Date: Dec 11, 2012 9:35 PM
Author: fom
Subject: Re: fom - 11 - CORRECTED - definition of proposition


The ortholattice





TRU

/ \
/ \
/ \
/ \
/ \
/ \

NO ALL

| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |

OTHER SOME

\ /
\ /
\ /
\ /
\ /
\ /

NOT




is a sublattice of the one constructed
from our line names.


Let A be some linguistic expression.


The expressions:


A
NOR(A,A)
NOR(NOR(A,NOR(A,A)),NOR(A,NOR(A,A)))
NOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A)))


label vertices in a lattice corresponding
to the free DeMorgan algebra on one
generator:



TRU

|
|
|

NOR(NOR(A,NOR(A,A)),NOR(A,NOR(A,A)))

/ \
/ \
/ \

A NOR(A,A)

\ /
\ /
\ /

NOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A)))

|
|
|

NTRU.




Then, A is a proposition if and only if all
maps from the free DeMorgan algebra generated
from A into the sublattice from our
20-element ortholattice has

TRU --> TRU
NOR(NOR(A,NOR(A,A)),NOR(A,NOR(A,A))) --> TRU

NTRU --> NOT
NOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A))) --> NOT

and one of

A --> ALL
NOR(A,A) --> OTHER

A --> SOME
NOR(A,A) --> NO

A --> OTHER
NOR(A,A) --> ALL

A --> NO
NOR(A,A) --> SOME



Note that the top and bottom correspond to
the ideal points significant to the topology
on the connectivity algebra.