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**Acronyms and symbols used in The Sophism Connections:**

MoQ - Metaphysics of Quality (Pirsig) SOM - Subject-Object Metaphysics

**The (Other) Sophism Connections:**

We may look at sophisms in an evolutionary
kind of perspective**:**

- As perceived by Sophists between 13,000b.c and 700b.c. (See
Pirsig's
*Birth of SOM*, for a historical perspective. Also research*Sophia*as Wisdom of Gnosis. Search Quantonics for Doug's**:**quantum*Light*,*Logos*,*pneuma*,*Sophia*, Gn and Gno.) - As perceived by Greeks starting approximately with Homer's Iliad through Aristotle
- Medieval sophistry using Buridan's work as an exemplar
- Modern sophistry as viewed by Western dialecticians in 20th century
- Quantum logic (sophistry; in Quantonics we now, Ceod
**E**2009, call it "coquecigrues") viewed by your reviewer, David Finkelstein at Georgia Tech, David J. Foulis at University of Massachusetts, et al.

Using classical definitae sophisms are self-referent, paradoxical,
and thus inconsistent classico-predicate-logical
propositions. For examples see Buridan's list of 20 sophisms under
a link titled Review of
Hughes' *John Buridan on Self Reference* in this review.

Again using classical definitae self-reference is a more general
concept than sophism. Not all self-referent classico-predicate-logical
(SOM) propositions are paradoxical and thus inconsistent. (Reader,
please remember we are speaking now from a classical SOM perspective.
From a quantum reality perspective and a Pirsigean MoQ perspective
of many truths and many contexts, sophisms are neither paradoxical
nor inconsistent. We are speaking SOMese here.) Examples of self-referent
SOM propositions which are not (according to SOM) paradoxical
nor inconsistent are tautologies. Aristotle's laws of syllogistic
logic are tautologies. According to SOM tautologies are always
true. Elsewhere in this review, we show you in quantum reality
tautologies are *not* always true,
and from any quantum~complementarospective *all* classical tautologies are sophisms.
"How can that be?" All quanta wave~stochastically change and occurrently evolve perpetually. Tautology depends upon both classical
'state' and classical 'identity.' Both 'state' and 'identity' are bogus in quantum~reality.
Doug - 13May2009.

In set theory, logicians sometimes speak of paradox. Examples
usually relate to what they call *the* power set of a set.
One excellent example of this kind of paradox in set theory is
one called Cantor's Paradox or *the* set of all sets paradox.
It goes something like this**:**

- C is
*the*set of all sets (implication**:**only one power set of all sets*exists*) - Given 1, then every subset of C is a member of C
- Given 1 & 2, then
*the*power set of C is a subset of C (power set is some base raised to C power) - But statements above imply
*the*cardinal value of*the*power set of C is less than or equal to*the*cardinal value of C - Cantor's theorem denies 4
- Thus 1 is a contradiction, a paradox

What is happening here? Classical logicians today (not quantum
logicians!) make fundamentally incomplete assumptions similar
to ones both Buridan and Hughes make. They assume one classical,
SOM reality. Note how use of 'the' above keeps pointing at and
implying a single reality. If we just eliminate all of '*the*'
in our list of SOM propositions, you begin to sense a new and
better realm awaiting.

Please read our June, 1999 Quantonic Question and Answer on mis-use, over-use of the. Also see thelogos.

Let us take a look, again, at differences between SOM reality
and quantum/MoQ reality**:**

SOM Reality |
Quantum Reality |

Assumes Reality is to objective Actuality (i.e., known) |
Assumes Reality is to both Actuality and Nonactuality |

Assumes a universal context for truth | Assumes both many contexts and
many truths |

appearance of any nonactuality paradox | there are no paradice |

As we can see in our table above, SOM reality is incomplete.
SOM reality denies any unknown part of actuality and it denies
nonactuality. (See our diagram of A
Map of a New Reality on our top page of this review.) So
when we execute steps of logic for *the* set of all sets
in SOM reality where reality is identical () only to what
is known, what exists, we achieve paradox. But when we execute
steps of logic for ** a** set of all sets in quantum/MoQ
reality we find no paradox. Why? Because quantum/MoQ reality
assumes reality is identical to

- C is a set of all sets (see comments about this below)
- Given 1, then every subset of C is a member of C
- Given 1 & 2, then a power set of C is a new context of C
- It is irrelevant (here) that a cardinal value of a power set of C is less than or equal to a cardinal value of C
- There is no contradiction, no paradox

Further, quantum/MoQ reality assumes a growable infinity of
contexts and truths. Quantum reality tells us if we view six
steps of logic above from a SOM perspective there appears to
be a paradox until one realizes that *the* power set simply
creates a new context different from *the* context of *the*
set of all (known, by presumption in SOM) sets. A context of
a power set of C is a different context from *the* context
of C. SOM sees this as a paradox because it assumes one universal
context when in any more general quantum reality there are many
contexts. In quantum reality there is no such thing as *the*
set of all sets because we can always make a more complex set
of all sets based on our latest claimed "...set of all sets."
If there were such a thing as a quantum set of all sets, it would
represent all of quantum reality which is complete but always
growable. Therefore in quantum reality *the* "...set
of all sets," would be inconsistent because of an uncertainty
relationship between quantum completeness and quantum consistency.

Clearly C and its power set used together in a SOM logical
proposition is an example of self-reference. Indeed, it is from
*the* SOM perspective a sophism.

Other logical paradoxes which appear in SOM's contrived and
constrained set theory, like Russell's Paradox, Burali-Forti
Paradox, *The* Set of All Cardinal Numbers Paradox, *the*
Family Paradoxes, etc., all arise from *the* blindered fundamental
axioms of SOM reality, but meet their demise under a more general
quantum reality.

Again, we see that *the* paradice of SOM sophisms evaporates
when we move from SOM reality to quantum
reality.

Self-reference, we could say here quantum "Gn¤sticism,"
is a powerful tool, and a powerful general logic caveat (a semaphore of *more*).
Modern computational symbolic formal logic usually calls self-reference
recursion. There
is a fine point of
difference twixt SOM perceptions of self-reference and recursion,
however. Recursion depends upon results of *the* (consider
possible amplified consequences of using 'a' here) prior iteration
for *the* (similar considerations here **—**
see Poincaré on *Chance*) next iteration. Recursive
processes must be initialized (primarily a SOMthink analytic
limitation, not a quantum limitation). There is no such restraint
on self-referent propositions or processes. They may (probably
do) prefer preconditions, but preconditions are not a requirement
as in (esp. SOM) recursive processes.

A good example of recursion in symbolic logic is calculation of factorial of a number. Other examples are generation of certain number series (e.g., Fibonacci), and calculation of sequential fractional digits of pi, e, or other natural numbers. Self reference or recursion works without (apparent) paradice in classical computational machines because a selected/preferred/conventional context for calculation is extremely constrained and controlled. Any recursive computational context is almost classically ideal, but it is contrived and because of its high consistency very incomplete.

Were we to allow our *real* world to intervene in workings
of any classical machine its consistency would diminish rapidly
as its unintended completeness increases. Examples of real world
intervention might be bombardment of sub-atomic particles changing
states of internal workings of a classical machine, or a more
familiar example of power failure intervention.

Apparently there are many other classes of SOM logical self-referent
forms than sophisms and tautologies. Two forms enlightened us
during our recent 20th century**:** chaos and fractals.

A marvelous thing about these two self-referent forms is they arrived on our now scene simultaneously with powerful computer graphics technologies. As a result we have been able to see chaos and an infinity of fractals graphically. When it happened SOMtalk about paradox in these two self-referent forms diminished. It is hard to call something paradoxical when you can see it graphed in a 2D mapping of an n-dimensional form generated using recursive functions.

We will not go further here other than to recommend more investigation
by you. Use your search engine to search for these terms**:**
chaos, fractal, recursion, self reference, etc. Read James Gleick's,
*Chaos*. Read Benoit Mandelbrot's, *The**
Fractal Geometry of Nature*. Read Patrick Grim's, *The**
Philosophical Computer*. Read works of Douglas Hofstadter,
too. Doug, although a well-trained SOMite, has done an immense
amount of work on topics relevant SOM's sophisms, self-reference,
and recursion. See his *Gödel, Escher, Bach*, and his
*Metamagical Themas*. Doug also has two other more recent
books (*Fluid Concepts and Creative Analogies*, and *Le**
Ton beau de Marot*) you can find if you search on his name
at your local online bookstore (we like Amazon.com). Doug had
a lot to do with an onslaught of computer viruses which depend
on self-reference to propagate their affects. Note that computer
viruses act in a much similar way nature does using self-reference
to create, evolve, commingle, and discreate life.

You will find it very worthwhile to use bibliographies of our recommended authors to find more information.

Be aware we just scratched surface on this broad subject. Our intent is not to be comprehensive on sophisms, but to show you SOM's limited, almost inutile perspective of reality in its dealings with self-reference.

Thanks for reading,

Doug.

**David Finkelstein of Georgia Tech:**

See Finkelstein's
page on the web. Also see references to his work in Nick Herbert's
*Quantum Reality* and on Rhett Savage's h
is for h-bar web site. Return.

**David J. Foulis of University of Massachusetts:**

Foulis presented a paper at a conference in May-June 1995,
i.e., *Einstein Meets Magritte* (*EMM*) conference.
Pirsig presented his *SODV*
paper at this same conference on June 1, 1995. Foulis' publications
on quantum logic and related subjects include**:**

*Mathematical Metascience*1998*Interval and Scale Effect Algebras*1996*A Half Century of Quantum Logic***—***What Have we Learned?*(Presented at the*EMM*conference.) 1995*Test Groups and Effect Algebras*1995*The Center of an Effect Algebra*1995*Effect Algebras and Unsharp Quantum Logics*1994*Filters and Supports in Orthoalgebras*1991*Superposition in Quantum and Classical Mechanics*1989

We requested Dr. Foulis' permission to publish or extensively quote his EMM paper here on our Quantonics site. As soon as we receive permission we will publish his paper or quote it extensively under our Quantum Connection part of this review. (We received Dr. Foulis' permission.) Return.

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(14Jul2000 rev - Repair minor typo ('an') in paragraph on chaos and fractals.)

(18Oct2000 rev - Add link to our recent Absoluteness as Quantum Uncertainty Interrelationship.)

(26Jan2004 rev - Substitute some GIFs and other symbols for incompatibilities.)

(29Jun2007 rev - Reformat. Minor red text updates.)

(9Dec2007 rev - Reformat slightly.)

(12-13May2009 rev - Add 'gnosis,' and 'wisdom' links.)