Macro vs Micro studies overcoming Cartesian Dilemma in a chosen System. Implicate order and explicate order are concepts coined by David Bohm to describe two different frameworks for understanding the same phenomenon or aspect of reality. He uses these notions to describe how the same phenomenon might look different, or might be characterized by different principal factors, in different contexts such as at different scales.

1. Based on Innovation by Dr. Barry Robson

2. The Grand Design
PROBABILSTIC SEMANTICS
Implicate to Explicate Order
and back again, via healthcare to worms to the simple theory of
everything
by
Barry Robson (Feb 4th 2014)
First presented at the DISCO Interuniversity Project Report
Workshop on Theoretical Semantics and the Web,
University of York March 18th 2014

3. Bioingine.com

4. Introduction
Bioingine.com

5. Explicate & Implicate Order
Implicate order and explicate order are concepts coined by
David Bohm to describe two different frameworks for
understanding the same phenomenon or aspect of reality. He
uses these notions to describe how the same phenomenon
might look different, or might be characterized by different
principal factors, in different contexts such as at different scales.
Macro vs Micro overcoming Cartesian Dilemma.
The implicate order, also referred to as the "enfolded" order,
is seen as a deeper and more fundamental order of reality.
In contrast, the explicate or "unfolded" order include the
abstractions that humans normally perceive.
– http://en.wikipedia.org/wiki/Implicate_and_explicate_order

6. The methods of theoretical physics should
be applicable to all those branches of
thought in which the essential features are
expressible with numbers. Paul A. M. Dirac,
Nobel Prize Banquet Speech, 1933
Dirac was also certainly modestly referring to his
extensions to physics via his notation and algebra as
further extensible to human language and thought,
because he explicitly considered poetry as emotional
and economics as subjective.
Probabilistic Semantics
Implicate to Explicate
From Quantum Mechanics to Language & Thought
System Dynamics to Systems Thinking

7. Contents
a. Principal Sources (Published Basis).
b. PART 1. Universal Exchange Language – a call
by the President’s Council.
c. PART 2. Quantification – Beyond Symbolic
Manipulation - Probabilistic Semantics.
d. PART 3. Information and Neuroscience
Aspects – Clues for A.I.?
e. PART 4. Is all of physics ultimately syllogistic
logic?

8. Principal Sources (Up to 2013)
• B. Robson, T. P. Caruso, and U. G. C. Balis (2013), “Suggestions for a Web-Based
Universal Exchange and inference Language for Medicine”, Computers in Biology and
Medicine, 2013 (12) 2297-2310.
• B. Robson, U. G. C. Balis, UGC , and T. P. Caruso (2012), “Considerations for a Universal Exchange Language for
Healthcare, IEEE Healthcom ’11 Conference Proceedings, June 13-15, 2011, Columbia, MO pp 173-176
• Robson, B, and TP Caruso. (2013) “A Universal Exchange Language for Healthcare”. MedInfo ’13: Proceedings of
the 14th World Congress on Medical and Health Informatics, Copenhagen, Denmark, Edited by CU Lehmann, E
Ammenwerth, and C Nohr. IOS Press, Washington, DC, USA.
• B. Robson, B. (2007) “The New Physician as Unwitting Quantum Mechanic: Is Adapting Dirac’s Inference System
Best Practice for Personalized Medicine, Genomics and Proteomics?”, J. Proteome Res. (Am. Chem. Soc.), Vol. 6,
No. 8: 3114 – 3126
• B. Robson (2009), “Towards Intelligent Internet-Roaming Agents for Mining and Inference from Medical Data”,
Studies in Health Technology and Informatics,
Vol. 149 pp 157-177
• B. Robson, (2009), “Links Between Quantum Physics and Thought” (for Medical A.I. Decision Support Systems),
Studies in Health Technology and Informatics, Vol. 149, pp 236-248
• B. Robson, B. (2012) “Towards Automated Reasoning for Drug Discovery and Pharmaceutical Business
Intelligence”, Pharmaceutical Technology and Drug Research, 2012 1: 3 ( 27 March 2012 )
• B. Robson, (2013), “Towards New Tools for Pharmacoepidemiology”, Advances in Pharmacoepidemiology and
Drug Safety, 1:6, http://dx.doi.org/10.4172/2167-1052.100012
• B. Robson (2013) “Rethinking Global Interoperability in Healthcare. Reflections and Experiments of an e-
Epidemiologist from Clinical Record to Smart Medical Semantic Web” Johns Hopkins Grand Rounds Lectures,
http://webcast.jhu.edu/Mediasite/Play/ 80245ac77f9d4fe0a2a2 bbf300caa8be1d
• http://quantalsemantics.com/papers/
• Three key preprints of submitted papers.

9. REPORT TO THE PRESIDENT REALIZING THE FULL
POTENTIAL OF HEALTH INFORMATION TECHNOLOGY
TO IMPROVE HEALTHCARE FOR AMERICANS:
THE PATH FORWARD
President’s Council of Advisors on Science and Technology (December 2010)
PART 1. UNIVERSAL EXCHANGE LANGUAGE

10. Origins of Q-UEL
• “In other sectors, universal exchange standards have resulted in new
products that knit together fragmented systems into a unified
infrastructure.”
• “The resulting ‘ network effect’ then increases the value of the
infrastructure for all, and spurs rapid adoption.”
• “By contrast, health IT has not made this transition.”
and so they call for an XML-like
“Universal Exchange Language”
UEL!
•“The market for new products and
services based on health IT remains
relatively small and undeveloped
compared with corresponding markets
in most other sectors of the economy,
and there is little or no network effect
to spur adoption.”
The Tower of Babel
The PCAST Report

11. Our Goal Is Also the Thinking Web WW4
Especially for Medicine.
• Language is Q-UEL - Quantum Universal Exchange
Language - a Semantic Web technology but based on Dirac
notation and algebra.
<Q-UEL-COMPLICATIONS context:=‘adverse drug reaction’
drug:=CODE:=SNOMED-CT :=‘=373270004/Penicillin - class of antibiotic
- (substance)’:=www.uel.org/drug/ Pfwd:=0.05 | causes:=‘causative
agent’:=www.qexl.org/causes/:=assoc:=1.5 |
allergy:=CODE:=SNOMED-CT:=2.16.840.1.113883.6.96:=
’106190000/Allergy /:246075003’ :=www.uel.org/allergy/
Pbwd:=0.001?:=www.uel.org/allergy_misdiagnosis/ Q-UEL-
COMPLICATIONS>

12. Current Set-Up and Sources
667,000 PATIENT RECORDS
Mullins, I. M., Siadaty, M. S., Lyman, J., Scully, K.,
Garrett, C. T., Miller, W. G., Robson, B., Apte, C.,
Weiss, S., Rigoutsos, Platt, D., Cohen, S., Knaus, W.
A. (2006) “Data mining and clinical data
repositories: Insights from a 667,000 patient data
set” Computers in Biology and Medicine,
36(12):1351-77
ALL US PATENTS
Robson, B., Li, J., Dettinger,
R., Peters, A., and Boyer, S.K.
(2011), “Drug discovery using
very large numbers of patents.
General strategy with
extensive use of match and
edit operations”. Journal of
Computer-Aided Molecular
Design 25(5): 427-441
AUTOMATIC WEB
SURFING (XTRACTS)
Robson, B. , Caruso, T.P, and
Balis, (2013), “Suggestions
for a Web-Based Universal
Exchange and inference
Language for
Medicine”, Computers in
Biology and Medicine, 2013
(12) 2297-2310.
“IN-HOUSE”
SEMANTIC
WEB AND
AUTOMATED
REASONING
SEMANTIC
WEB

13. Link to the Medical Semantic Web
- Bra-Operator-Ket as Linguisitic
< subject expression | relationship expression | object
expression>
• Dirac braket notation maps to S-V-O languages.
•The relation expression is in our system (as typically in QM) is always
a real or complex Hermitian operator/matrix expression (if real, it is
“trivially Hermitian”).
•The above is a bi-directed edge in a general graph of probabilistic
knowledge representation.
– It is one element in a general graph or net of knowledge representation.
• The above dual spinor can appear in nested form physicists' call the
twistor - corresponding to a parsed sentence structure or knowledge
graph representation.

14. Q-UEL: Both Raw Data and Data-Mined
Summaries Can Be Probabilistic
• Example data from medical record.
<Q-UEL-POPULATION:=#23 Pfwd:=0.95 | has:=www.qexl.org /has_7/ |
cardiovascular:=’blood pressure (mmHg)’:=www.qexl.org/ blood_
pressure/:=systolic :=’125+/-30CI(2012)(consented visible
cardiovascular and year)’ club:=Virginia Q-UEL-POPULATION>
• Statistical summary statement from data-mining such.
<Q-UEL-EBM 'Systolic BP(nearest 10) ':='over 140 (2012)' Pfwd:=0.875
| when:=www.qexl.org:=/when_1/:=assoc:=2.438 | Age:='at least
50 (2012)' and BMI :=‘30 (2012)‘ and 'Fat(%)(nearest 5) ' :=’over 30
(2012)' and female and club:=1 Pbwd:=0.703 observed:=17892
expected:= 7214.516 time:=Fri May 3 12:00:16 2013' Q-UEL-EBM>

15. Q-UEL: Medical Examples Can be Quite Complicated!
Here Is a Record of a Prescription
<Q-UEL-PRESCRIPTION:=‘order entry and results reporting’:=(meaning:=www.qexl.org/prescription_3/, source:=’VistA FMQL’:= http://vista.caregraf.info/fmql/:=
referrer :=’Tom Munnecke’:=www.osehra.org/users/tom-munnecke/, author:=’Barry Robson (Sep 21 10:01:18 2013 GMT)’:=www.qexl.org/Barry_Robson_1/,
comment:=’example transcription of example VistA FMQL entry’, warning:=nonuse:=fictitious:=comment:=’Do not use as input. Hand-crafted for discussion, specification,
example, research, development and test purposes only. May contain errors. This example contains RDF-style definitions and above tag-name qualification features not in the
original source.’)
patient:=‘Aiden Ataguy’:=www.qexl.org/US_Patient_Reg189958822/
and provider:=(center:=‘Outpatient Site FMQL Clinic’:= www.qexl.org/US_MedCenter_Reg8411/, (physician, prescriber):=’ James Kildare’ :=www.qexl.org/US_MD
_Reg74356/)
and Rx:=(simvastatin:=code:=(NDC:=000006-0749-54, VA:=4010153) :=www.qexl.org/simvastatin/, tablets(number):=90, tablet(mg):=40, ‘prescriber instruction’:=
(literally:=‘Take one tablet by mouth every evening’, formally:=(tablets:=1 by:=mouth with:=water(presumed) ‘when (local patient time 24 hour clock)’:=18.00+/-6 ) ))
and Rx#:=‘800018 (Mar 5 09:11:03 2002 local)’
and fills:=(‘earliest possible’:= ‘Mar 5 09:11:03 2002 local ‘, ‘next possible’:= ‘Apr 5 24:00:00 2002 local’, ‘last possible’:= ‘March 6 24:00:00 2003 local’)
and ‘patient status’:=code:=SC:=(‘not exempt from copayment’, ‘days supply’:=30 refills:=11, renewable):=www.qexl.org/Verify_Status_US_Patient_Reg189958822_ US_MD
_Reg74356_Rx#:=800018/
and order:=initiated
and ‘prescribing status’:=expired
and ‘GMT minus local time(hours)’:=7 and zone:=constant
| triggered:=www.qexl.org/ triggered_3/ |
dispensing:=(ordered:=10, ‘unit price($)’:=0.80, available, delivery:=’window pickup’) and times:=(login:=’ Mar 5 13:50:17 2002 local’, fill:= ‘Mar
5 13:51:02 2002 local’, ‘last dispensed’:= ‘Mar 5 14:13:17 2002 local)’, ‘label:= ‘Mar 5 13:50:27 2002 local’, release:= ‘Mar 5 13:50:42 2002 local’))
and copies:=1
and counseling:=(given, understood)
and (pharmacist, enterer, printer, counselor):=’Nancy Devillers’:=www.qexl.org/ US_Pharmacist_ Reg101740/,
and order:=converted
and ‘dispensing status’:=expired
and ‘refill status’:=open
and ‘GMT minus local time(hours)’:=7 and zone:=constant
and tagtime:=‘Mar 5 20:50:43 2002 GMT’
Pbwd:=0:=comment:=‘Process is not reversible, and forward direction is certain as a matter of record (Pfwd:=1 is the default)’
Q-UEL-PRESCRIPTION>

16. Q-UEL: Definition Example. Implied
Probabilities Here are 1 (the Default).
<Q-UEL-MOLECULE Ampicillin | means:=
www.qexl.org/means_2/ | code:=IUPAC:= ‘2S,5R,6R)-6-{[(2R)-
2-Amino-2-phenylacetyl]amino}-3,3-dimethyl-7-oxo-4-thia-1-
azabicyclo[3.2.0]heptane-2-carboxylic acid’ or code:=SMILES:=
O=C(O)[C@@H]2N3C(=O)[C@@H](NC(=O)[C@@H](c1ccccc1)N)[
C@H]3SC2(C)C or code:= InChI:= InChI=1S/C16H19N3O4S/c1-
16(2)11(15(22)23)19-13(21)10(14(19)24-16)18-12(20)9(17)8-6-4-
3-5-7-8/h3-7,9-11,14H,17H21-2H3,(H,18,20)(H,22,23)/t9-,10-
,11+,14-/m1/s1 and ‘empirical formula’:=C16H19N3O4S and
Monoisotopic mass:=349.109619 and ‘average mass (Da)’:=
349.404785 Q-UEL-MOLECULE>

17. Q-UEL: Autosurf-and-Spawn XTRACTs Parse and
Re-express Source Text
< Q-UEL-XTRACT-BIOLOGY "`The human _brain |^is `the center
of| `the human nervous _system
[0http://en.wikipedia.org/wiki/Nervous_system]; `The human
_brain |^has `the `same `general _structure as| `the _brains
|of| `other mammals [0http://en.wikipedia.org/wiki/Mammal];
`The human _brain |^is larger than ^expected on `the basis of|
_body _size |among| `other primates
[0http://en.wikipedia.org/wiki/Primate]
[1(0)http://www.ncbi.nlm.nih.gov/pubmed/17148188]
[2file:input.txt#cite_note-Brain-num-1]" | from |
source:='http://en.wikipedia.org/wiki/Human_brain' time:='Wed
Oct 3 14:02:19 2012' extract:=0 Q-UEL-XTRACT-BIOLOGY >

18. Q-UEL: Reasoning
• One issue relates to the IBM Watson computer, which beat human
champions at Jeopardy but thought O’Hare airport was in Toronto
[35]. A Q-UEL metastatement did already know that if A travels to B,
then A is not B. A key rule in that process was that below reached
by an automatically generated Google query:-
<Q-UEL-XTRACT “Chicago-O'Hare International (ORD) |to| Toronto; Chicago-
O'Hare International (ORD) |(concerning)| flights”
(source:=http://www.cheapflights.com/flights-to-toronto/chicago-ohare-intl’
| from | Extract:=0 presource:=
‘http://www.google.com/search?hl=en&source=hp&q=
O%27Hare+airport+Toronto&gbv=2&oq=O%27Hare+airport+Toronto&gs_l=he
irloom-hp.3...12891.24047.0.24750.22.19.0.3.3.0.531.3292.0j15j2j5-
1.18.0...0.0...1c.1.nX3-bh6US _AQuery:%3D+O%27Hare’
Query:=’O%27Hare+airport+Toronto’ AutoQuery:=’O’Hare airport Toronto’
Hits:=2,330,000 ‘Select (not advert)’:=1) Q-UEL-XTRACT>

19. Q-UEL: Thesaurus, Dictionary and
Encyclopedia Extracts Aid XTRACTs
<Q-UEL-THESAURUS cat | suggests | '2. Special Vitality':='366.
Animal.':=pbwd:=0.03502 or 'Section II. PRECURSORY CONDITIONS AND
OPERATIONS':='455. [The desire of knowledge.] Curiosity.':=pbwd:=0.03226 or
'(ii) SPECIFIC SOUNDS':='407. [Repeated and protracted sounds.]
Roll.':=pbwd:=0.00667 or '(ii) SPECIFIC SOUNDS':='412. [Animal sounds.]
Ululation.':=pbwd:=0.00632 or 'Present Events':='151.
Eventuality.':=pbwd:=0.00571 or '(iii) PERCEPTIONS OF LIGHT':='441.
Vision.':=pbwd:=0.00448 or 'SECTION III. ORGANIC MATTER 1. VITALITY 1. Vitality
in general':='359. Life.':=pbwd:=0.00392 or '3. Fluids in Motion':='348. [Water in
motion.] River.':=pbwd:=0.00356 or '3. PROSPECTIVE AFFECTIONS':='864.
Caution.':=pbwd:=0.00223 or '5. INSTITUTIONS':='975. [Instrument of
punishment.] Scourge.':=pbwd:=0.00151 or '3. Contingent Subservience':='668.
Warning.':=pbwd:=0.00142 or
…………(etc)…………………….. Q-UEL-THESAURUS>
Note the increasing
appearance of probabilities,
which brings us to…

20. PART 2. QUANTIFICATION
Beyond Symbolic Manipulation
(Bliss Symbolics)
I want to go
to the cinema
Are you
sure?!!

21. Quantification - Dual Notation
• Q-UEL statements have Pfwd and Pbwd attributes (default 1). They
can be most simply be described as a value by a dual. E.g.
<A| B> = <A| if |B> = {P(A|B), P(B|A)}
• You can treat it pretty much as a real two-element vector (but see
later) , but then you have to define the algebraic operations, to come
out classically, e.g. products as for the chain rule and syllogistic form
<A|C> ≈ <A|B><B|C>
= {P(A|B), P(B|A)} {P(B|C), P(C|B)}
= {P(A|B) P(B|C), (P(C|B)P(B|A)}
• This above is also a simplest example of a probabilistic knowledge
representation as an inference net, analogous to a Bayes Net, but
bidirectional.

22. Quantification - Iota (i) Notation
i = ½ (1+h), i* = ½ (1- h)
• h is the hyperbolic imaginary number such that hh = +1.
• From the known properties of h, all algebra can be defined.
• * indicates throughout complex conjugation, i.e. change the
sign of the imaginary part
• A dual is really a hyperbolic complex value, and the form is
<A|B> = iP(A|B) + i*P(B|A)
= (iP(A) + i*P(B) )eI(A; B)
• I(A; B) is Fano’s mutual information ln(P(A, B) / P(A)P(B)).

23. Advantages of Iota Algebra: Simplicity
• Idempotent rule: ii = i, i*i* = i*
• Annihilation rule: i*i = 0, ii* = 0
• Normalization rule: i + i* = 1
• Eigensolutions exist for {x, y} as x and y with eigenvalues h=+1,
h=-1 respectively.
• It follows from the above that
Selection rule: i{x, y} = ix, i*{x, y} = i*y
Exponent rules: 1/i = i, ix = i, ei = i, log i = i
• F.Y.I. It is the simplest algebra, except powerfully and less obviously, for example,
ehx = cosh(x) + hsinh(x) = ie+x + i*e−x
(link to quantum mechanics: case of mother form as conjugate symmetry)
z(x+hy, n) = i z(x-y, n) + i* z(x+y, n) (Riemann zeta function summed to n)
and note that hi = -ih (where ii = −1) is anticommutative,
so that ehix = cos(x) + hi sin(x) = ie+ix + i*e−ix
Note pervasive nature of the spinor projector form i…. i*! !!!

24. Hyperbolic Complex Algebra in Quantum Mechanics and
Classical Inferencing - Previous Work
• Here i = ½ (1+h), i * = ½ (1- h) are physicists’ spinor projectors,
quantum field operators, where h is the hyperbolic imaginary
number corresponding to physic’s g5 and relates to Dirac’s gtime such
that hh = +1.
• The Lorenz transform of the standard imaginary number i → h renders
wave mechanics classical (B. Robson, B. (2007) “The New Physician as Unwitting
Quantum Mechanic: Is Adapting Dirac’s Inference System Best Practice for Personalized Medicine,
Genomics and Proteomics?”, J. Proteome Res. (A.m. Chem. Soc.), Vol. 6, No. 8: 3114 – 3126)
• This hyperbolic number h has been used since the 1990s in neural
nets – solves the XOR problem in one neuron (e.g. Y. Kuroe, T. Shinpei , and
H. Iima, H. (2011), “Models of Hopfield-Type Clifford Neural Networks and Their Energy Functions - Hyperbolic
and Dual Valued Networks”, Lecture Notes in Computer Science Vol 7062, 560-569).
• Hyperbolic “wave” functions (probability amplitudes) have also been
proposed as a basis of natural neural nets and the mind (A. Khrennikov
(2004), “On Quantum-Like Probabilistic Structure of Mental Information”, Open Systems &
Information Dynamics, Vol. 11:3, 267-275).

25. Origin of Spinor Form - We can Construct a
Hermitian Commutator to Express Categorical Logic
<A | B> = <A| if |B>
= <B | are |A> (Categorical interpretation of conditional)
= ½[P(A|B) + P(B|A)] +½ h [P(A|B) - P(B|A)] (Hermitian Commutator)
where h is the hyperbolic number such that hh = + 1 so that we have
classical behavior of <A|B> like a dual (P(A|B), P(B|A), and
½[P(A|B) + P(B|A)] = <A| some are |B> = <B | some are |A>
is the existential or “some” part and
½ [P(A|B) - P(B|A)]
is the residual universal or “all” part on the interval -1/2 …..+1/2 for P(“All
A are B”) to P(“All B are A”) (we can always change the sign basis as a matter
of definition, but we will keep it this way, to “match <A|B>).

26. More Theory: Q-UEL comprises Dirac’s bras,
kets, brakets, ketbras and bra-operator-kets
• The bra <A| = [<A|X1>, <A|X2>, <A|X3>, ….]
• The ket |B> = [<X1|A>, <X2|B>, <X3|C>,…..]T
• These are probability distributions.
• The vector product is expressible in a law of composition of
probabilities in vector space:
<A|B> = SXA|X><X|B> meaning Si=1,2,3,…<A|Xi><Xi|B>
It holds classically in everyday space if SX P(A|B,X) – SXP(A|X) = 0,
SXP(B|A,X) – SXP(B|X) = 0, and we can express this as a covariance
law about direct independence by indirect dependence of A and B.
• Operator = Matrix = ketbra product = |A> (x) <B|
• This is a Hermitian probability density matrix.
• But < subject expression | object expression> or < subject
expression | relationship expression | object expression> can be
manipulated in their own right as a scalar real or scalar complex
object.

27. Linguistic Consequences of Inserting a Hermitian Operator
• Converse relationship (bidirected edge of directed graph)
<dogs | chase |cats>* = <cats | chase | dogs>
• Trivially Hermitian operator (real valued)
<Jack | marries | Jill> = <Jill | marries |Jack>
• Active-passive inverse
<cats | chase* |dogs> = <cats| ‘are chased by’ | dogs>
= <dogs | chase | cats>
• Orthogonal vectors (mutually exclusive arguments)
<dogs | (x) | cats > = <dogs | cats> = <dogs | if | cats>
= < cats | are | dogs> = 0;
chase |cats> or <dogs | chase inverts elements of one of the vectors , so
<dogs | chase | cats> ≠ 0
– But we can define are generally as a joining operator so that
< cats | are | mammals> ≠ 0. The extent of identity of mutual exclusivity is in the
real part of the value of <A| are |B> = <B|A>.
We can indeed
express all these
as matrices and
vectors!

28. Example Twistor Expressions
• By Dirac’s rules A R B in <A| R |B> can be algebraic expressions with
manipulation laws.
< Jack and Jill | went up and went down | the hill or the mountain>
– It remains scalar complex.
• Also by Dirac’s rules the arguments in those expressions can be real or
complex scalars, vectors, or operators/matrices, including brakets and
bra-reltor-kets etc., so subject to his reules we can construct e.g.
<< the rain | mainly ‘stays on’ | the plain> | in |Spain>
<< the rain | in | Spain> | mainly ‘stays on’ | the plain>
• With h-complex algebra alone, these contract under the
idempotent and annihilation rules, e.g. < the rain in Spain | mainly
‘stays on’ | the plain>.
• To keep a parsed tree structure, however, we encode clauses with
different flavors of h: h1, h2, h3, … (see later)

29. Deep Grammar: Metastatements
• Following Semantic Web terminology, the basic forms <A| R | B>
are called statements, and rules which we call metastatements act on
them as operators, using binding variables.
• They evolve the inference network, say converting one or more
statements to one or more others, as in a syllogism, or vice versa.
• They can be probabilistic too.
• Statements and metastatements can be input, but corresponding to
PROLOG data and program respectively.
• Metastaments are said to have extended twistor form from physics,
e.g. the syllogism template
< <$A| are | $C> |= | <$A | are | $B><$B | are | $C> >
• Language is defined that way as input . Notice reverse Ogden
reduction of vocabulary:
< <$A| pays | $B> |= | <$A | gives | money><money| to | $B> >
• The processes are reversible. The system can reason work forwards
and/or backwards by the above processes.

30. Statement Reconciliation
• In the course of inference net evolution, statements can be generated that
are detectable as semantically equivalent to one already existing
(therefore redundant information has been generated - literally as logs of
underlying probabilities).
• Similarly two identical statements can exist with different values.
• They are converted into a canonical form determined by a metastatment,
and the value is also reconciled.
<A| R | B> ←
<A| R | B> + <A| R | B> - <A| R | B><A| R | B>
• Repeated, this process is a bionomial expansion and independent of the
order in which the statements are processed.
• It represents an inclusive OR and the fact that statements are independent
and can be counted as independent observations of events. except that
we decide to reconcile them to one.
• However, we can count (“data mine”) and get our underlying probabilities
that way.

31. Some Features of Hyperbolic Dirac Nets for
Automated Reasoning (1)
• HDNs are networks composed of brakets and
bra-relator-kets.
– …and sometimes more complex sentences or knowledge
elements as twistors (nested brakets or bra-relator kets
analgous to a parsed sentence structure).
• The product of all the probability duals is a dual
representing their collective degree of truth in
two directions (e.g. of conditionality).
– To have further meaning, their must be a categorical or causal
relationships, or represent some kind of chain of effect.
– That implies logical AND between bra-relator-kets etc.
throughout, but other logical operators are possible.

32. Some Features of Hyperbolic Dirac Nets for
Automated Reasoning (2)
• The value is purely real (the imaginary part vanishes) for
networks, or parts of them, which
– represent a joint probability
– represent a cyclic path in the network (cyclic paths are no
problem as they are in other inference systems, e.g. Bayes
Nets!)
– represent a purely existential rather than universal statement,
or analogously represent a trivially Hermitian relationship, as in
<Jack | marries | Jill> = <Jill | marries | Jack>
• They may evolve under the action of metastatements,
representing deep grammar, syllogistic and higher order
logic, and language definition.

33. PART 3. Information and
Neuroscience Aspects (Clues for A.I.?)
Hyperbolic Complex Processing by Neurons
A. Khrennikov (2004), “On Quantum-Like Probabilistic Structure of Mental Information”, Open
Systems & Information Dynamics, Vol. 11:3, 267-275).
H. Iima, H. (2011), “Models of Hopfield-Type Clifford Neural Networks and Their Energy
Functions - Hyperbolic and Dual Valued Networks”, Lecture Notes in Computer Science Vol
7062, 560-569).
N. Chattergee and S. Sinha (2008) “Understanding the Mind of a Worm. Hierarchic Network
Structure Underlying Nervous System Function in C. Elegans”
http://www.ppls.ed.ac.uk/ppig/documents/ChatterjeeSinha.pdf

34. An Information-Theoretic Perspective
• It is often convenient to work with logs of probabilities and probability
ratios such as association constants.
• By hyperbolic complex algebra, if <A | R | B> = {0.9, 0.7) then log<A |
R | B> = {log 0.9, log 0.7}
• This directly gives insight into the information content and information
processing.
• For example, a knowledge or inference network of N brakets or bra-
relator-kets etc. can contain up to N bits of information in each
direction.
Proof: we can always re-express a statement with some kinds of markers to
indicate that one or both probabilities P less than 0.5 are now to be seen as 1-P,
and hence the information –log2P is between –log21 = 0 and
–log20.5 = 1 bits. <no cats |chase | dogs>, <cats | chase | no dogs>, and <no
cats | chase | no dogs> is one way example way of indicating the three
“negations”, though two would depart from normal English interpretation.

35. Neural Net Information-Theoretic Interpretation of
Generation of Associations and Dual Probabilities
Event
A
I(B
)
Event
B
I(A; B)
I(A
)
I(B|A) I(A|B)
Add
Information
to
Subtract
Information
from
Input layer
Dual self
Information
layer
Mutual
Information
layer
Dual
conditional
Information
layer
HDNs can render I(X)
as the Riemann
Partially Summated
Zeta function
z(X) = z(s=1, n[X])
that goes from 0 at
n[X] = 1 to ~ ln(X) as
n[X] increases.

36. Very Weak or Unused Associations may be deleted
(as in Sleep?)
I(B
)
I(A
)
Add
Information
to
Subtract
Information
from
Dual self
Information
layer
Mutual
Information
layer
Dual
conditional
Information
layer
Our “A.I. systems” using duals (and so hyperbolic complex by
implication) can be re-expressed in terms of many K(A; B; C;…) =
exp(I(A; B; C;…)) and self probabilities. They take up a lot of
memory. But we do not need those I(A; B; C;…) ≈ 0, i.e. K(A; B;
C;…) ≈ 1, that represent negligible associations. As apparently in
the human brain, we can (with some caveats) remove them.

37. Alternatively, Strong Associations Can Be Grown to
Include More Elaborate Elements
Event
A
I(B
)
Event
B
I(C
)
Event
C
I(A
)
I(A, B, C)
This example is “complexity 3”
(A, B, C), enough for a semantic
triple <A| R |B>, 3 constituents as
“words” or basic phrases.
What degree of
semantic complexity
can be achieved in a
nervous system?
It does not automatically
follow that the elements of
these diagrams are single
neurons, but they can be, and
we can address the matter for
a very simple animal with few
neurons.

38. Caenorhabditis elegans Nervous System Mapped
• Only 302 neurons,
average of roughly
18 synapses per
neuron (1-35)
Behavior
• Defecation
• Touch sensitivity
• Egg laying
• Thermotaxis
• Oxygen level sensitivity
• Chemosensitivity
• Feeding locomotion
• Conditioning to tap/touch
• Exploration

39. C. Elegans Nervous System Mapped
SENSORY
MOTOR
I
N
T
E
R
Diagrams below give a rough indication of number of synapses per neuron
This suggests majority have complexity
equivalent to braket <A|B> = <A| if |B>,
semantic triple <A| R |B>, and a few
with a sentence or knowledge graph
structure of up to about 10-12
constituents, i.e. “words” or basic
phrases, and a few higher (20?).

40. 4. WORK IN PROGESS
Is all of physics ultimately syllogistic logic?
The Octonion-related Lie Groups
http://en.wikipedia.org/wiki/An_Exceptionally_Simple_Theory_of_Everything

41. Dirac Grammar Assertion
• The idea is that we shouldn’t really need metastaments
for purely logical and abstract semantic manipulation.
• Dirac’s manipulative grammar as algebra should do the
job, along with defining semantic significance of
mathematical classes of operator/matrix etc.
• However , to handle sentence structure and knowledge
graphs as a single algebraic entity, especially syllogistic
and higher order logic, may need to be extended beyond
the Clifford-Dirac algebra to do so.
• Hence the octonion project described now.

42. Can we get rid of metastaments, e.g. can we express
syllogisms purely algebraically?
Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Datisi Calemes
Celarent Camestres Disamis Dimatis
Darii Festino Ferison Fresison
Ferio Baroco Bocardo Calemos
Barbari Cesaro Felapton Fesapo
Celaront Camestros Darapti Bamalip
• We went half way by being hyperbolic complex, but the classical
syllogisms come in packs of fours or eights.

43. Categorical Notation
Set /
categorical
Spatial analog Notes
A The set of A. The inside of A.
~A The set of non-
A.
The outside of A. Negative, complement.
<A|B> All B are A. The inside of A is
inside and outside
the inside of B.
The Dirac analogue of
P(A|B), but P(B|A) is
encoded too: <A|B> =
<B|A>*
<A|B>* All A are B. The inside of A is
inside the inside of
B.
The Dirac analogue of
P(B|A), but P(A|B) is
encoded too: <A|B>* =
<B|A>
<~A|B>* All non-A are B. The outside of A is
inside the inside of B.
B is the universe except
possibly for at least part of
A.
<~A|~B>* All non-A are The outside of A is <~A|~B>* = <B|A>* holds

44. We Can build Syllogisms as pairs of brakets, but it requires
metastatements to turn the answers (products) into language.
Linear
Right fork
Left fork
Anticlockwise
rotation
of “Feynman”
diagram
Clockwise
rotation
of “Feynman”
diagram
Inference net pair components are
like the propositions of syllogisms.
Feynman- spacetime-like rotations
of 8 chain-rule structures <A|C> ≈
<A|B><B|C> generates the other
16 possible configurations.
Related by the complex
conjugate – change sign
of imaginary part

45. Encoding Basic Categorical Logic in a Hermitian
Commutator Requires 8 Imaginary Numbers?
⅛ x (
½ [ P(A|B) + P(B|A)] + ½ h1 ½ [P(A|B)-P(B|A)]
+ ½ [ P(~A |B) + P(B| ~A]) + ½ h2 ½ [P(~A|B)-P(B|~A)]
+ ½ [ P(A |~B) + P(~B| A)] + ½ h3 ½ [P(A|~B)-P(~B|A)]
+ ½ [ P(~A| ~B) + P(~B|~A)] + ½ h4 ½ [P(~A|~B)-P(~B|~A)]
+ ½ [ P(A|B) + P(~B|~A)] + ½ h5 ½ [P(A|B)-P(~B|~A)]
+ ½ [ P(~A|B) + P(~B|A)] + ½ h6 ½ [P(~A|B) - P(~B|A)]
+ ½ [ P(A| ~B) + P(B| ~A)] + ½ h7 ½ [P(A|~B)-P(B|~A)]
+ ½ [ P(~A|~B) + P(B|A)] + ½ h7 ½ [P(~A|~B) -P(B|A)]
)
These 4 pairs are reverse conditionalities
– adjoints. When joint probabilities are
established by multiplying by priors, we
can think of the extent that they satisfy
Bayes rule or law P(A,B) = P(A|B)P(B)=
P(B|A)P(A), which is the first aspect of
coherence. They also relate to marginal
summation and the joint probability
distribution, which is the second aspect
of coherence.
These 4 pairs are the logical laws of the
contrapositive, P(A|B) = P(~B|~A) etc. as
in “All mammals are cats” ≡ “All non-cats
are non-mammals”. They also relate to
marginal summation and the joint
probability distribution, which is the
second aspect of coherence. They are not
adjoints and the laws hold only under
certainty.
The imaginary part measures departure from these laws. But there are no degrees
of freedom in the real part. The sum over these terms is 8, ½ after the normalization.

46. No! - Encoding Basic Categorical Logic in a Hermitian
Commutator Requires 7 Imaginary Numbers!
1/7 x (
½ [ P(A|B) + P(B|A)] + ½ h1 ½ [P(A|B)-P(B|A)]
+ ½ [ P(~A |B) + P(B|~A]) + ½ h2 ½ [P(~A|B)-P(B|~A)]
+ ½ [ P(A |~B) + P(~B| A)] + ½ h3 ½ [P(A|~B)-P(~B|A)]
+ ½ [ P(~A| ~B) + P(~B|~A)] + ½ h4 ½ [P(~A|~B)-P(~B|~A)]
+ ½ [ P(A|B) + P(~B|~A)] + ½ h5 ½ [P(A|B)-P(~B|~A)]
+ ½ [ P(~A|B) + P(~B|A)] + ½ hx ½ [P(~A|B) - P(~B|A)]
+ ½ [ P(A| ~B) + P(B| ~A)] + ½ h6 ½ [P(A|~B)-P(B|~A)]
+ ½ [ P(~A|~B) + P(B|A)] + ½ h7 ½ [P(~A|~B) -P(B|A)]
)
We can express P(A|B) and P(B|A) as
explicit variables if we take out P(~A|B)
and P(~B|A), since adding them to the
above gives 2 or 1/8 after normalization.
The sum of the remaining 4 terms here is
2, or 1/8 after normalization.
To balance this with the imaginary part,
we take out the corresponding terms in
the laws of the contrapositive. It deletes
one manifestation of the law, shown in
red. The sum of the real parts of the
remaining 6 terms here is 2+P(A|B)
+P(B|A) or (1 + (PA|B)+P(B|A))/7 after
normalization.

47. Such an octonion system is the most complete
description of the physical world
(E8 particle symmetry)
Note that ei ei = -1 (flavors of the well known imaginary number i)
× 1 e1 e2 e3 e4 e5 e6 e7
1 1 e1 e2 e3 e4 e5 e6 e7
e1 e1 −1 +e3 −e2 +e5 −e4 −e7 +e6
e2 e2 −e3 −1 +e1 +e6 +e7 −e4 −e5
e3 e3 +e2 −e1 −1 +e7 −e6 +e5 −e4
e4 e4 −e5 −e6 −e7 −1 +e1 +e2 +e3
e5 e5 +e4 −e7 +e6 −e1 −1 −e3 +e2
e6 e6 +e7 +e4 −e5 −e2 +e3 −1 −e1
e7 e7 −e6 +e5 +e4 −e3 −e2 +e1 −1

48. Hyperbolic Octonion Multiplication
× 1 h1 h2 h3 h4 h5 h6 h7
1 1 h1 h2 h3 h4 h5 h6 h7
h1 h1 +1 +h3 −h2 +h5 −h4 −h7 +h6
h2 h2 −h3 +1 +h1 +h6 +h7 −h4 −h5
h3 h3 +h2 −h1 +1 +h7 −h6 +h5 −h4
h4 h4 −h5 −h6 −h7 +1 +h1 +h2 +h3
h5 h5 +h4 −h7 +h6 −h1 +1 −h3 +h2
h6 h6 +h7 +h4 −h5 −h2 +h3 +1 −h1
h7 h7 −h6 +h5 +h4 −h3 −h2 +h1 +1
It is an octonion except that hi hi =+1, i.e. we applied the
Lorenz transform i → h as before.

49. Octonion Spinor Projector Multiplication
ii = ½ (1+hi), i=1,2,3,4,5,6,7
ii ii = ii
ii* ii = iiii* = 0
i1i2 = ¼ (1+h1+h2+h3)
i1i3 = ¼ (1+h1+h3- h2)
i1i4 = ¼ (1+h1+h4+h5)
i1i5 = ¼ (1+h1+h5-h4)
i1i6 = ¼ (1+h1+h6-h7)
i1i7 = ¼ (1+h1+h7+h6)
i2i1 = ¼ (1+h1+h2- h3)
i2i3= ¼ (1+h2+h3+ h1)
i2i4 = ¼ (1+h2+h4+h6)
i2i5 = ¼ (1+h2+h5+h7)
i2i6 = ¼ (1+h2+h6-h4)
i2i7 = ¼ (1+h2+h7-h1)
i3i1 = ¼ (1+h3+h1+h2)
i3i2 = ¼ (1+h3+h2- h1)
i3i4 = ¼ (1+h3+h4+h7)
i3i5 = ¼ (1+h3+h5-h6)
i3i6 = ¼ (1+h3+h6+h5)
i3i7 = ¼ (1+h3+h7-h4)
i4i1 = ¼ (1+h4+h1-h5)
i4i2= ¼ (1+h4+h2- h6)
i4i3 = ¼ (1+h4+h3-h7)
i4i5 = ¼ (1+h4+h5+h1)
i4i6 = ¼ (1+h4+h6+h2)
i1i7 = ¼ (1+h4+h7+h3)
i5i1 = ¼ (1+h5+h1+h4)
i5i2 = ¼ (1+h5+h2- h7)
i5i3 = ¼ (1+h5+h3+h6)
i5i4= ¼ (1+h5+h4-h1)
i5i5 = ¼ (1+h5+h6-h3)
i5i6 = ¼ (1+h5+h7+h2)
i7i1 = ¼ (1+h7+h1-h6)
i7i2 = ¼ (1+h7+h2+ h5)
i7i3 = ¼ (1+h7+h3+h4)
i7i4= ¼ (1+h7+h4-h3)
i7i5 = ¼ (1+h7+h6-h2)
i7i6 = ¼ (1+h7+h7+h1)
i6i1 = ¼ (1+h6+h1+h7)
i6i2 = ¼ (1+h6+h2+h4)
i6i3 = ¼ (1+h6+h3-h5)
i6i4= ¼ (1+h6+h4-h2)
i6i5 = ¼ (1+h6+h6-h3)
i6i7 = ¼ (1+h6+h7-h1)

50. QUESTIONS?