Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: fom - 11 - definition of proposition
Replies: 1   Last Post: Dec 11, 2012 9:35 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
fom - 11 - definition of proposition
Posted: Dec 8, 2012 6:22 AM

The ortholattice

TRU

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

NO ALL

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

OTHER SOME

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

THIS

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 --> THIS
NOR(NOR(A,A),NOR(NOR(A,A),NOR(A,A))) --> THIS

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.

Date Subject Author
12/8/12 fom
12/11/12 fom