fom
Posts:
1,969
Registered:
12/4/12


Re: LOGIC & MATHEMATICS
Posted:
May 30, 2013 6:07 PM


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 truthfunctional 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 onedimensional 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.

