1.1. Core doctrines in human pathology, reflected in:

1.1.1. Consensus conference proceedings (Kao, 2004; Epstein, 1998).
1.1.2. Review texts. (Sinard; Haber, 2000).
1.1.3. Specialty Texts (Bostwick; Miettinnen).
1.1.4. Tumor staging manuals (AJCC).
1.1.5. Tumor classifications (Berman, 2004).
1.1.6. Pathology reporting protocols. (CAP; ADASP).
1.1.7. Continuing Medical Education (Checkpath).

1.2. Organized as hierarchical lists.
1.3. Transferred to commercial spreadsheet programs (Excel VBA).
1.4. Displayed as rectangular tables. Analyzed with statistical tests.
1.5. Mathematical model and computer script (Excel VBA).
1.6. Nandset computing method: finds all solutions (mathematical completeness).
1.7. Determines the logical consistency of lists.
1.8. Computability in polynomial computing resources.
1.9. Modal logic theory of ethical data collection (Moore, 2003).
1.10. Classical logic has no ordering concept: a statement is either true or false or unknown.
1.11. Bayesian logic has too much ordering: exact probability numbers are required, and subject to arithmetic operations, often unjustified.
1.12. Order logic is ordinal only.

2.1 Zermelo-Frankel Set Theory. (Kleene; Suppes).

C = is a member of.
{} = Ø = empty set.
U = set-union.
/\ =set-intersection.
- = set-subtraction.
c = subset.

2.2 Commercial spreadsheet application. Microsoft® Excel®. (Visual Basic for Applications, Excel 2000).
2.3 Embedded programming language. Microsoft® Visual Basic® for Applications. (Visual Basic for Applications, Excel 2000).
2.4 Perl programming language. (Perl).

3.1. Spreadsheet, Ш: collection of sheets, or folios. Each folio: rectangular table; rows, columns, cells.
3.2. Patient-folio, П: individual patient data.
3.3. Disease-folio, Д: logic of disease.
3.4. Combined-folio, Ш = (П U Д).

4.1. Patient-folio, П:

4.1.1. Observed and inferred statements for each patient.
4.1.2. Quantitative, interval, ranked, categorical, and binary data allowed.
4.1.3. All data expressed as true, false, or missing-value statements.

4.1.4. Example:

П	İ	♀	♂
Mary	+İ	+♀	~♂
Bill	+İ	~♀	+♂
Pat	+İ	+♀	.

4.1.5. That is: Mary -> İ; Mary -> ♀; Mary -> ~♂; Bill -> İ, etc.

4.2. Disease-folio, Д:

4.2.1. Topmost-leftmost cell: origin, İ^[0], true for all patients.
4.2.2. Every filled-in cell: parent-concept, child-concept, or both.
4.2.3. Every parent-concept has one-or-more children-concepts.
4.2.4. First child-concept, one-step-down-one-step-right from its parent-concept.
4.2.4. Subsequent children of same parent, same column as siblings.
4.2.5. Each parent-cell and its parent-cells implies the inclusive-or of its children-cells.

4.2.6. Example:

Д	0	1	2
1	+İ	.	.
2	.	+♀	.
3	.	.	~♂
4	.	+♂	.
5	.	.	~♀

4.2.7. That is: İ -> (♀ | ♂); İ & ♀ -> ~♂; İ & ♂ -> ~♀.

4.3. Combined-folio, Ш:

4.3.1. Example:

Ш	0	1	2	3
1	Mary	.	.	.
2	.	+İ	.	.
3	.	.	+♀	.
4	.	.	.	~♂
5	Bill	.	.	.
6	.	+İ	.	.
7	.	.	+♂	.
8	.	.	.	~♀
9	Pat	.	.	.
10	.	+İ	.	.
11	.	.	+♀	.
12	+İ	.	.	.
13	.	+♀	.	.
14	.	.	~♂	.
15	.	+♂	.	.
16	.	.	~♀	.

5.1. SPREADSHEET-REPRESENTATION. Rectangular tables constructed according to the above rules:

Advantage: Easy-to-read.

Examples:

П	İ	♀	♂
Mary	+İ	+♀	~♂
Bill	+İ	~♀	+♂
Pat	+İ	+♀	.

Д	0	1	2
1	+İ	.	.
2	.	+♀	.
3	.	.	~♂
4	.	+♂	.
5	.	.	~♀

5.2. SYMBOLIC-LOGIC-REPRESENTATION. Set of statements of the form: it is true that X; or X implies Y; or X and Y; or X inclusive-or Y; or not-x; where x and y are true-false statements.

Advantage: Traditional logic notation.

Examples:

Mary -> İ.
Mary -> ♀.
Mary -> ~♂.
Bill -> İ.
Bill -> ♂.
Bill -> ~♀.
Pat -> İ.
Pat -> ♀.
İ -> (♀ | ♂).
İ & ♀ -> ~♂.
İ & ♂ -> ~♀.

5.3. NANDSET-REPRESENTATION. Collection of sets of the form: {X, Y, Z,...}, where not all of X, Y, Z,... are true at once.

Advantage: Transparent calculations, i.e., X [+] Y = Z, nandset addition, where X={a, b, ...}, Y={~a, c, ,,,}, and Z={b, c, ...}. Nandset addition, performed exhaustively, completely and consistently defines the system, and may be performed exhaustively after polynomial steps in a suitably constrained system (i.e., not an exponential number of logic expressions). For nandset, X, that belongs to either П or Д and anonymous completely described patient, A, then X is NOT a subset of A.
See: http://www.netautopsy.org/modlthry.htm

Examples:

1. {Mary, ~İ}.
2. {Mary, ~♀}.
3. {Mary, +♂}.
4. {Bill, ~İ}.
5. {Bill, ~♂}.
6. {Bill, +♀}.
7. {Pat, ~İ}.
8. {Pat, ~♀}.
9. {İ, ~♀, ~♂}.
10. {İ, +♀, +♂}.
11. {İ, +♂, +♀}.

Note that nandsets 10 and 11 are redundant (i.e., Zermelo-Frankel sets are order-insensitive).

Sample calculation:

{Pat, ~İ} [+] {İ, +♀, +♂} = {Pat, +♀, +♂}.
{Pat, ~♀} [+] {Pat, +♀, +♂} = {Pat, +♂}

Therefore, one concludes that Pat is not-male.

6.1. ATOMSET of distinct statements (atoms, A), each with definite true-false status.

6.2. Each atom, a C A, is either a unique PATIENT-IDENTIFIER, a DATUM, or a MEDICAL-ENTITY, i.e., A = (P U D U E), (P /\ D) = Ø, (P /\ E) = Ø, (D /\ E) = Ø.

6.3. No self-reference paradoxes, e.g., no patient can be named, "I am not a patient".

6.4. Special medical-entity: İMPORTANCE, İ^[0]. Every patient is important.

6.5. A patient-atom has no negation.

6.6. Each non-patient atom, a C (A-P), has an EXACT NEGATION, ~a C (A-P).

6.7. Set of anonymous completely-described patients, Ю. For every U C Ю:

6.7.1. U c (A-P).

6.7.2. For every U C Ю and u_k C U, ~u_k ~C U.

6.7.3. For every u=(u₁,...,u_Л) C U, there exists a unique INFLECTION POINT, I, at which:

6.7.3.1. For 0< j < I, u^j/u^Л = -1; and

6.7.3.2. For I < k < Л, u^k/u^I = +1.

6.8. Example. Two-level division of female/male, Л = 2, T -> (♂¹ | ♀²).

Ю	T0	♀1	♀2	♂1	♂2	Description
1	~	+	+	~	~	Usual female non-teamster.
2	~	+	~	~	+	Weak female non-teamster.
3	~	~	~	+	+	Usual male non-teamster.
4	~	~	+	+	~	Weak male teamster.
5	+	+	+	~	~	Usual female non-teamster.
6	+	+	~	~	+	Weak female teamster.
7	+	~	~	+	+	Usual male teamster.
8	+	~	+	+	~	Weak male teamster.

6.9. Example. Three-level division of female/male, Л = 3, T -> (♂¹ | ♀³).

Ю	T0	♀1	♀2	♀3	♂1	♂2	♂3	Description
1	~	+	+	+	~	~	~	Usual female non-teamster.
2	~	+	+	~	~	~	+	Weaker female non-teamster.
3	~	+	~	~	~	+	+	Weakest female non-teamster.
4	~	~	~	~	+	+	+	Usual male non-teamster.
5	~	~	~	+	+	+	~	Weaker male non-teamster.
6	~	~	+	+	+	~	~	Weakest male non-teamster.
7	+	+	+	+	~	~	~	Usual female teamster.
8	+	+	+	~	~	~	+	Weaker female teamster.
9	+	+	~	~	~	+	+	Weakest female teamster.
10	+	~	~	~	+	+	+	Usual male teamster.
11	+	~	~	+	+	+	~	Weaker male teamster.
12	+	~	+	+	+	~	~	Weakest male teamster.

6.10. Example. Three-level division of active colitis.

Ю	Fam Hx Crohn's.	Hx Foreign Travel.	Ulcerative colitis1	Ulcerative colitis2	Ulcerative colitis3	Crohn's colitis1	Crohn's colitis2	Crohn's colitis3	Infectious colitis1	Infectious colitis2	Infectious colitis3
1	+	~	+	~	~	+	+	~	+	~	~
2	~	+	+	~	~	+	~	~	+	+	~
3	+	+	+	~	~	+	±	~	+	±	~

7.1. NOTATION: {X^[i]} = {xⁱ}. {X^[i} = {xⁱ, xⁱ⁺¹, ..., x^Л}. {X^i]} = {x⁰, x¹, ..., xⁱ}.

7.2. Classical-logic:

Every (important) patient, İ, is either a female, ♀, or a male, ♂. If a patient is female, then the patient is not-male. If a patient is male, then the patient is not-female.

Д	0	1	2
1	+İ[0]	.	.
2	.	+♀[0]	.
3	.	.	~♂[0]
4	.	+♂[0]	.
5	.	.	~♀[0]

7.3. Order-logic:

Every patient, İ, is either a teamster, T or else not-a-teamster, ~T. Among teamsters, males are more frequent than females.

Д	0	1	2
1	+İ[0]	.	.
2	.	~T[0]	.
3	.	.	♀[1
4	.	.	♂[2
5	.	T[0]	.
6	.	.	♂[1
7	.	.	♀[2

The nandsets are:

{+i⁰, +t⁰, ~t⁰}.
{+i⁰, ~t⁰, ~♀¹, ~♂²}.
{+i⁰, ~t⁰, ~♀², ~♂²} (vacuous).
{+i⁰, +t⁰, ~♂¹, ~♀²}.
{+i⁰, +t⁰, ~♂², ~♀²} (vacuous).

That is, a patient is either a teamster or not; among non-teamsters, females are more frequent than males; and among teamsters, males are more frequent than females.

7.4. Subset table of anonymous completely-described patients, teamsters only, Л=2, T -> (♂¹ | ♀²).

Ю	T0	♀1	♀2	♂1	♂2	Description	Status
1	+	+	+	~	~	Usual female teamster.	OK
2	+	+	~	~	+	Weak female teamster.	Excluded
3	+	~	~	+	+	Usual male teamster.	OK
4	+	~	+	+	~	Weak male teamster.	OK

In this table, all males but only usual females take a job as a teamster, i.e., male teamsters (lines 2,3) are more frequent than female teamsters (line 1).

7.5. Subset table of anonymous completely-described patients, teamsters only, Л=3, T -> (♂¹ | ♀³):

Ю	T0	♀1	♀2	♀3	♂1	♂2	♂3	Description
1	+	+	+	+	~	~	~	Usual female teamster.
2	+	+	+	~	~	~	+	Weaker female teamster.
3	+	+	~	~	~	+	+	Weakest female teamster.
4	+	~	~	~	+	+	+	Usual male teamster.
5	+	~	~	+	+	+	~	Weaker male teamster.
6	+	~	+	+	+	~	~	Weakest male teamster.

8.1. THEOREM 1. Classical-logic inclusive-or.
The spreadsheet:

Д	0	1
1	+İ[0]	.
2	.	+A[0]
3	.	~A[0]

is vacuous.
Proof. The only nandset is: {i⁰, ~a⁰, +a⁰}, which is vacuous, since it excludes no anonymous completely-decribed patient.

8.2. THEOREM 2. Classical-logic and.
The spreadsheet:

Д	0	1	2
1	+İ[0]	.	.
2	.	+A[0]	.
3	.	.	~A[0]

is contradictory.
Proof. The nandsets are: {i⁰, ~a⁰} [+] {i⁰, +a⁰, +a⁰} = Ø, which is contradictory, since it excludes all anonymous completely-decribed patients.



8.3. THEOREM 3. Order-logic inclusive-or.
The spreadsheet:

Д	0	1
1	+İ[0]	.
2	.	+A[1
3	.	~A[2

is NOT vacuous.
Proof. The nandsets are: {+i⁰, ~a¹, +a²} and {+i⁰, ~a², +a²}. The first nandset is not vacuous, since it excludes the anonymous completely-decribed patient, {+i⁰, ~a⁰, ~a¹, +a²}



8.4. THEOREM 4. Order-logic and.
The spreadsheet:

Д	0	1	2
1	+İ[0]	.	.
2	.	+A[1	.
3	.	.	~A[2

is contradictory.
Proof. The nandsets are: {+i⁰, ~a¹}; {+i⁰, ~a²}; {+i⁰, +a¹, +a²}; and {+i⁰, +a², +a²} = {+i⁰, +a²}. Then {+i⁰, ~a²} [+] {+i⁰, +a²} = Ø, which is contradictory.

Д	0	1	2	3	4
0	+İ0	.	.	.	.
1	.	+İ0	.	.	.
2	.	.	♂	.	.
3	.	.	.	~♀	.
4	.	.	♀	.	.
5	.	.	.	~♂	.
6	.	.	.	.	□4~PRC
7	.	+İ0	.	.	.
8	.	.	~□kX	.	.
9	.	.	□kX	.	.
10	.	.	.	□k-1X	.
11	.	+İ0	.	.	.
12	.	.	~♂	.	.
13	.	.	♂	.	.
14	.	.	.	~>60Y	.
15	.	.	.	>60Y	.
16	.	.	.	.	□2+PRC
17	.	.	♂	.	.
18	.	.	.	~USX	.
19	.	.	.	+USX	.
20	.	.	.	.	□2+PRC
21	.	.	♂	.	.
22	.	.	.	~PFX	.
23	.	.	.	+PFX	.
24	.	.	.	.	□4+PRC
25	.	.	♂	.	.
26	.	.	.	~□2+PRC	.
27	.	.	.	□2+PRC	.
28	.	.	.	.	#PSA.
29	.	.	.	□3+PRC	.
30	.	.	.	.	#PBX.

Д	0	1	2
0	+İ	.	.
1	.	~colitis	.
2	.	+colitis	.
3	.	.	+infectious colitis.
4	.	.	+pseudomembranous colitis.
5	.	.	+radiation colitis.
6	.	.	+ischemic colitis.
7	.	.	+microscopic colitis.
8	.	.	+diversion colitis.
9	.	.	+self-limited colitis.
10	.	.	+focal active colitis.
11	.	.	+inflammatory bowel disease.

Д	0	1	2	3	4
0	+İ	.	.	.	.
1	.	~skin.	.	.	.
2	.	+skin.	.	.	.
3	.	.	non-neoplastic dermatoses.	.	.
4	.	.	.	inflammatory dermatoses.	.
5	.	.	.	.	acute inflammatory dermatoses.
6	.	.	.	.	chronic inflammatory dermatoses.
7	.	.	.	.	granulomatous inflammatory dermatoses.
8	.	.	.	.	infectious dermatoses.
9	.	.	.	vesiculobullous dermatoses.	.
10	.	.	.	follicular dermatoses.	.
11	.	.	.	atrophic dermatoses.	.
12	.	.	.	connective-tissue dermatoses.	.
13	.	.	.	reactive dermatoses.	.
14	.	.	neoplastic dermatoses.	.	.
15	.	.	.	keratinocytic neoplastic dermatoses.	.
16	.	.	.	appendageal neoplastic dermatoses.	.
17	.	.	.	fibroblastic neoplastic dermatoses.	.
18	.	.	.	melanocytic neoplastic dermatoses.	.

Д	0	1	2	3	4
0	+İ	.	.	.	.
1	.	<Three weeks.	.	.	.
2	.	+Three weeks.	.	.	.
3	.	.	+Amnionic cavity.	.	.
4	.	.	+Ectoderm.	.	.
5	.	.	.	+Lateral ectoderm.	.
6	.	.	.	+Neuro-ectoderm.	.
7	.	.	+Mesoderm.	.	.
7	.	.	.	+Paraxial mesoderm.	.
8	.	.	.	+Somite.	.
9	.	.	.	+Intermediate mesoderm.	.
10	.	.	.	+Lateral plate mesoderm.	.
11	.	.	.	.	+Somatic mesoderm.
12	.	.	.	.	+Splanchnic mesoderm.
13	.	.	+Entoderm.	.	.
14	.	.	+Intra-embryonic Coelom.	.	.
1	.	>Three weeks.	.	.	.

From Dr. Berman's Cancer Classification, based upon the embryologic origin of tumor stem cells.

embryonic
    primitive
        primitive_differentiating
           totipotent_or_multipotent_differentiating
           limited_differentiating
        germ cell
        primitive_non_differentiating
    non_primitive
        endoderm_or_ectoderm
            endoderm_or_ectoderm_surface
            endoderm_or_ectoderm_endocrine
            endoderm_or_ectoderm_parenchymal
            odontogenic_epithelium
        mesoderm
            mesenchyme
                connective_tissue
                    muscle
                    fibrous_tissue
                    vascular
                    adipose_tissue
                    bone_cartilage
                heme_lymphoid
            non_mesenchymal_mesoderm
                coelomic
                    coelomic_ductal
                    coelomic_cavities
                    coelomic_gonadal
                sub_coelomic
                    sub_coelomic_gonadal
                    sub_coelomic_endocrine
                    sub_coelomic_nephric
        neuroectoderm_neural_plate
            neural_tube
               neural_tube_parenchyma
               neural_tube_lining
            neural_crest
               peripheral_nervous_system
               neural_crest_endocrine
               neural_crest_melanocytic

Ю	0	1	2	3	4	5	6
0	+İ	.	.	.	.	.	.
1	.	+Embryonic.	.	.	.	.	.
2	.	.	+Primitive embryonic.	.	.	.	.
3	.	.	.	+Primitive differentiating embryonic.	.	.	.
4	.	.	.	.	+Totipotent or_multipotent primitive differentiating embryonic.	.	.
5	.	.	.	.	+Limited primitive differentiating embryonic.	.	.
6	.	.	.	+Germ_cell embryonic.	.	.	.
7	.	.	.	+Primitive non_differentiating embryonic.	.	.	.
8	.	.	+Non-primitive embryonic.	.	.	.	.
9	.	.	.	+Endoderm or_ectoderm.	.	.	.
10	.	.	.	.	+Endoderm or_ectoderm surface.	.	.
11	.	.	.	.	+Endoderm or_ectoderm endocrine.	.	.
12	.	.	.	.	+Endoderm or_ectoderm parenchymal.	.	.
13	.	.	.	.	+Odontogenic epithelium.	.	.
14	.	.	.	+Mesoderm.	.	.	.
15	.	.	.	.	+Mesenchyme.	.	.
16	.	.	.	.	.	+Connective tissue.	.
17	.	.	.	.	.	.	+Muscle.
18	.	.	.	.	.	.	+Fibrous_tissue.
19	.	.	.	.	.	.	+Vascular.
20	.	.	.	.	.	.	+Adipose_tissue.
21	.	.	.	.	.	.	+Bone_cartilage.
22	.	.	.	.	+Non-mesenchymal mesoderm.	.	.
23	.	.	.	.	.	+Coelomic mesoderm.	.
24	.	.	.	.	.	.	+Coelomic ductal mesoderm.
25	.	.	.	.	.	.	+Coelomic cavity mesoderm.
26	.	.	.	.	.	.	+Coelomic gonadal mesoderm.
27	.	.	.	.	.	+Subcoelomic mesoderm.	.
28	.	.	.	.	.	.	+Subcoelomic gonadal mesoderm.
29	.	.	.	.	.	.	+Subcoelomic endocrine mesoderm.
30	.	.	.	.	.	.	+Subcoelomic nephric mesoderm.

13.0. M = highest level of certainty. H = last interval in time.
ÇШ = all true logical consequences of spreadsheet Ш.
çШ = all computed consequences of spreadsheet Ш.
Theorem: çШ = ÇШ.

13.1. Double-negative rule. The double-negative of each atomic-statement equals the positive of that atomic-statement.

13.1.1. For every +a C A, -a C A, -a ~= a, ++a = +a, and --a = +a.
13.1.2. For every +a C A, $^ka = $^ka; #a=#-a; !a=!-a..

13.2. Progressive certainty rule. A more-certain atomic-statement implies a less-certain atomic-statement.

13.2.1. ($^ka->$^k-1a)@Д₀.
13.2.2. Nandset definition: {+$^ka,-$^k-1a} C Д₀ for every k, 1 < k < M-1 and a C A.

13.3. Data-absolute rule. An observed-datum is equally true or false or missing-value at all levels of certainty.

13.3.1. ($d->$^Md)@Д₀.
13.3.2. Nandset definition: {$d,-$^Md} C Д₀, for every d C D.

13.4. Hippocratic rule. Data should be not collected unless needed (Hippocratic).

13.4.1. (-#d->-!d)@Д₀.
13.4.2. Nandset definition: {-#d,+!d} C Д₀, for d C D.

13.5. Conative rule. Necessary data should be collected if the patient consents (conative).

13.5.1. ((-$d&#d)->!d)@Д₀.
13.5.2. Nandset definition: {-$d,+#d,-!d} C Д₀, for d C D.

13.6. Vexative rule. Data or demanded in order of medical need.

13.6.1. keð..d-vexative: (□^ke&□^kð, ...,&-$d)->(#d| □^k+1-e) @ Д₀.
13.6.2. Nandset definition: {+$^ke,e,+$^kð,ð,..,-$d,-#d, -$^k+1e} C Д₀, for 1 < k < M-2, ðC D, and e C E.

13.7. Ontology rule. Necessary data increase one's certainty of medical entities.

13.7.1. (□^kð..)-> (□^ke|□^k+1-e) @ Д₀, 1<k<M-2.
What is □^M-e?
Or, more interestingly, what is □^∞-e?
13.7.2. Nandset definition: {+$^kð,ð,..,-e,-$^k+1e} C Д₀, for 1 < k < M-2, d C D, ð c (D - {+d,-d}).

13.8. Ethical data collection rule. All ethical data are either positive, negative, failed-attempt, or not-attempted.

13.8.0. If d is d-Hippocratic, then there exists at most one I, 1 < I < H, such that:
13.8.1. (POS-DATA): +$d, +d, +!d true for Д_I, xor
13.8.2. (NEG-DATA): +$d, -d, +!d true for Д_I, xor
13.8.3. (FAIL-DATA): -$d, +!d true for Д_I, xor
13.8.4. (NOTRY-DATA): -$d, +$d, -#d, +#d, -!d, +!d not true for Д_I.

13.9. Cover Rule. Entities are re-calculated at each moment in time, based upon available evidence at the time; Data, once collected, are never changed or forgotten.

13.9.1. It is true that -$^ka @ C_I if and only if it is not true that $^ka @ ÇД_I
13.9.2. Nandset definition: {$^ka} C C_I if and only if {-$^ka} ~ C ÇД_I, for 1 < I < H, 1 < k < M, and a C A.

Evidence-based statistical measures for order-logic.

14.1. A 2×2 CONTINGENCY TABLE is a rectangular table in which one binary factor (example: sex, ♀ vs. ♂) is compared to a second binary factor (example: teamster, ~T vs. T):

Д	♀	♂	.
~T	a	b	v
+T	c	d	w
.	x	y	z

14.2. Now suppose that 250 female employees and 250 male employees from a particular shipping company are surveyed, and the OBSERVED VALUES for totals, are as follows:

Д	♀	♂	.
~T	245	205	450
+T	5	45	50
.	250	250	500

Is the result statistically significant? That is, are there disproportionately more male-teamsters than female-teamsters?

14.3. The raw-data values, 245=a, 205=b, 5=c, 45=d, are the cell-totals. The row-sum values, 245+205=450=v and 5+45=50=w are the row-totals. The column-sum values, 245+5=250=x and 205+45=250=y are the column-totals. The sum of all cell-totals, which equals the the sum of the row-totals, as well as the sum of the column-totals, is the grand-total, a+b+c+d=v+w=x+y=z.

14.4. The usual statistical analysis procedures are: the X² test and the Fisher exact test. In these tests, we obtain EXPECTED VALUES for totals, as follows:

Д	♀	♂	.
~T	A	B	V
+T	C	D	W
.	X	Y	Z

where V=v; W=w; X=x; Y=y; and Z=z (i.e., marginal and grand totals are equal to observed); and A=(V×X)/Z; B=(V×Y)/Z; C=(W×X)/Z; and D=(W×Y)/Z. For the example: A=225, B=225, C=25, D=25, V=450, U=50, W=50, Z=500.

14.5. In the TOKEN SWAP TEST, one starts with the expected value of, say, the upper left cell, A, and adds (or subtracts) tokens from A, one by one, until A -> A=1 -> A+2 -> -> a, in a manner that the marginal totals remain constant. That is, if A -> A+1, then B -> B-1, C -> C-1, D -> D+1, so that V = (A+1)+(B-1) = v, W = (C-1)+(D+1) = w, X = (A+1)+(C-1) = x, and Y = (B-1)+(D+1) = y, Thus, in general, if A -> -> A+q, then B -> B-q, C -> C-q, and D -> D+q.

14.6. When A+q -> A+q+1, then there are B-q tokens that may be drawn from the upper right cell and D-q tokens that may be drawn from the lower left cell. But when A+q -> A+q-1, B-q+1 tokens that may be drawn from the upper right cell and D-q+1 tokens that may be drawn from the lower left cell. Thus there are ... swaps that increase A+q to A+q+1, and ... swaps that decrease A+q to A+q-1. This generates a token swap distribution.....

14.7. The superiority of the token swap test over traditional contingency table analysis is that one can initialized the expected values at A+Q, ... , based upon your medical experience, where max(-A,-D) < Q < min(B,C). For example, if one expects that male teamsters would be twice as numerous as female teamsters, then one could force 2×C = D, and solve for Q. In traditional analyses, Q is required to be zero.



14.8. In this report, we suggest that the token swap test is a more appropriate analysis, since the expected values of the cell-totals are not preset by statistical theory, but can be manipulated according to medical experience.

15.1. Two intellectual pillars of anatomic pathology informatics: image-recognition; concept-management.
15.2. Concepts in in descending order of importance on spreadsheet.
15.3. Syllogistic quality of reasoning in anatomic pathology.
15.4. Some statements with greater weight than others.
15.5. Formalism is ordinal, not cardinal,
15.6. Stepwise character of medical intentions and therapies.
15.7. Reduces to classical symbolic logic if all superscripts are zero.
15.8. Supports theories of ethical data collection and contingency table analysis.
15.9. Mathematical theories can organize medical knowledge and patient data.
15.10. Can enhance clinicopathologic data collection and surveillance.