Logical Graphs

[Jon Awbrey]

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seqfans,

i'm in the process of "memoir-izing" my lifelong adventures
at the lambda point of algebra, geometry, and logic, a big
part of which is my exploration of peirce's logical graphs.

there's an intro in progress here:

http://www.getwiki.net/wiki.php?title=Logical_Graph

and a slightly more adventurous memoir, more trepidly in progress here:

http://www.getwiki.net/wiki.php?title=Futures_Of_Logical_Graphs

nota bene.  these are being written up at the GetMeta wiki site,
so the talk pages are available for questions and discussion.

i'm not really seeing the sequences the way i used to right now,
but i think there may be some musement for seqfans in this area.

the generic idea is that you have a syntax given by a formal language
that has a nice correspondence with graph-theoretical structures, say,
the correspondence between rooted trees and closed parenthesis strings.

then you have a semantics that partitions the syntactic elements into
"semantic equivalence classes" (SEC's).  enumerative aspects come in
here when you think of counting the syntactic elements by classes.

very often, the SEC's are induced by "rules of equivalence".

for example, with rooted trees expressed as closed parentheses strings,
we may have the rules:

"()()" = "()" and "(())" = "" = <empty expression>

these rules partition the space of rooted trees
into the SEC of "()" and the SEC of "".

that is what some folks (after Spencer-Brown)
call the "arithmetic" of the system at hand.

the syntax of the "algebra" is obtained by introucing indeterminates:
a, b, c, ..., p, q, r, ..., x, y, z, ...

the semantics of the algebra is obtained by observing consequential
rules of equivalence, for example, the following:

"a()" = "()" and "a((b)(c))" = "((ab)(ac))"

et sic deinceps ...

jon awbrey

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