SUO: *Date 03 Apr 2002 -- Logical Confoundations
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Logical Confoundations
Today I will begin in earnest to try and explain what is so wrong
about any formulation of classical logic that proposes to employ
the quintain of logical operators {and, or, not, implies, equals}
as its basis, treating them as "primitives", whether explicitly,
by postulation, or implicitly, by default.
Up til now, I have felt forced to explain my concerns by way of
analogy and even metaphor, saying that this basis is as bad for
logic as roman numerals are for arithmetic, and these similes
are far more apt than many may presently understand, but the
reasons that I say all this, and persistently, are supported
by mathematical reasonings of the "utmost severity", and it
is past time that I began to lay it out in category detail.
To express the matter in the most precise of technical terms,
the pentad {and, or, not, implies, equals} -- and, of course,
by "equals" I mean "logical equivalence" -- is algebraically,
formally, posing as logical basis, "confoundedly degenerate".
By way of one last analogy, but still a bit nearer to the crux of
the problem, calling this fabulous five a set of "primitives" is
every bit as ill-founded for the purposes of classical logic as
calling {1, 2, 3, 4, 5} a set of "primes" would be for the ends
of number theory.
The closest I have come to explaining this clearly was in this comment:
| I personally associate it with questions about the "presentation"
| of mathematical structures in terms of "generators and relations",
| but I don't know if that is a familiar enough resource hereabouts.
|
| If I could put it in those terms, I would say that
| the pentalphabet {~, &, v, =>, <=>} does not make
| a good set of "generators" because there are too
| many "relations" to be found among its elements.
|
| But I know that you know what it means to present an axiom system
| in the form of independent axioms. Even though it can be hard to
| find such a presentation and tough to prove it when you've got it,
| most people sense that it's more than just elegance, but a matter
| of cognitive economy or conceptual efficiency to go for that form.
|
| And that style of elect élan is just what the pentaglyphic
| pseudo-basis {~, &, v, =>, <=>} fails to achieve for logic,
| because the independence of these operators is compromised
| by the excessive complicity of their insider entanglements.
|
| At the outset this factor affects only that part of logic that
| has to do with indicating objects of the type q : %B%^k -> %B%.
| But whatever affects the complexion of that face of indication
| also redounds on the complex of descriptions like p : X -> %B%.
|
| http://suo.ieee.org/email/msg07840.html
So let me start again by trying to clear up
the residual obscurities in that statement.
But Tomorrow,
Jon Awbrey
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