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Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: LOGIC & MATHEMATICS
Posted: May 30, 2013 6:07 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 5/30/2013 11:20 AM, Julio Di Egidio wrote:
> "fom" <fomJUNK@nyms.net> wrote in message
> news:_-2dnT_Ah67vJTvMnZ2dnUVZ_oOdnZ2d@giganews.com...

>> On 5/29/2013 7:13 AM, Julio Di Egidio wrote:
> <snipped>
>

>> I almost forgot that I put these
>> in the thread. Assuming this is
>> to what you had been referring,
>> I have taken the time to find the
>> hypertext links.

>
> Yes, and thanks very much, you are awesome.
>

>> I am not trying to "purport" a logical
>> system as much as I am trying to
>> represent an analysis of it.

>
> I can parse what you write, but the technical relevance is quite above
> my head.


Suppose you are just a young person beginning a mathematics
curriculum. You keep running into parenthetical references
to "(choose your favorite set theory)".

In like fashion, one has "choices" concerning logic. There is
classical, realist logic and there is intuitionist logic in which
excluded middle is treated differently. Of course, there are many
"logics" besides these.

What exactly is involved with choosing a logic? In particular,
why might classical logic be "classical"?

Somewhere in Wittgenstein (TLP) is the comment that what
can be eliminated should be eliminated. I am not certain if
that need be a guiding principle, but, what happens if negation
is eliminated? It can certainly be done. Everyone knows what
a NAND and NOR are capable of doing since those tend to be the
primary logic gates for constructing computer circuits.

Two questions immediately arise. First, is there a criterion
for deciding whether either of the complete connectives should
be viewed as the canonical complete connective? That is, is
there an asymmetry in the dyslexia we call truth-functional
logic?

Second, the complete connectives have the capabilities they
do under interpretation precisely because of a "system". Although
the semantics of propositional connectives had not been apparent
in the original analyses leading to axiomatic representations
of classical logic, the effective decision procedure provided
by truth tables has clarified the nature of the compositionality
that arose from Frege's analyses. Given this, can one understand
a system without negation that expresses the classical bivalence?

I found this system in the 16 points of an affine geometry. The
transformations, of course, occur in the associated 21 point
geometry. The essential theorem involved is as follows:

"If 1, 2, 3, 4 are any four distinct elements of a one-dimensional
primitive form, there exist projective transformations which will
transform 1234 into any one of the following permutations of
itself: 1234, 2143, 3412, 4321"

"Projective Geometry"
Veblen and Young

So, let

1=AND
2=NAND
3=OR
4=NOR

Then the identity map (at this point I prefer to call it "recitation"
because of the overuse of "identity") is obvious,

1234 :=> 1234

Negation is

1234 :=> 2143

Conjugation (De Morgan) is

1234 :=> 3412

Contraposition is

1234 :=> 4321

This is easily seen using truth tables IF you disassociate your
notion of a truth function from some fixed representation for
expressing the system of truth functions. This is discussed in
the post

https://groups.google.com/group/sci.logic/msg/05d458e86b5c363a?dmode=source


I will stop here because it is quite a bit all at once. But,
what I see in my mind's eye is a structure without 'T' or 'F'
and without negation (in the affine points) that expresses
classical bivalence. The entire system of truth functions
is needed to express the compositionality (the intensional
equational axiomatization) that came from the Fregean
analysis.

And, it is compatible with Frege's retraction of logicism:

"The more I have thought the matter
over, the more convinced I have become
that arithmetic and geometry have
developed on the same basis -- a
geometrical one in fact -- so that
mathematics in its entirety is
really geometry"

Frege


It is also compatible with a certain Russellian view:

"..., I shall deal first with projective
geometry. This, I shall maintain, is
necessarily true of any form of
externality, and is, since some such
form is necessary to experience,
completely a priori."

Russell

> Spare thoughts: by looking at your tables, you've made me
> think back to a prototype logical calculator I have concocted together
> last year: I could get all logical connectives on some basic lattice
> from one single primitive, but to get any arithmetic at all I had to
> start fixing points on the lattice.
>


Well, my ideas are actually motivated by considering
set theory. So, arithmetic, for me, is inside of a
theory of classes...


>> However you come to view these posts, please
>> try to keep in mind that it is hard to see
>> things differently from others. And, since
>> I received no inquiries, there is little to
>> no explanation.

>
> As an outsider to that specific research, I'd surely benefit from an
> explicit problem statement: namely, "what was that for?" would be the
> first question. In fact, I won't insist for more explanations: if you
> keep posting, I'll keep learning...
>


As long as you are interested, it is a benefit
to me to make explanations. I owe you some
thanks for your questions.






Date Subject Author
5/26/13
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5/28/13
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fom
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