Helmut, List ...
Short on time tonight so let me just
respond to the next point you raised:
HR:
> A simple triadic or n-adic relation, I think, belongs
> to secondness, and has only two modes, the quality, eg.
> function or caprice (intension), and the resulting set
> of tuples (extension).
There is a kind of secondness involved in any use of set theory,
indeed, there are several kinds of dyadic relations in the mix,
all intimately related. Letting X be the universe of discourse,
there is the dyadic elementhood or membership relation x ∈ X,
there is the subset relation A ⊆ X, and every subset A ⊆ X has
a “characteristic” or “indicator” function f_A : X → {0, 1} with
f_A(x) = 1 if x ∈ A and f_A(x) = 0 if x ∉ A. So one could say,
if one wishes, there is secondness afoot in the extensions of
whatever symbols one uses to demarcate or distinguish portions
of the universe. As it usually turns out, though, if you know
enough to invoke secondness, you usually know enough to say
something more specific about the dyadic relation you have
in mind.
This is a very old theme. The very word “existence”, whether by
way of folk etymology or not, is said to mean “standing out”, the
way a subset stands out against its ground. It's a nice image if
nothing else. In another connection, some take the prevalence of
these set-theoretic dyadic relations, along with their assumption
of set theory's foundational status, as proving all structure to
be ultimately dyadic.
Well, I have my reasons to doubt that,
but that's all the time I have tonight.
Regards,
Jon
On 4/21/2017 4:59 PM, Helmut Raulien wrote:
> Jon, List,
> I am not so sure, if thirdness is about any triadic relation.
> The categories in Peirce's "new list" of them are quality, relation,
> representation. Maybe "representation" is a very special kind of
> triadic relation. A simple triadic or n-adic relation, I think,
> belongs to secondness, and has only two modes, the quality, eg.
> function or caprice (intension), and the resulting set of tuples
> (extension). Example: The triadic function "x_1 + x_2 = x_3",
> with the three sets X_1, X_2, X_3 not being classes of any kind,
> at least not of the special kind (whatever that is), that would
> allow representation, and make it having to do with the third
> category. I guess, that a difference between Peirce's relation
> theory, and his semiotics and category theory, is, that the
> first is about all triadic relations, and the latter only
> about sign relations or representational relations (the
> special kind of triadic relations).
> Best,
> Helmut
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