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Topic: A possible role for geometry in foundtions
Replies: 26   Last Post: Aug 26, 2013 7:32 PM

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 fom Posts: 1,968 Registered: 12/4/12
A possible role for geometry in foundtions
Posted: Aug 1, 2013 10:48 PM

In one of the matheology threads I asked ZG if
he would be interested in certain geometric
ideas I have formulated. This thread is a
presentation of those ideas.

It will have repetition of statements made
elsewhere.

The title of the initial post is meant to
convey the idea that certain fundamental
relations of propositional semantics can
be modeled geometrically. There had been a
time when I would have made foundational
claims. There comes a time when one learns
that such things are always a matter of
belief. Enough said. It is not about
belief.

This post has, as its primary purpose, the
presentation of a projective geometry and its
corresponding affine geometry. There are actually
two projective geometries presented. The
geometry corresponding to the main topic has
labels corresponding to the connectives of
propositional logic. The second geometry is
the original source of the labeling using
natural numbers. The basis of this construction
is what is called a difference set.

The table on page 94 (pdf page 106) of the

http://carlossicoli.free.fr/C/Coxeter_H.S.M.-Projective_Geometry,_2nd_Edition__-Springer%282003%29.pdf

serves as an an example from the literature
of the underlying construction involved.

What is relevant to this particular form
of presentation is the idea that a single
label is used for both points and lines.

This is the "math trick" used to relate
the relations of propositional logic to
extensions of propositional logic in a
"geometric" manner. If one numerically
orders the lines of the second geometry,
one will have a presentation similar to
that in the link.

The 16-point affine geometry is labeled
with names corresponding to propositional
connectives. But, it has 20 lines. Since
the lines have the same names as the points,
the system of 20 lines incorporates the
possibility of extension.

Consider, for example, extension by modal
operators. Using a unary negation, one would
have

[], []~, <>, <>~

or

[], []~, ~[]~, ~[]

But without using unary negation, the
extension introduces a quadruple with
specific relations.

For a variable of quantificational
logic one has,

Ex, Ex~, ~Ex~, ~Ex

or, respectively,

~Ax~, ~Ax, Ax, Ax~

So, the 20 lines of the affine geometry
extend the 16 labels by 4. In this sense,
it makes possible the notion of extension
interpretable relative to geometric structure.

There is another part of the "trick"
that will be discussed later. It involves
a substitution in the labels. The
substitution will be discussed here, but
the "trick" also involves relations in a
20-element ortholattice with a 16-element
Boolean sub-block.

In the presentation of labels, these 4
extension symbols are given Aristotelian
labels. There is

ALL and NO

In addition, there is

SOME and OTHER

Technically, one would not see 'OTHER'
in a syllogistic argument. The mnemonic
here is that OTHER is what is left after
SOME is chosen as the fixed species.

Relative to the affine structure, these
four labels are located on the line at
infinity of the projective structure.

Relative to the affine structure, the
remaining label on the line at infinity
of the projective structure is

NOT

An early motivating principle of this
work had been to develop an understanding
of logical structure without a unary
negation. It seemed natural to use this
label to complete the system of labels.

The second part of the "math trick" is
to replace the label NTRU corresponding
to an affine point with the label NTRU
in the set of line labels. This is
necessary by virtue of the construction.

One could compare this with Frege's use
of self-contradictory descriptions to
ground his quantificational logic or Kant's
use of self-contradictory descriptions
to ground the modal nature of his epistemological
investigations. What I mean here is that
something more is involved with the extension
from propositional logic to its extensions.
Given the necessary relations of the labeling,
the substitution mentioned above may be
given an interpretation along some such lines.

But, what is important at this point is
merely that a namespace is given having
certain relations determined by geometric
structure.

Here are the two geometries. I shall
add explanations to the thread a little
at a time.

===============================

NTRU: {NOT, NO, OTHER, ALL, SOME}

NOT: {NOT, NTRU, LEQ, XOR, TRU}
LEQ: {NOT, IF, NIMP, IMP, NIF}
XOR: {NOT, OR, NAND, AND, NOR}
TRU: {NOT, FLIP, LET, DENY, FIX}

NO: {NO, NTRU, IF, OR, FLIP}
IF: {NO, LEQ, NIMP, NAND, LET}
OR: {NO, XOR, IMP, AND, DENY}
FLIP: {NO, TRU, NIF, NOR, FIX}

OTHER: {OTHER, NTRU, IMP, NAND, FIX}
IMP: {OTHER, LEQ, NIF, OR, DENY}
NAND: {OTHER, XOR, IF, NOR, LET}
FIX: {OTHER, TRU, NIMP, AND, FLIP}

ALL: {ALL, NTRU, NIMP, NOR, DENY}
NIMP: {ALL, LEQ, IF, AND, FIX}
NOR: {ALL, XOR, NIF, NAND, FLIP}
DENY: {ALL, TRU, IMP, OR, LET}

SOME: {SOME, NTRU, NIF, AND, LET}
NIF: {SOME, LEQ, IMP, NOR, FLIP}
AND: {SOME, XOR, NIMP, OR, FIX}
LET: {SOME, TRU, IF, NAND, DENY}

===========================================

18: {0, 9, 11, 4, 3}

0: {0, 18, 8, 6, 1}
8: {0, 10, 19, 13, 14}
6: {0, 16, 12, 2, 15}
1: {0, 7, 17, 5, 20}

9: {9, 18, 13, 12, 20}
13: {9, 8, 14, 16, 5}
12: {9, 6, 10, 15, 17}
20: {9, 1, 19, 2, 7}

11: {11, 18, 10, 16, 7}
10: {11, 8, 19, 12, 17}
16: {11, 6, 13, 2, 5}
7: {11, 1, 14, 15, 20}

4: {4, 18, 14, 2, 17}
14: {4, 8, 13, 15, 7}
2: {4, 6, 19, 16, 20}
17: {4, 1, 10, 12, 5}

3: {3, 18, 19, 15, 5}
19: {3, 8, 10, 2, 20}
15: {3, 6, 14, 12, 7}
5: {3, 1, 13, 16, 17}

Date Subject Author
8/1/13 fom
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8/11/13 fom
8/10/13 fom
8/11/13 Sandy
8/11/13 fom
8/13/13 Shmuel (Seymour J.) Metz
8/13/13 fom
8/14/13 Shmuel (Seymour J.) Metz
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8/15/13 Shmuel (Seymour J.) Metz
8/23/13 fom
8/24/13 Shmuel (Seymour J.) Metz
8/25/13 fom
8/25/13 fom
8/26/13 Shmuel (Seymour J.) Metz
8/26/13 fom
8/13/13 Tucsondrew@me.com
8/13/13 fom