SUO: Re: In Praise of Zeroth Order Logic
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ZOL SIG:
In my qualified response (or hedged bet) with regard to Jim's question
about "What Language to Use", I showed how the partition constraints
that are implicit in John Sowa's "Top Level Categories" (TLC) can
be represented as a single proposition in a particular language
for propositional, sentential, or zeroth order logic (ZOL).
I have been calling this particular representation of ZOL
by the name of the "reflective extension of logical graphs",
or "RefLog", for short.
At the present time I am still not certain that I have read
the intentions of John's lattice representation correctly,
but, as far as its use in providing us with an example of
a certain level of complexity, it does not really matter.
I will now elaborate on the meanings of these types of expressions,
as a way of illustrating the properties of the formal language
in which they are written.
Let us give this proposition, the one that expresses the
feature constraints of TLC, a shorter name, say, "Tolc".
Thus, we have:
| "Tolc" =
|
| "
| (( 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 ))
|
| ))
| "
Just by way of review:
> This expression makes use of two basic forms:
>
> 1. An expression of the form "(( X1 ),( X2 ), ... ,( Xk ))"
> says that the universe of discourse is partitioned into
> k "mutually exclusive and exhaustive categories" (MEEC's),
> those for which the propositions X1, X2, ..., Xk, respectively,
> are true.
>
> 2. An expression of the form "( Y ,( Y1 ),( Y2 ), ... ,( Yk ))"
> says that the part of the universe of discourse where Y is true
> is partitioned into k MEEC's, those for which the propositions
> Y1, Y2, ..., Yk, respectively, are true.
>
> The "recessing" of the larger part of the expression within
> a logically otiose double negation "(( ... ))" is merely a trick
> that makes the processing more efficient for a particular program.
Here is the basic set-up.
Let A be a formal alphabet (lexicon, vocabulary) of 25 logical features,
which may be given nominal and verbose spellings in the following way:
| x<01> = Abstract,
| x<02> = Actuality,
| x<03> = Continuant,
| x<04> = Description,
| x<05> = Form,
| x<06> = History,
| x<07> = Independent,
| x<08> = Intention,
| x<09> = Juncture,
| x<10> = Mediating,
| x<11> = Nexus,
| x<12> = Object,
| x<13> = Occurrent,
| x<14> = Participation,
| x<15> = Physical,
| x<16> = Prehension,
| x<17> = Process,
| x<18> = Proposition,
| x<19> = Purpose,
| x<20> = Reason,
| x<21> = Relative,
| x<22> = Schema,
| x<23> = Script,
| x<24> = Situation,
| x<25> = Structure.
Given an arbitrary alphabet of logical features A = {x1, ..., xk},
let us employ the following notations to talk about various sorts of
objects that occur in and around the associated universe of discourse:
1. <<A>> = <<x1, ..., xk>> =
The "set of interpretations" (cells, points, vectors)
in the universe of discourse that is generated by A.
2. A^ = (<<A>> -> B) =
The "set of propositions" (boolean functions) on A.
3. [[A]] = [[x1, ..., xk]] =
The "universe of discourse" that is generated by A,
comprising both of the previous sets, and accordingly
comprehended as the ordered pair (<<A>>, A^), made up
of all logical points and all logical functions that
are associated with the alphabet A.
As far as the types of these spaces go --
geometric, functional, and integrated --
we have the following specifications:
1. <<A>> : B^k, in other words, <<A>> is isomorphic to B^k.
2. A^ : (B^k -> B), the space of functions {f : B^k -> B}.
3. [[A]] : (B^k, (B^k -> B)), notated as (B^k +-> B) or [[B^k]].
In this setting, each proposition f in A^ has two kinds of typings:
a. The "concrete typing" of the form f : <<A>> -> B
elicits the full richness of the qualitative features in A.
b. The "abstract typing" of the form f : B^k -> B
demotes the concrete units to their quantitative codes in B^k.
Given all this, then, the proposition Tolc is (or can be interpreted as)
a function Tolc : <<A>> -> B, where A = {x<01>, ..., x<25>} as above, or
a function Tolc : B^25 -> B, up to a level of "isotypical abstraction".
Next time I will discuss how a proposition like Tolc specifies an embedding --
not in the least bit procrustean! -- of a lattice like TLC within a certain
type of boolean lattice, as I briefly suggested, a little bit inaccurately,
in the following statement:
> This whole expression effectively tells one how to embed the lattice
> in B^25, where B = {0, 1}, a 25-dimensional universe of discourse --
> binary cube, truth table, venn diagram, or however you want to view it --
> 25 being the number of terms in the vocabulary, which are here interpreted
> as binary features or boolean variables.
Until next time,
Jon Awbrey
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