SUO: Re: Examples! Examples! Examples!
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EEE. Note 3
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Example 1. John Sowa's "Top Level Categories" (cont.)
Recall the alphabet !A! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"}
that we enumerated last time. These terms form what some people call the
"non-logical vocabulary" of the subject matter in question, and I suppose
that's a useful enough term for the moment. When we view things purely
graph-theoretically, these brands of labels will be called "paints".
In the Cactus Language, which can be interpreted in a couple of different ways
for zeroth order logic, the "logical vocabulary", if you can even call it that,
consists of just four symbols, specifically, the blank space character " ", the
left parenthesis "(", the comma ",", and the right parenthesis ")". When I use
the graph-theoretic idiom, I tend to call these "marks", as in punctuation marks,
or "measures". The two different interpretations that I mentioned correspond to
what C.S. Peirce called the "Entitative" and the "Existential" interpretations
of his graphical syntax. I will use the existential reading for this example.
In the existential interpretation, the blank symbol stands for two things:
1. Taken by itself it represents a boolean value of 1 or true.
2. Placed between terms it serves as a conjunctive connective.
The other symbols operate in tandem as a bracket of the form "( , , , )",
with any finite number k = 0, 1, 2, 3, ... of intervening argument places.
In the existential interpretation, this so-called "boundary bracket" has
the following basis for grounding its meaning:
1. For k = 0. The expression "()" means "false".
2. For k > 0. The expression "(e_1, ..., e_k)" means that
just one of the k expressions e_j is false.
This gives us enough to write out an axiom for TLC.
The first version I got just by looking at the
lattice diagram and trying to figure out what
it meant in terms of logical constraints on
the logical features that are represented
by the terms of the alphabet !A!.
Table 1. TLC in Cactus Language (Version 1)
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| |
| (( 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 )) |
| |
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To read this, one can think this way:
1. An expression of the form (x_1, x_2, ..., x_k), if you take it
as asserting or imposing a logical constraint on the argument
features x_1, ..., x_k, means that just one of the x_j is false.
2. In particular, (x) means that x is false.
3. Consequently, an expression of the form ((x_1),(x_2), ..., (x_k))
means that exactly one of the expressions (x_1),(x_2), ..., (x_k)
is false, which means that just one of the x_j is true. This is
precisely equivalent to saying that the universe is partitioned
into mutually exclusive and exhaustive classes each of which
falls under exactly one term of the list x_1, x_2, ..., x_k.
4. Now look at expressions of the form (x, (x_1),(x_2), ..., (x_k)),
where the term "x" appears at depth one and all the rest of the
terms x_1, ..., x_k appear at depth two. Consider the cases:
a. Suppose x is true. Then the expression reduces to
the form of a partition ((x_1),(x_2), ..., (x_k)).
b. Suppose x is false. Then it's the only one that is,
and the expression reduces to (x_1)(x_2) ... (x_k),
the conjunction of the negations of the other terms.
In effect, one has said that the things for which
the "genus" term x is true are partitioned among
the "species" terms x_1, x_2, ..., x_k, and the
the things for which x is false can have none
of the terms x_1, x_2, ..., x_k apply to them.
To sum it up, we can read a clause like:
| (( Object ),( Process ),( Schema ),( Script ),
| ( Juncture ),( Participation ),( Description ),( History ),
| ( Structure ),( Situation ),( Reason ),( Purpose ))
as asserting or imposing an absolute partition on the universe,
and we can read each one of the following clauses:
| ( 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 ))
as asserting or imposing a relative partition on the universe.
Jon Awbrey
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