SUO: Re: IFF LOT Glossary
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Robert & All,
Assuming for the moment that my last correction is correct,
let's see if we have enough data to make an example of TLC.
| Let's say we start with a formal language L c !A!*.
| Written another way, L in Pow(!A!*). Thus, Pow(!A!*)
| is our first candidate for a "lattice of languages" (LOL)
| over the alphabet !A!. Let's write LOL(!A!) = Pow(!A!*).
|
| A theory T is a just a subset of L, in symbols, T c L.
| Written another way, T in Pow(L) where L in Pow(!A!*).
| We need to know the relation now between T and !A!.
|
| * Pow
| !A! ----- !A!* ----- LOL(!A!)
| | |
| c elt
| | | Pow
| L ===== L ----- Pow(L)
| | | |
| c c elt
| | | |
| T ===== T ===== T
To formalize John Sowa's "Top Level Categories" (TLC), we take
a (descriptive or ontological) alphabet or lexicon of 25 terms,
!TLC! = {a_1, ..., a_25} = {"Abstract", ..., "Structure"}.
JA: We think of the alphabet as providing us with a "codebook", a filter,
or a template, that we use to code arbitrary elements of experience
that come to us from a source or space that we may call, without
too much loss of generativity, "X". So !TLC! determines a map,
code : X -> TLC = <|!TLC!|> ~=~ B^25. For any "predicate" f
about the world X, that is, any f : X -> B, the code map
induces a coded predicate code(f) : TLC -> B given by
the equation (code(f))(code(x)) = code(f(x)) = f(x),
given that the code map acts as the identity on B.
Given the open nature of X, this is the sort of equation
that can hold approximately at best, but I do not know
how to formalize that, or even whether it can be done.
f
X o----------->o B
\ ^
\ /
code \ / code(f)
\ /
v /
o
TLC ~=~ B^25
(code(f))(code(x)) = f(x)
code
X o----------->o TLC ~=~ B^25
| |
| |
f | | code(f)
| |
v v
B o============o B
code
(code(f))(code(x)) = code(f(x)) = f(x)
To get a language with the power of ZOL over !TLC!, we adjoin the extra marks:
!M! = {m_1, m_2, m_3, m_4}
m_1 = " " = blank
m_2 = "(" = links
m_3 = "," = comma
m_4 = ")" = right
Take the augmented alphabet !A! = !A!(!TLC!) = !M! |_| !TLC!.
There is a formal language L = !C!(!TLC!) c !A!*, a grammar for which is known,
and a parser for which has been written. Each expression of L can be read to
denote one of the functions f : TLC -> B, conveniently called a "proposition".
For any f : TLC -> B, we also write: f in TLC^ = (TLC -> B) = {f : TLC -> B}.
And, of course, by virtue of the fact that f : TLC -> B "indicates" a subset
of TLC, we can also read each expression of L as denoting such a subset.
* Pow
!A! ----- !A!* ----- LOL(!A!) = Pow(!A!*)
| |
c elt
| | Pow
L ===== L ----- Pow(L)
| | |
c c elt
| | |
T ===== T ===== T
As examples of theories about TLC, we spent some time looking at a couple, both
of which can be expressed as singleton theories, that is, as sets that consist
of single expressions from L. We examined T_!a! = {!a!} and T_!c! = {!c!},
where !a! is shown in Table 20 and !c! is shown in Table 21, below:
Table 20. TLC in Cactus Language: Axiom !a!
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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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Table 21. TLC in Cactus Language: Axiom !c!
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| |
| (( Independent ),( Relative ),( Mediating )) |
| |
| (( Physical ),( Abstract )) |
| |
| (( Continuant ),( Occurrent )) |
| |
| (( Actuality , Independent Physical )) |
| (( Form , Independent Abstract )) |
| (( Prehension , Relative Physical )) |
| (( Proposition , Relative Abstract )) |
| (( Nexus , Mediating Physical )) |
| (( Intention , Mediating Abstract )) |
| |
| (( Object , Independent Physical Continuant )) |
| (( Process , Independent Physical Occurrent )) |
| (( Schema , Independent Abstract Continuant )) |
| (( Script , Independent Abstract Occurrent )) |
| (( Juncture , Relative Physical Continuant )) |
| (( Participation , Relative Physical Occurrent )) |
| (( Description , Relative Abstract Continuant )) |
| (( History , Relative Abstract Occurrent )) |
| (( Structure , Mediating Physical Continuant )) |
| (( Situation , Mediating Physical Occurrent )) |
| (( Reason , Mediating Abstract Continuant )) |
| (( Purpose , Mediating Abstract Occurrent )) |
| |
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Jon Awbrey
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