to objective Actuality (i.e., known) to both Actuality and Nonactuality
appearance
of any nonactuality 
paradox there are no
paradice
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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:
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:
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:
|
|
|
| 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:
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:
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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