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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:

  1. 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.)
  2. As perceived by Greeks starting approximately with Homer's Iliad through Aristotle
  3. Medieval sophistry using Buridan's work as an exemplar
  4. Modern sophistry as viewed by Western dialecticians in 20th century
  5. Quantum logic (sophistry; in Quantonics we now, CeodE 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:

  1. C is the set of all sets (implication: only one power set of all sets exists)
  2. Given 1, then every subset of C is a member of C
  3. Given 1 & 2, then the power set of C is a subset of C (power set is some base raised to C power)
  4. But statements above imply the cardinal value of the power set of C is less than or equal to the cardinal value of C
  5. Cantor's theorem denies 4
  6. 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 both an actual realm and a nonactual realm. Where SOM reality is incomplete, quantum/MoQ reality is complete. Let us show a quantum reality dual of SOM's steps:

  1. C is a set of all sets (see comments about this below)
  2. Given 1, then every subset of C is a member of C
  3. Given 1 & 2, then a power set of C is a new context of C
  4. 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
  5. 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 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,


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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©Quantonics, Inc., 1998-2011 Rev. 12-13May2009  PDR Created: 17Dec1998  PDR
(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.)