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.