Inquiry Into Inquiry

Semiotics, Semiosis, Sign Relations • Discussion 13

Posted on August 30, 2021 by Jon Awbrey

Re: Category Theory • Peiyuan Zhu • Henry Story

Dear Peiyuan, Henry …

Way back during my first foundations + identity crisis I explored every alternative, deviant, non-standard version of logic and set theory I could scrape up — I remember saying to one of my professors, “How come we’re still talking about logical atoms in the quantum era?” — and he sent me off to read about quantum logics, which had apparently already fallen out of fashion at the time. Remarkably enough, I did find one Peircean scholar who had done a lot of work on them, but they didn’t seem to be what I needed right then.

My present, still pressing applications require me to start from much more elementary grounds, stuff I can build up from boolean sources and targets, universes with coordinate spaces of type $(\mathbb{B}^k, \mathbb{B}^k \to \mathbb{B}).$

Regards,

Jon

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Posted in C.S. Peirce, Category Theory, Logic, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations | Tagged C.S. Peirce, Category Theory, Logic, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations | 9 Comments

Animated Logical Graphs • 81

Posted on August 26, 2021 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • A Negative Comment On Negations

Minsky and Papert’s Perceptrons was the work that nudged me over the line from gestalt psychology, psychophysics, relational biology, etc. and made me believe AI could fly. I later found out a lot of people thought it had thrown cold water on the subject but that was not my sense of it.

The real reason Rosenblatt’s perceptrons short-shrift XOR and EQ among the sixteen boolean functions on two variables is the adoption of a particular role for neurons in the activity of the brain and a particular model of how neurons serve computation, namely, as threshold activation devices. It is as if we tried to do mathematics using only the inequality $\le$ instead of using equations. Sure, we can express equations in roundabout ways but why tolerate the resulting inefficiency? As a final observation, $x \le y$ for boolean variables $x, y$ is equivalent to $x \Rightarrow y$ so this fits right in with the weakness of implicational inference compared to equational inference rules.

But there are other models for the role neurons play in the activity of the brain and the work they do in computation.

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Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | 7 Comments

Semiotics, Semiosis, Sign Relations • Discussion 12

Posted on August 25, 2021 by Jon Awbrey

Re: Peirce List • Robert Marty (quoted)

RM:: I persist in the idea that in your six combinations [O, S, I] only one is relevant for semiotics, the others being out of the field […] On the projections, there is also matter for discussion … but to discuss well one must reserve a rather large agenda … I thus wait for your reply dealing with semiosis to resume a debate well-centered on the essential …

Dear Robert,

A bit of calm today — and feeling slaked after a day spent minding Voltaire’s advice and pulling weeds from our garden — I’ll take up one of your last problems first as it may be the one most quickly resolved.

I take it you are referring to the section of the Sign Relation article titled “Six Ways of Looking at a Sign Relation” which begins as follows.

In the context of 3-adic relations in general, Peirce provides the following illustration of the six converses of a 3-adic relation, that is, the six differently ordered ways of stating what is logically the same 3-adic relation:

So in a triadic fact, say, for example

$A ~\text{gives}~ B ~\text{to}~ C$

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:

These six sentences express one and the same indivisible phenomenon.
(C.S. Peirce, “The Categories Defended”, MS 308 (1903), EP 2, 170–171).

“These six sentences express one and the same indivisible phenomenon.”

It’s a statement telling of the difference between affairs of grammar and affairs of logic, mathematics, and phenomena.

To be continued …

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Semiotics, Semiosis, Sign Relations • Comment 4

Posted on August 23, 2021 by Jon Awbrey

ah, what do mathematicians know of life’s exigency?
proof is our rock and our soul necessity.
we don’t just make abstractions, we are abstractions.
it’s coffee and doughnuts all the way down …
no one disturbs our vain diagrams
till human voices wake us, and we drown.

🙞 also sprach 0*
— 23 august 2021

Cf: Apology : T.S. Eliot | Context : Ironic
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Semiotics, Semiosis, Sign Relations • Discussion 11

Posted on August 22, 2021 by Jon Awbrey

Re: Peirce List • Robert Marty

RM:

You evoke many concepts with their relations, the explanation of which would take a considerable amount of time, to the point that you are reduced to answering yourself. I want to question you on the point that interests me particularly, which concerns your entry into Peirce’s semiotics. I found it among all your links here:

Sign Relation

You will tell me if this is the right reference. If it is so, then I think you have made a bad choice, and of course, I explain myself. To be clear and precise, I must reproduce the entirety of your “Definition” paragraph:

Definition

Dear Robert,

I’m just beginning to get out from under the deluge of tasks put off by the pandemic … I think I can finally return to your remarks of August 12 on my sketch of Peirce’s theory of signs for the general reader interested in semiotics.

Your message to the List had many detailed quotations, so I’m in the process of drafting an easier-on-the-eyes blog version. When I get done with that — it may be a day — I’ll post my reply on the thread dealing with Semiotics, Semiosis, Sign Relations, so as to keep focused on signs.

Regards,

Jon

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Relations & Their Relatives • Discussion 20

Posted on August 18, 2021 by Jon Awbrey

Re: Information = Comprehension × Extension
Re: Category Theory • Morgan Rogers

MR:: Care to make any of this more precise?
[The above] formula, for example?

Yes, it will take some care to make it all more precise, and I’ve cared enough to work on it when I get a chance. I initially came to Peirce’s 1865–1866 lectures in grad school from the direction of graph-&-group theory in connection with a 19th century device called a “table of marks”, out which a lot of work on group characters and group representations developed.

A table of marks for a transformation group $(G, X)$ is an incidence matrix with $1$ in the $(g, x)$ cell if $g$ fixes $x$ and $0$ otherwise. I could see Peirce’s formula was based on a logical analogue of those incidence matrices so that gave me at least a little stable ground to inch forward on.

The development of Peirce’s information formula is discussed in my ongoing study notes, consisting of selections from Peirce’s 1865–1866 Lectures on the Logic of Science and my commentary on them.

Information = Comprehension × Extension

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Posted in C.S. Peirce, Category Theory, Control, Cybernetics, Dyadic Relations, Information, Inquiry, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiosis, Semiotics, Sign Relations, Systems Theory, Triadic Relations | Tagged C.S. Peirce, Category Theory, Control, Cybernetics, Dyadic Relations, Information, Inquiry, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiosis, Semiotics, Sign Relations, Systems Theory, Triadic Relations | 9 Comments

Relations & Their Relatives • Discussion 19

Posted on August 18, 2021 by Jon Awbrey

Re: Category Theory • Henry Story

HS:: Could one not say that Frege also had a three part relation? I guess: for singular terms their Sense and Reference. […] His argument could be explained very simply. Imagine you start with a theory of language where words only have referents. Then since in point of fact Hesperus = Phosphorus, The Morning Star = The Evening Star, the simple theory of meaning would not allow one to explain how the discovery that they both were the planet Venus, came to be such a big event. So sense cannot be reduced to reference. Equalities can have informational content.

Peirce’s take on semiotics is often compared with Frege’s parsing of Sinn und Bedeutung. There’s a long tradition concerned with the extension and intension of concepts and terms, also denotation and connotation, though the latter tends to be somewhat fuzzier from one commentator to the next. The following paper by Peirce gives one of his characteristically thoroughgoing historical and technical surveys of the question.

C.S. Peirce (1867) • Upon Logical Comprehension and Extension

The duality, inverse proportion, or reciprocal relation between extension and intension is the generic form of the more specialized galois correspondences we find in mathematics. Peirce preferred the more exact term comprehension for a compound of many intensions. In his Lectures on the Logic of Science (Harvard 1865, Lowell Institute 1866) he proposed his newfangled concept of information to integrate the dual aspects of comprehension and extension, saying the measures of comprehension and extension are inversely proportional only when the measure of information is constant. The fundamental principle governing his “laws of information” could thus be expressed in the following formula.

$\mathrm {Information} = \mathrm {Comprehension} \times \mathrm {Extension}$

Information = Comprehension × Extension

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Relations & Their Relatives • Discussion 18

Posted on August 17, 2021 by Jon Awbrey

Re: Relations & Their Relatives • 4
Re: Category Theory • Morgan Rogers

MR:: So a “sign process” would be a subset $L \subseteq O \times S \times I \times T$ , where $T$ is a time domain?

There are a couple of ways we usually see the concept of a sign relation $L \subseteq O \times S \times I$ being applied.

There is the translation scenario where $S$ and $I$ are two different languages and a large part of $L$ consists of triples $(o, s, i)$ where $s$ and $i$ are co-referent or otherwise equivalent signs.
There is the transition scenario where $S = I$ and we have triples of the form $(o, s, s^\prime)$ where $s^\prime$ is the next state of $s$ in a sign process. As it happens, a concept of process is more basic than a concept of time, since the latter involves reference to a standard process commonly known as a clock.

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Relations & Their Relatives • 4

Posted on August 15, 2021 by Jon Awbrey

From Dyadic to Triadic to Sign Relations

Peirce’s notation for elementary relatives was illustrated earlier by a dyadic relation from number theory, namely, the relation written $``{i|j}"$ for $``{i} ~\text{divides}~ {j}".$

Cf: Relations & Their Relatives • 3

Table 1 shows the first few ordered pairs of the relation on positive integers corresponding to the relative term, “divisor of”. Thus, the ordered pair ${i\!:\!j}$ appears in the relation if and only if ${i}$ divides ${j},$ for which the usual mathematical notation is $``{i|j}".$

Table 2 shows the same information in the form of a logical matrix. This has a coefficient of ${1}$ in row ${i}$ and column ${j}$ when ${i|j},$ otherwise it has a coefficient of ${0}.$ (The zero entries have been omitted for ease of reading.)

Just as matrices of real coefficients in linear algebra represent linear transformations, matrices of boolean coefficients represent logical transformations. The capacity of dyadic relations to generate transformations gives us part of what we need to know about the dynamics of semiosis inherent in sign relations.

Cf: Relations & Their Relatives • Discussion 1

The “divisor of” relation $x|y$ is a dyadic relation on the set of positive integers $\mathbb{M}$ and thus may be understood as a subset of the cartesian product $\mathbb{M} \times \mathbb{M}.$ It forms an example of a partial order relation, while the “less than or equal to” relation $x \le y$ is an example of a total order relation.

The mathematics of relations can be applied most felicitously to semiotics but there we must bump the adicity or arity up to three. We take any sign relation $L$ to be subset of a cartesian product $O \times S \times I,$ where $O$ is the set of objects under consideration in a given discussion, $S$ is the set of signs, and $I$ is the set of interpretant signs involved in the same discussion.

One thing we need to understand is the sign relation $L \subseteq O \times S \times I$ relevant to a given level of discussion may be rather more abstract than what we would call a sign process proper, that is, a structure extended through a dimension of time. Indeed, many of the most powerful sign relations generate sign processes through iteration or recursion or similar operations. In that event, the most penetrating analysis of the sign process or semiosis in view is achieved through grasping the generative sign relation at its core.

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Posted in C.S. Peirce, Category Theory, Dyadic Relations, Logic, Logic of Relatives, Logical Graphs, Mathematics, Nominalism, Peirce, Pragmatism, Realism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged C.S. Peirce, Category Theory, Dyadic Relations, Logic, Logic of Relatives, Logical Graphs, Mathematics, Nominalism, Peirce, Pragmatism, Realism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | 8 Comments

Semiotics, Semiosis, Sign Relations • Discussion 10

Posted on August 14, 2021 by Jon Awbrey

Re: Semiotics, Semiosis, Sign Relations • Discussion 8
Re: Category Theory • Morgan Rogers

MR: Please clearly state at least one “distinctive quality of sign relations”.

Sign relations are triadic relations.

Can any triadic relation be a sign relation?

I don’t know. I have pursued the question myself whether any triadic relation could be used somehow or other in a context of communication, information, inquiry, learning, reasoning, and so on where concepts of signs and their meanings are commonly invoked — there’s the rub — it’s not about what a relation is, “in itself”, intrinsically or ontologically, but a question of “suitability for a particular purpose”, as they say in all the standard disclaimers.

What Peirce has done is to propose a definition intended to capture an intuitive, pre-theoretical, traditional concept of signs and their uses. To put it on familiar ground, it’s like Turing’s proposal of his namesake machine to capture the intuitive concept of computation. That is not a matter to be resolved by à priori dictates but by trying out candidate models in the intended applications.

I gave you what I consider Peirce’s best definition of a sign in relational terms and I pointed out where it needs filling out to qualify as a proper mathematical definition, most pointedly in the further definitions of correspondence and determination.

That is the current state of the inquiry as it stands on this site …

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Inquiry Into Inquiry

Semiotics, Semiosis, Sign Relations • Discussion 13

Animated Logical Graphs • 81

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Semiotics, Semiosis, Sign Relations • Discussion 12

Semiotics, Semiosis, Sign Relations • Comment 4

Semiotics, Semiosis, Sign Relations • Discussion 11

Relations & Their Relatives • Discussion 20

Relations & Their Relatives • Discussion 19

Relations & Their Relatives • Discussion 18

Relations & Their Relatives • 4

From Dyadic to Triadic to Sign Relations

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