ONT Re: Apposite Purposes Of Logical Languages Objectified (APOLLO)
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Here is a rough introduction to the RefLog Syntax:
Define the "alphabet of punctuation marks" $M$ = {" ", ",", "(", ")"}.
The members of $M$ are vocalized as "blank, "comma", "links", "right".
1. There is a parametric family $L$ of formal languages of character strings,
where the parameter in question is a set $X$ of so-called "variable names",
such that, for each set $X$ = {"x_1", ..., "x_k"} of variable names, there
is a formal language L($X$) in $L$ over the alphabet A($X$) = $M$ |_| $X$.
The grammar of these languages can be given in all of its boring detail,
but most folks will get the gist of it easily enough from looking at
a representative sample of strings in a typical language L($X$).
| Examples. If $X$ = {"x", "y"}, then these are typical strings in L($X$):
|
| " ", ( ), x, y, (x), (y), x y, (x y), (x, y), ((x)(y)), ((x, y)), ...
|
| Nota Bene. When you start to get spots in front of your eyes,
| you will tend to get lax about putting in all the quote marks.
2. There is a parallel family of formal languages of graphical structures,
generically known as "painted and rooted cacti" (PARC's), that exist in
a one-to-one correspondence with these string expressions, being more or
less roughly, at a suitable level of abstraction, their parse graphs as
data structures in the computer. The PARC's for the above formulas are:
| Examples.
| x y x y
| o o o---o
| x y x y x y \ / \ /
| o o o o o---o o o
| | x y | | x y | \ / | |
| @, @, @, @, @, @, @, @, @, @, @, ...
|
| " ", ( ), x, y, (x), (y), x y, (x y), (x, y), ((x)(y)), ((x, y)), ...
Together, these two families of formal languages constitute a system
that is called the "reflective extension of logical graphs" (RefLog).
Strictly speaking, RefLog is an abstract or "uninterpreted" formal system,
but its expressions enjoy, as a rule, two dual interpretations that assign
them the meanings of propositions or sentences in "zeroth order logic" (ZOL),
to wit, what Peirce called the "alpha level" of his systems of logical graphs.
For example, the string expression "(x (y))" parses into the following graph:
| x y
| o---o
| |
| @
You can "deparse" the string from the graph by traversing
it like so, reading off the marks and variables as you go:
| o---"x"-"("-"y"-->o
| ^ |
| | x ( y |
| | o-------o v
| | | ) ")"
| | | |
| "(" (|) ")"
| ^ | |
| | | |
| | @ |
| | v
| Start Finish
| Traverse Traverse
In the "existential interpretation" of RefLog,
in which I do my own thinking most of the time,
concatenation of expressions has the meaning of
logical conjunction, while "(x)" has the meaning
of "not x", and so the above string and graph have
a meaning of "x => y", "x implies y", "if x then y",
"not x without y", or anything else that's equivalent.
The blank expression is assigned the value of "true".
Hence, the expression "()" takes the value of "false".
The bracket expression "(x_1, x_2, ..., x_k)" is given
the meaning "Exactly one of the x_j is false, j=1...k".
Therefore, "((x_1),(x_2), ...,(x_k))" partitions the
universe of discourse, saying "Just one x_j is true".
Jon Awbrey
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