Adic Versus Tomic

[Jon Awbrey]

Re: Multiple-Valued Logic
At: https://list.iupui.edu/sympa/arc/peirce-l/2020-11/thrd1.html#00022

HR: As Peircean semiotics is a three-valued logic,
I think it bears relevance for the discussion
about multiple-valued logic.

Helmut & All,

The distinction between “k-adic” (involving a span of k dimensions)
and “k-tomic” (involving a range of k values) is one of the earliest
questions I can remember discussing on the Peirce List, along with the
host of other lists we often cross-posted on in those heady surfer days.
It is critical not to confuse the two aspects of multiplicity. In some
cases it is possible to observe what mathematicians call a “projective”
relation between the two aspects, but that does not make them identical.

I'm adding a lightly edited excerpt from one of those earlier discussions
as I think it introduces the issues about as well as I could manage today.

Regards,

Jon

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[Arisbe] Re: Inquiry Into Isms • k-adic versus k-tomic
Jon Awbrey ari...@stderr.org
Tue, 21 Aug 2001 00:34:30 -0400

https://web.archive.org/web/20141005035422/http://stderr.org/pipermail/arisbe/2001-August/000878.html

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Here is an old note I've been looking for since we started on this bit about isms,
as I feel like I managed to express in it somewhere my point of view that the key
to integrating variant perspectives is to treat their contrasting values as axes
or dimensions rather than so many points on a line to be selected among, each in
exclusion of all the others. To express it briefly, it is the difference between
k-tomic decisions among terminal values and k-adic dimensions of extended variation.

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Subj: Re: Dyads
Date: Fri, 08 Dec 2000 00:48:18 -0500
From: Jon Awbrey <jaw...@oakland.edu>
To: Stand Up Ontology <standard-up...@ieee.org>

Jon Awbrey wrote (JA):
Tom Gollier wrote (TG):

JA: I think that we also need to distinguish "dichotomous thinking" ...
from "dyadic thinking" (DT). One has to do with the number of values,
{0, 1}, {F, T}, {evil, good}, and so on, that one imposes on the cosmos,
the other has to do with the number of dimensions that a persona puts on
the face of the deep, that is, the number of independent axes in the frame
of reverence that one projects on the scene or otherwise puts up to put the
cosmos on.

TG: Your transmission [above] kind of faded out after the "number of values",
but do you mean a difference between, say, two values of truth and falsity
on the one hand, and all things being divided into subjects and predicates,
functions and arguments, and such as that on the other? If so, I'd like
to second the notion, as not only are the two values much less odious,
if no less rigorous, in their applications, but they're often maligned
as naive or simplistic by arguments which actually should be applied
to the idea, naive and simplistic in the extreme, that there are
only two kinds of things.

JA: There may be a connection -- I will have to think about it --
but "trichotomic", "dichotomic", "monocotyledonic", whatever,
refer to a number of values, 3, 2, 1, whatever, as in the range
of a function. In contrast, "triadic", "dyadic", "monadic",
as a series, refer to the number of independent dimensions that
are involved in a relation, which you could represent as axes
of a coordinate frame or as columns in a data table. As the
appearance of the word "independent" should clue you in,
this will be one of those parti-colored woods in which
the interpretive paths of mathematicians and normal
folks are likely to diverge.

JA: There is a typical sort of phenomenon of misunderstanding that often
arises when people imbued in the different ways of thinking try to
communicate with each other. Just to illustrate the situation for
the case where n = 2, let me draw the following picture:

| Dyadic Span of Dimensions
| ^ ^
| \ /
| \ /
| o o
| |\ /|
| | \ / |
| | \ / |
| | \ / |
| v \ / v
| <-----o-----o-----o----->
| Dichotomic Spectrum of Values

JA: This is supposed to show how the "number of values" (NOV) thinker
will project the indications of the "number of axes" (NOA) thinker
onto the straight-line spectrum of admitted directions, oppositions,
or values, tending to reduce the mutually complementing dimensions
into a tug-of-war of strife-torn exclusions and polarizations.

JA: And even when the "tomic" thinker tries to achieve a balance,
a form of equilibrium, or a compromising harmony, whatever,
the distortion that is due to this manner of projection
will always render the resulting system untenable.

JA: Probably my bias is evident.

JA: But I think that it is safe to say, for whatever else
it might be good, tomic thinking is of limited use in
trying to understand Peirce's thought.

JA: Just to mention one of the settings where this theme
has arisen in my studies recently, you may enjoy the
exercise of reading, in the light of this projective
template, Susan Haack's 'Evidence & Inquiry', where
she strives to achieve a balance or a compromise
between foundationalism and coherentism, that is,
more or less, objectivism and relativism, and
with some attempt to incorporate the insights
of Peirce's POV. But a tomic thinker, per se,
will not be able to comprehend what the heck
Peirce was talking about.

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[Cliff Joslyn]

Jon: I think what you have is sound, and can be described in a number of ways. In years past in seeking ways to both qualify and quantify variety in systems I characterized this distinction as between "dimensional variety" and "cardinal variety". Thankfully, this seems straightforward from a mathematical perspective, namely in a standard relational system S = \times_{i=1}^k X_i, where the X_i are dimensions (something that can vary), typically cast as sets, so that \times here is Cartesian product. Here k is the "dimensional variety" (number of dimensions, k-adicity), while n_i = |X_i| is the "cardinal variety" (cardinality of dimension i, n_i-tomicity (n_i-tonicity, actually??)). One might think of the two most classic examples:

*) Multiadic diatom/nic: Maximal (finite) dimensionality, minimal non-trivial cardinality: The bit string < b_1, b_2, ..., b_k > where there are k Boolean dimensions X_i = { 0,1 }. One can imagine k \goesto \infty, an infinite bit string, even moreso.

*) Diadic infini-omic: Minimal non-trivial dimensionality, maximal cardinality: The Cartesian plane \R^2, where there are 2 real dimensions.

There's another quantity you didn't mention, which is the overall "variety" or size of the system, so \Prod_{n=1}^k n_i, which is itself a well-formed expression (only) if there are a finite number of finite dimensions.

--
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| Cliff Joslyn, Cybernetician at Large
V cajo...@gmail.com