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Dec 28, 2020, 3:35:52 PM12/28/20

to Peirce List, Cybernetic Communications, Ontolog Forum, Structural Modeling, SysSciWG

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

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

[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

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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.

o~~~~~~~~~o~~~~~~~~~o~~~~~~ARCHIVE~~~~~~o~~~~~~~~~o~~~~~~~~~o

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.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

[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

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

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.

o~~~~~~~~~o~~~~~~~~~o~~~~~~ARCHIVE~~~~~~o~~~~~~~~~o~~~~~~~~~o

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.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Dec 28, 2020, 7:39:40 PM12/28/20

to Jon Awbrey, cyb...@googlegroups.com, cajo...@gmail.com

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.

--

O------------------------------------->

| Cliff Joslyn, Cybernetician at Large

V cajo...@gmail.com

*) 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.

O------------------------------------->

| Cliff Joslyn, Cybernetician at Large

V cajo...@gmail.com

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