∀	skin_specimen s	Inflammation i	Neoplasia n
1	+∀	+s	+i	+n
2	+∀	+s	+i	-n
3	+∀	+s	-i	+n
4	+∀	+s	-i	-n
5	+∀	-s	+i	+n
6	+∀	-s	+i	-n
7	+∀	-s	-i	+n
8	+∀	-s	-i	-n

In the above possible_patient_description table, row #1 is a skin_specimen that shows inflammation and neoplasia; row #2 is a skin_specimen that shows inflammation and no neoplasia; row #3 is a skin_specimen that shows no inflammation and neoplasia; row #4 is a skin_specimen that shows no inflammation and no neoplasia; row #1 is no skin_specimen that shows inflammation and neoplasia, etc.

Now suppose that we confine our attention exclusively to those anypatients who actually undergo a skin_specimen. That is: anypatient IMPLIES skin_specimen, or +∀ ⇒ +skin_specimen.

In classical logic, NAND (=not_and) is the logical operator.... A NANDSET is...... By the Whitehead-Russell formulation in Principia Mathematica (1911), for any true_false statement, X, +X ⇒ +Y is equivalent to the nandset, {+X, -Y}. That is, if +X IMPLIES +Y, then the statements, +X and -Y cannot coexist. For the example in which +∀ ⇒ +s, we have the nandset, {+∀, -s}. This nandset operates on the possible_patient_description_table by REMOVING those possible_patient_description_elements which are supersets of {+∀, -s}, as follows:

+∀ +human disease pathogenesis +humoral_human_disease_pathogenesis +phlegm +blood +urine +bile +pathoanatomic_human_disease_pathogenesis +anatomy +physiology +pathologic_anatomy

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀, -s} Supersets Removed.

Row:	Anypatient ∀	skin_specimen s	Inflammation i	Neoplasia n
1	+∀	+s	+i	+n
2	+∀	+s	+i	-n
3	+∀	+s	-i	+n
4	+∀	+s	-i	-n
5	▒	▒	▒	▒
6	▒	▒	▒	▒
7	▒	▒	▒	▒
8	▒	▒	▒	▒

Next, let us impose the additional condition that every skin_specimen on anypatient must be inflamed or neoplastic, i.e., (+∀∧+s)⇒(+i∨+n). This expression corresponds to the nandset, {+∀, +s, -i, -n}. This additional nandset operates on the possible_patient_description_table by removing those possible_patient_description_elements which are supersets of {+∀,+s,-i,-n}, as well as the previously removed {+∀, -s}, as follows:

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀,+s,+i,+n}, {+∀,+s,-i,-n}, {+∀, -s} Supersets Removed.

Row:	Anypatient ∀	skin_specimen s	Inflammation i	Neoplasia n
1	+∀	+s	+i	+n
2	+∀	+s	+i	-n
3	+∀	+s	-i	+n
4	▒	▒	▒	▒
5	▒	▒	▒	▒
6	▒	▒	▒	▒
7	▒	▒	▒	▒
8	▒	▒	▒	▒

Finally, let us impose still another condition, namely that every skin_specimen on anypatient must be not neoplastic if it is inflamed, and not inflamed if it is neoplastic, i.e.,(+∀ ∧ +s ∧ +i) ⇒ -n and ( +∀ ∧ +s ∧ +n) ⇒ -i This expression corresponds to the nandsets, { +∀, +s, +i, +n} and { +∀, +s, +n, +i} . Since order is irrelevant in a set, these are equal expressions. This still additional nandset operates on the possible_patient_description_table by removing those possible_patient_description_elements which are supersets of {+∀,+s,+i,+n} as well as the previously removed {+∀,+s,-i,-n} and {+∀, -s}, as follows:

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀,+s,+i,+n}, {+∀,+s,-i,-n}, {+∀, -s} Supersets Removed.

Row:	Anypatient ∀	skin_specimen s	Inflammation i	Neoplasia n
1	▒	▒	▒	▒
2	+∀	+s	+i	-n
3	+∀	+s	-i	+n
4	▒	▒	▒	▒
5	▒	▒	▒	▒
6	▒	▒	▒	▒
7	▒	▒	▒	▒
8	▒	▒	▒	▒

For every parent-row in the hierarchy, i.e., row with children, there is exactly one implication, and correspondingly exactly one nandset. The above example is represented in hierarchy as follows:

  Row   Hierarchy
    1   +∀       
    2      +s          
    3         +i       
    4            +n    
    5         +n       
    6            +i    
    7      -s

In this seven-row hierarchy, the four boldface rows (1, 2, 3, 5) have children, and the three non-boldface rows (4, 6, 7) do not have children. These childed rows in the hierarchy have a corresponding nandset.

Row Parent Logic Expression Nandset 1 +∀ +∀⇒(+s∨-s) {+∀,-s,-s} 2 +s; (+∀∧+s)⇒(+i∨+n) {+∀,+s,-i,-n} 3 +i; (+∀∧+s∧+i)⇒(-n) {+∀,+s,+i,+n} 5 +n; (+∀∧+s∧+n)⇒(-i) {+∀,+s,+n,+i}

There are many powerful mathematical properties of the hierarchy formalism. A nandset is VACUOUS if for some X, it contains both +X and -X. For example, the assertion that anypatient implies either skin_specimen or not skin_specimen, namely, +∀⇒(+s∨-s), with corresponding nandset, +∀,-s,-s}, is vacuous. Proof: Since every possible_patient_description_element contains either +X or -X but not both, there exists no possible_patient_description_element that has {+X,-X} as a subset. Therefore, {+X,-X,...} removes no possible_patient_description_element from the possible_patient_description_table.

By convention, we assume that every possible patient is "anypatient". Therefore, a nandset that eliminates the truth of +&8704; is INCONSISTENT. For example, consider a possible_patient_description_table in which skin_specimen is both TRUE and FALSE. Then:

+∀ +∀ +s +∀ -s

has nandsets: {+∀, -∀, -∀} (vacuous), {+∀, +∀, -s}, and {+∀, +∀, +s}. These nandsets are subsets of ANY possible_patient_description_element.

INTERCALATION is the process of inserting a new theory into an existing hierarchy. The INTERCALATION THEOREM asserts that intercalation does not disturb the existing structure. That is, every additional nandset contains at least one element from the new theory. Let:

+∀ +∀ +x₀ +x₁ +x₂ ...

be the original theory (x-theory), and intercalate a new theory (y-theory), as follows:

+∀ +∀ +x₀ +x₁ +x₂ ... +∀ +y₀ +y₁ +y₂ ...

Proof: The nandsets for the x-theory are: {+∀, -∀} (vacuous); {+∀, +∀ -x₀}; and {+∀, +∀, +x₀, -x₁, -x₂, ...}.

The nandsets for the (combined) y-theory are: {+∀, -∀} (vacuous); {+∀, +∀ -x₀}; and {+∀, +∀, +x₀, -x₁, -x₂, ...}; {+∀, +∀ -y₀}; and {+∀, +∀, +y₀, -y₁, -y₂, ...}. Therefore, every newly-introduced nandset contains at least one y-element.

When intercalating a new theory, one must take care not to take over/assume terminology from the old theory, in case the old terms mean something different in the new theory. For example, Hippocrates employed terms such as bronchitis (=βρον#&967;ιτις) and nephritis (=ν∈φριτις), but he clearly had a different understanding of these terms than we do today (Kuhn, 1962).

RETIREMENT is the process of removing an old theory into an existing hierarchy. The RETIREMENT THEOREM asserts that removal does not disturb the remaining structure.

+∀ +∀ +x₀ +x₁ +x₂ ... +∀ +y₀ +y₁ +xysub>2 ...

be the unretired theory (y-theory), and remove subtheory (x-theory), as follows:

+∀ +∀ +y₀ +y₁ +y₂ ...

Proof: The nandsets for the (unretired) y-theory are: {+∀, -∀} (vacuous); {+∀, +∀ -x₀}; and {+∀, +∀, +x₀, -x₁, -x₂, ...}; {+∀, +∀ -y₀}; and {+∀, +∀, +y₀, -y₁, -y₂, ...}.

Removal of subtheory (x-theory) yields nandsets: {+∀, -∀} (vacuous); {+∀, +∀ -y₀}; and {+∀, +∀, +y₀, -y₁, -y₂, ...}. Q.E.D.

FUZZY THEORY. Many concepts in medicine are known with different levels of certainty, for a particular patient, and have corresponding levels of therapy and prognosis. Diagnoses with a good prognosis may require nothing more than the patient return at the next regular screening interval; whereas diagnoses of the same pathologic process with a bad prognosis may require heroic measures. For example, a patient with a skin_specimen showing solar_elastosis (a¹) might return to the clinic in one year (r¹); the same patient with a skin_specimen showing an ordinary actinic keratosis (a²) might return to the clinic in six months; (r²); the same patient with an incompletely excised bowenoid actinic keratosis (a³) might return to the clinic in three months for a complete excision; (r³); and the same patient with an incompletely excised, superficially invasive squamous cell carcinoma (a⁴) might return to the clinic in one month for a complete excision. (r⁴); All these four conditions lie along a continuum leading to invasive squamous cell carcinoma, but each step along the progression requires more immediate attention than its predecessors. This progression corresponds to the logic expressions aⁿ ⇊ a^n-1 and rⁿ ⇊ r^n-1; or to the nandsets, {+aⁿ, -a^n-1 and {+rⁿ, -r^n-1, The ultimate condition, (a^∞) might be premature death from widely metastatic squamous cell carcinoma. The hierarchy is:

+∀ +∀ +a₁ +r₁ +a₂ +r₂ +a₃ +r₃ +a₄ +r₄ ... +a_N +r_N ... +a_∞

where +a_N might represent, say, squamous cell carcinoma with extensive lymph node involvement (TNM class N2) and widespread metastases (TNM class MX); and a^∞ represents premature death from widely metastatic squamous cell carcinoma.

In CLASSICAL SET THEORY, set membership is on-or-off, i.e., either x ∈ X (x is a member of set X) or x ~∈ X (x is not a member of set X). In FUZZY SET THEORY the membership relation, μ_v, assumes values v, where 0<v<1, with v=1 for complete membership and v=0 for complete non-membership in classical set theory. That is, classical set theory is a special case of fuzzy set theory, which inherits all the properties and theorems of classical set theory when v=1 or v=0.

In the present model, v=(1-2ⁿ). That is, a skin_specimen showing an ordinary actinic keratosis, a², has fuzzy-membership-value v=(1-2²)=3/4 for a² μ_3/4 a^∞; a skin_specimen showing bowenoid actinic keratosis, a³, has fuzzy-membership-value v=(1-2³)=7/8 for a³ μ_7/8 a^∞, etc.; and corresponding clinic appointment schedule memberships, v=(1-2²)=3/4 for r² μ_3/4 r^∞; v=(1-2³)=7/8 for a³ μ_7/8 a^∞, etc.;

THEOREMS.

CLASSICAL LOGIC serves as a powerful tool for organizing the concepts of dermatopathology, because much of dermatopathology has already been assembled as chapters and lists (Sinard, Barnhill, Weedon). Mathematical models are then used to detect inconsistencies in terminology and branching structure. Every hierarchy begins with the origin, +∀ = anypatient. +∀ may have one or more children, say, +a, +b, +c, indicated by placing the children right one, down one, from its parent:

+∀ +a +b +c

Each of +a, +b, +c,... , may have their own children, say:

+∀ +a +d +e +b +f +g +c +h +i

And so forth. In a resource description framework, +∀ represents the unique, ULTIMATE CLASS, and each of the descendent classes appears exactly once. Thus RDF is purely a hierarchy of nomenclature.

One can employ NEGATION and FUZZY THEORY to extend the power of a simple class hierarchy. A possible_patient_description is a list of binary (true_false) features, such as irregular macule, pearly papule, interface separation, perivascular inflammation, etc., that may be true for a particular location on a particular patient at a given time.

A complete list of features, assigned either+ (=true) or - (=false), not both and not neither, is a POSSIBLE_PATIENT_DESCRIPTION_ELEMENT, or truth_table in classical logic. Some combinations in a possible_patient_description_element are impossible, such as a malignant seborrheic keratosis; or unlikely, such as a sarcoma arising in a hypertrophic actinic keratosis. We can declare the impossibility by an IMPLICATION, e.g., +seborrheic_keratosis IMPLIES -malignant. Unlikely events (fuzzy) are discussed further below. Another way to declare an impossibility is to create a NANDSET of things that cannot coexist, e.g., {+seborrheic_keratosis,+malignant}. In general, any implication of the form, +a IMPLIES +b, corresponds to the nandset, {+a, -b}.

One of the important advances in medical theory over the past half-century, fuzzy set theory, has suffered in part because of its unfortunate name. We prefer: concentric sets, contour sets, certainty sets, or progressive sets. Fuzzy set theory is a GENERALIZATION of classical set theory, and allows the representation of partial knowledge about set membership. Many processes in medicine have this character, as for example, the likelihood that sun-damaged skin will lead to metastatic squamous cell carcinoma and premature death. Let us denote NO SIGNIFICANT PATHOLOGIC ABNORMALITY as a⁰ and PREMATURE DEATH as a^∞ Then the intermediate stages might be:

a¹ = solar elastosis
a² = actinic keratosis
a³ = bowenoid actinic keratosis
a⁴ = Bowen's disease
a⁵ = superficially invasive squamous cell carcinoma
a⁶ = deeply invasive squamous cell carcinoma
a⁷ = N1 lymph node involvement
a⁷ = N2 lymph node involvement
a⁷ = M1 distant metatases.
...
a^∞ = premature death.

In general, all processes leading to possible premature death fall into such a progression.

STEPS.

1. English declarative sentences, electronic form: already available in dermatopathology texts, although not publicly/generally available (Barnhill, 2004; Macdonald et al 2007; Brehmer-Andersson, 2006; Weedon, 2002; McKee et al, 2005; Farmer, 1999; Venkataram, 2006).

2. RDF model.
3. Logic model.
4. Fuzzy theory.
5. Proofs: Mathematical set theory.

4.1. INTRODUCTION TO SET THEORY.

SET THEORY is the mathematical theory of collections of abstract objects, or SETS. Although the roots of set theory lie in Aristotelian (384-322 BC) logic, there was a great period growth in late 19th and early 20th century Central Europe, in which set theory was used to address fundamental questions about infinity, consistency, and provability in mathematics. Many of the concepts of set theory carried over into the development of the digital computer, by John von Neumann (1903-1957) and others. In the latter 20th century, Lotfi A. Zadeh introduced the ideas of fuzzy set theory, later applied to the biomedical context by Kazem Sadegh-zadeh. Its unfortunate name notwithstanding, FUZZY SET THEORY is a more general and precise formulation than classical set theory, since it allows partial membership instead of only complete-membership or complete-non-membership of classical set theory.

There are two primary objects of classical set theory (i.e., set-membership, denoted ∈; and the empty set or null set, denoted {} or Ø); and a handful of rules. We shall confine our attention to finite (although possibly very large) sets. The set of greatest interest to us in dermatopathology shall be the set of skin biopsies.

For finite sets, is it often convenient of denote a set as the list of elements that belong to, or are members of, that set. The elements belonging to the set are enclosed in curly brackets, {,,,...} separated by commas. for example, consider the set, Ô, of six general outcomes of a skin biopsy:

Ô = {+inflammation, +major_tissue_reaction_noninflammatory, +minor_tissue_reaction_noninflammatory, +dysplastic_neoplastic, +unremarkable, +unsatisfactory}

The order of elements is irrelevant, and repeated elements are redundant. Thus, the same set may be represented as:

Ô = {+major_tissue_reaction_noninflammatory, +minor_tissue_reaction_noninflammatory, +inflammation, +dysplastic_neoplastic, +unremarkable, +inflammation, +inflammation, +unremarkable, +inflammation, +unsatisfactory}

The null set is the set that contains no members, denoted {} or Ø.

A set is defined exactly by its members. That is, two sets are considered equal if they contain the same members, regardless of how these members were determined. For example, the set of married bachelors and the set of 1850 Chevrolets are equal, because they contain the same members, namely, no members.

Several notations are commonly used in elementary set theory:

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.

We shall pay particular attention to SETS OF BINARY (TRUE_FALSE) STATEMENTS, i.e., statements with a true_false value. When true_false statement X is true, then it is denoted +X; when statement X is false, then it is denoted -X.

The logic_operator, NAND (= not_and) is the most fundamental operation in classical propositional logic, because all other operators in classical propositional logic (e.g., not, and, inclusive_or, exclusive_or, implies, ...) can be formed from a combination of NAND_operators. This fact was discovered by the late Jan Łukasiewicz (1878-1957), inventor of so-called Polish (parenthesis-free) logic. The fundamental building block of the digital computer, the transistor, is the mathematical equivalent of a NAND_GATE.

Any set of true_false statements, A,B,C,..., form a NANDSET, {+A,+B,+C,...}, if and only if not all the statements in the set can be true at once. Thus, for example, (+A ∨ +B) (i.e., +A or +B) corresponds to the nandset, {-A,-B}, i.e., it is not true that both -A and -B are true at once. Nandsets were introduced as NULLITIES by the late Harvard professor of philosophy, Prof. Willard Van Orman Quine, a member of the famed Vienna Circle of Exact Logic.

Nandsets have special properties. The nandset NULL SET implies that the system is INCONSISTENT. A nandset that contains ± any element, as for example, {+A,-A,+B,+C,...}, is VACUOUS, i.e., removal of this nandset does not change the logical status of the system.

NANDSET SUBSET OPERATION: If X is a nandset and X ⊆ Y, then Y is a nandset.

NANDSET UNION OPERATION: Finally, if X, Y are nandsets, and there exists a unique +z ∈ X such that -z ∈ Y, then Z = ((X ∪ Y) - {+z,-z}) is a nandset (possibly vacuous).

Exhaustive application of the nandset subset and union operations suffices to find all and only the nandsets for the system.

Of particular interest in building an implication hierarchy is the fact that (+A ⇒ +B) (i.e., +A implies +B) corresponds to the nandset, {+A,-B}, i.e., it is not true that both +A and -B are true at once.

We propose that all of skin biopsy interpretation can be delineated as a sufficiently rich hierarchy of implied concepts. By convention, the ORIGIN of this universe of concepts is ANYTHING, denoted ∀. We say that "anypatient" implies either a skin biopsy or it does not:

+∀ +skin_specimen -skin_specimen

In classical logic, we write +∀ ⇒ (+skin_specimen ∨ -skin_specimen), where ⇒ denotes IMPLIES and ∨ denotes INCLUSIVE_OR. That is, anything implies either skin_specimen or not skin_specimen, corresponding to the nandset, {+∀, -skin_specimen, +skin_specimen}. This is a vacuous nandset (not very interesting).

We further propose that all skin biopsies fall under one (or more) of the following six subcategories:

+∀ +skin_specimen +inflammation +major_tissue_reaction_noninflammatory +minor_tissue_reaction_noninflammatory +dysplastic_neoplastic +unremarkable +unsatisfactory -skin_specimen

In classical logic, we write (+∀ ∧ +skin_specimen) ⇒ (+inflammation ∨ +major_tissue_reaction_noninflammatory ∨ +minor_tissue_reaction_noninflammatory ∨ +dysplastic_neoplastic ∨ +unremarkable ∨ +unsatisfactory), where ∧ denotes AND. That is, (+anything and +skin_specimen) implies either +inflammation or +major_tissue_reaction_noninflammatory or +minor_tissue_reaction_noninflammatory or +dysplastic_neoplastic or +unremarkable or+unsatisfactory), or some combination of these.

LDIP MAJOR CLASSES:

1. Person.
2. Event.
3. Data_object.
4. Specimen.
5. Reagent.
6. Instrument.
7. Terminology.

SPELLING CONVENTIONS.

1. American-English spelling.
2. All lower case.
3. Blank_space and hyphen replaced with underline (_).
4. No apostrophe or apostrophe_s. All nouns singular.
5. Inheritance denoted by carriage-return and indentation.
6. Existing standards used, where applicable and freely available.
7. Date/time convention: International Standards Organization, ISO 8601.
8. For names of stains and methods, all stopwords or barrierwords (prepositions, conjunctions, articles, pronouns, auxiliary verbs, and method, modified, stain, solution, and technique) are removed or placed at the end of the phrase.

Skin, including epidermis, dermis, subcutis, and appendages is the largest organ of the human body, by weight or by volume. Because of its exposure to environmental insults and relative ease of observation and biopsy, skin is subject to many different diseases, that are well-described histopathologically. In the USA, for example, there is an incidence of over one million (0.3%) new skin malignancies per year.

As with many pathologic entities, skin diseases may be organized hierarchically, in outline form. Such outlines may be found in the Tables of Contents of standard textbooks; and are repeated in arbitrary (i.e., alphabetical) order in the corresponding subject indexes (Barnhill, 2004; Macdonald et al 2007; Brehmer-Andersson, 2006; Weedon, 2002; McKee et al, 2005; Farmer, 1999; Venkataram, 2006).

After editing for repeats, synonyms, and minor consistencies, these sources may be formulated as an RDF.

As an example, we consider a general approach to skin biopsy evaluation. The hierarchy/outline begins at the origin, +∀ ("anypatient"). For mathematical completeness, we use the convention that each level in the hierarchy implies an exhaustive list of alternatives. At the outset, "anypatient" (+∀) is either a skin_specimen or it is not. (It is convenient to use exclusively lower-case letters; exclusively singular nouns; and to replace blanks with underline (_), since blanks are used for many purposes.) We write skin_specimen=true as +skin_specimen; and skin_specimen=false as -skin_specimen. Then the universe of skin_specimen interpretation is:

+∀ +skin_specimen -skin_specimen

More specifically, skin_specimen evaluation may be broadly classified as:

+∀ +skin_specimen +inflammation +major_tissue_reaction_noninflammatory +minor_tissue_reaction_noninflammatory +dysplastic_neoplastic +unremarkable +unsatisfactory -skin_specimen

Even more specifically:

+∀ +skin_specimen +inflammation +superficial_perivascular +superficial_deep_dermal +folliculitis_perifolliculitis +panniculitis +major_tissue_reaction_noninflammatory +lichenoid_reaction +psoriasiform_reaction +spongiotic_reaction +vesiculobullous_reaction +granulomatous_reaction +vasculopathic_reaction +minor_tissue_reaction_noninflammatory +epidermolytic_hyperkeratosis +acantholytic_dyskeratosis +cornoid_lamellation +papillomatosis +acral_angiofibroma +eosinophilic_cellulitis +transepithelial_elimination +dysplastic_neoplastic +unremarkable +unsatisfactory -skin_specimen

At this point, the outline has become so large that one can no longer look it over at a glance. We can dissect out a smaller subset of the original hierarchy, as follows:

Within each terminal class of a skin disease RDF, one may attribute any number of properties, including demographics, clinical histories, physical findings, and histopathology. For example:

6. MATHEMATICAL MODEL.

The mathematical model for RDF presented here, employs the single-parent feature of RDF classes. Mathematical proofs, using elementary set theory, demonstrate the existence of significant features of RDF classes: INTERCALATION and RETIREMENT. INTERCALATION is the ability of the RDF class structure to accept new branches without disturbing the pre-existing structure. RETIREMENT is the ability of the RDF class structure to mask existing branches without disturbing the remaining structure.

As an example, let us consider an existing RDF structure that sketches out Hippocrates' (460-379 BC) four-humor theory of human disease pathogenesis, namely: phlegm, blood, urine, and bile. This RDF structure might be diagrammed as:

+∀ +human_disease_pathogenesis +phlegm +blood +urine +bile -human_disease_pathogenesis

where ∀ denotes the origin-point of the hierarchy, read as: "anypatient".

In the seventeenth century, Giovanni Battista Morgagni (1682-1771) and others proposed a theory of human disease pathogenesis based upon normal anatomy (gross and eventually microscopic), physiology, and pathologic (altered) anatomy. This RDF structure of pathoanatomy might be diagrammed as:

+∀ +human_disease_pathogenesis +anatomy +physiology +pathologic_anatomy -human_disease_pathogenesis

In the historical period when the two theories stood side-by-side in European medicine, it must have become clear that the patho-anatomic model of human disease pathogenesis showed greater explanatory power than the humoral model.

T. S. Kuhn's (1922-1996) theory of scientific revolutions suggests that the medical-scientific community makes sudden "paradigm shifts", and then younger scientists wait while the older proponents of the legacy theory retire or die off....

One way to compare theories side-by-side is to INTERCALATE the new theory into the existing (legacy) theory. (Kuhn's theory doubts that this is possible.) Assuming that both theories can be characterized as RDF class structures, then this intercalation might appear as follows:

+∀ +human disease pathogenesis +humoral_human_disease_pathogenesis +phlegm +blood +urine +bile +pathoanatomic_human_disease_pathogenesis +anatomy +physiology +pathologic_anatomy

RETIREMENT is then a formally-graceful/polite way to remove the humoral model, once it has clearly become defunct. The transition from Theory A to Theory B might be expressed as:

+∀ +A +C₁ +C₂ ... +B +D₁ +D₂ ...

More generally, each theory might have an arbitrary depth of detailed descriptions and predictions:

+C₁ +C₁₁ +C₁₂ ... +C₂ +C₂₁ +C₂₂ ... ... +B +D₁ +D₁₁ +D₁₂ ... +D₂ +D₂₁ +D₂₂ ... ...

We shall define INTERCALATION mathematically as the appending of additional branches on an existing RDFH.

RETIREMENT is defined mathematically as the removal of branches from an existing RDFH, by one of two methods:

Method ONE:

+∀ +A +C₁ +C₂ ... +∀ +B +D₁ +D₂ ...

retires all of the children of A at once, with an added child of +∀

Method TWO:

+∀ +A +C₁ -∀ +C₂ -∀ ... +B +D₁ +D₂ ...

also retires all the children of +A, by adding -∀ as the lone grandchild for each child of +A.

Method 1 is more convenient, but Method 2 allows one to remove children of A selectively.

A convenient way to represent this structure in classical symbolic logic is as a series of IMPLICATIONS, as follows:

+∀ implies (+A or +B)
(+∀ and +A) implies (+C₁ or +C₂ or ...)
(+∀ and +A and +C₁) implies +C₁₁ or +C₁₂ or ...)
(+∀ and +A and +C₂) implies +C₂₁ or +C₂₂ or ...)
...
(+∀ and +B) implies (+D₁ or +D₂ or ...)
(+∀ and +B and +D₁) implies +D₁₁ or +D₁₂ or ...)
(+∀ and +B and +D₂) implies +D₂₁ or +D₂₂ or ...)
...

where

+X means that X is true;
-X means that X is false; and
--X equals +X.

and or is understood as inclusive_or.

Whitehead-Russell implication, say, +X ⇒ +Y, is defined as: it is true that (-X or +Y). More generally, +X ⇒ (+Y or +Z or ) is defined as: it is true that (-X or +Y or +Z or...). Alternatively, a statement of the form: it is true that (-X or +Y or +Z or...) may be represented as the NANDSET, {-X,+Y,+Z}, i.e., the list of statements that CANNOT all be true. In classical set theory, the ELEMENTS of a set are enclosed in CURLY BRACKETS, {...}. The set containing no members, called the NULL SET or EMPTY SET, is denoted {} or Ø. Commas are superfluous so long as the separation among set-elements is clear. Repeated elements are redundant. The order of elements is arbitrary. A nandset that contains both +X and -X is VACUOUS.

The INTERCALATION THEOREM states that for I:

+∀ +A +C₁ +C₂ ... +B

expanded to II:

+∀ +A +C₁ +C₂ ... +B +D₁ +D₂ ...

no logic expressions not containing B are introduced.
Proof. The nandsets for I are: {+∀-A-B} and {+∀+A-C₁-C₂}. The nandsets for II are: {+∀-A-B}, {+∀+A+C₁+C₂}, and {+∀+B-D₁-D₂}. Q.E.D.

The RETIREMENT THEOREM states that I:

+∀ +A +C₁ +C₂ ... +B +D₁ +D₂ ...

may be transformed into II:

+∀ +B +D₁ +D₂ ...

by Retirement Method 1 or Retirement Method 2.
Proof.. The nandsets for I are: {+∀-A-B}, {+∀+A-C₁-C₂}, and {+∀+B-D₁-D₂}.

Retirement Method 1. The revised nandsets for I are: {+∀-A-B}, {+∀+A-C₁-C₂-∀}, and {+∀+B-D₁-D₂}. Since {+∀+A-C₁-C₂-∀} is vacuous, Q.E.D.

Retirement Method 2. The revised nandsets for I are: {+∀-A-B}, {+∀+A-C₁-C₂}, {+∀+A-C₁+∀}, {+∀+A-C₂+∀}, and {+∀+B-D₁-D₂}. Performing nandset additions yields:...........FALSE PROOF.

7. FUZZY THEORY.

8. DISCUSSION.

There are a lot of silly books, written sometimes by very smart or famous people, who suggest that humans can do without computers; or that computers can do without humans.

Traditionally, biomedical knowledge has been accorded a lower academic status than, say, physics or chemistry. Physics and chemistry deal in exact, measured quantities and equations; whereas many significant areas of biomedicine deal in complaints, palpations, and visual pattern recognition, gross or microscopic. Nonetheless, well-trained biomedical scientists can often narrow a particular set of such observations down to a short list of probable diagnoses, given sufficiently detailed history, physical findings, and laboratory test results. An experienced diagnostician asks the right questions first, and goes directly to the short list. How does he/she do this?

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34. Wilson RE, Crocker DW, Fairgrieve J, Bartholomay AF, Emerson K Jr, Moore FD.
Adrenal structure and function in advanced carcinoma of the breast. II. The relation of steroid excretion to adrenal morphology and the outcome of adrenalectomy, with description of a new discriminant function.
JAMA. 1967 Feb 13;199(7):474-482.
PMID: 6071280.
PubMed Entry

35. Shea SM, Bartholomay AF.
In numero studies in a cell renewal system. Periodically adjusted cell renewal process.
J Theor Biol. 1965 Nov;9(3):389-413.
PMID: 5875183.
PubMed Entry

36. Bartholomay AF.
Enzymatic reaction-rate theory: a stochastic approach.
Ann N Y Acad Sci. 1962 Mar 2;96:897-912.
PMID: 13865428.
PubMed Entry

37. Bartholomay AF.
A stochastic approach to statistical kinetics with application to enzyme kinetics.
Biochemistry. 1962 Mar;1:223-230.
PMID: 13865427
PubMed Entry

38. Bartholomay AF.
The sequential probability ratio test applied to the design of clinical experiments.
N Engl J Med. 1957 Mar 14;256(11):498-505.
PMID: 13419030.
PubMed Entry

39. Barnhill R.
Textbook of Dermatopathology. Second Edition.
New York: McGraw-Hill Professional. 2004;:.
ISBN: 0071396608, 912 pages.

40. Macdonald D, Ben-Gashir M, Robson A.
Dermatopathology.
New York: Blackwell Publishing, Inc. 2007;:.
ISBN: 1405107790, 528 pages.

41. Brehmer-Andersson E.
Dermatopathology. First Edition.
New York: Springer. 2006;:.
ISBN: 354030245X, 254 pages.

42. Weedon D.
Skin Pathology. Second Edition.
New York: Churchill Livingstone. 2002;:.
ISBN: 0443070695, 1100 pages.

43. McKee PH, Calonje JE, Granter SR.
Pathology of the Skin. With CD-ROMs and with Clinical Correlations. Third Edition.
New York: Mosby. 2005;:.
ISBN: 0323036724, 2112 pages.

44. McKee PH.
Essential Skin Pathology (Paperback)
New York: Mosby. 1999;:.
ISBN: 0723430675, 256 pages.

45. Farmer E, Hood AF.
Pathology of the Skin (Hardcover)
Publisher: McGraw-Hill Professional; 2 edition. 1999;:.
ISBN: 0838580793, 987 pages.

46. Venkataram M.
Fundamentals of Pathology of Skin.
New York: Anshan Ltd; 2nd Rev edition. 2006;:.
ISBN: 1904798691, 214 pages.

46a. Murphy GF.
Dermatopathology. A Practical Guide to Common Disorders.
Philadelphia: W. B. Saunders Company. 1995;:.
ISBN 0-7216-2418-9, 505 pages.

47. Wang X, Gorlitsky R, Almeida JS.
From XML to RDF: how semantic web technologies will change the design of 'omic' standards.
Nature Biotechnology 23, 1099 - 1103 (2005).
Published online: 7 September 2005; | doi:10.1038/nbt1139. From XML to RDF: how semantic web technologies will change the design of 'omic' standards.
Xiaoshu Wang, Robert Gorlitsky & Jonas S Almeida
"....A data standard is more than just a medium to uniform data representation. By laying out the overall structure of relationships of the encoded data, a data standard will effectively define a schema for a particular area of domain knowledge. In this account, a data standard resembles the basic 'form of intuition', which, in a Kantian interpretation, conditions human perception during knowledge generation. In addition, a data standard, once accepted, will become the lingua franca for the respective community. Indeed, linguists have long postulated that language is not a mere label but the very origin of thought. As a recent study on the numerical cognition of a Brazilian tribe has forcefully demonstrated, human cognition itself is constrained by the language formalism. Finally, standardization in the information age has a unique characteristic in that it is often carried out prior to or in parallel with technology development. As history has demonstrated that a standard developed with incomplete knowledge could indeed hamper innovation, additional care must be taken to ensure that a designed data standard can evolve and adapt to a changing paradigm."
See: Kant I. Kritik der reinen Vernunft.

48. Zeman J.
Modal Logic, The Lewis-Modal Systems.
Oxford: Oxford at the Clarendon Press. 1973;:.
ISBN not stated, 302 pages.

49. Moore GW, Miller RE, Hutchins GM.
Microcomputer translator for medical text: Theorem verification for Chapter Two of Zeman's Modal Logic.
Adv Math Comput Med. 7:1621-1633, 1986.

50. Association for Pathology Informatics.
http://www.pathologyinformatics.org
Site last tested: 12/21/2006.

51. Laboratory Digital Imaging Project.
http://www.ldip.org
Site last tested: 12/21/2006.

52. von Hansemann D.
Ueber das konditionale Denken in der Medizin und seine Bedeutung fuer die Praxis.
Berlin: August Hirschwald Verlag. 1912:;1-31.

53. Sadegh-zadeh K.
Grundlagenproblem einer Theorie der klinischen Praxis. Teil 1: Explikation des medizinischen Diagnosebegriffs.
Metamed 1:76-102, 1977.

54. Sadegh-Zadeh K.
Advances in fuzzy theory.
Artif Intell Med. 1999 Mar;15(3):309-323.
PMID: 10206113
PubMed Entry

55. Zadeh LA.
A note on prototype theory and fuzzy sets.
Cognition. 1982 Nov;12(3):291-297.
PMID: 6891312
PubMed Entry

56. Moore GW, Hutchins GM.
Effort and demand logic in medical decision making.
Metamedicine. 1980;1:277-304.

57. Łukasiewicz J.
Elementy Logiki Matematycznej.
Warsaw. 1929;:.
as cited in Zeman, 1973.

58. Łukasiewicz J.
A system of Modal Logic.
The Journal of Computing Systems. 1953;1:111-149.
as cited in Zeman, 1973.

59. Łukasiewicz J.
Arithmetic and Modal Logic.
The Journal of Computing Systems. 1953;1:213-219.
as cited in Zeman, 1973.

60. Łukasiewicz J.
On a Controversial Problem of Aristotle's Modal Syllogistic.
Dominican Studies. 1954;7:114-123.
as cited in Zeman, 1973.

61. Garson J.
Modal Logic.
The Stanford Encyclopedia of Philosophy (Winter 2001 Edition). Edward N. Zalta, ed.
http://plato.stanford.edu/archives/win2001/entries/logic-modal/
Site last tested: 12/21/2006.

62. Sadegh-zadeh K.
Fuzzy Health, Illness, and Disease.
J Med Phil 2000;25(5):605-638.

63. Sadegh-zadeh K.
Fuzzy Genomes.
Artif Intell Med. 2000;18:1-28.

64. Sadegh-zadeh K.
The Fuzzy Revolution: Goodbye to the Aristotelian Weltanschauung.
Artif Intell Med. 2001;21:1-25.

65. Steimann F.
Fuzzy Set Theory in Medicine. Editorial.
Artif Intell Med. 1997;11:1-7.

66. Zadeh LA.
Fuzzy sets.
Information and Control. 1965;8:338-353.

67. Parfrey NA, Moore GW, Hutchins GM.
Is pain crisis a cause of death in sickle cell disease?
Am J Clin Pathol. 1985 Aug;84(2):209-212.
PMID: 4025226.
PubMed Entry

68. Moore GW, Hutchins GM, Miller RE.
Token swap test of significance for serial medical data bases.
Am J Med. 1986 Feb;80(2):182-190.
PMID: 3511687.
PubMed Entry

69. Moore GW, Hutchins GM, Miller RE.
A new paradigm for hypothesis testing in medicine, with examination of the Neyman Pearson condition.
Theor Med. 1986 Oct;7(3):269-282.
PMID: 3798393.
PubMed Entry

70. Heckerling PS.
Token swap test revisited.
Comput Methods Programs Biomed. 2003 Mar;70(3):265-269.
PMID: 12581559.
PubMed Entry
Heckerling PS. Department of Medicine (M/C 787), University of Illinois, 840 South Wood Street, Chicago, IL 60612, USA. pshecker@uic.edu
"The token swap test measures the association between row and column variables of a 2 x 2 table in sample misclassification space, and makes no assumptions about repeated, random sampling from a source population."

71. Sinard JH.
Outlines in Pathology.
Philadelphia: W.B.Saunders Company. A Harcourt Health Sciences Company. 1996.
ISBN 0-7216-6341-9, 229 pages.

72. Barnhill RL, ed, Crowson AN, assoc ed, Busam KJ, asst ed, Grantner SR, asst ed.
Textbook of Dermatopathology.
New York: McGraw-Hill. Health Professions Division. 1998;:.
ISBN 0-07-005726-5, 909 pages.

73. Westra WH, Hruban RH, Phelps TH, Isacson C.
Surgical Pathology Dissection. An Illustrated Guide. Second Edition. With a Forward by Askin FB.
New York: Springer. 2002.
ISBN 0-387-95559-3, 258 pages.

74. Lester SC.
Manual of Surgical Pathology.
New York: Churchill Livingstone. A Harcourt Health Sciences Company. 2001.
ISBN 0-443-07918-8, 336 pages.

75. Weidner N, ed.
Modern Surgical Pathology (2 Volume Set).
New York: Churchill Livingstone; 1st edition (January 15, 2001).
ISBN: 0443079188, 336 pages.

76. Sternberg SS, ed. Antonioli DA, Carter D, Eggleston JC, Mills SE, Oberman H, assoc eds.
Diagnostic Surgical Pathology.
New York: Raven Press. 1989;:.
ISBN 0-88167-442-7, 1776 pages, 2 vols.

77. Rosai J, Ackerman LV.
Ackerman's Surgical Pathology (2 Vol Set). Eighth Edition.
St Louis: CV Mosby; 8th edition (January 15, 1996)
ISBN: 0801670047, 2732 pages.

78. American Joint Committee on Cancer.
AJCC Cancer Staging Manual. Sixth Edition.
New York: Springer. 2002.
ISBN 0-387-95271-3, 421 pages.

79. Ormsby A, Bergfeld WF, Tubbs RR, Hsi ED.
Evaluation of a new paraffin-reactive CD7 T-cell deletion marker and a polymerase chain reaction-based T-cell receptor gene rearrangement assay: Implications for diagnosis of mycosis fungoides in community clinical practice.
J Am Acad Dermatol. 2001;45:405-413.

80. Sutton's Law: Go where the money is. The most likely explanation is best. The late Willie Sutton was a bank robber who, when asked why he always robbed banks, allegedly state: "Because that's where the money is". (In his published autobiography Sutton denies it.) The idea is: don't waste your effort on low-yield crime; go where the money is. The idea entered the peer-reviewed medical literature with the publication of:

Petersdorf RG, Beeson PB.
Fever of unexplained origin: report on 100 cases.
Medicine (Baltimore). 1961 Feb;40:1-30.
PMID: 13734791.

The issue was the presumptive treatment of a patient with fever of unknown origin based upon clinical evidence, prior to the return of culture results.

Zebra Rule: "When you hear hoofbeats in the street, think of horses, not zebras." Because (at least in America), the hoofbeats are much more likely to be from horses. An academic, tertiary care hospital is sometimes called a "zebra farm".

82. Bundy A, ed.
Artificial Intelligence Techniques. A comprehensive catalogue. Fourth, revised edition.
Berlin: Springer Verlag. 1997
ISBN 3-540-59323-3, 141 pages.

83. Moore GW, Miller RE, Hutchins GM.
Microcomputer translator for medical text: Theorem verification for Chapter Two of Zeman's Modal Logic.
Adv Math Comput Med. 7:1621-1633, 1986.

84. Moore GW, Riede UN, Polacsek RA, Miller RE, Hutchins GM.
Automated translation of German to English medical text.
Am J Med. 1986 Jul;81(1):103-111.

85. Moore GW, Hutchins GM, Miller RE.
Examination of disease names using non-Abelian symbolic logic.
Methods Inf Med. 1986 Apr;25(2):109-115.

86. Moore GW, Riede UN, Polacsek RA, Miller RE, Hutchins GM.
Group theory approach to computer translation of medical German.
Methods Inf Med. 1986 Jul;25(3):176-182.

87. Moore GW, Miller RE, Hutchins GM, Erozan YS, de la Monte SM, Riede UN, Polacsek RA.
Multilingual translation techniques in the analysis of narrative medical text.

88. Offerhaus GJA, Tersmette AC, Moore GW, Hershey J, Polacsek RA.
Dutch respelling rules for English and German medical word lists.
Methods Inf Med. 1987 Jul;26(3):99-103.

89. Moore GW, Miller RE, Hutchins GM.
Determining cause of death in 45,564 autopsy reports.
Theor Med. 1988 Jun;9(2):179-186.

90. Moore GW, Boitnott JK, Miller RE, Eggleston JC, Hutchins GM.
Integrated pathology reporting, indexing, and retrieval system using natural language diagnoses.
Mod Pathol. 1988 Jan;1(1):44-50.

91. Moore GW, Wakai I, Satomura Y, Giere W.
TRANSOFT: Medical translation expert system.
Artif Intell Med 1:149-157, 1989.

92. Moore GW, Berman JJ.
Performance analysis of manual and automated systematized nomenclature of medicine (SNOMED) coding.
Am J Clin Pathol. 1994 Mar;101(3):253-256.

93. Berman JJ, Moore GW, Donnelly WH, Massey, JK, Craig B.
SNOMED Analysis of 40,124 Surgical Pathology Cases.
Am J Clin Pathol 102:539-540, 1994

94. Berman JJ, Moore GW.
SNOMED-encoded surgical pathology databases: A tool for epidemiologic investigation.
Mod Pathol. 1996 Sep;9(9):944-950.

95. Moore GW, Polacsek RA, Casanova MF, Erozan YS, Hershey J, Miller RE, Hutchins GM.
Multilingual respelling rules for an English medical word list.
MEDINFO-86. (R Salamon, B Blum, M Jorgensen, eds.) Elsevier Science Publishers B.V. (North Holland), 1986. Proc. Fifth World Congress on Medical Informatics, October 26-30, 1986, Washington, D.C., pp 1106-1110.

96. Yu CC-Y, Moore GW, Unschuld PU.
Romanized Chinese respelling rules for an English medical word list.
Proc Annu Symp Comput Appl Med Care. 1987;11:xxx-xxx. Washington DC, November 1-4, 1987.

97. Moore GW, Hutchins GM, Boitnott JK, Miller RE, Polacsek RA.
Word root translation of 45,564 autopsy reports into MeSH titles.
Proc Annu Symp Comput Appl Med Care. 1987;11: Washington DC, November 1-4, 1987.

100. Moore GW, Berman JJ.
Automatic SNOMED coding.
Proc Annu Symp Comput Appl Med Care. 1994;18:225-229.

101. Berman JJ, Moore GW, Donnelly WH, Massey JK, Craig B.
A SNOMED analysis of three years accessioned cases (40,124) of a surgical pathology department: implications for pathology- based demographic studies.
J Amer Med Informatics Assn (JAMIA), Symposium Supplement, 1994, and Proceedings of the 18th Annual Symposium on Computer Applications in Medical Care 18:188-192, 1994.

102. Dreyfus HL.
What Computers Still Can't Do.
Paperback, 429 pages (October 30, 1992)
Cambridge, MA: MIT Press;
ISBN: 0262540673, 429 pages.

103. Stampp KM.
The Peculiar Institution : Slavery in the Ante-Bellum South.
Paperback Reissue edition (December 1989)
New York: Vintage Books.
ISBN: 0679723072.

104. Asimov I.
I, Robot.
Mass Market Paperback, 272 pages, Reprint edition (July, 1994). New York: Bantam Books.
ISBN: 0553294385, 272 pages.

105. Asimov I.
The Robots of Dawn.
Mass Market Paperback Reprint edition (April, 1994). New York: Spectra.
ISBN: 0553299492.

106. Asimov I.
The Complete Robot.
Paperback, 512 pages (1983). New York: Acacia Press, Inc.
ISBN: 0586057242, 512 pages.

... Asimov I.
The End of Eternity.

When I first read this book as an adolescent, I accepted Asimov's assumption without question that scientists could not imagine the technology for time travel without seeing an actual time machine; and therefore by destroying the prototype for this time machine, the heroes of the book were able to end time-travel, and its eventual negative consequences as described in the book. Having seen the marvels of human imagination during my own lifetime, over the past 40 years, and having studied what our forefathers (Archimedes, Hippocrates, Plato, etc.) were able to imagine without any available technology, I no longer believe that humans lack the imagination to build a time machine.

On the other hand, I fully accept the idea, promulgated in Asimov's short story, The Dead Past, that some government bureaucracies might lack the imagination and political will to fund such research into time-travel.

107. Seife C.
Zero. The Biography of a Dangerous Idea.
London: Penguin Books. 2000.
ISBN: 0-670-88457-X, 248 pages.
This book includes an account of the execution of Hippasus of Metapontum, a member of the Pythagorean cult, who had dared to reveal the existence of irrational numbers to persons outside the cult.

108. Stewart I.
Flatterland. Like Flatland. Only More So.
Cambridge, MA: Perseus Publishing. 2001.
ISBN 0-7382-0442-0, 301 pages.
This book is a sequel of Edwin A. Abbott's FLATLAND, published in 1884, and cited in Stephen Hawking's A BRIEF HISTORY OF TIME.

109. Casti JL, DePauli W.
Gödel. A Life of Logic.
Cambridge, MA: Perseus Publishing. 2000.
ISBN 0-7382-0274-6, 210 pages.

111. Scarborough D, Sternberg S.
Methods, Models, and Conceptual Issues. An Invitation to Cognitive Science. Volume 4.
Cambridge, MA: MIT Press. 1998.
ISBN 0-262-65946-0, 950 pages.

112. Changeux J-P, Connes A.
Conversations on Mind, Matter, and Mathematics
Ed & Transl: DeBevoise MB. Princeton, NJ: Princeton University Press. 1995.
ISBN 0-691-08759-8, 260 pages.

113. Hockey S.
A Guide to Computer Applications in the Humanities.
Chapter 8. Sound Patterns. pp. 168-188.
Baltimore: The Johns Hopkins Univ Press. 1980.
ISBN 0-8018-2891-0, 248 pages.
Cited: Ott W. Metrical Analysis of Latin Hexameter by Computer. Revue 4:7-24, 1966.
Cited: Greenberg NA. Scansion Purement Automatique de l'Hexamère Dactylique. Revue 1967;3:1-25.

114. Woodger JH.
The Axiomatic Method in Biology.
Out of Print.
Classic book by the grandfather of biomathematics and bioinformatics.

115. Woodger JH.
Techniques of Theory Construction.
Out of Print.
Classic book by the grandfather of biomathematics and bioinformatics.

116. Born M.
The Restless Universe. Second Edition.
Authorized Transl: Winifred M. Deans, MA, BsC.
New York: Dover Publications, Inc. 1951.
315 pages.

117. Cate FH.
Privacy in the Information Age.
Washington: Brookings Institution Press. 1997.
ISBN 0-8157-1316-9

118. Davis M.
Computability and Unsolvability.
New York: Dover Publications, Inc. 1958.
ISBN 0-486-61471-9, 248 pages.
A short description of the major issues in the field of Computability and Unsolvability. A nice appendix, with a review of the major theorems of Number theory, and Matiyasevic's demonstration that Hilbert's Tenth Problem is insoluble. Good reference section.

119. Snow CP.
The Two Cultures.
With an Introduction by Stephan Collini.
Cambridge: Canto. Cambridge University Press. 1959.
ISBN 0-521-45730-0, 107 pages.
The 1960s cult book that launched a generation of academic debate about the separation of science and humanities education.

120. Stevenson J.
The Complete Idiot's Guide to Philosophy.
New York: Alpha Books. A Division of Macmillian Reference USA. A Simon and Schuster Macmillan Company. 1998.
ISBN 0-02-861981-1, 266 pages.
A quick romp through the history of philosophy. Good for the amateur. A lot of serious omissions, including Gödel, in my opinion.

121. Farmer R, Miller D, Lawrenson R.
Lecture Notes on Epidemiology and Public Health Medicine. Fourth Edition.
Oxford: Blackwell Science. 1996.
ISBN 0-86542-611-2, 288 pages.

122. Orwant J, Hietaniemi J, Macdonald J.
Mastering Algorithms with Perl.
Cambridge: O'Reilly. 1999.
ISBN 1-56592-398-7, 684 pages.

123. Greek Mathematics.
Goold GP, ed. Thomas I transl. Loeb Classical Library. #335. Cambridge, MA: Harvard University Press. 1939.
ISBN 0-674-99369-1, 511 pages.
The Loeb formula of facing page translations, Greek and English. Includes: Pythagoras, Thales, Plato, Aristotle, Euclid.

124. Hippocrates.
Hippocrates. Volume I.
Jones WHS, transl. Loeb Classical Library. Cambridge, MA: Harvard University Press. 1923.
ISBN 0-674-99162-1, 361 pages.
Includes Hippocrates' Oath, with explanatory notes.

125. Singh S.
Fermat's Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Division of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.
The on-again, off-again, on-again proof by Sir Andrew Wiles of Fermat's Last Theorem. A brief run-through of the history of Diophantus's famous conjecture, with Fermat's notorious inscription:

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.

The conjecture is:

a^k + b^k = c^k ,

where a, b, c, and k, are all integers. There are no solutions where k > 2.

126. Giere W.
Dokumentation und Informationsaufbereitung fuer den Arzt. Beitraege zur Medizischen Informatik.
Kirsten K, ed. Darmstadt: Epilog Verlag. 1996.
ISBN 3-9803214-7-9, 437 pages.
Collected works of Germany's foremost medical informatician.

127. Hofstadter DR.
Gödel, Escher, Bach. An Eternal Golden Braid.
New York: Basic Books. A member of the Perseus Books Group. 1979.
ISBN 0-465-02656-7, 777 pages.
Pulitzer Prize-Winning book, that introduced the general educated public to Gödel, Time Magazine's mathematician of the twentieth century. A really dynamite set of references, including notes, in the field of mathematical logic.

128. Unseld DW.
Medizinisches Wörterbuch der deutschen und englischen Sprache. Achte, neubearbeitete Auflage.
Medical Dictionary of the English and German Languages. Eighth Edition, revised and enlarged.
Stuttgart: Wissenschaftliched Verlagsgesellschaft mbH. 1982.
ISBN 3-8047-0661-4, 593 pages.

129. Lewis GL.
Turkish Grammar.
Oxford: Oxford at the Clarendon Press. 1967.
ISBN 0-19-815838-6, 303 pages.

130. Andrews GL.
Number Theory.
New York: Dover Publications, Inc. 1971.
ISBN 0-486-68252-8, 259 pages.

131. Croxton FE.
Elementary Statistics. With Applications in Medicine and the Biological Sciences.
New York: Dover Publications, Inc. 1953.
376 pages.

132. Aldenderfer MS, Blashfield RK.
Cluster Analysis.
A Sage University Paper #44. Newbury Park: Sage Publications.
ISBN 0-8039-2376-7, 88 pages.

133. Arabie P, Carroll JD, DeSarbo WS.
Three-Way Scaling and Clustering.
A Sage University Paper #65. Newbury Park: Sage Publications.
ISBN 0-8039-3068-2, 92 pages.

134. Eco U.
A Theory of Semiotics.
Bloomington, IN: A Midland Book. Indiana University Press. 1979;:.
ISBN 0-253-35955-4, 354 pages.
Serious academic work by the beloved author of The Name of the Rose.

135. Ulam SM.
Adventures of a Mathematician.
Berkley, CA: University of California Press. 1991.
ISBN 0-52007154-9, 329 pages.
My favorite chapter is chapter 15, random reflections, in which much of the lore of mathematics is discussed.

136. Bernstein PL.
Against the Gods. The Remarkable Story of Risk.
New York: John Wiley & Sons, Inc. 1996.
ISBN 0-471-29563-9, 383 pages.
A fantastic excursion through the history of probability and chance, starting with the ancient Egyptians and ending with modern worldwide business practices. Probability was originally studied in order to INCREASE BENEFITS, as in winning at gambling or staying alive longer. Now, probability has its most important applications in order to DECREASE LOSSES.

137. DeCew JW.
In Pursuit of Privacy. Law, Ethics, and the Rise of Technology.
Ithaca, NY: Cornell University Press. 1997.
ISBN 0-8014-3380-0, 199 pages.

138. Sandritter W.
Histopathologie. Lehrbuch und Atlas fuer Studierende und Aerzte. Sechste, verbesserte Auflage.
Stuttgart: F. K. Schattauer Verlag. 1975.
ISBN 3-7945-0454-2, 309 pages.

139. Simpson A.
HTML publishing Bible. WIndows 95 Edition.
Foster City, CA: IDG Books Worldwide, Inc. An International Data Group Company. 1996.
ISBN 0-7645-3009-7, 774 pages.
HTML is the display language of the Internet. Forget ADOBE, JavaScript, Java, etc. HTML is king, because HTML is everywhere.

140. Chametzky RA.
A Theory of Phrase Markers and the Extended Base.
Albany, NY: State University of New York Press. 1996.
ISBN 0-7914-2972-5, 206 pages.

141. Scarborough D, Sternberg S.
An Invitation to Cognitive Science. Second Edition. Methods, Models, and Conceptual Issues. Volume 4.
Cambridge, MA: A Bradford Book. The MIT Press. 1998.
ISBN 0-262-65046-0, 950 pages.

142. von Neumann.
The Computer and the Brain.
New Haven, CT: Yale University Press. 1957;:.

143. Boole G.
An Investigation of the Laws of Thought. On which are founded the Mathematical Theories of Logic and Probabilities.
New York: Dover Publications, Inc. 1954.

144. Lewis CI, Langford CH.
Symbolic Logic. Second Edition.
New York: Dover Publications, Inc. 1932.

145. Borkowski L.
Formale Logik. Logische Systeme. Einführung in die Metalogik. Ein Lehrbuch.
München: C. H. Beck. 1977.
ISBN 3-406-06802-2, 578 pages.
Formal logic. Logical systems. Introduction to metalogic.
Polish Original Title: Logika Formalna. Systemy logiczne. Wstep do metalogiki.
© 1968. Panstwowe Wydawnictwo Naukowe. Warszawa.
Translators: Beschoner K-H, Meyer KJ.

147. Haack S.
Deviant Logic, Fuzzy Logic: Beyond the Formalism.
Chicago: University of Chicago Press. 1996;:.
ISBN: 0226311341, 292 pages.

148. Nguyen HT, Walker EA.
A First Course in Fuzzy Logic.
Hardcover, 300 pages, 2 edition, July 1999.
New York: CRC Press.
ISBN: 0849316596.

149. Schneier B.
Cryptography. Second Edition.
New York: John Wiley and Sons. 1997.

150. Backus J.
Turing award article.
Comm ACM, Vol. 21, No. 8, august 1978.

151. Naur P.
Revised Report on the Algorithmic Language ALGOL 60.
Comm ACM, Vol. 3 No.5, pp. 299-314, May 1960.

152. Courant R, Robbins H, Stewart I.
What is Mathematics? An Elementary Approach to Ideas and Methods. Second Edition.
Oxford: Oxford University Press. 1996.

153. Asimov I.
Realm of Numbers.
Out of Print.

154. Agnew RP.
Calculus and Analytic Geometry.
1962.

155. Tymoczko T.
New Directions in the Philosophy of Mathematics.
Princeton, NJ: Princeton University Press. 1998;:.
ISBN 0-691-03498-2, 436 pages.

156. Greek Mathematics.
Goold GP, ed. Thomas I transl. Loeb Classical Library. #335.
Cambridge, MA: Harvard University Press. 1939.
ISBN 0-674-99369-1, 511 pages.

157. Singh S.
Fermat's Enigma. The Epic Quest to Solve the World's Greatest Mathematical Problem.
New York: Anchor Books. A Division of Random House, Inc. 1997.
ISBN 0-385-49362-2, 315 pages.

158. Suppes P.
Introduction to Logic.
New York: Van Nostrand. 1957.

159. Bernays P.
Axiomatic Set Theory.
New York: Dover Publications. 1968.

160. Suppes P.
Axiomatic Set Theory.
New York: Dover Publications. 1972.
ISBN 0486616304.

161. Davis M.
Computability and Unsolvability.
New York: Dover Publications, Inc. 1958.
ISBN 0-486-61471-9, 248 pages.

162. Andrews GL.
Number Theory.
New York: Dover Publications, Inc. 1971.
ISBN 0-486-68252-8, 259 pages.

163. Schneier B.
Applied Cryptography, Second Edition. Protocols, Algorithms, and Source Code in C.
New York: John Wiley & Sons, 1996.

164. Tarjan RE.
Data Structures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Mathematics.
Paperback, December, 1983.
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ISBN: 0898711878.

165. Worldwide Web Consortium.
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166. U.S. National Library of Medicine.
Unified Medical Language System.
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167. U. S. National Library of Medicine.
UMLS Knowledge Sources. Twelfth Edition. Unified Medical Language System.
U. S. Department of Health and Human Services. National Institutes of Health. National Library of Medicine. 2001.

168. Haber, Gattuso, Spitz, David.
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169. Light R.
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170. Prophet EB, Mills B, Arrington JB, Sobin LH.
Laboratory Methods in Histotechnology. 1992: Armed Forces Institute of Pathology,
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ISBN: 1-881041-00-X 1992.

171. Moore GW, Berman JJ.
Performance Analysis of Manual and Automated Systematized Nomenclature of Medicine (SNOMED) Coding.
Am J Clin Pathol 101:253-256, 1994.

172. Berman JJ, Moore GW.
Image analysis software for the detection of preneoplastic and early neoplastic lesions.
Cancer Letters 77:103-109, 1994.

173. Moore GW, Berman JJ, Hanzlick RL, Buchino JJ, Hutchins GM.
A prototype international autopsy database: 1625 consecutive fetal and neonatal autopsy facesheets spanning twenty years.
Arch Pathol Lab Med. 1996;120:782-785.

174. Berman JJ, Moore GW, Donnelly WH, Massey, JK, Craig B.
SNOMED Analysis of 40,124 Surgical Pathology Cases.
Am J Clin Pathol 102:539-540, 1994

175. Berman JJ, Moore GW.
SNOMED-Encoded surgical pathology databases: a tool for epidemiologic investigation.
Modern Pathology 9:944-950, 1996.

176. Moore GW, Hutchins GM, Miller RE.
Natural language text searches in autopsy pathology: Role of non-Abelian symbolic logic.
2nd Ann. Conf. A.A.M.S.I., Baltimore, MD, October, 1983.

177. Moore GW, Miller RE, Hutchins GM, Riede UN, Polacsek RA.
Multilingual translation techniques in the analysis of narrative medical text.
Baltimore, MD: Proc 9th Ann Symp Comp Appl Med Care. November 10-13, 1985;9:.

178. Moore GW, Polacsek RA, Casanova MF, Erozan YS, Hershey J, Miller RE, Hutchins GM.
Multilingual respelling rules for an English medical word list.
MEDINFO-86. (R Salamon, B Blum, M Jorgensen, eds.) Elsevier Science Publishers B.V. (North Holland), 1986. Proc. Fifth World Congress on Medical Informatics, October 26-30, 1986, Washington, D.C., pp 1106-1110.

179. Moore GW, Hutchins GM, Boitnott JK, Miller RE, Polacsek RA.
Word root translation of 45,564 autopsy reports into MeSH titles.
Proc SCAMC, Symposium on Computer Applications in Medical Care, Washington DC, November 1-4, 1987.

180. Moore GW, Berman JJ.
Automatic SNOMED Coding.
Symposium on Computer Applications in Medical Care 1994;18:.

181. Berman JJ, Moore GW, Donnelly WH, Massey JK, Craig B.
A SNOMED analysis of three years accessioned cases (40,124) of a surgical pathology department: implications for pathology-based demographic studies.
J Amer Med Informatics Assn (JAMIA), Symposium Supplement, 1994, and Proceedings of the 18th Annual Symposium on Computer Applications in Medical Care 18:188-192, 1994.

182. Moore GW, Berman JJ.
Automatic SNOMED Coding.
Journal of the American Medical Informatics Association (JAMIA), Symposium Supplement 1994 and the Proceedings of the 18th Annual Symposium for Computer Applications in Medicine (SCAMC), pp 225-229, 1994

183. Berman JJ, Moore GW, Donnelly WH, Massey JK, Craig B.
A SNOMED analysis of three years' accessioned cases of (40,124) of a surgical pathology department: implications for pathology-based demographic studies.
J Am Med Informat Assn (JAMIA), Symposium Supplement 1994 and the Proceedings of the 18th Annual Symposium for Computer Appllications in Medicine (SCAMC), pp 188-192, 1994

184. Lilienfeld DE, Stolley PD.
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185. MacMahon B, Trichopoulos D.
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186. Barker DJP, Cooper C, Rose G.
Epidemiology in Medical Practice. Fifth Edition.
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187. Gordis L.
Epidemiology.
Philadelphia: W. B. Saunders Co. 1996;:.
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188. Farmer R, Miller D, Lawrenson R.
Lecture Notes on Epidemiology and Public Health Medicine. Fourth Edition.
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189. Hutchins GM, McLendon W W. (eds.).
College of American Pathologists Conference XXIX on Restructuring Autopsy Practice for Health Care Reform.
Arch Pathol Lab Med 120. 733-781, 1996.

190. Berman JJ, Moore GW, Hutchins GM.
Maintaining patient confidentiality in the public domain Internet Autopsy Database (IAD).
Proc AMIA Annu Fall Symp. 1996;:328-32.

191. Berman JJ, Moore GW, Hutchins GM.
U.S. Senate Bill 422. the Genetic Confidentiality and Nondiscrimination Act of 1997.
Diagn Mol Pathol. 1998 Aug;7(4):192-6. Review.

192. Court C.
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193. Lemay L, Tyler D.
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194. Clayton EW, Steinberg KK, Khoury MJ, Thomson E, Andrews L, Kahn MJ, Kopelman LM, Weiss JO.
Informed consent for genetic research on stored tissue samples.
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195. James DS, Leadbeatter S.
Confidentiality, death and the doctor.
J Clin Pathol. 1996;49:1-4.

196. Moore GW, Wakai I, Satomura Y, Giere W.
TRANSOFT: Medical translation expert system.
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197. Hutchins GM, McLendon WW (eds.).
College of American Pathologists Conference XXIX on Restructuring Autopsy Practice for Health Care Reform.
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198. Friedman BA.
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199. Schneier B.
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201. Sweeney L.
Replacing Personally-Identifying Information in Medical Records, the Scrub System.
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202. Barrows RC, Cimino JJ, Clayton PD.
Mapping Clinically Useful Terminology to a Controlled Medical Vocabulary.
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203. Bell DS, Cimino JJ, Evans DA, Hersh WR, Huff ST.
Toward a Medical-Concept Representation Language.
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204. Hall PA, Lemoine NR.
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205. Brannigan VM.
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208. Frey CM, McMillen MM, Cowan CD, Horm JW, Kessler LG.
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209. Ashworth TG.
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211. Kircher T, Nelson J, Burdo H.
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212. Kircher T, Anderson RE.
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213. Slater DN.
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214. Hanzlick R, ed.
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215. Svendsen E, Hill RB.
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216. McFarlane MJ, Feinstein AR, Wells CK, Chan CK.
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217. McFarlane MJ.
Clinical diagnosis is not a source of bias in selection for necropsy.
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218. Goldman L, Sayson R, Robbins S, Cohn LH, Bettmann M, Weisberg M.
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219. Goldman L.
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220. Cameron HM, McGoogan E.
A prospective study of 1152 hospital autopsies. 1. Inaccuracies in death certification.
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221. Tymoczko T.
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222. Greek Mathematics.
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223. U. S. Defense Advanced Research Projects Agency (DARPA).
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224. U. S. Defense Advanced Research Projects Agency (DARPA). Agent Markup Language.
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225. U. S. Defense Advanced Research Projects Agency (DARPA). Ontology Inference Layer.
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226. Simpson A. HTML Publishing Bible, Windows 95 Edition. Foster City, CA: IDG Books Worldwide, Inc., 1996. International Data Group Company, 919 East Hillsdale Blvd, Suite 400, Foster City, CA 94404. 1-415-655-3200.

227. Till D. Teach yourself PERL 5 in 21 days, Second Edition. Indianapolis: SAMS Publishing, 1996. SAMS Publishing, 201 West 103rd Street, Indianapolis, IN 46290. 1-800-428-5331.

228. U. S. Code of Federal Regulations. 1995. 45 CFR Subtitle A (10-1-95 Edition), part 46.101 (b) (4).
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The complete Common Rule document (45CFR46), at URL:
http://www.uaf.edu/oar/irb/45cfr46.html
or at URL:
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229. U. S. Code of Federal Regulations. 1999. 45 CFR Parts 160 - 164. Standards for Privacy of Individually Identifiable Health Information; Proposed Rule.
Department of Health and Human Services. Office of the Secretary.
Fed Regist. 1999 Nov 3;64(212):59917-59966. http://aspe.hhs.gov/admnsimp/

230. National Cancer Institute's Confidentiality Brochure, at URL:
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231. Moore GW, Berman JJ.
Anatomic Pathology Data Mining.
In: Cios KJ, ed. Medical Data Mining and Knowledge Discovery.
2001. XVIII, 502 pp. 98 figs., 98 tabs. Hardcover.
ISBN: 3-7908-1340-0.
Copyright Springer-Verlag: Berlin/Heidelberg 1999.

232. Sweeney L.
Computational Disclosure Control: A Primer on Data Privacy Protection.
PhD Thesis. Massachusetts Institute of Technology. Spring, 2001. Draft.
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233. Sweeney L.
Privacy and medical-records research.
N Engl J Med. 1998 Apr 9;338(15):1077.
PMID: 9537887; UI: 98181820.

234. Sweeney L.
Guaranteeing anonymity when sharing medical data, the Datafly System.
Proc AMIA Annu Fall Symp. 1997;:51-55.
PMID: 9357587; UI: 98020458.

235. Gödel K.
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236. Gödel K.
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Trsl: Meltzer B. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. [German].
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237. Nagel E, Newman JR.
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238. Hofstadter DR.
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239. Smith B.
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240. Quine WVO.
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241. U. S. Defense Advanced Research Projects Agency (DARPA). Agent Markup Language.
http://www.daml.org

242. U. S. Defense Advanced Research Projects Agency (DARPA). Ontology Inference Layer.
http://www.ontoknowledge.org/oil

243. Boole G.
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244. Lewis CI, Langford CH.
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245. Nguyen HT, Walker EA.
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246. Quine WVO, Ullian JS.
The Web of Belief. Second Edition.
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247. Moore GW, Hutchins GM, Bulkley BH.
Certainty levels in the nullity method of symbolic logic: application to the pathogenesis of congenital heart malformations.
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248. Moore GW, Hutchins GM.
Effort and demand logic in medical decision making.
Metamedicine 1980;1:277-304.

249. Cios KJ, Moore GW. 2000.
Medical Data Mining and Knowledge Discovery: An Overview.
In: Cios KJ, ed. Medical Data Mining and Knowledge Discovery.
2001. XVIII, 502 pp. 98 figs., 98 tabs. Hardcover.
ISBN: 3-7908-1340-0.
Copyright Springer-Verlag: Berlin/Heidelberg 1999.

250. College of American Pathologists.
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251. Collaborative Prostate Cancer Tissue Resource.
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252. Suppes P.
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253. Bernays P.
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254. Suppes P.
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255. Davis M.
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256. Andrews GL.
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257. Schneier B.
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258. Tarjan RE.
Data Structures and Network Algorithms. CBMS-NSF Regional Conference Series in Applied Mathematics.
Paperback, December, 1983.
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ISBN: 0898711878.

259. Worldwide Web Consortium.
http://www.w3.org

260. Light R.
Presenting XML.
Sams.net Publishing. 1997.

261. U.S. National Library of Medicine.
Unified Medical Language System.
http://www.nlm.nih.gov/research/umls/

262. U. S. National Library of Medicine.
UMLS Knowledge Sources. Twelfth Edition. Unified Medical Language System.
U. S. Department of Health and Human Services. National Institutes of Health. National Library of Medicine. 2001.

263. Vigorita VJ, Moore GW, Hutchins GM.
Absence of correlation between coronary arterial atherosclerosis and severity or duration of diabetes mellitus of adult onset.
Am J Cardiol. 1980;46:535-542.

264. Moore GW, Hutchins GM.
Consistency versus completeness in medical decision making: Application to 155 patients autopsied after coronary artery bypass graft surgery.
Proc 6th Annu Symp Comput Appl Med Care. 1982;6:805-811.
This paper has an amusing example of the misuse of statistics. In a study of 155 patients autopsied after coronary bypass surgery, it was discovered that short stature had a high correlation with severity of diabetes mellitus. After an initial euphoria about possibly uncovering a fundamental new insight into diabetes pathogenesis, the authors' further investigation revealed that many of the "short statured" patients with severe diabetes were in fact lower-extremity amputees, due to diabetic peripheral vascular disease. The autopsy prosectors had simply reported body-lengths from the head to wherever the body ended!

265. Moore GW, Brown LA, Miller RE.
Set Theory Definition and Algorithm for Medical De-Identification.
Arch Pathol Lab Med. 2001;:in press.
http://www.netautopsy.org/apep00st.htm

266. Johns Hopkins Autopsy Resource.
http://www.netautopsy.org/

267. Moore GW, Berman JJ, Hanzlick RL, Buchino JJ, Hutchins GM. 1996.
A prototype Internet autopsy database. 1625 consecutive fetal and neonatal autopsy facesheets spanning 20 years.
Arch Pathol Lab Med. 1996;120:782-785.

268. Hornung J.
Kritik der Signifikanztests.
Metamed 1977;1:325-345.
If you can read German and find this journal article, it's a nice summary of the popular statistical significance tests.

269. Moore GW, Hutchins GM.
The persistent importance of autopsies.
Mayo Clin Proc. 2000 Jun;75(6):557-558.

270. Wilbur WJ.
Overview of Books at NCBI.
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271. Berman J.
Shared Pathology Informatics Network.
http://grants.nih.gov/grants/guide/rfa-files/RFA-CA-01-006.html
2001;:.
This was an ambitious project involving pathology informatics, which in my opinion did not live up to its promise.

272. Nelson SJ, Cole WG, Tuttle MS, Olson NE, Sherertz DD.
Recognizing new medical knowledge computationally.
Proc Annu Symp Comput Appl Med Care. 1993;17:409-413.

273. Moore GW, Polacsek RA, Erozan YS, de la Monte SM, Miller RE, Hutchins GM, Riede UN.
Multilingual translation techniques in the analysis of narrative medical text.
Comput Methods Programs Biomed. 1986 Mar;22(1):35-42.

274. Chomsky N.
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276. Singer KP, Jones TJ, Breidahl PD.
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277. Gootman EC.
Calculus.
Hauppauge NY: Barron's Educational Series, Inc. 1997;:.
Textbook by a professor, who according to one of his students, "could teach calculus to a cat."

278. Berman JJ.
Tumor classification: molecular analysis meets Aristotle.
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279. Upton G, Cook I.
A Dictionary of Statistics.
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280. Merriam-Webster.
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      208. Borkowski A, Lee DH, Sydnor DL, Johnson RJ, Rabinovitch A, Moore GW.
Intranet-based quality improvement documentation at the Veterans Affairs Maryland Health Care System.
Mod Pathol. 2001 Jan;14(1):1-5.

      209. Moore GW, Hutchins GM.
The persistent importance of autopsies.
Mayo Clin Proc. 2000 Jun;75(6):557-558.

      211. Stewart I.
Flatterland. Like Flatland. Only More So.
Cambridge, MA: Perseus Publishing. 2001.
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Reviewed in: Neurocomputing. 2001;:in press.

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An Introduction to Neural Computing. Second Edition.
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Conversations on Mind, Matter, and Mathematics
Ed & Transl: DeBevoise MB. Princeton, NJ: Princeton University Press. 1995.
ISBN 0-691-08759-8, 260 pages.
Reviewed in: Neurocomputing. 2001;:in press.

      216. Moore GW, Berman JJ.
Anatomic Pathology Data Mining.
Chapter 4. In: Cios KJ. Medical Data Mining and Knowledge Discovery. Berlin: Springer Verlag. 2000;4:61-107.
ISBN: 3-7908-1340-0, 502 pages.
Published within the series: "Studies in Fuzziness and Soft Computing", Physica-Verlag Heidelberg, a Springer-Verlag Company.

      217. Greenberger NJ, Berntsen MS, Velakaturi VN.
Differential Diagnosis in Internal Medicine: Medical Book of Lists.
Paperback, 347 pages 5th edition (January 15, 1998)
St Louis: Mosby-Year Book;
ISBN: 0323001319, 347 pages.

      218. Gödelization of a Pathology Database: Re-identification by Inference.
G. William Moore, MD, PhD Lawrence A. Brown, MD, Robert E. Miller, MD.
Arch Pathol Lab Med. 2002;:in press.

      219. Goethe University Autopsy Register: Anonymized Bilingual Database.
W. Giere, MD. G. William Moore, MD, PhD Grover M. Hutchins, MD.
Arch Pathol Lab Med. 2002;:in press.

      220. Set Theory Definition and Algorithm for Medical De-Identification.
G. William Moore, MD, PhD Lawrence A. Brown, MD, Robert E. Miller, MD.
Arch Pathol Lab Med. 2001;:in press.

301. Web-based Free-Text Query System for Surgical Pathology Reports with Automatic Case De-Identification.
Robert E. Miller, MD, John K. Boitnott, MD, G. William Moore, MD, PhD.
Arch Pathol Lab Med. 2001;:in press.

302. UMLS Concordance for a Comprehensive Pathology Text.
John H. Sinard, MD, PhD, G. William Moore, MD, PhD.
Arch Pathol Lab Med. 2001;:in press.

303. Linguistic Inventory of the Johns Hopkins Surgical Pathology Database.
G. William Moore, MD, PhD, Robert E. Miller, MD.
Arch Pathol Lab Med. 2001;:.

304. Hippocrates.
Hippocrates. Volume I.
Jones WHS, transl. Loeb Classical Library. Cambridge, MA: Harvard University Press. 1923.
ISBN 0-674-99162-1, 361 pages.
Includes Hippocrates' Oath, with explanatory notes.
According to:
http://www.geocities.com/everwild7/noharm.html
"First, Do No Harm" Is Not in the Hippocratic Oath, but rather in Epidemics, Bk. 1, Sect. 11. One translation reads: 'Declare the past, diagnose the present, foretell the future; practice these acts. As to diseases, make a habit of two things: to help, or at least to do no harm.'
This note supplied by Harris G. Yfantis, MD.

305. Irvine AD.
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306. Berry paradox.
From Wikipedia, the free encyclopedia.
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307. Sutton W, Linn E.
Where the Money Was. The Memoirs of the World's Greatest Bank Robber.
New York: Ballantine Books. 1976;:.
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Part Two: Breaking Out. Sutton's Law, pp. 148-150.

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Remark about Sutton's Law on p. 27.

309. Quine WV.
Theory of Deduction.
Cambridge, MA: Harvard Cooperative Society. 1948;:65-81.
Prof. Quine, Harvard Department of Philosophy, whose career included participation in the famous Vienna Circle of Exact Philosophy, and who continued to be productive in the twenty-first century, introduced the idea of nandsets, which Quine designated as nullities, because they are lists of elements that cannot all be true.

310. Quine WV.
Methods of Logic.
New York: Henry Holt & Co. 1950;:.

311. Quine WV.
The problem of simplifying truth functions.
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312. Quine WV.
A way to simplify truth functions.
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313. McCluskey EJ jr.
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314. Smith B.
Mereotopology: A Theory of Parts and Boundaries.
Data and Knowledge Engineering. 1996;20:287-303.

315. Quine WVO.
Ontological relative, and other essays.
New York: Columbia University Press. 1969;:.

316. Zadeh IA, Kacprzyk J, Zadeh LA, eds.
Computing with Words in Information/Intelligent Systems. 1: Foundations (Studies in Fuzziness and Soft Computing).
Berlin: Springer Verlag GmbH; 1 edition (November 13, 2006)
ISBN-10: 379081217X, 517 pages.
ISBN-13: 978-3790812176, 517 pages.

317. Professor Lotfi A. Zadeh Professor of Computer Science University of California, Berkeley

Plato laid the foundation of what is now known as fuzzy logic indicating that there was a third region beyond true and false. It was Jan Łukasiewicz who first proposed a systematic alternative to the bi-valued logic of Aristotle and described the 3-valued logic, with the third value being ‘possible’. Lotfi Zadeh, in his theory of fuzzy logic, proposed making the membership function operate over the range of real numbers [0,1]. He proposed new operations for the calculus of logic and showed that fuzzy logic was a generalization of classical logic. Lotfi Zadeh with John R. Ragazzini, developed the z- transform method in discrete time signal processing and analysis, which is now standard in digital signal processing, digital control, and other discrete- time systems used In industry and research. Professor Zadeh's latest work includes computing with words and perceptions which he will discuss at his BCIG lecture entitled …

A New Frontier in Computation --- Computation with Information Described In Natural Language.

Thursday, February 8, 2007, 3:00 to 4:30 PM at the National Institutes of Health Clinical Center (Building 10) Medical Board Room (Room 2C116) Bethesda, MD.

This lecture is sponsored by the NIH Biomedical Computing Interest Group www.nih-bcig.org For additional information call 301-496-0191.

318. Zadeh LA.
From computing with numbers to computing with words. From manipulation of measurements to manipulation of perceptions.
Ann N Y Acad Sci. 2001 Apr;929:221-252. Review.
PMID: 11357866.
PubMed Entry

319. Zadeh LA.
Linguistic variables, approximate reasoning and dispositions.
Med Inform (Lond). 1983 Jul-Sep;8(3):173-186.
PMID: 6600041.
PubMed Entry

320. Zadeh LA.
A note on prototype theory and fuzzy sets.
Cognition. 1982 Nov;12(3):291-297.
PMID: 6891312.
PubMed Entry

321. Zadeh LA.
Nonlinear Multipoles.
Proc Natl Acad Sci U S A. 1953 Apr;39(4):274-280.
PMID: 16589260
PubMed Entry

322. Berman JJ.
Biomedical Informatics.
Sudbury, MA: Jones and Bartlett Publishers. 2007;:.
ISBN 978-0-7637-4135-8, 459 pages.
ISBN 0-7637-4135-3, 459 pages.
This book is a must-read for biomedical professionals, on-the-job or in-training, with a serious interest in biomedical data management. The author has broad training and experience in biomedicine and informatics. You can digest the highlights of the book in three hours. The book embraces all modern biomedical informatics. There are chapters devoted to the definition of biomedical databases; the availability of biomedical databases; privacy and confidentiality; database standard formats; programming; biomedical nomenclatures; automated translation and de-identification methods; cryptography; clinical trials; and how to apply for a grant. Each chapter begins with the existing history and background of the chapter's subject matter, followed by a complete description of the area, and copious examples. The author has a flair for biomedical informatics history, including such historical persons as Hippocrates, Brahe, Kepler, Newton, Morgagni, Jenner, Pasteur, Wunderlich, and Ehrlich. The annotated literature references and the glossary, both filled with the author's mordant comments and opinions, are worth reading on their own.

323. Kleene SC.
Mathematical Logic.
Mineola, NY: Dover Publications, Inc. 2002;:.
ISBN 0-486-4233-9, 398 pages.

324. Whitehead AN.
An Introduction to Mathematics.
Oxford: Oxford University Press. 1948;:.
ISBN 0-19-500211-3, 191 pages.
Prof. Whitehead, along with Prof. Bertrand Russell, are the coauthors of the famous Principia Mathematica, which sought to subsume all known mathematics within a comprehensive proof system. The belief that all true theorems could be proved in such a system is known as logical positivism, a belief shown to be false by Gödel's famous paper.

325. Murphy EA.
A Companion to Medical Statistics.
Baltimore: The Johns Hopkins University Press. 1985;:.
ISBN 0-8018-2612-8, 303 pages.

326. Shortliffe EH, Perreault LE, eds. Wiederhold G, Fagan LM, assoc eds.
Medical Informatics. Computer Applications in Health Care and Biomedicine. Second Edition.
New York: Springer. 2001;:.
ISBN 0-387-98472-0, 854 pages.

327. Nahikian HM.
A Modern Algebra for Biologists.
Chicago: The University of Chicago Press. 1964;:.
ISBN not stated, 236 pages.

328. Stewart I.
Concepts of Modern Mathematics.
New York: Dover Publications, Inc. 1995;:.
ISBN 0-486-28424-7, 339 pages.

329. Rescher N, Urquhart A.
Temporal Logic.
Volume 3. Library of Exact Philosophy.
New York: Springer Verlag. 1971;:.
ISBN 0-387-80995-3, 273 pages.
In: Chapter 2.
Topological Logic, pp.

330. Moore GW, Hutchins GM.
Effort and demand logic in medical decision making.
Metamedicine 1:277-304, 1980.

331. Moore GW, Hutchins GM.
A Hintikka possible worlds model for certainty levels in medical decision making.
Synthese 1981;48:87-119.

332. Rescher N, Garson J.
Topological Logic.
J Symbol Logic 1968;33:537-548.
Wittgenstein-Carnap style truth table.

333. Taleb NN.
The Black Swan: The Impact of the Highly Improbable.
New York: Random House. 2007;:.
ISBN-10: 1400063515, 400 pages.
ISBN-13: 978-1400063512, 400 pages.
Impact of highly improbable events, such as violation of Sutton's Law.

334.

335.

336.

337.

338.

339.

340.

341.

342.

343.

344.

345.

346.

347.

348.

349.

20. APPENDIX A.

Identifier:ldip:Terminology
Class Label: Terminology
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: All terminology used in composing and coding the
  Electronic_medical_record, including free-text-English (ugh!),
  syntactically-controlled English, SNOMED, UMLS, MeSH,....
subClassOf:Class
Contributor:Bill Moore
Date_of_contribution:11-13-2006
                       
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

12. APPENDIX B. ANATOMIC NAMES.

1. Body Regions.
2. Surface Anatomy.
3. Cardiovascular System.
4. Respiratory System.
5. Tubular gastrointestinal system.
6. Hepatobiliary System.
7. Genitourinary System.
8. Integumentary System.
9. Muscular System.
10. Skeletal System.
11. Endocrine System.
12. Lymphoreticular System.
13. Central Nervous System.
14. Sense Organ.
15. Peripheral Nervous System.
16. Placenta.

 1. BODY REGIONS.
                                
 Gross Anatomy: 
    1.1. Head_region.
    1.2. Facial_region.
    1.3. Neck_region.
    1.4. Thoracic_region.
    1.5. Abdominal_region.
    1.6. Dorsal_region.
    1.7. Perineal_region.
    1.8. Upper_extremity_region.
    1.9. Lower_extremity_region.
                              


 2. SURFACE ANATOMY.
                                 
 Gross Anatomy: 
 2.1. Face_ 
 2.1.1. Face_frontal_eminence 
 2.1.2. Face_glabella 
 2.1.3. Face_zygomatic_arch 
 2.1.4. Face_mental_protuberance 
 2.1.5. Face_mandibular_angle 
 2.1.6. Face_mandibular_inferior_border 
 2.1.7. Face_mastoid_process 
 2.2. Eye_ 
 2.2.1. Eye_pupil 
 2.2.2. Eye_iris 
 2.2.3. Eye_palpebral_fissure_superior 
 2.2.4. Eye_palpebral_fissure_inferior 
 2.2.5. Eye_semilunar_fold 
 2.2.6. Eye_conjunctiva 
 2.2.7. Eye_lacrimal_caruncle 
 2.2.8. Eye_medial_angle 
 2.2.9. Eye_lateral_angle 
 2.3. Ear_ 
 2.3.1. Ear_tragus 
 2.3.2. Ear_antitragus 
 2.3.3. Ear_intertragic_incisure 
 2.3.4. Ear_lobule 
 2.3.5. Ear_acoustic_meatus_external 
 2.3.6. Ear_helix 
 2.3.7. Ear_antihelix 
 2.4. Oral_cavity_ 
 2.4.1. Oral_cavity_uvula 
 2.4.2. Oral_cavity_palatopharyngeal_notch 
 2.4.3. Oral_cavity_palatine_tonsil 
 2.4.4. Oral_cavity_palatoglossal_arch 
 2.4.5. Oral_cavity_tongue_vallate_papilla 
 2.4.6. Oral_cavity_tongue_fungiform_papilla 
 2.5. Neck_anterior_ 
 2.5.1. Neck_anterior_hyoid_bone 
 2.5.2. Neck_anterior_thyroid_cartilage 
 2.5.3. Neck_anterior_cricoid_cartilage 
 2.5.4. Neck_anterior_thyroid_gland 
 2.5.5. Neck_anterior_carotid_triangle 
 2.5.6. Neck_anterior_submental_triangle 
 2.5.7. Neck_anterior_submandibular_triangle 
 2.5.8. Neck_anterior_anterior_triangle 
 2.5.9. Neck_anterior_posterior_triangle 
 2.5.10. Lymph_node_cervical_ 
 2.5.11. Lymph_node_cervical_preauricular 
 2.5.12. Lymph_node_cervical_submental 
 2.5.13. Lymph_node_cervical_anterior 
 2.5.14. Lymph_node_cervical_posterior 
 2.5.15. Lymph_node_cervical_supraclavicular 
 2.6. Chest_ 
 2.6.1. Chest_jugular_notch 
 2.6.2. Chest_clavicle 
 2.6.3. Chest_sternal_angle 
 2.6.4. Chest_sternal_manubrium 
 2.6.5. Chest_costal_margin 
 2.6.6. Chest_xiphoid_process 
 2.6.7. Lymph_node_axillary_ 
 2.6.8. Lymph_node_axillary_lateral 
 2.6.10. Lymph_node_axillary_central 
 2.6.11. Lymph_node_axillary_apical 
 2.6.12. Lymph_node_axillary_anterior 
 2.6.13. Lymph_node_axillary_posterior 
 2.7. Abdomen_anterior_ 
 2.7.1. Abdomen_anterior_hypochondriac_left 
 2.7.2. Abdomen_anterior_epigastric 
 2.7.3. Abdomen_anterior_hypochondriac_right 
 2.7.4. Abdomen_anterior_lumbar_left 
 2.7.5. Abdomen_anterior_periumbilical 
 2.7.6. Abdomen_anterior_lumbar_right 
 2.7.7. Abdomen_anterior_iliac_left 
 2.7.8. Abdomen_anterior_hypogastric 
 2.7.9. Abdomen_anterior_iliac_right 
 2.7.10. Abdomen_anterior_linea_alba 
 2.7.11. Abdomen_anterior_mcburney_line 
 2.7.12. Abdomen_anterior_arcuate_line 
 2.7.13. Abdomen_anterior_inguinal_ligament 
 2.7.14. Abdomen_anterior_superior_iliac_spine 
 2.7.15. Abdomen_anterior_pubic_tubercle 
 2.7.16. Abdomen_posterior_iliac_crest 
 2.8. Back_ 
 2.8.1. Back_external_occipital_protuberance 
 2.8.2. Back_mastoid_process 
 2.8.3. Back_scapular_acromion 
 2.8.4. Back_scapular_spine 
 2.8.5. Back_spina_prominens 
 2.8.6. Back_vertebral_spinous_process 
 2.8.7. Back_sacral_dorsum 
 2.8.8. Back_iliac_crest 
                                                                 
                              


 3. CARDIOVASCULAR SYSTEM.
                                   
 Gross Anatomy: 
    3.1. Heart. 
       3.1.1. Atrium 
          3.1.1.1. Right_atrium 
          3.1.1.2. Left_atrium 
       3.1.2. Ventricle 
          3.1.2.1. Right_ventricle 
          3.1.2.2. Left_ventricle 
       3.1.3. Atrioventricular_valve 
          3.1.3.1. Tricuspid_valve 
          3.1.3.2. Mitral_valve 
       3.1.4. Semilunar_valve 
          3.1.4.1. Pulmonic_valve 
          3.1.4.2. Aortic_valve 
             3.1.4.2.1. Aortic_valve_right_coronary_cusp 
             3.1.4.2.2. Aortic_valve_left_coronary_cusp 
             3.1.4.2.3. Aortic_valve_non_coronary_cusp 
       3.2. Great_vessel 
          3.2.1. Great_artery 
             3.2.1.1. Aorta 
             3.2.1.2. Pulmonary_artery 
          3.2.2. Great_vein 
             3.2.2.1. Inferior_vena_cava 
             3.2.2.2. Superior_vena_cava 
             3.2.2.3. Pulmonary_vein 
                3.2.2.3.1. Pulmonary_vein_left_superior 
                3.2.2.3.2. Pulmonary_vein_left_inferior 
                3.2.2.3.3. Pulmonary_vein_right_superior 
                3.2.2.3.4. Pulmonary_vein_right_inferior 
       3.3. Artery 
       3.4. Capillary 
       3.5. Vein 
Microanatomy: 
1. Heart. 
1.1. Lumen 
1.2. Endocardium 
1.3. Cardiac_myocyte 
1.4. Epicardium 
1.5. Pericardium 
3. Artery 
3.1. Tunica_intima:Endothelium 
3.2. Internal_elastic_lamina 
3.3. Tunica_media 
3.4. External_elastic_lamina 
3.5. Tunica_Adventitia 
                                 
                                                               


 4. RESPIRATORY SYSTEM.
                                 
Gross Anatomy: 
    4.1. Nose
       Nasal_cavity
       Paranasal_sinus
          Ethmoid_sinus
          Maxillary_sinus
          Frontal_sinus
    4.2. Larynx
       4.2.1. Laryngeal_cartilage
       4.2.2. Thyroid_cartilage
       4.2.3. Cricoid_cartilage
       4.2.4. Arytenoid_cartilage
       4.2.5. Corniculate_cartilage
       4.2.6. Cuneiform_cartilage
       4.2.7. Epiglottis
       4.2.8. Muscle_laryngeal
       4.2.9. Cavity_laryngeal
    4.3. Trachea. 
    4.4. Mainstem_bronchus 
    4.5. Main_bronchus 
       4.5.1. Left_main_bronchus 
       4.5.2 Right_main_bronchus 
    4.6. Lung 
       4.6.1. Right_lung 
          4.6.1.1. Right_lung_upper_lobe 
          4.6.1.2. Right_lung_middle_lobe 
          4.6.1.3. Right_lung_lower_lobe 
       4.6.2. Left_lung 
          4.6.2.1. Left_lung_upper_lobe 
          4.6.2.2. Left_lung_lingula 
          4.6.2.3. Left_lung_lower_lobe 



                                 
Microanatomy: 
    4.3. Trachea. 
       4.3.1. Lumen 
       4.3.2. Mucosa_respiratory
       4.3.3. Submucosa_respiratory
       4.3.4. Adventitia_respiratory
                                                


  5. TUBULAR GASTROINTESTINAL SYSTEM.
                                 
Gross Anatomy: 
 1. Oral Cavity. 
1.1. Lip. 
1.2. Buccal_region 
1.3. Hard_Palate 
1.4. Tongue 
1.5. Soft_palate 
1.6. Oropharynx 
2. Esophagus 
2.1. Esophagus_upper_third 
2.2. Esophagus_middle_third 
2.3. Esophagus_lower_third 
2.4. Gastroesophageal_junction 
3. Stomach 
3.1. Gastroesophageal_junction 
3.2. Gastric_cardia 
3.3. Gastric_fundus 
3.4. Gastric_body 
3.5. Gastric_antrum 
3.6. Gastric_pylorus 
4. Duodenum 
4.1. Duodenum_first_part 
4.2. Duodenum_second_part 
4.3. Duodenum_ampulla_Vater 
4.4. Duodenum_third_part 
5. Jejunum 
6. Ileum 
7. Cecum 
8. Appendix 
9. Ascending_colon
10. Hepatic_flexure
11. Transverse_colon
12. Splenic_flexure
13. Descending_colon
14. Sigmoid_colon
15. Rectum
16. Anus
                                 
Microanatomy: 
3. Stomach 
3.1. Gastric_lumen 
3.2. Gastric_mucosa 
3.2.1. Gastric_mucosa_epithelium 
3.2.2. Gastric_mucosa_pit 
3.2.2. Gastric_mucosa_neck 
3.2.2. Gastric_mucosa_chief_cell 
3.2.2. Gastric_mucosa_parietal_cell 
3.2.2. Gastric_mucosa_antral_cell 
3.3. Gastric_lamina_propria 
3.4. Gastric_lymphoid_nodule 
3.5. Gastric_muscularis_mucosae 
3.4. Gastric_lamina_propria 
4. Duodenum 
4.1. Lumen 
4.1. Mucosa 
4.1.1. Mucosa_villus 
4.1.2. Mucosa_crypt 
4.2. Lamina_propria 
4.3. Muscularis_mucosae 
4.4. Submucosa 
4.4.1. Brunner_gland 
4.6. Muscularis_propria_circular 
4.7. Muscularis_propria_longitudinal 
4.8. Subserosa 
4.9. Serosa 
5. Jejunum 
5.1. Lumen 
5.1. Mucosa 
5.1.1. Mucosa_villus 
5.1.2. Mucosa_crypt 
5.2. Lamina_propria 
5.3. Muscularis_mucosae 
5.4. Submucosa 
5.7. Muscularis_propria_longitudinal 
5.6. Muscularis_propria_circular 
5.8. Subserosa 
5.9. Serosa 
                                                          


 6. HEPATOBILIARY SYSTEM.
                                 
Gross Anatomy: 
    6.1. Liver. 
    6.2. Biliary_tree. 
    6.3. Bile_duct. 
       6.3.1. Right_hepatic_duct. 
       6.3.2. Left_hepatic_duct. 
       6.3.3. Common_bile_duct. 
    6.4. Gallbladder. 
    6.5. Pancreas_exocrine. 
    6.6. Pancreatic_duct. 
    6.7. Ampulla_Vater. 
                                 
Microanatomy: 
    6.1. Liver. 
       6.1.1. Liver_parenchyma 
       6.1.2. Liver_sinusoid
       6.1.3. Liver_hepatic_artery 
       6.1.4. Liver_bile_ductule
       6.1.5. Liver_portal_vein 
    6.2. Biliary_tree. 
       6.2.1. Lumen
       6.2.2. Mucosa
       6.2.3. Muscularis_mucosae
       6.2.4. Submucosa
       6.2.5. Muscularis_propria
          6.2.5.1. Muscularis_propria_longitudinal
          6.2.5.2. Muscularis_propria_circular
       6.2.6. Subserosa
       6.2.7. Serosa
    6.4. Gallbladder. 
       6.4.1. Lumen
       6.4.2. Mucosa
       6.4.3. Muscularis_mucosae
       6.4.4. Submucosa
       6.4.5. Muscularis_propria
          6.4.5.1. Muscularis_propria_longitudinal
          6.4.5.2. Muscularis_propria_circular
       6.4.6. Subserosa
       6.4.7. Serosa
    6.5. Pancreas_exocrine. 
    6.6. Pancreatic_duct. 
    6.7. Ampulla_Vater. 
                                           


  7. GENITOURINARY SYSTEM.
                                 
Gross Anatomy: 
                                 
1. Kidney 
1.1. Renal_cortex 
1.1.1. Renal_cortex_upper_pole 
1.1.2. Renal_cortex_middle 
1.1.3. Renal_cortex_lower_pole 
1.2. Renal_medulla 
1.3. Renal_pelvis 
2. Ureter 
2.1. Ureter_right 
2.2. Ureter_left 
3. Urinary_bladder. 
3.1. Urinary_bladder_dome. 
3.2. Urinary_bladder_trigone. 
3.3. Urinary_bladder_left_ureteral_orifice. 
3.4. Urinary_bladder_right_ureteral_orifice. 
3.5. Urinary_bladder_urethral_orifice. 
4. Urethra 
4.1. Distal_urethra:female 
4.2. Prostatic_urethra:male 
4.3. Penile_urethra:male 
5. Internal_genitalia_female 
5.1. Ovary:female 
5.2. Epoophoron:female 
5.3. Uterian_tube:female 
5.3.1. Uterian_tube_fimbria:female 
5.3.2. Uterian_tube_main:female 
5.4. Ovarian_ligament:female 
5.5. Uterus:female 
5.5.1. Uterus_cornu_uteri:female 
5.5.2. Uterus_corpus_uteri:female 
5.5.3. Uterus_cervix_uteri:female 
5.5.4. Uterus_gravida:female 
5.6. Vagina:female 
5.6.1. Vagina_upper_third:female 
5.6.2. Vagina_lower_two_thirds:female 
6. External_genitalia_female:female. 
6.1. Vulva:female. 
6.1.1. Vulva_clitoris:female. 
6.1.2. Vulva_labium_majus:female. 
6.1.3. Vulva_labium_minus:female. 
7. Internal_genitalia_male:male. 
7.1. Testis:male. 
7.3. Epididymis:male. 
7.4. Spermatic_cord:male. 
7.5. Scrotum:male. 
7.6. Seminal_vesicle:male. 
7.7. Vas_deferens:male. 
7.8. Prostate:male. 
8. External_genitalia_male. 
8.1. Penis:male. 
8.1.1. Penile_urethra:male. 
8.1.2. Corpus_cavernosum_left:male 
8.1.3. Corpus_cavernosum_right:male 
8.1.3. Corpus_cavernosum_urethrae:male 
Microanatomy: 
3. Urinary_bladder 
3.1. Urinary_bladder_lumen 
3.2. Urinary_bladder_urothelium 
3.3. Urinary_bladder_lamina_propria 
3.4. Urinary_bladder_inner_longitudinal_smooth_muscle 
3.5. Urinary_bladder_middle_circular_smooth_muscle 
3.6. Urinary_bladder_outer_longitudinal_smooth_muscle 
3.7. Urinary_bladder_tunica_adventitia 
5.1. Ovary:female 
5.1.1. Ovary_liquor_folliculi 
5.1.2. Ovary_membrana_granulosa 
5.1.3. Ovary_theca_interna 
5.1.4. Ovary_theca_externa 
5.1.5. Ovary_cumulus_oophorus 
5.1.6. Ovary_tunica_albuginea 
5.3. Uterian_tube:female 
5.3.1. Uterian_tube_lumen 
5.3.2. Uterian_tube_mucosa 
5.3.3. Uterian_tube_submucosa 
5.3.4. Uterian_tube_circular_muscularis 
5.3.5. Uterian_tube_longitudinal_muscularis 
5.3.6. Uterian_tube_subserosa 
5.3.7. Uterian_tube_serosa 
5.5. Uterus:female 
5.5.0.1. Uterus_lumen 
5.5.0.2. Uterus_endometrium 
5.5.0.2.1. Uterus_endometrium_compacta 
5.5.0.2.2. Uterus_endometrium_spongiosa 
5.5.0.2.3. Uterus_endometrium_basalis 
5.5.0.3. Uterus_myometrium 
5.5.0.4. Uterus_subserosa 
5.5.0.5. Uterus_serosa 
7.8. Prostate:male. 
7.8.1. Prostate_acinar_lumen 
7.8.2. Prostate_acinar_epithelium 
7.8.3. Prostate_ductule 
                              


   8. INTEGUMENTARY SYSTEM.
                                 
Gross Anatomy: 
      8.1. Skin_Scalp 
      8.2. Skin_Face 
      8.3. Skin_Neck 
      8.4. Skin_Upper_extremity 
      8.5. Skin_Chest 
      8.6. Skin_Abdomen 
      8.7. Skin_Back 
      8.8. Skin_Perineum 
      8.9. Skin_Lower_extremity 
                                 
Microanatomy: 
      8.1. Epidermis 
         8.1.1. Epidermis_stratum_corneum 
         8.1.2. Epidermis_stratum_granulosum 
         8.1.3. Epidermis_stratum_spinosum 
         8.1.4. Epidermis_stratum_basale 
         8.1.5. Epidermis_stratum_ 
      8.2. Dermis 
         8.2.1. Dermis_papillary_dermis 
         8.2.2. Dermis_reticular_dermis 
         8.2.3. Dermis_deep_dermis 
      8.3. Appendage_Hair
         8.3.1. Hair_shaft
         8.3.2. Hair_ostium
         8.3.3. Hair_infundibulum
         8.3.4. Hair_arrector_pili
         8.3.5. Hair_outer_root_sheath
         8.3.6. Hair_inner_root_sheath
         8.3.7. Hair_papilla
         8.3.8. Hair_matrix_cell
      8.4. Eccrine_sweat 
      8.5. Apocrine_sweat 
      8.6. Nail 
                                                       


  9. MUSCULAR SYSTEM.
     Head_muscles.
     Neck_muscles.
     Back_muscles.
     Thoracic_muscles.
     Abdominal_muscles.
     Upper_extremity_muscles.
     Lower_extremity_muscles.
                                                       


  10. SKELETAL SYSTEM.
     Axial_skeleton
        Cranial_bones
        Vertebral_column
        Thoracic_bones
     Appendicular_skeleton
        Upper_extremity_bones
        Lower_extremity_bones
     Joints
        Fibrous_joints
        Cartilaginous_joints
        Synovial_joints
                                                       
                       


  11. ENDOCRINE SYSTEM. 
     Pituitary
        Pituitary_anterior=adenohypophysis
        Pituitary_posterior=neurohypophysis
     Pineal_body
     Thyroid
     Parathyroid
     Pancreas_endocrine
     Adrenal_gland
        Adrenal_gland_right
        Adrenal_gland_left
                                                       


  12. LYMPHORETICULAR SYSTEM. 
     Spleen
     Thymus
     Lymph_nodes_regional
     Lymphatic_duct
     Lymphatic_trunk
     Lymphatic_vessel
                                                       


  13. CENTRAL NERVOUS SYSTEM.
    Meninges
       Dura_mater_brain
       Dura_mater_spinal_cord
       Arachnoid_mater_brain
       Arachnoid_mater_spinal_cord
       Pia_mater_brain
       Pia_mater_spinal_cord
    Medulla_spinal
       Gray_columns
       Substantia_alba
    Encephalon
       Medulla_oblongata=myelencephalon
    Pons=metencephalon
    Fourth_ventricle
    Cerebellum
    Mesencephalon
       Cerebral_peduncle
       Tectum_mesencephalic
       Aqueduct_mesencephalic
       Base_cerebral_peduncle
       Substantia_nigra
       Tegmentum_mesencephalic
       Tectum_mesencephalic
    Prosencephalon
    Diencephalon
       Epithalamus
       Dorsal_thalamus
       Metathalamus
       Ventral_thalamus
       Hypothalamus
       Third_ventricle
    Telencephalon
    Cerebrum
       Frontal_lobe
       Parietal_lobe
       Temporal_lobe
       Occipital_lobe
       Corpus_callosum
       Lateral_Ventricle
                                                       


  14. SENSE ORGANS.
     Eye
     Ear
        Inner_ear
        Middle_ear
        External_ear
     Olfactory_organ
     Taste_organ
        Taste_organ_calyx
        Taste_organ_pore
                                                       


 15. PERIPHERAL NERVOUS SYSTEM. 
    Cranial_nerve
       Olfactory_nerve
       Optic_nerve
       Oculomotor_nerve
       Trochlear_nerve
       Trigeminal_nerve
       Abducens_nerve
       Facial_nerve
       Auditory_nerve
       Glossopharyngeal_nerve
       Vagus_nerve
       Spinal_accessory_nerve
       Hypoglossal_nerve
    Spinal_nerve
       Cervical_nerve
       Brachial_plexus
       Thoracic_nerve
       Lumbar_nerve
       Sacral_nerve
       Coccygeal_nerve
       Lumbosacral_plexus
                                                       


 16. PLACENTA. 
    Amnion
    Umbilical_cord
       Mucoid_connective tissue
       Umbilical_artery
       Umbilical_vein
       Vitelline_duct
       Allantoic_duct
       Umbilical_celom
    Chorio_allantois
       Chorion_laeve
       Chorion_frondosum
       Chorionic_lamina
       Intervillous_space
    Fetal_part
       Intimal_zone
       Chorioallantoic villus
       Trophoblastic_trabecula
       Intervillous_space
       Subchorionic_sapce
       Marginal_sinus
       Lobe=cotyledon
       Trophoblastic_tubule
       Trophospongium
       Absorbent_areas
    Maternal_part
       Endometrium
          Decidual_cell
          Decidua
             Decidua_basalis
                Endometrial_gland
                Endometrial_cupula
                Endometrial_crypt
                Endometrial_caruncle
             Decidua_capsularis
             Decidua_parietalis
          Fibrinoid_substance

22. APPENDIX C. SKIN ALTERATIONS.

Clinical Features:

 Macule: Flat discoloration <1 cm. 
 Patch: Flat discoloration >1 cm. 
 Papule: Solid elevation <1 cm. 
 Plaque: Flat, solid elevation >1 cm. 
 Nodule: Round, solid elevated lesion >1 cm. 
 Vesicle: Small fluid-filled blister <1 cm. 
 Bulla: Fluid-filled blister >1 cm.

Pathologic Features:

 Acanthosis: Thickening of the epidermal layer 
 Acantholysis: Loss of cell-cell adhesion in epidermis. 
 Ballooning: Intracellular edema. 
 Crust: Serum within scale. 
 Dyskeratosis: Premature keratinization In spinous layer. 
 Exocytosis: . 
 Hypergranulosis: Thick granular cell layer 
 Hyperkeratosis: Increased thickness of cornified layer. 
 Hypogranulosis: Thin granular cell layer 
 Lichenification:  Thickening of skin from chronic rubbing.  
 Parakeratosis: Hyperkeratosis with residual pyknotic nuclei. 
 Papillomatosis: Elongation of dermal papillae. 
 Scale: Visibly cornified cells. 
 Spongiosis: Intercellular edema. 
 Telangiectasia: Dilated vessels. 
 Vacuolar Change: Small separations above/below basement membrane.

23. APPENDIX D. DERMATOPATHOLOGIC CLUES.

+∀ +skin_specimen +dermatopathologic_clue +process_feature +photosensitivity_process +rubbing_process +drug_reaction_process +elastic_tissue_process +deficiency_process +fungal_infection_process +folliculitis_process -process_feature +tissue_feature +superficial_inflammation +deep_inflammation +busy_dermis +stratum_corneum_absent +filled_papillary_dermis +papillary_microabscess +basement_membrane_thickening +middermal_infiltrate +middermal_mucin +epidermotropism +exocytosis +epidermal_follicular_vacuum_cleaner +parakeratosis +parakeratosis_follicular_lipping +parakeratosis_spire +interstitial_eosinophil +bottom_heavy_infiltrate +bare_underbelly_sign +intraluminal_giant_cell +intravascular_leukocyte +apoptotic_keratinocyte +collagen_bundle_vertical +fibrillary_collagen_loose_pink +erythrocyte_extravasation +epidermal_pallor +clear_cell_tumor -tissue_feature +disease_feature +herpes_folliculitis +grover_disease +cicatricial_pemphigoid +mycosis_fungoides +pautrier_microabscess +epidermotropism +atypia_cytologic +alopecia_areata +androgenic_alopecia +bullous_late +granuloma_annulare +necrobiosis_lipoidica +trichoepithelioma +kaposi_sarcoma +bacillary_angiomatosis -disease_feature -dermatopathologic_clue -skin_specimen

24. APPENDIX E. LARGE SPECIMEN CHECKLISTS.

SKIN SPECIMEN


 ...
 +skin_specimen
    +location
       +surface anatomy
    +procedure
       +biopsy
       +excision_simple
       +excision_radical
    +size
       +long_axis
       +short_axis
       +depth
    +shape
       +circular
       +ellipse
       +irregular
    +epidermal_surface
       +scar
       +color
       +ulcer
       +size
    +cut_surface

LARGE SPECIMEN CHECKLISTS:

 +∀
    +∀
       +length.
          (cm intervals).
          (nm intervals).
       -length.
    +∀
       +mass
          (mg intervals).
          (g intervals).
       -mass
    +∀
       +time.
       -time.
    +∀
       +large_specimen_checklist.
          +procedure.
             +skin_procedure.
                +biopsy.
                +excision.
             +colon_procedure.
                +right_hemicolectomy.
                +transverse_colectomy.
                +left_hemicolectomy.
                +abdominoperineal_resection.
                +rectosigmoid_resection.
                +total_abdominal_colectomy.
                +sigmoidectomy.
             +lung_procedure.
             +kidney_procedure.
             +prostate_procedure.
          +size.
             +dimension.
                +dimension_greatest.
                +dimension_middle.
                +dimension_least.
             +tumor_quantitation.
          +tumor_configuration
             +plaque_like.
             +annular.
             +ulcerating.
             +infiltrative.
          +site
             +skin.
             +colon.
                +cecum.
                +ascending_colon.
                +hepatic_flexure.
                +transverse_colon.
                +splenic_flexure.
                +descending_colon.
                +sigmoid_colon.
                +rectosigmoid.
                +rectum.
             +lung.
             +kidney.
             +prostate.
          +laterality.
             +left.
             +right.
             +bilateral.
             +midline.
             +laterality_not_specified.
          +focality
             +unifocal
             +multifocal
          +histologic_type.
             +adenocarcinoma.
             +mucinous_adenocarcinoma.
             +signet_ring_adenocarcinoma.
             +small_cell_carcinoma.
             +squamous_cell_carcinoma.
             +medullary_carcinoma.
             +sarcomatoid_carcinoma.
             +papillary_carcinoma.
             +sarcomatoid_carcinoma.
             +clear_cell_renal_cell_carcinoma.
             +chromophil_renal_cell_carcinoma.
             +chromophobe_renal_cell_carcinoma.
             +urothelial_cell_carcinoma.
             +undifferentiated_carcinoma.
          +histologic_grade.
             +low_grade.
             +high_grade.
             +gleason_score.
             +fuhrman_nuclear_grade.
          +surgical_margin.
             +proximal_surgical_margin.
                +involved_by_tumor.
                -involved_by_tumor.
             +distal_surgical_margin.
                +involved_by_tumor.
                -involved_by_tumor.
             +circumferential_surgical_margin.
                +involved_by_tumor.
                -involved_by_tumor.
          +invasion.
             +lymphatic_invasion.
                -present.
                +present.
             +venous_invasion.
                -present.
                +present.
             +perineural_invasion.
                -present.
                +present.
             +seminal_vesicle_invasion.
                -present.
                +present.
          +pathologic_staging.
             +pathologic_staging_tumor.
                +pT0.
                +pTIS.
                +pT1.
                +pT2.
                +pT3.
                +pT4.
             +pathologic_staging_lymph_node.
                +pN0.
                +pN1.
                +pN2.
             +pathologic_staging_metastasis.
                +pM0.
                +pM1.
                +pMX.
    -large_specimen_checklist.

25. APPENDIX F. SPECIMEN ACCESSIONING.

26. APPENDIX G. QUALITY ASSURANCE. FOLLOWUP.

27. APPENDIX H.
THEOREMS, FROM
ZEMAN'S MODAL LOGIC.
CHAPTERS 1,2.

NOTES. Zeman's logic expressions are written in parenthesis-free, so-called Polish notation (Łukasiewicz,.......). where operators:

LOGIC OPERATORS Zeman Our notation notation Description C ⇒ IMPLIES K & AND A | INCLUSIVE_OR, ALTERNATION E ⇔ EQUALS K - NOT

In traditional symbolic logic notation, the operators are INFIXed between the subject and object, i.e., subject OPERATOR object. For example, p IMPLIES q, denoted p ⇒ q; or p AND q, denoted p & q.

In Polish notation, the operators PREFIX the subject and object, i.e., OPERATOR subject object. For example, IMPLIES p q, denoted ⇒pq; or AND p q, denoted & p q.

Thus, expression 1.2 is represented as

p ⇒ (q ⇒ p).

in traditional notation, and

CpCqp

in Polish notation.

Proofs consist of translating Polish notation into in traditional notation (for easier readability); then translating the traditional notation into NANDSETs. If all nandsets are vacuous, then the theorem is proved.

Proof tricks:

--p ⇔ +p (Double Negative).
(+p ⇒ +q) ⇔ nandset {+p,-q} (Nandset).
(+p ⇒ +q) ⇔ (-p | q) (Whitehead_Russell Transformation).
(+p = +q) ⇔ (+p ⇒ +q) and (+q ⇒ +p) (Equivalence).
(+p | +q) ⇔ -(-p & -q) (DeMorgan's Rule).
(+p & +q) ⇔ -(-p | -q) (DeMorgan's Rule).
((+p & +q) | +r) ⇔ ((+p & +r) | (+p & +r)) (Distributive Rule).
((+p | +q) & +r) ⇔ ((+p | +r) & (+p & +r)). (Distributive Rule).

EXPANSION THEOREM.

+p +p +q ⇔ +q +q ...

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

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

BACKTRACK THEOREM.

+p +p +q ⇔ +s ... +r +s +q ...

Proof: It suffices to show:
(1) that ((+p ⇒ +q) & (+p ⇒ +s) & ((+p & +s) ⇒ +r)) ⇒ (+p ⇒ +s)
(2) that ((+p ⇒ +q) & (+p ⇒ +s) & ((+p & +s) ⇒ +r)) ⇒ ((+p & +s) ⇒ +r)
and (3) that ((+p ⇒ +q) & (+p ⇒ +s) & ((+p & +s) ⇒ +r)) ⇒ ((+p & +s & +r) ⇒ +q)

Proof (1): Nandsets


 {-p,-p,-p,+p,-s},
 {-p,-p,-s,+p,-s},
 {-p,-p,+r,+p,-s},
 {-p,+s,-p,+p,-s},
 {-p,+s,-s,+p,-s},
 {-p,+s,+r,+p,-s},
 {+q,-p,-p,+p,-s},
 {+q,-p,-s,+p,-s},
 {+q,-p,+r,+p,-s},
 {+q,+s,-p,+p,-s},
 {+q,+s,-s,+p,-s}, and
 {+q,+s,+r,+p,-s}

are vacuous.

Proof (2): Nandsets


 {-p,-p,-p,+p,+s,-r},
 {-p,-p,-s,+p,+s,-r},
 {-p,-p,+r,+p,+s,-r},
 {-p,+s,-p,+p,+s,-r},
 {-p,+s,-s,+p,+s,-r},
 {-p,+s,+r,+p,+s,-r},
 {+q,-p,-p,+p,+s,-r},
 {+q,-p,-s,+p,+s,-r},
 {+q,-p,+r,+p,+s,-r},
 {+q,+s,-p,+p,+s,-r},
 {+q,+s,-s,+p,+s,-r}, and
 {+q,+s,+r,+p,+s,-r}

are vacuous.

Proof (3): Nandsets


 {-p,-p,-p,+p,+s,+r,-q},
 {-p,-p,-s,+p,+s,+r,-q},
 {-p,-p,+r,+p,+s,+r,-q},
 {-p,+s,-p,+p,+s,+r,-q},
 {-p,+s,-s,+p,+s,+r,-q},
 {-p,+s,+r,+p,+s,+r,-q},
 {+q,-p,-p,+p,+s,+r,-q},
 {+q,-p,-s,+p,+s,+r,-q},
 {+q,-p,+r,+p,+s,+r,-q},
 {+q,+s,-p,+p,+s,+r,-q},
 {+q,+s,-s,+p,+s,+r,-q}, and
 {+q,+s,+r,+p,+s,+r,-q}

are vacuous.

INTERCALATION THEOREM.

+p +p +q ⇔ +s +r +r ... ... +q +s ...

Proof: It suffices to show that: (((+p & +q) ⇒ +r) & ((+p & +q) ⇒ +s)) ⇒ ((+p & +q) ⇒ +r)
Nandsets


 {-p,-p,+p,+q,-r},
 {-p,-q,+p,+q,-r},
 {-p,+s,+p,+q,-r},
 {-q,-p,+p,+q,-r},
 {-q,-q,+p,+q,-r},
 {-q,+s,+p,+q,-r},
 {+r,-p,+p,+q,-r},
 {+r,-q,+p,+q,-r}, and
 {+r,+s,+p,+q,-r}

are vacuous.

RETIREMENT THEOREM.

+p +p +q ⇔ +s +r +r +q +s +q -s

Proof. IF. It suffices to show that: (((+p & +q) ⇒ +r) & ((+p & +q) ⇒ +s) & ((+p & +q) ⇒ -s)) ⇒ ((+p & +q) ⇒ +r).
Nandsets


 {-p,-p,-p,+p,+q,-r},
 {-p,-p,-q,+p,+q,-r},
 {-p,-p,-s,+p,+q,-r},
 {-p,-q,-p,+p,+q,-r},
 {-p,-q,-q,+p,+q,-r},
 {-p,-q,-s,+p,+q,-r},
 {-p,+s,-p,+p,+q,-r},
 {-p,+s,-q,+p,+q,-r},
 {-p,+s,-s,+p,+q,-r},
 {-q,-p,-p,+p,+q,-r},
 {-q,-p,-q,+p,+q,-r},
 {-q,-p,-s,+p,+q,-r},
 {-q,-q,-p,+p,+q,-r},
 {-q,-q,-q,+p,+q,-r},
 {-q,-q,-s,+p,+q,-r},
 {-q,+s,-p,+p,+q,-r},
 {-q,+s,-q,+p,+q,-r},
 {-q,+s,-s,+p,+q,-r},
 {+r,-p,-p,+p,+q,-r},
 {+r,-p,-q,+p,+q,-r},
 {+r,-p,-s,+p,+q,-r},
 {+r,-q,-p,+p,+q,-r},
 {+r,-q,-q,+p,+q,-r},
 {+r,-q,-s,+p,+q,-r},
 {+r,+s,-p,+p,+q,-r},
 {+r,+s,-q,+p,+q,-r}, and
 {+r,+s,-s,+p,+q,-r}

are vacuous.

Proof. ONLY IF. It suffices to show that: ((+p & +q) ⇒ +r) ⇒ (((+p & +q) ⇒ +r) & ((+p & +q) ⇒ +s) & ((+p & +q) ⇒ -s)).
Nandsets


 {-p,-q,+r,+p,+p,+p},
 {-p,-q,+r,+p,+p,+q},
 {-p,-q,+r,+p,+p,+s},
 {-p,-q,+r,+p,+q,+p},
 {-p,-q,+r,+p,+q,+q},
 {-p,-q,+r,+p,+q,+s},
 {-p,-q,+r,+p,-p,+p},
 {-p,-q,+r,+p,-p,+q},
 {-p,-q,+r,+p,-p,+s},
 {-p,-q,+r,+q,+p,+p},
 {-p,-q,+r,+q,+p,+q},
 {-p,-q,+r,+q,+p,+s},
 {-p,-q,+r,+q,+q,+p},
 {-p,-q,+r,+q,+q,+q},
 {-p,-q,+r,+q,+q,+s},
 {-p,-q,+r,+q,-p,+p},
 {-p,-q,+r,+q,-p,+q},
 {-p,-q,+r,+q,-p,+s},
 {-p,-q,+r,-r,+p,+p},
 {-p,-q,+r,-r,+p,+q},
 {-p,-q,+r,-r,+p,+s},
 {-p,-q,+r,-r,+q,+p},
 {-p,-q,+r,-r,+q,+q},
 {-p,-q,+r,-r,+q,+s},
 {-p,-q,+r,-r,-p,+p},
 {-p,-q,+r,-r,-p,+q}, and
 {-p,-q,+r,-r,-p,+s}

are vacuous.

STRICT ALTERNATIVE THEOREM. In a simplified model of cytogenetics, let there be exactly two genders, female and male (i.e., no gender variants); and consider the presence_or_absence of the y_chromosome. Then:

+cytogenetics +male +y_chromosome -male -y_chromosome

Alternatively:

+cytogenetics +y_chromosome +male -y_chromosome -male

According to the Strict Alternative Theorem, these two expressions are equal. In a more general expression:

+p +p +q +r +r ⇔ +q -q -r -r -q

Proof. IF. It suffices to show that
(1A): (((+p & +q) ⇒ +r) & ((+p & -q) ⇒ -r)) ⇒ ((+p & +r) ⇒ +q)); and that
(1B): (((+p & +q) ⇒ +r) & ((+p & -q) ⇒ -r)) ⇒ ((+p & -r) ⇒ -q)).

Proof. 1A. Nandsets


 {-p,-p,+p,+r,-q},
 {-p,+q,+p,+r,-q},
 {-p,-r,+p,+r,-q},
 {-q,-p,+p,+r,-q},
 {-q,+q,+p,+r,-q},
 {-q,-r,+p,+r,-q},
 {+r,-p,+p,+r,-q},
 {+r,+q,+p,+r,-q}, and
 {+r,-r,+p,+r,-q}

are vacuous.

Proof. 1B. Nandsets


 {-p,-p,+p,-r,+q},
 {-p,+q,+p,-r,+q},
 {-p,-r,+p,-r,+q},
 {-q,-p,+p,-r,+q},
 {-q,+q,+p,-r,+q},
 {-q,-r,+p,-r,+q},
 {+r,-p,+p,-r,+q},
 {+r,+q,+p,-r,+q}, and
 {+r,-r,+p,-r,+q}

are vacuous.

Proof. ONLY IF. It suffices to show that (2A): (((+p & +r) ⇒ +q) & ((+p & -r) ⇒ -q)) ⇒ ((+p & +q) ⇒ +r)); and that (2B): (((+p & +r) ⇒ +q) & ((+p & -r) ⇒ -q)) ⇒ ((+p & -q) ⇒ -r))

Proof. 2A. Nandsets


 {-p,-p,+p,+q,-r},
 {-p,+r,+p,+q,-r},
 {-p,-q,+p,+q,-r},
 {-q,-p,+p,+q,-r},
 {-q,+r,+p,+q,-r},
 {-q,-q,+p,+q,-r},
 {+r,-p,+p,+q,-r},
 {+r,+r,+p,+q,-r}, and
 {+r,-q,+p,+q,-r}

are vacuous.

Proof. 2B. Nandsets


 {-p,-p,+p,-q,+r},
 {-p,+r,+p,-q,+r},
 {-p,-q,+p,-q,+r},
 {-q,-p,+p,-q,+r},
 {-q,+r,+p,-q,+r},
 {-q,-r,+p,-q,+r},
 {+r,-p,+p,-q,+r},
 {+r,+r,+p,-q,+r}, and
 {+r,-q,+p,-q,+r}

are vacuous.

Here are Zeman's theorems and nandset proofs:

1.1. CCpCqrCCpqCpr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r)). Nandsets {-p,-p,+p,-r}, {-p,+q,+p,-r}, {-q,-p,+p,-r}, {-q,+q,+p,-r}, {+r,-p,+p,-r}, and {+r,+q,+p,-r}, are vacuous.
Live Proof. Click on the SUBMIT button (below).
1.2. CpCqp. Proof: +p ⇒ (+q ⇒ +p). Nandset {+p,+q,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
1.3. CCpqCpp. Proof: (+p ⇒ +q) ⇒ (+p ⇒ +p). Nandset {+p,-q,+p,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
1.4. Cpp. Proof: (+p ⇒ +p). Nandset {+p,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
1.5. CCpqCCqrCpr. Proof: (+p ⇒ +q) ⇒ ((+q ⇒ +r) ⇒ (+p ⇒ +r)). Nandsets {-p,-q,+p,-r}, {-p,+r,+p,-r}, {+q,-q,+p,-r}, and {+q,+r,+p,-r} are vacuous.
Live Proof. Click on the SUBMIT button (below).
1.6. CCCCqrCprsCCpqs. Proof: ((+q ⇒ +r) ⇒ ((+p ⇒ +r) ⇒ +s)) ⇒ ((+p ⇒ +q) ⇒ +s). Nandsets are vacuous. ...............................
1.7. CCpCqrCCsqCpCsr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+s ⇒ +q) ⇒ (+p ⇒ (+s ⇒ +r))). Nandsets are vacuous. ...............................

C1. CCpCqrCCpqCpr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r)). Nandsets {-p,-p,+p,-r}, {-p,+q,+p,-r}, {-q,-p,+p,-r}, {-q,+q,+p,-r}, {+r,-p,+p,-r}, and {+r,+q,+p,-r} are vacuous.
Live Proof. Click on the SUBMIT button (below).
C2. CpCqp. Proof: +p ⇒ (+q ⇒ +p). Nandset {+p,+q,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
2.1. CCpqCpp. Proof: (+p ⇒ +q) ⇒ (+p ⇒ +p). Nandset {+p,-q,+p,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
2.2. Cpp. Proof: (+p ⇒ +p). Nandset {+p,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
2.3. CpCqr, Cpq, p ├ r. Proof: ((+p ⇒ (+q ⇒ +r)) & (+p ⇒ +q)) ⇒ (+p ⇒ +r). Nandsets {-p,-p,+p,-r}, {-p,+q,+p,-r}, {-q,-p,+p,-r}, {-q,+q,+p,-r}, {+r,-p,+p,-r}, and {+r,+q,+p,-r} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.4. p, q ├ p. Proof: +p ⇒ (+q ⇒ +p). Nandset {+p,+q,-p} is vacuous.
Live Proof. Click on the SUBMIT button (below).
2.5. CsCCpCqrCCpqCpr. Proof: (+s ⇒ (+p ⇒ (+q ⇒ +r))) ⇒ ((+q ⇒ +r) ⇒ (+q ⇒ +r)). Nandsets are vacuous. ...............................
2.6. CCsCpCqrCsCCpqCpr. Proof: (+s ⇒ (+p ⇒ (+q ⇒ +r))) ⇒ (+s ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r))). Nandsets are vacuous. ...............................
2.7. CCqrCpCqr. Proof: (+q ⇒ +r) ⇒ (+p ⇒ (+q ⇒ +r)). Nandsets {-q,+p,+q,-r} and {+r,+p,+q,-r} are vacuous.
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2.8. CCqrCCpqCpr. Proof: (+q ⇒ +r) ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r)). Nandsets {-q,-p,+p,-r}, {-q,+q,+p,-r}, {+r,-p,+p,-r}, and {+r,+q,+p,-r} are vacuous.
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2.9. CCCqrCpqCCqrCpr. Proof: ((+q ⇒ +r) ⇒ (+p ⇒ +q)) ⇒ ((+q ⇒ +r) ⇒ (+p ⇒ +r)). Nandsets are vacuous. ...............................
2.10. CCpqCCqrCpq. Proof: (+p ⇒ +q) ⇒ ((+p ⇒ +q) ⇒ (+p ⇒ +r)). Nandsets {-p,-q,+p,-q}, {-p,+r,+p,-q}, {+q,-q,+p,-q}, and {+q,+r,+p,-q} are vacuous.
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2.11. CCpqCCqrCpr. Proof: (+p ⇒ +q) ⇒ ((+q ⇒ +r) ⇒ (+p ⇒ +r)). Nandsets {-p,-q,+p,-r}, {-p,+r,+p,-r}, {+q,-q,+p,-r}, and {+q,+r,+p,-r} are vacuous.
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2.12. CCCpqrCqr. Proof: ((+p ⇒ +q) ⇒ +r) ⇒ (+q ⇒ +r). Nandsets {+p,+r,+q,-r} and {-q,+r,+q,-r}, are vacuous.
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2.13. CCCCqrCprsCCpqs. Proof: (((+q ⇒ +r) ⇒ (+p ⇒ +r)) ⇒ +s) ⇒ ((+p ⇒ +q) ⇒ +s). Nandsets are vacuous. ...............................
2.14. CCpCqrCCsqCpCsr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+s ⇒ +q) ⇒ (+p ⇒ (+s ⇒ +r))). Nandsets are vacuous. ...............................
2.15. CCpCqrCqCpr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ (+q ⇒ (+p ⇒ +r)). Nandsets are vacuous. ...............................
2.16. CpCCpqq. Proof: p ⇒ (p ⇒ q) ⇒ q)). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.
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2.17. CCpqCCpCqrCpr. Proof: (+p ⇒ +q) ⇒ ((+p ⇒ (+q ⇒ +r)) ⇒ (+q ⇒ +r)). Nandsets are vacuous. ...............................
2.18. CCpCpqCpq. Proof: (+p ⇒ (+p ⇒ +q)) ⇒ (+p ⇒ +q). Nandsets {-p,+p,-q}, {-p,+p,-q}, and {+q,+p,-q} are vacuous.
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2.19. CCCpqqCCqpCCpqp. Proof: ((+p ⇒ +q) ⇒ +q) ⇒ ((+q ⇒ +p) ⇒ ((+p ⇒ +q) ⇒ +p)). Nandsets are vacuous. ...............................
2.20. CCqpCCCqppp. Proof: (+q ⇒ +p) ⇒ (((+q ⇒ +p) ⇒ +p) ⇒ +p). Nandsets are vacuous. ...............................
2.21. CCqpCCCqpCCpqpp. Proof: (+q ⇒ +p) ⇒ ((+q ⇒ +p) ⇒ (((+p ⇒ +q) ⇒ +p)) ⇒ +p). Nandsets are vacuous. ...............................
2.22. CCCqpCCpqpCCqpp. Proof: ((+q ⇒ +p) ⇒ ((+p ⇒ +q) ⇒ +p)) ⇒ ((+q ⇒ +p) ⇒ +p). Nandsets are vacuous. ...............................
2.23. CCCpqqCCqpp. Proof: ((+p ⇒ +q) ⇒ +q) ⇒ ((+q ⇒ +p) ⇒ +p). Nandsets are vacuous. ...............................
2.24. CCqpCCCpqqp. Proof: ((+q ⇒ +p) ⇒ (((+p ⇒ +q) ⇒ +q) ⇒ +p). Nandsets are vacuous. ...............................
K1. CpCqKpq. Proof: p ⇒ (q ⇒ (p & q)). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.
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K2. CKpqp. Proof: (p & q) ⇒ p. Nandset {+p,+q,-p} is vacuous.
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K3. CKpqq. Proof: (p & q) ⇒ q. Nandset {+p,+q,-q} is vacuous.
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2.25. CKqpCKqpKpq. Proof: (+q & +p) ⇒ ((+q & +p) ⇒ (+p & +q)). Nandsets are vacuous. ...............................
2.26. CKqpKpq. Proof: (q & p) ⇒ (p & q). Nandsets {+q,+p,-p} and {+q,+p,-q} are vacuous.
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2.27. CpKpp. Proof: p ⇒ (p & p). Nandsets {+p, -p} and {+p, -p} are vacuous.
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2.28. CCpqCpCrKqr. Proof: (+p ⇒ +q) ⇒ (+p ⇒ (+r ⇒ (+q & +r))). Nandsets are vacuous. ...............................
2.29. CCpqCKprCKprKqr. Proof: (+p ⇒ +q) ⇒ ((+p & +r) ⇒ ((+p & +r) ⇒ (+q & +r))). Nandsets are vacuous. ...............................
2.30. CCpqCKprKqr. Proof: (+p ⇒ +q) ⇒ ((+p & +r) ⇒ (+q & +r)). Nandsets {-p,+p,+r,-q}, {-p,+p,+r,-r}, {+q,+p,+r,-q}, and {+q,+p,+r,-r} are vacuous.
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2.31. CCpqCKrpKrq. Proof: (+p ⇒ +q) ⇒ ((+r & +p) ⇒ (+r & +q)). Nandsets {-p,+r,+p,-r}, {-p,+r,+p,-q}, {+q,+r,+p,-r}, and {+q,+r,+p,-q} are vacuous.
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2.32. CCqrCKqqKqr. Proof: (+q ⇒ +r) ⇒ ((+q & +q) ⇒ (+q & +r)). Nandsets {-q,+q,-q}, {-q,+q,-r}, {+r,+q,-q}, and {+r,+q,-r} are vacuous.
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2.32a. CCqrCqKqr. Proof: (+q ⇒ +r) ⇒ (+q ⇒ (+q & +r)). Nandsets {-q,+q,-q}, {-q,+q,-r}, {+r,+q,-q}, and {+r,+q,-r} are vacuous.
2.33. CCqpCCpqCCCqrCqKqrCCprCpKqr. Proof: (+q ⇒ +p) ⇒ (+p ⇒ +q) ⇒ ((+q ⇒ +r) ⇒ ((+q ⇒ (+q & +r))) ⇒ ((+p ⇒ +r) ⇒ (+p ⇒ (+q & +r)))) Nandsets are vacuous. ...............................
2.34. CCCqrCqKqrCCqpCCpqCCprCpKqr. Proof: ((+q ⇒ +r) ⇒ (+q ⇒ (+q & +r))) ⇒ ((+q ⇒ +p) ⇒ ((+p ⇒ +q) ⇒ ((+p ⇒ +r) ⇒ (+p ⇒ (+p & +r))))). Nandsets are vacuous. ...............................
2.35. CCqpCCpqCCprCpKqr. Proof: (+q ⇒ +p) ⇒ ((+p & +q) ⇒ ((+p & +r) (+p ⇒ (+q & +r)))). Nandsets are vacuous. ...............................
2.36. CpCCpqCCprCpKqr. Proof: +p ⇒ ((+p ⇒ +q) ⇒ ((+p ⇒ +r) ⇒ (+p ⇒ (+q & +r)))). Nandsets are vacuous. ...............................
2.37. CCpqCCprCpKqr. Proof: (+p ⇒ +q) ⇒ ((+p ⇒ +r) ⇒ (+p ⇒ (+q & +r))). Nandsets are vacuous. ...............................
2.38. CCpCqrCpCqr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ (+p ⇒ (+q ⇒ +r)). Nandsets {-p,+p,+q,-r}, {-q,+p,+q,-r}, and {+r,+p,+q,-r} are vacuous. ...............................
2.39. CCpCqrCKpqCKpqr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p & +q) ⇒ ((+p & +q) ⇒ +r)). Nandsets are vacuous. ...............................
2.40. CCpCqrCKpqr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p & +q) ⇒ +r). Nandsets {-p,+p,+q,-r}, {-p,+p,+q,-r}, and {+r,+p,+q,-r} are vacuous.
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2.41. CCrsCCpCqrCpCqs. Proof: (+r ⇒ +s) ⇒ ((+p ⇒ (+q ⇒ +r)) ⇒ (+p ⇒ (+q ⇒ +s))). Nandsets are vacuous. ...............................
2.42. CCpCqrCCrsCpCqs. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+r ⇒ +s) (+p ⇒ (+q ⇒ +r))). Nandsets are vacuous. ...............................
2.43. CKpqCCKpqrr. Proof: (+p & +q) ⇒ (((+p & +q) ⇒ +r) ⇒ +r). Nandsets {+p,+q,-p,-r}, {+p,+q,-q,-r}, and {+p,+q,+r,-r} are vacuous.
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2.44. CpCqCCKpqrr. Proof: (+p ⇒ (+q ⇒ (((+p & +q) ⇒ +r) ⇒ +r))). Nandsets {+p,+q,-p,-r}, {+p,+q,-q,-r}, and {+p,+q,+r,-r} are vacuous.
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2.45. CCKpqrCpCqr. Proof: ((+p & +q) ⇒ +r) ⇒ (+p ⇒ (+q ⇒ +r)). Nandsets {-p,+p,+q,-r}, {-q,+p,+q,-r}, and {-r,+p,+q,-r} are vacuous.
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2.46. ECpCqrCKpqr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p & +q) ⇒ +r) and ((+p & +q) ⇒ +r) ⇒ (+p ⇒ (+q ⇒ +r)). Nandsets are vacuous. ...............................
2.47. CKpKqrp. Proof: (p & (q & r)) ⇒ p. Nandset {+p,+q,+r,-p} is vacuous.
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2.48. CKpKqrKqr. Proof: (+p & (+q & +r)) ⇒ (+q ⇒ +r). Nandsets {+p,+q,+r,-q} and {+p,+q,+r,-r} are vacuous.
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2.49. CKpKqrq. Proof: (p & (q & r)) ⇒ q. Nandset {+p,+q,+r,-q} is vacuous.
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2.50. CKpKqrr. Proof: (p & (q & r)) ⇒ r. Nandset {+p,+q,+r,-r} is vacuous.
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2.51. CKpKqrKpq. Proof: (p & (q & r)) ⇒ (p & q). Nandsets {+p,+q,+r,-p} and {+p,+q,+r,-q} are vacuous.
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2.52. CKpKqrKKpqr. Proof: (p & (q & r)) ⇒ ((p & q) & r). Nandsets {+p,+q,+r,-p}, {+p,+q,+r,-q}, and {+p,+q,+r,-r} are vacuous.
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2.53. CKKpqrKpKqr. Proof: ((+p & +q) & +r)) ⇒ (p & (q & r)). Nandsets {+p,+q,+r,-p}, {+p,+q,+r,-q}, and {+p,+q,+r,-r} are vacuous.
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2.54. EKpKqrKKpqr. Proof: (+p & (+q & +r)) ⇒ ((+p & +q) & +r); and ((+p & +q) & +r) ⇒ (+p & (+q & +r)). Nandsets are vacuous. ...............................
2.55. EKpqKqp. Proof: ((+p & +q) = (+q & +p)) is ((p & q) ⇒ (q & p)) and ((q & p) ⇒ (p & q)). Nandsets {+p,+q,-q}, {+p,+q,-p}, {+q,+p,-p}, and {+q,+p,-q}, are vacuous.
2.56. EpKpp. Proof: (+p = (+p & +p)) is (+p ⇒ (+p & +p)) and ((+p & +p) ⇒ +p). Nandsets {+p,-p}, {+p,-p}, {+p,+p,-p}, and {+p,+p,-p} are vacuous.

A1. CpApq. Proof: +p ⇒ (+p | +q). Nandset {+p,-p,-q} is vacuous.
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A2. CqApq. Proof: +q ⇒ (+p | +q). Nandset {+q,-p,-q} is vacuous.
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A3. CCprCCqrCApqr. Proof: (+p ⇒ +r) ⇒ ((+q ⇒ +r) ⇒ ((+p | +q) ⇒ +r)). Nandsets are vacuous. ...............................
2.57. CAppp. Proof: (+p | +p) ⇒ +p. Nandsets {+p,-p} and {+p,-p} are vacuous.
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2.58. CApqAqp. Proof: (+p | +q) ⇒ (+q | +p). Nandsets {+p,-q,-p} and {+q,-q,-p} are vacuous.
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2.59. CApqAApqr. Proof: (+p | +q) ⇒ ((+p | +q) | +r). Nandsets {+p,-p,-q,-r} and {+r,-p,-q,-r} are vacuous.
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2.60. CpAApqr. Proof: +p ⇒(+p | (+q | +r)). Nandset {+p,-p,-q,-r} is vacuous.
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2.61. CqAApqr. Proof: +q ⇒(+p | (+q | +r)). Nandset {+q,-p,-q,-r} is vacuous.
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2.62. CAqrAApqr. Proof: (+q | +r) ⇒ ((+p | +q) | +r) Nandsets {+q,-p,-q,-r} and {+r,-p,-q,-r} are vacuous.
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2.63. CApAqrAApqr. Proof: (+p | (+q | +r)) ⇒ ((+p | +q) | +r). Nandsets {+p,-p,-q,-r}, {+q,-p,-q,-r}, and {+r,-p,-q,-r} are vacuous.
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2.64. CAApqrApAqr. Proof: ((+p | +q) | +r) ⇒ (+p | (+q | +r)). Nandsets {+p,-p,-q,-r}, {+q,-p,-q,-r}, and {+r,-p,-q,-r} are vacuous.
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2.65. CCsqCsAqr. Proof: (+s ⇒ +q) ⇒ (+s ⇒ (+q ⇒ +r)). Nandsets {-s,+s,-q,-r} and {+q,+s,-q,-r} are vacuous.
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2.66. CpCCpqAqr. Proof: +p ⇒ ((+p ⇒ +q) ⇒ (+q | +r)). Nandsets {+p,-p,-q,-r} and {+p,+q,-q,-r} are vacuous.
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2.67. CrCsAqr. Proof: +r ⇒ (+s ⇒ (+q | +r)). Nandset {+r,+s,-q,-r} is vacuous.
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2.69. CCpqCAprAqr. Proof: (+p ⇒ +q) ⇒ ((+p | +r) ⇒ (+q | +r)). Nandsets {-p,+p,-q,-r}, {-p,+r,-q,-r}, {+q,+p,-q,-r}, and {+q,+r,-q,-r} are vacuous.
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2.70. CCpqCArpArq. Proof: (+p ⇒ +q) ⇒ ((+r | +p) ⇒ (+r | +q)). Nandsets {-p,+r,-r,-q}, {-p,+p,-r,-q}, {+q,+r,-r,-q}, and {+q,+p,-r,-q} are vacuous.
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2.71. CApqCCpqq. Proof: (+p | +q) ⇒ ((+p ⇒ +q) ⇒ +q). Nandsets are vacuous. ...............................
2.72. EAppp. Proof: (+p | +p) = +p is ((+p | +p) ⇒ +p) and (+p ⇒ (+p | +p)). Nandsets {+p,-p}, {+p,-p}, and {+p,-p,-p} are vacuous.
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2.73. EApqAqp. Proof: ((+p | +q) = (q | p)) is ((+p | +q) ⇒ (+q | +p)) and ((+q | +p) ⇒ (+p | +q)). Nandsets {+p,-q,-p}, {+q,-q,-p}, {+q,-p,-q}, and {+p,-p,-q} are vacuous.
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2.74. EApAqrAApqr. Proof: (+p | (+q | +r)) ⇒ ((+p | +q) | +r); and ((+p | +q) | +r) ⇒ (+p | (+q | +r)). Nandsets are vacuous. .................................
2.75. CKqrApq. Proof: (+q & +r) ⇒ (+p | +q). Nandset {+q,+r,-p,-q} is vacuous.
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2.76. CApKqrApq. Proof: (+p | (+q & +r)) ⇒ (+p | +q). Nandsets {+p,+p,-p,-q}, {+p,+r,-p,-q}, {+q,+p,-p,-q}, and {+q,+r,-p,-q} are vacuous.
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2.77. CKqrApr. Proof: (+q & +r) ⇒ (+p | +q). Nandset {+q,+r,-p,-q} is vacuous.
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2.78. CApKqrApr. Proof: (+p | (+q & +r)) ⇒ (+p | +r). Nandsets {+p,+p,-p,-r}, {+p,+r,-p,-r}, {+q,+p,-p,-r}, and {+q,+r,-p,-r} are vacuous.
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2.79. CApKqrKApqApr. Proof: (+p | (+q & +r)) ⇒ ((+p | +q) & (+p | +r)). Nandsets are vacuous. ...............................
2.80. CrCArqArKpq. Proof: +r ⇒ ((+r | +q) ⇒ (+r | (+p & +q))). Nandsets {+r,+r,-r,-p}, {+r,+r,-r,-q}, {+r,+q,-r,-p}, and {+r,+q,-r,-q} are vacuous.
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2.81. CpCqArKpq. Proof: (+p ⇒ (+q ⇒ (+r | (+p & +q)))). Nandsets are vacuous. ...............................
2.82. CpCrArKpq. Proof: (+p ⇒ (+r ⇒ (+r | (+p & +q)))). Nandsets are vacuous. ...............................
2.83. CpCArqArKpq. Proof: +p ⇒ ((+r | +q) ⇒ (+r | (+p & +q))). Nandsets are vacuous. ...............................
2.84. CArpCArqArKpq. Proof: (+r | +p) ⇒ ((+r | +p) ⇒ (+r | (+p & +q))). Nandsets are vacuous. ...............................
2.85. CKApqAprApKqr. Proof: ((+p | +q) & (+p | +r)) ⇒ (+p | (+q & +r)). Nandsets are vacuous. ...............................
2.86. EApKqrKApqApr. Proof: (+p | (+q & +r)) ⇒ ((+p | +q) & (+p | +r)); and ((+p | +q) & (+p | +r)); ⇒ (+p | (+q & +r)). Nandsets are vacuous. ...............................
2.87. EKpAqrAKpqKpr. Proof: (+p & (+q | +r)) ⇒ ((+q & +r)| (+q & +r)); and ((+q & +r)| (+q & +r)) ⇒ (+p & (+q | +r)). Nandsets are vacuous. ...............................
2.88. CCprCCCCrprrCCCpqrr. Proof: (+p ⇒ +r) ⇒ ((((+r ⇒ +p) ⇒ +r) ⇒ +r) ⇒ (((+p ⇒ +q) ⇒ +r) ⇒ +r)). Nandsets are vacuous. ...............................
2.89. CCprCCCpqrr. Proof: (+p ⇒ +r) ⇒ (((+p ⇒ +q) ⇒ +r) ⇒ +r). Nandsets are vacuous. ...............................
2.90. ApCpq. Proof: -p | (+p | +q). Nandset {+p,-p,-q} is vacuous.
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2.91. CCCpqqApq. Proof: ((+p ⇒ +q) ⇒ +q) ⇒ (+p | +q). Nandsets are vacuous. ...............................
2.92. EApqCCpqq. Proof: (+p | +q) ⇒ ((+p ⇒ +q) ⇒ +q); and ((+p ⇒ +q) ⇒ +q) ⇒ (+p | +q). Nandsets are vacuous. ...............................
2.93. CCprCCCqrCCCrqqrCCqrCCCpqqr. Proof: (+p ⇒ +r) ⇒ (((+q ⇒ +r) ⇒ (((+r ⇒ +q) ⇒ +q) ⇒ +r)) ⇒ (((+q ⇒ +r) ⇒ (((+p ⇒ +q) ⇒ +q) ⇒ +r)))). Nandsets are vacuous. ...................................
2.94. CCprCCqrCCCpqqr. Proof: (+p ⇒ +r) ⇒ ((+q ⇒ +r) ⇒ (((+p ⇒ +q) ⇒ +q) ⇒ +r)). Nandsets are vacuous. ...............................
N1. CCpNpNp. Proof: ((p ⇒ -p) ⇒ -p). Nandset {+p,+p,-p} is vacuous.
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N2. CNpCpq. Proof: (-p ⇒ (p ⇒ -q)). Nandset {-p,+p,-q} is vacuous.
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2.95. CCpqCCqNpCpNp. Proof: (+p ⇒ +q) ⇒ ((+q ⇒ -p) ⇒ (+p ⇒ -p)). Nandsets are vacuous. ...............................
2.96. CCpqCNqCpNp. Proof: (-p ⇒ q) ⇒ (-q ⇒ (p ⇒ -p)). Nandsets {-p,-q,+p} and {+q,-q,+p} are vacuous.
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2.97. CCpqCNqNp. Proof: (p ⇒ q) ⇒ (-q ⇒ -p). Nandsets {-p,-q,+p} and {+q,-q,+p} are vacuous.
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2.98. CCpqCCpCqNpCpNp. Proof: (+p ⇒ +q) ⇒ (+p ⇒ ((+q ⇒ -p)) ⇒ (+p ⇒ -p)). Nandsets are vacuous. ...............................
2.99. CCpqCCpNqNp. Proof: (+p ⇒ +q) ⇒ ((+p ⇒ -q) ⇒ -p). Nandsets {-p,-p,+p}, {-p,-q,+p}, {+q,-p,+p}, and {+q,-q,+p} are vacuous.
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2.100. CqCCpNqNp. Proof: q ⇒ ((p ⇒ -q) ⇒ -p). Nandsets {+q,-p,+p} and {+q,-q,+p} are vacuous.
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2.101. CCpNqCqNp. Proof: (p ⇒ -q) ⇒ (q ⇒ -p). Nandsets {-p,+q,+p} and {-q,+q,+p} are vacuous.
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2.102. CpNNp. Proof: (+p ⇒ --p). Nandset {+p,-p} is vacuous.
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2.103. CNNNpNp. Proof: (---p ⇒ -p). Nandset {-p,+p} is vacuous.
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2.104. ENpNNNp. Proof: (-p = ---p) is (-p ⇒ ---p) and (---p ⇒ -p). Nandsets {-p,+p} and {-p,+p} are vacuous.
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2.105. CNpCNNpq. Proof: -p ⇒ (--p ⇒ q). Nandset {-p,+p,-q} is vacuous.
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2.106. CANpqCNNpq. Proof: (-p | +q) ⇒ (--p ⇒ +q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
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2.107. CCNNpqCpq. Proof: (--p ⇒ +q) ⇒ (+p ⇒ +q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
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2.108. CpCNqNCpq. Proof: p ⇒ (-q ⇒ -(+p ⇒ +q)). Nandsets {+p,-q,-p} and {+p,-q,+q} are vacuous.
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2.109. CKpNqNCpq. Proof: (p & -q) ⇒ -(+p ⇒ +q). Nandsets {+p,-q,-p} and {+p,-q,+q} are vacuous.
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2.110. CCpqNKpNq. Proof: (p ⇒ -q) ⇒ -(+p & +q). Nandset {-p,+q,+p,-q} is vacuous.
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2.111. CNNCpqNKpNq. Proof: --(+p ⇒ +q) ⇒ -(+p & -q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
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2.112. CpCNqKpNq. Proof: +p ⇒ (-q ⇒ (+p & -q)). Nandsets {+p,-q,-p} and {+p,-q,+q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.113. CpCNKpNqNNq. Proof: ((+p ⇒ -(+p & -q)) ⇒ --q). Nandsets are vacuous. ...............................
2.114. CNKpNqCpNNq. Proof: -(+p & -q) ⇒ (+p ⇒ --q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.115. CCpCqrCqCNNpNNr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ (+q ⇒ (--p ⇒ --r)). Nandsets are vacuous. ...............................
2.116. CCpCqrCNNpCNNqNNNNr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ (--p ⇒ (--q ⇒ ----r)). Nandsets are vacuous. ...............................
2.117. CCpCqrCNNpCNNqNNr. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ (--p ⇒ (--q ⇒ --r)). Nandsets are vacuous. ...............................
2.118. CNNCpqCNNpNNq. Proof: --(+p ⇒ +q) ⇒ (--p ⇒ --q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
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2.119. CCNNpNNqCpNNq. Proof: (--p ⇒ --q) ⇒ (+p ⇒ --q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.120. CCpCqrCCpCrsCpCqs. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p ⇒ (+r ⇒ +s)) ⇒ (+p ⇒ (+q ⇒ +s))). Nandsets are vacuous. .................................
2.121. CCpCqrCCpNrCpNq. Proof: (+p ⇒ (+q ⇒ +r)) ⇒ ((+p ⇒ -r) ⇒ (+p ⇒ -p)). Nandsets are vacuous. ...............................
2.122. CNCpqNNp. Proof: -(+p ⇒ +q) ⇒ --p. Nandset {+p,-q,-p} is vacuous.
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2.123. CCpNqCCNqNpCpNp. Proof: (+p ⇒ -q) ⇒ ((-q ⇒ -p) ⇒ (+p ⇒ -p)). Nandsets {-p,+q,+p,+p}, {-p,-p,+p,+p}, {-q,+q,+p,+p}, and {-q,-p,+p,+p} are vacuous.
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2.124. CCpNqCCNqNpNp. Proof: (+p ⇒ -q) ⇒ ((-q ⇒ -p) ⇒ -p). Nandsets {-p,+q,+p}, {-p,-p,+p}, {-q,+q,+p}, and {-q,-p,+p} are vacuous.
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2.125. CNCpqNq. Proof: -(+p ⇒ +q) ⇒ -q. Nandset {+p,-q,+q} is vacuous.
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2.126. CNCpqCCNqNpNp. Proof: -(+p ⇒ +q) ⇒ ((-q ⇒ -p) ⇒ -p). Nandsets {+p,-q,+q,+p} and {+p,-q,-p,+p} are vacuous.
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2.127. CNCpqNCNqNp. Proof: -(+p ⇒ +q) ⇒ -(-q ⇒ -p). Nandsets {+p,-q,+q} and {+p,-q,-p} are vacuous.
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2.128. CCpNNqNNCpq. Proof: (+p ⇒ --q) ⇒ --(+p ⇒ +q). Nandsets {-p,+p,-q} and {+q,+p,-q} are vacuous.
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2.129. CKpqKNNpq. Proof: (+p & +q) ⇒ (--p & +q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.
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2.130. CCNNpNqNKNNpNNq. Proof: (--p ⇒ -q) ⇒ -(--p & --q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.131. CCpNqNKNNpq. Proof: (+p ⇒ -q) ⇒ -(--p & +q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.
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2.132. CKNNpqNCpNq. Proof: (--p & +q) ⇒ -(+p ⇒ -q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.133. CANpNqCpNq. Proof: (-p | -q) ⇒ (+p ⇒ -q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.
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2.134. CNCpNqNANpNq. Proof: -(+p ⇒ -q) ⇒ -(-p | -q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.
Live Proof. Click on the SUBMIT button (below).
2.135. CCNpNNpNNp. Proof: (-p ⇒ --p) ⇒ --p. Nandsets {+p,-p} and {+p,-p} are vacuous.
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2.136. CCNppNNp. Proof: (-p ⇒ +p) ⇒ --p. Nandsets {+p,-p} and {+p,-p} are vacuous.
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2.137. CCpqCCANpqrCpr. Proof: (+p ⇒ +q) ⇒ (((-p | +q) ⇒ +r) ⇒ (+p ⇒ +r)). Nandsets are vacuous. ...............................
2.138. CCpqCNANpqNp. Proof: (+p ⇒ +q) ⇒ (-(-p | +q) ⇒ +p). Nandsets {+p,+p,-q,-p} and {-q,+p,-q,-p} are vacuous.

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

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

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

2.142. CpCNKpqNq. Proof: +p ⇒ (-(+p & +q) ⇒ -q). Nandsets {+p,-p,+q}ÿ20and {+p,-q,+q} are vacuous.

2.143. CNKpqCpNq. Proof: -(+p & +q) ⇒ (+p ⇒ -q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.

2.144. CNANpNqNNKpq. Proof: -(-p | -q) ⇒ --(+p & -q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.

2.145. CNNKpqNNp. Proof: --(+p & -q) ⇒ --p. Nandset {+p,+q,-p} is vacuous.

2.146. CNNKpqNNq. Proof: --(+p & -q) ⇒ --q. Nandset {+p,+q,-q} is vacuous.

2.147. CNNKpqKNNpNNq. Proof: --(+p & -q) ⇒ (--p & --q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.

2.148. CCpNqCNNpNq. Proof: (+p ⇒ -q) ⇒ (--p ⇒ -q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.

2.149. CNNpCCpNqNq. Proof: --p ⇒ ((+p ⇒ -q) ⇒ -q). Nandsets {+p,-p,+q} and {+p,-q,+q} are vacuous.

2.150. CNNpCNNqNCpNq. Proof: --p ⇒ (--q ⇒ (--p ⇒ -q)). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.

2.151. CKNNpNNqNCpNq. Proof: (--p & --q) ⇒ -(+p ⇒ -q). Nandsets {+p,+q,-p} and {+p,+q,-q} are vacuous.

2.152. ENNEpqKNNCpqNNCqp. .....................
2.153. CCpNqNKpq. Proof: (+p ⇒ -q) ⇒ -(+p & +q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.

2.154. CANpNqNKpq. Proof: (-p | -q) ⇒ -(+p & +q). Nandsets {-p,+p,+q} and {-q,+p,+q} are vacuous.

2.155. CNApqNp. Proof: -(+p | +q) ⇒ -p. Nandset {-p,-q,+p} is vacuous.

2.156. CNApqNq. Proof: -(+p | +q) ⇒ -q. Nandset {-p,-q,+q} is vacuous.

2.157. CNApqKNpNq. Proof: -(+p | +q) ⇒ (-p & -q). Nandsets {-p,-q,+p} and {-p,-q,+q} are vacuous.

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

2.159. CNpCNqNApq. Proof: -p ⇒ (-q ⇒ -(+p | +q)). Nandsets {-p,-q,+p} and {-p,-q,+q} are vacuous.
2.160. CKNpNqNApq. Proof: ((-p & -q) ⇒ (+p | +q). Nandset {+p,+q,-p,-q} is vacuous.

2.161. ENApqKNpNq. (DeMorgan's Law.) Proof: -(+p | +q) ⇒ (-p & -q); and (-p & -q) ⇒ -(+p | +q). Nandsets {-p,-q,+p} and {-p,-q,+q}; and {-p,-q,+p} and {-p,-q,+q} are vacuous.

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

2.163. CCCpqqCCpqq. Proof: ((+p ⇒ +q) ⇒ +q) ⇒ ((+p ⇒ +q) ⇒ +q). Nandsets are vacuous. ...............................
2.164. CCCpqqCNpq. Proof: ((+p ⇒ +q) ⇒ +q) ⇒ (-p ⇒ +q). Nandsets {+p,+q,-p,+q} and {-q,+q,-p,+q} are vacuous.

2.165. CCNpqNKNpNq. Proof: (-p ⇒ +q) ⇒ -(-p & -q). Nandsets {+p,-p,-q} and {+q,-p,-q} are vacuous.

2.166. CCNppp. Proof: ((-p ⇒ +p) ⇒ +p). Nandset {+p,+p,-p} is vacuous.

2.167. CNNpp. Proof: (--p ⇒ +p). Nandset {+p,-p} is vacuous.

2.168. EpNNp. Proof: (+p = --p) is (+p ⇒ --p) and (--p ⇒ p). Nandsets {+p,-p} and {+p,-p} are vacuous.

2.169. CCNpNqCNNqNNp. Proof: (-p ⇒ -q) ⇒ (--q ⇒ --p). Nandsets {+p,+q,-p} and {-q,+q,-p} are vacuous.

2.170. CCNpNqCqp. Proof: (-p ⇒ -q) ⇒ (+q ⇒ +p). Nandsets {+p,+q,-p} and {-q,+q,-p} are vacuous.

2.171. CCNpNNqCNqp. Proof: (-p ⇒ --q) ⇒ (-q ⇒ +p). Nandsets {+p,-q,-p} and {-q,-q,-p} are vacuous.

2.172. CCNpqCNqp. Proof: (-p ⇒ +q) ⇒ (-q ⇒ +p). Nandsets {+p,-q,-p} and {-q,-q,-p} are vacuous.

2.173. ApNp. Proof: (+p | -p). Nandset {-p,+p} is vacuous.
2.174. CCpqANpq. Proof: (+p ⇒ +q) ⇒ (-p | +q). Nandsets {-p,+p,-q} and {-q,+p,-q} are vacuous.

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

2.176. CNNKpqKpq. Proof: --(+p & +q) ⇒ (+p & +q). Nandsets {+p,-p,-q} and {+q,-p,-q} are vacuous.

2.177. CNKNpNqApq. Proof: -(-p & -q) ⇒ (+p | +q). Nandsets {+p,-p,-q} and {+q,-p,-q} are vacuous.

2.178. EANpNqNKpq. (DeMorgan's Law.) Proof: (-p | -q) ⇒ -(+p & +q); and -(+p & +q) ⇒ (-p | -q). Nandsets {-p,+p,+q} and {-q,+p,+q}; and {-p,+p,+q} and {-q,+p,+q} are vacuous.

CN1. CCpqCCqrCpr. Proof: (+p ⇒ +q) ⇒ ((+q ⇒ +r) ⇒ (+p ⇒ +r)). Nandsets {-p,-q,+p,-r}, {-p,+r,+p,-r}, {+q,-q,+p,-r}, and {+q,+r,+p,-r} are vacuous.

CN2. CpCNpq. Proof: (+p ⇒ (-p ⇒ +q)). Nandset {+p,-p,+q} is vacuous.

CN3. CCNppp. Proof: ((-p ⇒ +p) ⇒ +p). Nandsets {+p,-p} and {+p,-p} are vacuous.

KN1. CpKpp. Proof: (+p ⇒ (+p & +p)). Nandsets {+p,-p} and {+p,-p} are vacuous.

KN2. CKpqp. Proof: ((+p & +q) ⇒ +p). Nandset {+p,+q,-p} is vacuous.

KN3. CCpqCNKqrNKrp. Proof: (+p ⇒ +q) ⇒ (-(+q & +r) ⇒ -(+r & +p)). Nandsets are vacuous. ...............................
5.72. CLKpqLp. Proof: (+□(+p&+q)) ⇒ +□+p. Nandsets {+$p,+$q,+p,+q,-$p} and {+$p,+$q,+p,+q,-p} are vacuous.

5.73. CLKpqLp. Proof: (+□(+p&+q)) ⇒ +□+q. Nandsets {+$p,+$q,+p,+q,-$q} and {+$p,+$q,+p,+q,-q} are vacuous.

5.74. CLKpqKLpLq. Proof: (+□(+p&+q)) ⇒ (+□+p & +□+q). Nandsets {+$p,+$q,+p,+q,-$p}, {+$p,+$q,+p,+q,-p}, {+$p,+$q,+p,+q,-$q}, and {+$p,+$q,+p,+q,-q} are vacuous.

5.78. CLpCLqLKpq. Proof: (+□+p) ⇒ ((+□+q) ⇒ (+□(+p&+q)). Nandsets {+$p,+p,+$q,+q,-$p}, {+$p,+p,+$q,+q,-$q}, {+$p,+p,+$q,+q,-p}, and {+$p,+p,+$q,+q,-q} are vacuous.

5.79. CKLpLqLKpq. Proof: (+□+p &+□+q) ⇒ (+□(+p &+q)). Nandsets {+$p,+p,+$q,+q,-$p}, {+$p,+p,+$q,+q,-$q}, {+$p,+p,+$q,+q,-p}, and {+$p,+p,+$q,+q,-q} are vacuous.

5.80. ELKpqKLpLq. Proof: +□(+p&+q) ⇔ (+□+p & +□+q). Nandsets {+$p,+$q,+p,+q,-$p}, {+$p,+$q,+p,+q,-p}, {+$p,+$q,+p,+q,-$q}, {+$p,+$q,+p,+q,-q}, {+$p,+p,+$q,+q,-$p}, {+$p,+p,+$q,+q,-$q}, {+$p,+p,+$q,+q,-p}, and {+$p,+p,+$q,+q,-q} are vacuous.

5.83. CALpLqLApq. Proof: (+□+p|+□+q) ⇒ +□(+p|+q). I CAN'T MAKE THIS PROOF WORK, ALTHOUGH I CAN PROVE THE CONVERSE.

NOTE: CpLp is not necessarily true, but CpMp is always true in GWM modal logic. However (Zeman, p. 89), there are some modal logics where CpMp is not necessarily true.

ADDITIONAL GWMML THEOREMS AND PROOFS.

GWM1. CpMp. +p ⇒ +◇+p.
Proof: Nandset {+p,+$p,-p} is vacuous.

GWM2. CLpLp. +□+p ⇒ +□+p.
Proof: Nandsets {-$p,-p,+$p} and {-$p,-p,+p} are vacuous.

GWM3. CLpMp. +□+p ⇒ +◇+p.
Proof: Nandset {-$p,-p,+$p,+p} is vacuous.

GWM4. CCLpMNpLNp. ((+□+p ⇒ +◇+p) ⇒ +◇+p).
Proof: Nandsets {-$p,+$p,+p} and {-p,+$p,+p} are vacuous.

SUMMARY: RULES OF ETHICAL DATA COLLECTION.

0. The logic in this report is based upon classical logic, with the following three complementizers: try/harm/cost(!); need/value(#); and knowledge/certainty($). That is, the harm created by achieving higher levels of knowledge/certainty must be balanced by the need/value in obtaining that knowledge/certainty.

1. Rule 1. Double-negative = positive. Complementizer-negative = complementizer-positive. That is: negative-negative-x equals x; know-negative-x equals know-x; try-negative-x equals try-x; need-negative-x equals need-x.

2. Rule 2. Certainty ordinal. More-certain implies less-certain. Certain^k+1ω implies certain^kω.
Nandset definition: {-$^kω,+$^k+1ω} ∈ S₀, for every k, 1 < k < M-1 and ω ∈ Ω.

3. Rule 3. Data-absolute. You either know a datum or not. There is no data-ladder: you are either at the bottom of the ladder or at the top of the ladder. Know-datum implies know^M-datum.
Nandset definition: {+$d,-$^Md} ∈ S₀, for every d ∈ D.

4. Rule 4. Hippocrates. First do no harm. That is, harm-datum implies need-datum. (Contrapositively: no-need-datum implies no-harm-datum.)
Nandset definition: {-#d,+!d} ∈ S₀, for d ∈ D.

5. Rule 5. Conative. Try if you must. Not-know-datum and need-datum implies harm-datum.
Nandset definition: {-$d,+#d,-!d} ∈ S₀, for d ∈ D.

6. 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.
Nandset definition: {+$^ke,e,+$^kδ,δ,..,-$d,-#d, -$^k+1e} ∈ S₀, for 1 < k < M-2, δ∈ D, and e ∈ E.

7. Rule 7. Ontology. 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.
Nandset definition: {+$^kδ,δ,..,-e,-$^k+1e} ∈ S₀ and {+$^kδ,δ,..,-$^ke} ∈ S₀, for 1 < k < M-2, d ∈ D, δ ⊆ (D - {+d,-d}).

8. Rule 8. Ethical Data Registration. 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. That is, for d ∈ D, there exists at most one J, 1 < J < H, at which (8.1.1) +$d, +d, +!d are true; or else (8.1.2) +$d, -d, +!d are true; or else (8.1.3) -$d, +!d are true. (8.2) Otherwise, for every J, 1 < J < H, -$d, +$d, -#d, +#d, -!d, +!d are all not entered into (S_J - S_J-1). The nandsets for Rule 8 are: (8.1.1) {-$d}, {-d}, {-!d} ∈ (S_J - S_J-1); or else (8.1.2) {-$d}, {+d}, {-!d} ∈ (S_J - S_J-1); or else (8.1.3) {+$d}, {-!d} ∈ (S_J - S_J-1). (8.2) Otherwise, {+$d}, {-$d}, {+#d}, {-#d}, {+!d}, {-!d} ~∈ S_J - S_J-1).

9. Rule 9. Schrödinger's Rule. At data-collection-step J, we create a set, O_J, the SCHRÖDINGER OPENING. The nandset for -$^kω, namely, {+$^kω}, is placed in O_J if and only if the nandset for +$^kω, namely, {-$^kω}, is NOT a member of the logical consequences of the data-collection-step. That is, anything that is uncertain at data-collection-step J is declared uncertain in O_J. If the cat's life is uncertain at data-collection-step J, then it is declared uncertain in O_J. However, the cat may spring alive again at data-collection-step (J+1). Watch closely: the reasoning is a little tricky.

Rule 9, Schrödinger's Rule: It is true that -$^kω for O_J if and only if +$^kω is not a logical consequence of S_J. The nandset for Rule 9 is: {+$^kω} ∈ O_J if and only if {-$^kω} ~∈ ∫S_J, where ∫ represents LOGICAL CONSEQUENCES OF.

10. Harm/Cost(!). Every datum has a harm/cost: inconvenience, needle-stick, financial payment,....

11. Value/Need(#). Hippocrates [99]: only collect data that need/valued to be collected.

12. Know($). What to collect, what not to collect: Vexation Ladder; Ontology Ladder.

POSSIBLE PATIENT DESCRIPTION TABLE.

Counter- Example	Rule 4: Hippocratic	Rule 5: Conative	Rule 6: Vexative	Rule 7a: Ontologic	Rule 7b: Ontologic	Rule 8a: Data Registration	Rule 8b: Data Registration	Rule 8c: Data Registration	Rule 8d: Data Registration	Rule 9: Schrödinger Opening
{-$^∞p,	┅	{-$^∞p,	{-$^∞p,	{+$^∞p,	{+$^∞p,	┅	┅	┅	┅	{+$^∞p}
-#p,	{-#p,	+#p,	-#p,	┅	┅	┅	┅	┅	┅	┅
-!p,	+!p}	-!p}	┅	┅	┅	┅	┅	┅	┅	┅
±p,	┅	┅	┅	±p,	±p,	┅	┅	┅	┅	┅
+e,	┅	┅	+e,	-e,	┅	┅	┅	┅	┅	┅
+$^ke,	┅	┅	+$^ke,	┅	-$^ke}	┅	┅	┅	┅	{-$^ke}
-$^k+1e,	┅	┅	-$^k+1e}	-$^k+1e}	┅	┅	┅	┅	┅	{+$^k+1e}
.	.	.	.	.	.	.	.	.	.	.
-$^∞q,	┅	{-$^∞q,	{-$^∞q,	{+$^∞q,	{+$^∞q,	┅	{+$^∞q}	┅	┅	{+$^∞q}
+#q,	{-#q,	+#q,	-#q,	┅	┅	┅	┅	┅	┅	┅
+!q,	+!q}	-!q}	┅	┅	┅	┅	{-!q}	┅	┅	┅
±q,	┅	┅	┅	±q,	±q,	┅	┅	┅	┅	┅
+e,	┅	┅	+e,	-e,	┅	┅	┅	┅	┅	┅
+$^ke,	┅	┅	+$^ke,	┅	-$^ke}	┅	┅	┅	┅	{-$^ke}
-$^k+1e,	┅	┅	-$^k+1e}	-$^k+1e}	┅	┅	┅	┅	┅	{+$^k+1e}
.	.	.	.	.	.	.	.	.	.	.
+$^∞r,	┅	{-$^∞r,	{-$^∞r,	{+$^∞r,	{+$^∞r,	┅	┅	{-$^∞r}	┅	{+$^∞r}
+#r,	{-#r,	+#r,	-#r,	┅	┅	┅	┅	┅	┅	┅
+!r,	+!r}	-!r}	┅	┅	┅	┅	┅	{-!r}	┅	┅
-r,	┅	┅	┅	-r,	-r,	┅	┅	{+r}	┅	┅
+e,	┅	┅	+e,	-e,	┅	┅	┅	┅	┅	┅
+$^ke,	┅	┅	+$^ke,	┅	-$^ke}	┅	┅	┅	┅	{-$^ke}
-$^k+1e,	┅	┅	-$^k+1e}	-$^k+1e}	┅	┅	┅	┅	┅	{+$^k+1e}
.	.	.	.	.	.	.	.	.	.	.
+$^∞s,	┅	{-$^∞s,	{-$^∞s,	{+$^∞s,	{+$^∞s,	┅	┅	┅	{-$^∞s}	{+$^∞s}
+#s,	{-#s,	+#s,	-#s,	┅	┅	┅	┅	┅	┅	┅
+!s,	+!s}	-!s}	┅	┅	┅	┅	┅	┅	{-!s}	┅
+s,	┅	┅	┅	+s,	+s,	┅	┅	┅	{-s}	┅
+e,	┅	┅	+e,	-e,	┅	┅	┅	┅	┅	┅
+$^ke,	┅	┅	+$^ke,	┅	-$^ke}	┅	┅	┅	┅	{-$^ke}
-$^k+1e,	┅	┅	-$^k+1e}	-$^k+1e}	┅	┅	┅	┅	┅	{+$^k+1e}

28. APPENDIX I.
EMBRYOGENESIS.


 +embryogenesis
    +preimplantation
       +gametogenesis
          +primary_meiotic_division
             +primary_meiotic_prophase
             +primary_meiotic_metaphase
             +primary_meiotic_anaphase
             +primary_meiotic_telophase
             +primary_meiotic_daughter_cell
          +secondary_meiotic_division
             +secondary_meiotic_prophase
             +secondary_meiotic_metaphase
             +secondary_meiotic_anaphase
             +secondary_meiotic_telophase
             +secondary_meiotic_daughter_cell
       +ovulation
       +fertilization
       +morula
    +implantation
       +blastocyst
          +inner_cell_mass=+embryoblast
          +outer_cell_mass=+trophoblast
          +blastocyst_cavity
    +bilaminar_germ_disc
       -embryo
       +embryo
          +extracelomic_cavity=+primary_yolk_sac
          +endoderm
          +ectoderm
          +amnionic_cavity
          +amnioblast
    +trilaminar_germ_disc
       -embryo
       +embryo
          +amnionic_cavity
          +ectoderm
          +ordinary_ectoderm
          +neurectoderm
             +primitive_streak
             +primitive_pit
             +notochord
             +neurenteric_canal
             +neural_fold
             +neural_plate
          +mesoderm
             +mesoderm_intraembryonic
             +mesoderm_extraembryonic
             +mesoderm_intermediate
                +somite
                   +dermatome
                      +dermis
                      +connective_tissue
                      +cartilage
                      +bone
                   +myotome
                      +skeletal_muscle
                      +smooth_muscle
                      +myocardial_muscle
                   +nephrotome
                      +nephric_tubule
                      +internal_glomerulus
                      +external_glomerulus
                      +pronephros
                         +vestigial
                      +mesonephros
                         +ovary_stroma
                         +testis_stroma
                      +metanephros
                         +kidney
          +endoderm
       +body_system
          +ectoderm
             +ordinary_ectoderm
                +epidermis
                +hair
                +nail
                +eccrine_sweat_gland
                +apocrine_sweat_gland
                +sebaceous_gland
                +tooth enamel
                +epithelial_lining
             +neurectoderm
                +central_nervous_system
                +peripheral_nervous_system
                +sensory_epithelium
                +hypophysis
                +adrenal_medulla
          +mesoderm
             +dermis
             +connective_tissue
             +cartilage
             +bone
             +muscle
                +skeletal_muscle
                +smooth_muscle
                +myocardial_muscle
             +blood_element
             +lymph_element
             +adrenal_cortex
             +spleen
         +endoderm
             +blood_vessel_endothelium
             +allantois
                +allantoic_duct_remnant
             +respiratory_epithelium
                +respiratory_epithelium_pharyngeal
                +respiratory_epithelium_tracheobronchial
                +respiratory_epithelium_alveolar
             +tonsillar_parenchyma
             +thyroid_parenchyma
             +parathyroid_parenchyma
             +thymus_parenchyma
             +hepatic_parenchyma
             +pancreatic_parenchyma
             +urothelium
                +urinary_bladder_urothelium
                +urethral_urothelium
             +tympanic_membrane
             +eustachian_tube
             +esophageal_epithelium
             +gastric_epithelium
             +small_intestinal_epithelium
             +colonic_epithelium
         +primitive_yolk_sac
            +omphalomesenteric_duct
               +omphalomesenteric_duct_remnant

29. APPENDIX J.
FOUNDATIONS OF DERMATOPATHOLOGY DIAGNOSIS.

+∀ +skin_disease_recognition +major_tissue_reaction_noninfectious_inflammatory_pattern +lichenoid_pattern <A> +lichen_planus_lp +civatte_body_prominent +band_like_inflammation +wedge_hypergranulosis +hypertrophic_form +lichen_simplex_chronicus +orthokeratosis_compact +hypergranulosis +parakeratosis_focal +psoriasiform_hyperplasia +papillomatosis_mild +papillary_dermis_thickening +lichen_planus_variant_lp_variant +lichen_planus_atrophic +lichen_planus_hypertrophic +lichen_planus_linear +lichen_planus_ulcerative +lichen_planus_erythematosus +lichen_planus_erythema_dyschromicum_perstans +lichen_planus_actinicus +lichen_planus_planopilaris_lpp +lichen_planus_pemphigoides +lichen_planus_keratosis_chronica =+lichenoid_benign_keratosis_lbk +lichen_planus_lupus_erythematosus_overlap_lp_le +lichen_nitidus +papular_lichenoid +acanthosis +lichen_striatus +linear +irregular_lichenoid_reaction +follicular_infiltrate +eccrine_infiltrate +lichen_planus_like_keratosis =+lichenoid_benign_keratosis_lbk +lichen_planus_like_keratosis +solitary +civatte_body_prominent +solar_lentigo +lichenoid_drug_eruption +parakeratosis_focal +eosinophilia +plasmacytosis +melanin_incontinence +fixed_drug_eruption_lichenoid +interface_infiltrate_obscuring +neutrophilia +cell_death_above_stratum_basale +interface_dermatitis_pattern <B> +erythema_multiforme_em +interface_infiltrate_obscuring +subepidermal_vesiculation +cell_death_variable +graft_vs_host_disease_gvhd +basal_vacuolation +apoptotic_keratinocyte +satellite_cell_necrosis +apoptotic_keratinocyte +lymphocytic_infiltrate_variable +lupus_erythematosus_le +basal_vacuolation +cell_death_minimal +mucin +follicular_plugging +basement_thickening +dermatomyositis +basal_vacuolation +epidermal_atrophy +dermal_mucin +interface_infiltrate_superficial +poikiloderma +basal_vacuolation +telangiectasia +pigment_incontinence +dermal_sclerosis_late +pityriasis_lichenoides +lymphocytic_vasculitis +epidermal_cell_death +interface_infiltrate_obscuring +hemorrhage_focal +parakeratosis_focal +paraneoplastic_pemphigus +interface_infiltrate_obscuring +subepidermal_vesiculation +cell_death_variable +acantholysis_surabasal +clefting +clefting_subepidermal +eruption_lymphocyte_recovery +maculopapular_eruption +cytoreductive_therapy +acute_myelogenous_leukemia +microscopic +upper_dermal_t_cell_infiltrate +vascular_dilation +apoptotic_keratinocyte +satellite_cell_necrosis +apoptotic_keratinocyte +pautrier_microabscess +aids_interface_dermatitis +hiv_disease +microscopic +interface_infiltrate_obscuring +subepidermal_vesiculation +cell_death_variable +lupus_erythematosus +microscopic +interface_infiltrate +dermatomyositis +poikiloderma_congenitale +hereditary_sclerosing_poikiloderma +kindler_syndrome +congenital_telangiectatic_erythema +lichen_sclerosus_atrophicus_lsa +dyskeratosis_congenita +pityriasis_lichenoides_acuta +pityriasis_lichenoides_chronica +persistent_viral_reaction +perniosis +paraneoplastic_pemphigus +primary_neoplasm +very_rare +secondary_syphilis +prominent_plasma_cells +lichenoid_purpura +lichenoid_contact_dermatitis +secondary_syphilis_late +very_rare +prominent_plasma_cells +porokeratosis +very_rare +drug_eruption +phototoxic_dermatitis_photoallergic +prurigo_pigmentosa +erythroderma +very_rare +mycosis_fungoides +very_rare +pautrier_microabscess +mycosis_fungoides +pautrier_microabscess +epidermotropism +atypia_cytologic +regressing_verucca +regressing_tumor +lichen_amyloidosus +vitiligo +lichenoid_tattoo_reaction +psoriasiform_pattern <C> +psoriasis +psoriasis_treated +aids_associated_psoriasiform_dermatitis =+pruritic_papular_eruption_aids +pustular_psoriasis +reiter_syndrome +pityriasis_rubra_pilaris_prp +orthokeratosis_diffuse +follicular_plugging +parakeratosis_mild +acanthosis_spotty +hypergranulosis +spongiosis_mild +perivascular_infiltrate +very_rare +acantholytic_dyskeratosis_focal +very_rare +parapsoriasis +lichen_simplex_chronicus_lsc +orthokeratosis_compact +hypergranulosis +parakeratosis_focal +psoriasiform_hyperplasia +papillomatosis_mild +papillary_dermis_thickening +subacute_chronic_spongiotic_dermatitis +very_rare +subacute_chronic_spongiotic_dermatitis +spongiosis +eosinophilia +plasmacytosis +erythroderma +very_rare +erythroderma +psoriasiform_hyperplasia +spongiosis_mild +mycosis_fungoides +very_rare +pautrier_microabscess +mycosis_fungoides +pautrier_microabscess +epidermotropism +atypia_cytologic +candidosis +dermatophytosis +inflammatory_linear_verrucous_epidermal_nevus_ilven +norwegian_scabies +bowen_disease_psoriasiform_variant +clear_cell_acanthoma +lamellar_ichthyosis +pityriasis_rosea_pr +pellagra +psoriasiform_acanthosis +orthokeratosis +parakeratosis_focal +acrodermatitis_enteropathica +glucagonoma_syndrome +migratory_erythema_migratory +secondary_syphilis +spongiotic_pattern <D> +neutrophilic_spongiosis +pustular_psoriasis +iga_pemphigus +infantile_acropustulosis +acute_generalized_exanthematous_pustulosis +palmoplantar_pustulosis +dermatophytosis +candidosis +beetle_dermatitis +insect_bite_reaction +arthropod_bite_reaction +pustular_contact_dermatitis +eosinophilic_spongiosis +pemphigus_precursor +very_rare +herpetiform_pemphigus +pemphigus_vegetans +bullous_pemphigoid +idiopathic_eosinophilic_spongiosis +eosiniphilic_polymorphic_pruritic_eruption +allergic_contact_dermatitis +protein_contact_dermatitis +arthropod_bite +eosinophilic_folliculitis =+ofugi_disease +incontinentia_pigmenti +miliarial_spongiosis +miliaria +follicular_spongiosis +infundibulofolliculitis +atopic_dermatitis_follicular +apocrine_miliaria +eosinophilic_folliculitis =+ofugi_disease +irregular_spongiosis +irritant_contact_dermatitis +allergic_contact_dermatitis +nummular dermatitis +sulzberger_garbe_syndrome_??????????????? +seborrheic_dermatitis +atopic_dermatitis_eczema +papular_dermatitis +pompholyx_dyshidrotic_eczema +hyperkeratotic_dermatitis_hand +very_rare +juvenile_plantar_dermatosis +vein_graft_donor_site_dermatitis +stasis_dermatitis +autoeczematization +pityriasis_rosea +papular_acrodermatitis_childhood +spongiotic_drug_reaction +chronic_superficial_dermatitis +very_rare +blaschko_dermatitis_????????????????????? +psoriasis_spongiotic +light_reaction_photoallergic_phototoxic +dermatophytosis +arthropod_bite +grover_disease_spongiotic_transient_acantholytic_dermatosis_tad +toxic_shock_syndrome +pruritic_urticarial_papule_plaque_pregnancy_puppp +pregnant +erythema_annulare_centrifugum_eac +pigmented_purpuric_dermatosis +very_rare +pityriasis_alba +eruption_lymphocyte_recovery +lichen_striatus +erythroderma +very_rare +mycosis_fungoides +very_rare +pautrier_microabscess +mycosis_fungoides +pautrier_microabscess +epidermotropism +atypia_cytologic +mycosis_fungoides +pautrier_microabscess +very_rare +vesiculobullous_pattern <E> +intracorneal_subcorneal_blister +skin_peeling_syndrome +impetigo +staphylococcal_scalded_skin_syndrome_ssss +dermatophytosis +pemphigus_foliaceus +pemphigus_erythematosus +herpetiform_pemphigus +subcorneal_pustular_dermatosis +iga_pemphigus +very_rare +infantile_pustular_dermatosis +acute_generalized_exanthematous_pustulosis +miliaria_crystallina +intraepidermal_blister +spongiotic_blister_disease +palmoplantar_pustulosis +amicrobial_pustulosis_autoimmune +viral_blister_disease +epidermolysis_bullosa +friction_blister +suprabasilar_blister +pemphigus_vulgaris_pv +pemphigus_vegetans_pve +paraneoplastic_pemphigus_parane_pemphigus +hailey_hailey_disease_hhd +darier_disease_keratosis_follicularis +grover_disease_tad +acantholytic_solar_keratosis +subepidermal_blister_pauciinflammatory +epidermolysis_bullosa_eb +porphyria_cutanea_tarda_pct +pseudoporphyria +bullous_pemphigoid_cell_poor +burn +burn_second_degree +burn_third_degree +cryotherapy +toxic_epidermal_necrolysis_ten +suction_blister +blister_over_scar +bullous_solar_elastosis +bullous_amyloidosis +waldenstrom_macroglobulinemia +drug_reaction +kindler_syndrome_?????????????????? +subepidermal_blister_lymphocytic +erythema_multiforme_em +paraneoplastic_pemphigus +bullous_fixed_drug_reacdtion +lichen_sclerosus_et_atrophicus_lsa +lichen_planus_pemphigoides +polymorphous_light_eruption +fungal_infection +dermal_allergic_contact_dermatitis +bullous_leprosy +very_rare +foamy_macrophages_afb_positive +bullous_mycosis_fungoides +very_rare +pautrier_microabscess +bullous_mycosis_fungoides +pautrier_microabscess +epidermotropism +atypia_cytologic +subepidermal_blister_eosinophilic +wells_syndrome +bullous_pemphigoid_bp +pemphigoid_gestationis +pregnant +herpes_gestationis +pregnant +arthropod_bite +drug_reaction +epidermolysis_bullosa_eb +subepidermal_blister_neutrophilic +dermatitis_herpetiformis_dh +linear_iga_bullous_dermatosis +cicatricial_pemphigoid +ocular_cicatricial_pemphigoid +very_rare +localized_cicatricial_pemphigoid +very_rare +deep_lamina_lucida_pemphigoid +anti_p200_pemphigoid +bullous_urticaria +bullous_acute_vasculitis +bullous_lupus_erythematosus_ble +erysipelas +sweet_syndrome =+acute_neutrophilic_dermatosis +epidermolysis_bullosa_acquisita +subepidermal_blister_mast_cell +bullous_urticaria_pigmentosa +irregular_blister_disease +drug_overdose_bulla +methyl_bromide_bulla +etretinate_bulla +puva_bulla +very_rare +puva +cancer_bulla +lymphatic_bulla +diabetic_bulla +granulomatous_pattern_noncaseating_epithelioid <F> +sarcoidal_granuloma +sarcoidosis +blau_syndrome +foreign_body +foreign_body_polarizing +foreign_body_nonpolarizing +syphilis_secondary +very_rare +sezary_syndrome +very_rare +herpes_zoster_scar +systemic_lymphoma +very_rare +variable_immunodeficiency +very_rare +tuberculoid_granuloma +tuberculosis +tuberculid +leprosy_indeterminate +foamy_macrophages_afb_positive +syphilis_late +leishmaniasis +protothecosis +rosacea +perioral_dermatitis +lupus_miliaris_disseminatus_faciei +crohn_disease +necrobiotic_granuloma +granuloma_annulare_ga +necrobiosis_lipoidica_diabeticorum_nld +diabetes_mellitus +necrobiotic_xanthogranuloma_nxd +rheumatoid_nodule_rn +foreign_body_reaction +vaccine_reaction +suppurative_granuloma +chromomycosis +pheohyphomycosis +sporotrichosis +mycobacteriosis_nontuberculous_atypical +blastomycosis +paracoccidioidomycosis +coccidioidomycosis +blastomycosis_like_pyoderma +nocardiosis +actinomycosis +cat_scratch_disease +lymphogranuloma_venereum_lgv +pyoderma_gangrenosum +ruptured_cyst +ruptured_follicle +foreign_body_granuloma +foreign_body_exogenous +polarizable +suture +silica +glass +talc +injection_granuloma +very_rare -polarizable +beryllium +zirconium +very_rare +foreign_body_endogenous +epidermal_inclusion_cyst_ruptured +pilar_cyst_ruptured +irregular_granuloma +melkerson_rosenthal_syndrome +very_rare +cutaneous_histiocytic_lymphangitis +very_rare +elastolytic_granuloma +very_rare +annular_granuloma_ochronosis +very_rare +granuloma_immunodeficiency +very_rare +interstitial_granulomatous_dermatitis +very_rare +interstitial_granulomatous_drug_reaction +very_rare +granulomatous_t_cell_lymphoma +very_rare +vasculopathic_pattern <G> +noninflammatory_purpura +traumatic_purpura +psychogenic_purpura +very_rare +purpuric_drug_reaction +bleeding_diathesis +senile_purpura +vascular_occlusive_disease +fibrin +warfarin_necrosis +livedoid_vasculopathy +disseminated_intravascular_coagulopathy_dic +purpura_fulminans +thrombotic_thrombocytic_purpura_ttp +thrombocythemia +cryoglobulinemia +cholesterol_embolism +antiphospholipid_syndrome_sle +factor_5_leiden_mutation +very_rare +sneddon_syndrome +meningococcemia +gonococcemia +cerebral_autosomal_dominant_arteriopathy_subcortical_infarct_leukoencephalopathy_cadasil +very_rare +acute_vasculitis +leukocytoclastic_vasculitis_lct +henoch_schoenlein_purpura +eosinophilic_vasculitis +very_rare +rheumatoid_vasculitis +urticarial_vasculitis +mixed_cryoglobulinemia +hypergammaglobulinemic_purpura +hypergammaglobulinemic_d_syndrome +septic_vasculitis +erythema_elevatum_diutinum +granuloma_faciale +localized_chronic_fibrosing_vasculitis +microscopic_polyangiitis +polyarteritis_nodosa +kawasaki_disease +superficial_thrombophlebitis +irregular_condition +neutrophilic_dermatosis +sweet_syndrome +bowel_associated_dermatosis_arthritis_syndrome +rheumatoid_neutrophic_dermatosis +acute_generalized_pustulosis +behcet_disease +abscess_forming_neutrophilic_dermatosis +chronic_lymphocytic_vasculitis +toxic_erythema +connective_tissue_disease +pruritic_urticarial_papule_plaque_pregnancy_puppp +pregnant +prurigo_pregnancy +pregnant +gyrate_erythema +annular_erythema +pityriasis_lichenoides +pigmented_purpuric_dermatosis +malignant_atrophic_papulosis_degos +perniosis +rickettsial_infection +viral_infection +pyoderma_gangrenosum +polymorphous_light_eruption +tumor_necrosis_factor_receptor_associated_periodic_syndrome_traps +sclerosing_lymphangitis_penis +leukemic_vasculitis +vasculitis_granulomatosis +crohn_disease +drug_reaction +herpes_zoster +infectious_granulomatous_disease +wegener_granulomatosis +lymphomatoid_granulomatosis +allergic_granulomatosis +lethal_midline_granuloma +giant_cell_arteritis +takayasu_arteritis +irregular_vasculitis +vascular_calcification +pericapillary_fibrin_cuff +vascular_aneurysm +erythermalgia +cutaneous_necrosis +cutaneous_ulceration +minor_tissue_reaction_noninflammatory +epidermolytic_hyperkeratosis +bullous_ichthyosiform_erythroderma +epidermal_nevus +palmoplantar_keratoderma +epidermolytic_acanthoma +disseminated_epiderolytic_acanthoma +nevoid_follicular_hyperkeratosis +actinic_keratosis +acantholytic_dyskeratosis +darier_disease +epidermal_nevus +grover_disease +warty_dyskeratoma +acantholytic_actinic_keratosis +cornoid_lamellation + +papillomatosis <H> +seborrheic_keratosis_sk +acrokeratosis_verruciformis +verruca_vulgaris_vv +epidermodysplasia_verruciformis_ev +verruca_plana_vp +stucco_keratosis +tar_keratosis +very_rare +bowen_disease_sccis +arsenical_keratosis_arker +actinic_keratosis_hypertrophic_hak +acanthosis_nigricans +reticulated_papillomatosis +epidermal_nevus +verrucous_carcinoma +keratosis_follicularis_spinulosa +digitate_keratosis +hyperkeratosis_lenticularis +rubbed_skin +very_rare +scratched_skin +very_rare +acral_angiofibroma <I> +adenoma_sebaceum +angiofibroma_men1 +very_rare +angiofibroma_neurofibromatosis +subungual_fibroma +periungual_fibroma +acquired_acral_fibrokeratoma +fibrous_papule_face +pearly_penile_papule +familial_myxovascular_fibroma +eosinophilic_cellulitis <J> +wells_syndrome +arthropod_bite +internal_cancer +bullous_pemphigoid +dermatitis_herpetiformis +trichophyton_rubra +transpithelial_elimination <K> +necrobiosis_lipoidica_nld +necrobiotic_xanthogranuloma_nxg +perforating_folliculitis +pseudoxanthoma_elasticum_pxe +elastosis_perforans_serpiginosa_eps +reactive_perforating_collagenosis +calcaneal_petechia +very_rare +amyloidosis +chondrodermatatis_nodularis_helicis_chronica_cnhc +urate_crystal =+gouty_tophus +calcinosis_cutis +osteoma_cutis +deep_mycosis +cutaneous_tuberculosis +blastomycosis_like_pyoderma +very_rare +granuloma_inguinale +sarcoidosis +very_rare +foreign_body_granuloma +suture_material +very_rare +lichen_nitidus +very_rare +papular_mucinosis +very_rare +acne_keloidalis_nuchae +very_rare +solar_elastosis +very_rare +cryotherapy_injury +very_rare +calcaneous_tumor +very_rare +absent_stratum_corneum +staphylococcal_scalded_skin_syndrome +pemphigus_foliacius +peeling_skin_syndrome +psoriatic_erythroderma +processing_artefact +dermis_focally_hypercellular_busy +granuloma_annulare_incomplete +granulomatous_dermatitis +granulomatous_drug_reaction +vasculitis_resolving +photodermatosis_chronic +folliculitis +breast_carcinoma_metastatic +desmoplastic_melanoma +kaposi_sarcoma_early_patch +filled_papillary_dermis +lichenoid_reaction +purpuric_dermatosis_pigmented +cutaneous_t_cell_lymphoma +parapsoriasis +mastocytoma +lichen_sclerosus_et_atrophicus_early +basement_membrane_thickened +lupus_erythematosus +dermatomyositis +lichen_sclerosus_et_atrophicus +middermal_infiltrate_mucin +cutaneous_lupus_erythematosus +reticular_erythematous_mucinosis +apoptotic_keratinocyte +lichenoid_tissue_reaction +drug_reaction +light_reaction +resolving_viral +aids_sebopsoriasis +incontinentia_pigmenti +bowen_disease +excoriation +glucagonoma_syndrome +bazex_syndrome +extravasated_erythrocyte +vasculitis +purpuric_eruption_pigmented +drug_eruption +viral_infection +rickettsial_infection +septicemia +erysipelas +arthropod_bite +pityriasis_rosea +bleeding diathesis +scurvy +kaposi_sarcoma +lichen_sclerosus_et_atrophicus +biopsy_trauma +epidermal_cell_pallor +pellagra +acrodermatitis_enteropathica +glucagonoma +hartnup_disease +lactate_dehydrogenase_deficiency_m_subunit +acroerythema +spongiotic_disease +clear_cell_acanthoma +clear_cell_acanthosis +clear_cell_papulosis +pagetoid_dyskeratosis +colloid_keratosis +clear_cell_tumor +epidermal_clear_cell_tumor +clear_cell_acanthoma +bowen_disease +basal_cell_carcinoma +squamous_cell_carcinoma +adnexal_clear_cell_tumor +paget_disease +clear_cell_syringoma +clear_cell_hidradenoma +clear_cell_hidradenocarcinoma +clear_cell_eccrine_adenocarcinoma +clear_cell_myoepithelioma +tricholemmal_carcinoma +sebaceous_adenoma +sebaceous_carcinoma +nevomelanocytic_clear_cell_tumor +balloon_cell_nevus +balloon_cell_melanoma +clear_cell_melanoma +clear_cell_sarcoma +mesenchymal_clear_cell_tumor +clear_cell_dermatofibroma +clear_cell_atypical_fibroxanthoma +clear_cell_fibrous_papule +salivary_gland_clear_cell_tumor +acinic_cell_carcinoma +hyalinizing_clear_cell_carcinoma +clear_cell_mucoepidermoid_carcinoma +metastatic_clear_cell_tumor +renal_cell_carcinoma +breast_carcinoma +hepatocellular_carcinoma +adenocarcinoma_lung +mesothelioma +tissue_reaction_pattern_inflammation +papillary_microabscess +dermatitis_herpetiformis +linear_iga_disease +cicatricial_pemphigoid +cicatricial_pemphigoid_localized +bullous_lupus_erythematosus +epidermolysis_bullosa_acquisita +drug_reaction +hypersensitivity_vasculitis +rare +rheumatoid_neutrophilic_dermatosis +pemphigoid_gestationis +pregnant +eosinophilia +deep_lamina_lucida_pemphigoid +exanthematous_pustulosis_generalized +rare +interstitial_eosinophil +arthropod_bite +cnidarian_contact_dermatitis +parasite_infestation +drug_reaction +toxic_erythema_pregnancy +pregnant +annular_erythema_infancy +infant +wells_syndrome +dermal_hypersensitivity +hypereosinophilic_syndrome +urticaria +bullous_pemphigoid +pemphigoid_gestationis +pregnant +internal_malignancy +rare +intraluminal_giant_cell +melkersson_rosenthal_syndrome +genitourinary_infection +cutaneous_histiocytic_lymphangitis +rosai_dorfman_disease +intravascular_leukocyte +urticaria +lymphomatoid_papulosis +superficial_deep_inflammation +eight_ls +light_reaction +lymphoma +leprosy +foamy_macrophages_afb_positive +lues +lichen_striatus +lupus_erythematosus +necrobiosis_lipoidica +arthropod_bite +drugs +dermatophyte +reticular_erythomatous_mucinosis +urticaria +gyrate_erythema +scleroderma_localized +drug_reaction +superficial_perivascular_inflammation +drug_reaction +dermatophytosis +viral_exanthem +chronic_urticaria +erythrasma +superficial_annular_erythema +pigmented_purpuric_dermatosis +resolving_dermatosis +superficial_deep_dermal_inflammation +folliculitis +perifolliculitis +panniculitis +epidermis +maturation +keratinization +pigmentation +dermis +collagen +elastic_tissue +cutaneous_mucinosis +generalized_myxedema +pretibial_myxedema +reticular_erythematous_mucinosis +scleredema +scleromyxedema +papular_mucinosis +acral_persistent_papular_mucinosis +cutaneous_mucinosis_infancy +focal_mucinosis +digital_mucous_cyst +mucocele +nevus_mucinosus +dermal_mucinosis_secondary +follicular_mucinosis +mucopolysaccharidosis +deposit +appendage +cyst +epidermal_cyst +epidermal_keratinization +keratohyaline_granule +flattened_surface_epithelium +location_gross +trunk +neck +face +location_micro +middermis +lower_dermis +epidermal_cyst_proliferating +multiloculated_cystic_space +proliferating_epithelium +squamous_eddy +frequent +epidermal_type +epidermal_type_keratinization +tricholemmal_type +tricholemmal_type_keratinization +hpv_cyst +hypergranulosis +papillated +intracytoplasmic_inclusion +vacuolar_keratin +digitated +verrucous +tricholemmal_cyst +tricholemmal_type_keratinization +cholesterol_cleft +calcification -keratohyaline_granule +hybrid_cyst +tricholemmal_type_keratinization +inner +epidermal_type_keratinization +outer +hair_matrix_cyst +pigmented_follicular_cyst +cutaneous_keratocyst +vellus_cyst +steatocystoma +milium_cyst +eccrine_hydrocystoma +apocrine_cystadenoma +bronchogenic_cyst +branchial_cyst +thymic_cyst +cutaneous_ciliated_cyst +median_raphe_cyst +dermoid_cyst +subcutis +panniculitis +panniculitis_septal +erythema_nodosum +necrobiosis_lipoidica +scleroderma +panniculitis_lobular +erythema_induratum +nodular_vasculitis +subcutanous_fat_necrosis_newborn +sclerema_neonatorum +cold_panniculitis +weber_christian_disease +alpha1_antitrypsin_deficiency +cytophagic_histiocytic_panniculitis +poststeroid_panniculitis +connective_tissue_panniculitis +lipodystrophy_syndrome +membranous_lipodystrophy +lipodermatosclerosis +factitial_panniculitis +infective_panniculitis +noninfective_pannicul

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	+	-	-	-	+
16	+	-	-	-	-

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	▒	▒	▒	▒	▒
16	▒	▒	▒	▒	▒

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	-	+	+	+
10	-	+	+	-
11	-	+	-	+
12	-	+	-	-
13	-	-	+	+
14	-	-	+	-
15	-	-	-	+
16	-	-	-	-

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	▒	▒	▒	▒
10	-	+	+	-
11	▒	▒	▒	▒
12	-	+	-	-
13	▒	▒	▒	▒
14	-	-	+	-
15	▒	▒	▒	▒
16	-	-	-	-

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	+	-	-	-	+
16	+	-	-	-	-

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	▒	▒	▒	▒	▒
16	▒	▒	▒	▒	▒

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	-	+	+	+
10	-	+	+	-
11	-	+	-	+
12	-	+	-	-
13	-	-	+	+
14	-	-	+	-
15	-	-	-	+
16	-	-	-	-

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	▒	▒	▒	▒
10	-	+	+	-
11	▒	▒	▒	▒
12	-	+	-	-
13	▒	▒	▒	▒
14	-	-	+	-
15	▒	▒	▒	▒
16	-	-	-	-

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

1. DISCLAIMER.

2. TABLE OF CONTENTS.

3. ABSTRACT: APIII #324.

OUTLINE OF EPOSTER.

Tumors of mucosal surfaces.

Resource Description Framework (RDF).

Technology.

Methods.

Logic model.

Namespace.

Resource Description Framework (RDF).

Laboratory Digital Imaging Project.

Event subclass.

Example: LDIP for Class Patient.

Monoparentality assumption.

Biomedical Semantic Content (Meaning).

Sample Ordered Triples.

Spelling, Punctuation Conventions.

Online Embryology RDF Resources.

Online Anatomy RDF Resources.

Relationspace.

Fuzzyspace.

Sutton's Law.

Zebra Rule.

Zebra Rule: Modal Logic.

Classical/Crisp Set Theory.

Set Theory Operations.

Fuzzy Set Theory.

Fuzzy Set Theory: Example.

Propositional Logic.

Rules of Propositional Logic.

Nand.

Nandset.

Possible Worlds / Possible Patients Model.

Mathematical Theorem.

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.

Mathematical Theorems: Proof Strategies.

Theorems of Classical/Modal Logic.

Live Computer Demonstration.

Public-domain Perl Source Code.

Model for Ethical Data Collection.

Complementizers for Ethical Data Collection.

Model for Ethical Data Collection.

Summary of Results.

Conclusions.

4. INTRODUCTION.

5. NAMESPACE.

6. RELATIONSPACE.

7. FUZZYSPACE.

8. PROPOSITIONAL LOGIC, SET THEORY.

9. MATHEMATICAL THEOREMS.

INTERCALATION.

RETIREMENT.

10. LIVE COMPUTER DEMONSTRATION.

11. MODEL FOR ETHICAL DATA COLLECTION.

12. EXAMPLES.

14. DERMATOPATHOGENESIS.

15. DISCUSSION.

NOTES

RELATIONSPACE.

Theorem §9.5. BACKTRACKING.

Theorem §9.6. INTERCALATION.

Theorem §9.7. RETIREMENT.

Small_round_blue_cell_tumor.

MATHEMATICAL MODEL.

Possible Patient Description Table / Small_round_blue_cell_tumor.

Possible Patient Description Table / SHADED Small_round_blue_cell_tumor.

PROPOSITIONAL LOGIC.

Truth Table / Possible Patient Descriptions.

Possible Patient Description Table.

Possible Patient Description Table / Anatomically Possible.

SUTTON'S LAW / ZEBRAS.

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

Possible Patient Description Table /
Small_round_blue_cell_tumor.

Possible Patient Description Table /
SHADED Small_round_blue_cell_tumor.

Truth Table /
Possible Patient Descriptions.

Possible Patient Description Table /
Anatomically Possible.

Skin Biopsy
Possible Patient
Descriptions Model.

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀, -s} Supersets Removed.

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀,+s,+i,+n}, {+∀,+s,-i,-n}, {+∀, -s} Supersets Removed.

Skin Biopsy:
Possible Patient Descriptions Model.
{+∀,+s,+i,+n}, {+∀,+s,-i,-n}, {+∀, -s} Supersets Removed.

27. APPENDIX H.
THEOREMS, FROM
ZEMAN'S MODAL LOGIC.
CHAPTERS 1,2.

28. APPENDIX I.
EMBRYOGENESIS.

29. APPENDIX J.
FOUNDATIONS OF DERMATOPATHOLOGY DIAGNOSIS.

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	+	-	-	-	+
16	+	-	-	-	-

Number	p Small_round blue_cell_tumor	r Retinoblastoma	q Merkel_cell	s Small_cell carcinoma_lung	t Heavy_smoker
1	+	+	+	+	+
2	+	+	+	+	-
3	+	+	+	-	+
4	+	+	+	-	-
5	+	+	-	+	+
6	+	+	-	+	-
7	+	+	-	-	+
8	+	+	-	-	-
9	+	-	+	+	+
10	+	-	+	+	-
11	+	-	+	-	+
12	+	-	+	-	-
13	+	-	-	+	+
14	+	-	-	+	-
15	▒	▒	▒	▒	▒
16	▒	▒	▒	▒	▒

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	-	+	+	+
10	-	+	+	-
11	-	+	-	+
12	-	+	-	-
13	-	-	+	+
14	-	-	+	-
15	-	-	-	+
16	-	-	-	-

Number	Female	>18 years	Fair skin	Uterus
1	+	+	+	+
2	+	+	+	-
3	+	+	-	+
4	+	+	-	-
5	+	-	+	+
6	+	-	+	-
7	+	-	-	+
8	+	-	-	-
9	▒	▒	▒	▒
10	-	+	+	-
11	▒	▒	▒	▒
12	-	+	-	-
13	▒	▒	▒	▒
14	-	-	+	-
15	▒	▒	▒	▒
16	-	-	-	-