[Inquiry] Re: Examples! Examples! Examples!

[Jon Awbrey]

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EEE.  Note 21

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There are many ways of probing the space TLC = <|a_1, ..., a_25|>
to find the "models" of a given proposition in TLC^ = (TLC -> B),
that is, the assignments of {false, true} values or {0, 1} values
to the 25 boolean variables that are associated with the alphabet
!TLC! = {a_1, ..., a_25} that render the value of the proposition,
say, q : TLC -> B, true, or equal to 1 in B.  However one does it,
doing this amounts to computing the truth table of the function q,
a truth table that spans 2^25 values x = <x_1, ..., x_25> for the
independent variable x in TLC ~=~ B^25, and for each one of these
a determinate value of the dependent variable, given by q(x) in B.

Obviously, we do not want to generate this truth table in a literal fashion
if we can find a smarter, more virtual way to acquire the same information.

As a manifestoly pertinent example of a proposition q : TLC -> B, let's
go back to my initial version of a TLC axiom !a! = !a!_1 : TLC -> B that
I got from poring over the Figure of the Top Level Category lattice here:

http://www.jfsowa.com/ontology/toplevel.htm

Now, having never received any feedback from the author as to whether
my axiom !a!_1 is a faithful representation of the intended structure,
I will simply go forward with it as one example of a proposition that
is worth examining for its implications, its impliers, and its models.

Here again is the expression of the proposition !a!_1.

Table 1.  TLC in Cactus Language (Version 1)
o-----------------------------------------------------------------------o
|                                                                       |
|   (( Object      ),( Process       ),( Schema      ),( Script  ),     |
|    ( Juncture    ),( Participation ),( Description ),( History ),     |
|    ( Structure   ),( Situation     ),( Reason      ),( Purpose ))     |
|                                                                       |
|   ( Independent   ,( Actuality  ),( Form        ))                    |
|   ( Relative      ,( Prehension ),( Proposition ))                    |
|   ( Mediating     ,( Nexus      ),( Intention   ))                    |
|                                                                       |
|   ( Physical      ,( Actuality ),( Prehension  ),( Nexus     ))       |
|   ( Abstract      ,( Form      ),( Proposition ),( Intention ))       |
|                                                                       |
|   ( Continuant    ,( Object      ),( Schema    ),( Juncture      ),   |
|                    ( Description ),( Structure ),( Reason        ))   |
|                                                                       |
|   ( Occurrent     ,( Process     ),( Script    ),( Participation ),   |
|                    ( History     ),( Situation ),( Purpose       ))   |
|                                                                       |
|   ( Actuality     ,( Object      ),( Process       ))                 |
|   ( Form          ,( Schema      ),( Script        ))                 |
|   ( Prehension    ,( Juncture    ),( Participation ))                 |
|   ( Proposition   ,( Description ),( History       ))                 |
|   ( Nexus         ,( Structure   ),( Situation     ))                 |
|   ( Intention     ,( Reason      ),( Purpose       ))                 |
|                                                                       |
o-----------------------------------------------------------------------o

A more symbolic, hopefully smarter way to tease out the models of a proposition
q : TLC -> B, as denoted by and expressed in a particular syntactic formula !q!,
for example, an expression in the cactus language, !q! in L(!TLC!) = !C!(!TLC!),
is to compute the "disjunctive normal form" (DNF) of the given proposition q or
its given expression !q!.  Well, it's only smarter if one does it in a smarter
way, otherwise one is pretty much doing exactly the same amount of work as
it takes to generate the whole truth table, not to mention the overhead
expense that is needed to maintain the whole motley, if dynamic array
of symbolic indirections that ultimately refer to logical values.

Jon Awbrey

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