## Semiotics, Semiosis, Sign Relations • Comment 4

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

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:

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:

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

Re: Information = Comprehension × Extension
Re: Category TheoryMorgan 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.

## Relations & Their Relatives • Discussion 19

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.

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.

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

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.

## Relations & Their Relatives • Discussion 18

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.

## Relations & Their Relatives • Review 1

Peirce’s notation for elementary relatives was illustrated earlier by a dyadic relation from number theory, namely ${i|j}$ for ${i}$ being a divisor of ${j}.$

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 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 in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations.

The capacity of relations to generate transformations gives us a clue to the dynamics of sign relations.

The divisor of relation signified by $x|y$ 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 is an example of a partial order, while the less than or equal to relation signified by $x \le y$ 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.

## Semiotics, Semiosis, Sign Relations • Discussion 10

MR:  Please clearly state at least one “distinctive quality of sign 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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## Semiotics, Semiosis, Sign Relations • Discussion 9

MR:
Okay, I may have mixed up the meanings of “object” and “interpretant” in my plain language translations above?  Re determination, I read “B is determined by A” as meaning the conjunction of

$\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, R(a,b,c)$

and

$\forall a \in A, \, \forall c \in C, \, R(a,b,c) \wedge R(a,b',c) \Rightarrow b = b'$   ?

Whether this is right depends on the answers to my previous questions.

Dear Morgan,

Let’s look at the gloss I gave for Determination under the Definition of a Sign Relation.

Determination.  Peirce’s concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal-temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.

Other words for this general order of determination are structure, pattern, law, form, and the one arising especially in cybernetics and systems theory, constraint.  It’s what happens when not everything that might happen actually does.  (The stochastic mechanic or the quantum technician will probably quip at this point, At least, not with equal probability.)

Regards,

Jon

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## Semiotics, Semiosis, Sign Relations • Discussion 8

RM:
Thank you for reminding me of the definition of a group that I have taught for 45 years … I think you work with the permutations of symmetrical groups that do not fit well with the interdependence of categories and which make us go out of the Peircian theory, which is not forbidden as long as we point it out.  I’ll look at the use you make of them when you’ve answered my previous questions with something other than a stream of links and the definition of a group!  (my Ph.D. Math is on Abelian Groups) … formulating my questions correctly takes me time, especially to grasp your thought … I would like a reciprocal … I always thought that you had the capacity to do it without giving up your certainties, but I must say that today I am disappointed …

Dear Robert,

Auld acquaintance is not forgot 🍻 I will convey your thanks to one who reminded me.

My reason for encoring mathematical groups as a class of triadic relations and elsewhere casting divisibility in the role of a dyadic relation was not so much for their own sakes as for the critical exercise my English Lit teachers used to style as “Compare and Contrast”.  For the sake of our immediate engagement, then, we tackle that exercise all the better to highlight the distinctive qualities of triadic relations and sign relations.

A critical point of the comparison is to grasp sign relations as collections of ordered triples — collections endowed with collective properties extending well beyond the properties of individual triples and their components.

Regards,

Jon

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## Semiotics, Semiosis, Sign Relations • Discussion 7

MR:
Okay, this is hard to parse, but I’ve looked at it a few times now framed with discussion from a few different sources, and it seems that if we fix some sets $A$ of signs, $B$ of interpretants and $C$ of objects, and treating the sign relation as $R \subseteq A \times B \times C,$ there are some reasonable restrictions/assumptions we could place on $R.$  For example:

1a.
$\forall a \in A, \, \forall b \in B, \, \exists c \in C, \, (a,b,c) \in R,$
“every sign means something to every interpretant”,
1b.
$\forall a \in A, \, \exists b \in B, \, \exists c \in C, \, (a,b,c) \in R,$ a weaker alternative,
“every sign means something to some interpretant”,
2a.
$\forall c \in C, \, \forall b \in B, \, \exists a \in A, \, (a,b,c) \in R,$
“every interpretant has a name for every object”,
2b.
$\forall c \in C, \, \exists b \in B, \, \exists a \in A, \, (a,b,c) \in R,$ a weaker alternative,
“every object has at least one name assigned to it by each interpretant,”

and so on.

However, none of these seem strictly necessary to me;  there could be meaningless symbols or nameless objects.  Does Peirce assume any of these things or similar?  If not, I suspect the answer to my question regarding mathematical distinguishing features of sign relations is that there aren’t any:  that any ternary relation can be understood as a sign relation if one squints hard enough.

As far as meaningless signs go, we do encounter them in theoretical analysis (“resolving conundra” and “steering around nonsense”) and empirical or computational applications (“missing data” and “error types”).  The defect of meaning can affect either denotative objects or connotative interpretants or both.  In those events we have to generalize sign relations to what are called sign relational complexes.

Signless objects are a different matter since cognitions and concepts count as signs in pragmatic semiotics and Peirce maintains we have no concept of inconceivable objects.

If you fancy indulging in a bit of cosmological speculation you could imagine the whole physical universe to be a sign of itself to itself, making $O = S = I,$ but this far downstream from the Big Bang we mortals usually have more pressing business to worry about.

In short, what we need sign relations for is not for settling big questions about cosmology or metaphysics but for organizing our thinking about object domains and constructing models of what goes on and what might go better in practical affairs like communication, inquiry, learning, and reasoning.

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