Jeff,
There's a rather long history of people borrowing Peirce's terminology while leaving his definitions
and methods behind. I see that fine old tradition has now been extended to category theory, graph
theory, and who knows what else.
The full title of Peirce's 1870 Logic of relatives is “Description of a Notation for the Logic of
Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”. From
the very beginning Peirce is clear about the distinction in semiotic roles between what plays the
role of a "notation" (a calculus, a language, or any brand of syntactic system) and what plays the
role of the object domain that is thereby denoted. Relative terms (rhemes or rhemata) are pieces of
notation in the sign domain. As are logical graphs (entitative or existential). The structure of a
syntactic expression does convey information about the structure of the relation it denotes, and
maybe its okay to call the formula "iconic" on that account, so long as you don't forget the great
differences between a single string of syntax and a whole collection of tuples that constitutes the
extension of the relation in question.
A good notation is a very handy thing, but only so long as you remember that it
is notation.
Otherwise nothing but confusion reigns ...
Regards,
Jon
Jeffrey Brian Downard wrote:
Sung, Jerry, List,
I understand that we are using somewhat different language and graph systems as bases for
understanding Peirce's arguments in his phenomenology and semiotics. Jon has suggested
that Peirce's different ways of diagramming these relations are hardly more than hints of
ideas, so we should be careful about how much we derive from one or another manner of
representing these things graphically. I want to resist Jon's suggestion. As such, let
me try to restate the question. Given your way of putting things, let me ask the
following question: is there anything that is obscured if you assume that there is an
"internal node" at the heart of every elementary relation? I am suggesting
that any graphical system, including the one that you have formulated or a bipartite
system of graphs that allows nodes to be connected to several lines (i.e. edges,
relations, etc.) will run the risk of obscuring things that Peirce wanted to analyze
further.
I recognize that you and Jerry seem to have different dispositions towards the
way Peirce is using graph theory to analyze these things. You are saying that
your system is equivalent to Peirce's and that it expresses the heart of what
he is really trying to say. Jerry, on the other hand, seems to suggest that
Peirce is hamstrung by limitations that have their source in outmoded 19th
century ways of thinking about and graphing chemical relations.
So, let me try to state the question I've been trying to push as a challenge.
For those who think that the most elemental relations can be understood as a
node with one, two or three relations jutting out from it (such as you have
characterized them in your diagrams, and as the nodes and edges are
characterized in the example of the bipartite graph on the WikiPedia site I
referred to earlier), are you able to articulate the basic points Peirce is
making against Kempe's analysis of mathematical form in your terms? My hunch
is that Peirce's arguments are valid, that they can be used against alternate
analyses of the reasonings--and that these alternate analyses fail because they
tend to obscure points about the elemental character of a triadic relation that
Peirce wanted to make as clear as possible.
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
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