## Survey of Differential Logic • 4

This is a Survey of blog and wiki posts on Differential Logic, material I plan to develop toward a more compact and systematic account.

## Sign Relational Manifolds • Discussion 3

AS:
I see — “sign relation” is a special term for triadic relations of some kind (with some properties);  like this:  thing in first position and thing in second position must refer to the thing in third position.  Where “refer” is an unary partial function from one thing to another.  Am I on a right direction?

Hi Alex,

It is not uncommon in practice to find a sign $s$ having many interpretant signs $i$ and many referent objects $o.$  Generally speaking, then, we start out with a sign relation $L$ as a subset of a cartesian product $L \subseteq O \times S \times I,$ where $O, S, I$ are sets called the object domain, sign domain, interpretant sign domain, respectively.  A definition of a sign relation — there are a few canonical ones we find useful in practice — will specify what sort of constraint is involved in forming that subset.

Regards,

Jon

## Sign Relational Manifolds • Discussion 2

AS:
Not on a narrow topic, but maybe you have a desire to answer.
Hypothesis.  Any material something can be a sign.
Is it possible to give an example of something material that cannot be a sign?

Hi Alex,

Sign relations are mathematical relations we can use to model processes of communication, learning, reasoning, just plain talking and thinking in general.  Anytime we can imagine a triadic relation where one thing, material or otherwise, is related to a second thing in such a way that both refer to a third thing, and that whole relationship is useful in modeling one of the above mentioned processes, then we have a candidate which may be suitable for serving the purpose of a sign relation in the pragmatic conception of the term.

Regards,

Jon

## Sign Relational Manifolds • Discussion 1

### Semiotic Orbits, Manifolds, Arcs

The arc of the semiotic universe is long but it bends towards universal harmony.

Semiotic manifolds, like physical and mathematical manifolds, may be finite and bounded or infinite and unbounded but they may also be finite and unbounded, having no boundary in the topological sense.  Thus unbounded semiosis does not imply infinite semiosis.

Here are two points in previous discussions where the question of infinite semiosis came up.

## Sign Relational Manifolds • 5

Let me try to say in intuitive terms what I think is really going on here.

The problem we face is as old as the problem of other minds, or intersubjectivity, or even commensurability, and it naturally involves a whole slew of other old problems — reality and appearance, or reality and representation, not to mention the one and the many.  One way to sum up the question might be “conditions on the possibility of a mutually objective world”.

Working on what oftentimes seems like the tenuous assumption that there really is a real world causing the impressions in my mind and the impressions in yours — more generally speaking, that there really is a real world impressing itself in systematic measures on every frame of reference — we find ourselves pressed to give an account of the hypothetical unity beneath the manifest diversity — and how it is possible to discover the former in the latter.

Manifold theory proposes one type of solution to that host of problems.

## Sign Relational Manifolds • 4

Another set of notes I found on this theme strikes me as getting to the point more quickly and though they read a little rough in places I think it may be worth the effort to fill out their general line of approach.

## Sign Relational Manifolds • 3

I’m not sure when it was I first noticed the relationship between manifolds and semiotics but I distinctly recall the passage in Serge Lang’s Differential and Riemannian Manifolds which brought the triadic character of tangent vectors into high relief.  I copied out a set of excerpts highlighting the point and shared it with the Inquiry, Ontology, and Peirce lists.

### Chapter 2.  Manifolds

Using the concepts and terminology from Lang’s text, I explained the connection between manifold theory and semiotics in the following way.

## Sign Relational Manifolds • 2

A sense of how manifolds are applied in practice may be gleaned from the set of excerpts linked below, from Doolin and Martin (1990), Introduction to Differential Geometry for Engineers, which I used in discussing differentiable manifolds with other participants in the IEEE Standard Upper Ontology Working Group.

What brought the concept of a manifold to mind in that context was a set of problems associated with perspectivity, relativity, and interoperability among multiple ontologies.  To my way of thinking, those are the very sorts of problems manifolds were invented to handle.

### Reference

• Doolin, Brian F., and Martin, Clyde F. (1990), Introduction to Differential Geometry for Engineers, Marcel Dekker, New York, NY.

## Sign Relational Manifolds • 1

Riemann’s concept of a manifold, especially as later developed, bears a close relationship to Peirce’s concept of a sign relation.

I will have to wait for my present train of thought to stop at a station before I can hop another but several recent discussions of geometry have brought the subject back to mind and I thought it might serve to drop off a few mail bags of related letters in anticipation of the next pass through this junction.

Here is a set of excerpts from Murray G. Murphey (1961), The Development of Peirce’s Philosophy, discussing Peirce’s reception of Riemann’s philosophy of geometry.

Later developments of the manifold concept, looking to applications on the one hand and theory on the other, are illustrated by excerpts in the next two posts.

## Zeroth Law Of Semiotics • Discussion 2

Paradoxes star among my first loves in logic.  So enamored was I with tricks of the mind’s eye I remember once concocting the motto, “Only what is paradoxical is ornery enough to exist”.  These days my less precocious self tends to suspect all our nominal paradoxes will gradually dissolve on sufficient inspection and placement in the proper light.  There I find the pragmatic spectrum of C.S. Peirce, stretching from the theory of triadic sign relations to the mathematical forms underlying logic, brings a full range of lights to the purpose.

It was by those lights, Peirce’s semiotic and logical graphs, I came to see through the fog of misdirection surrounding the so-called Liar Paradox, inscribing my epitaph to Epimenides under the heading “All Liar, No Paradox”.  More than that it became possible to see how the apparent paradox derives its appearance from unexamined assumptions about the relation between signs and objects.

That much prologue brings us up to speed with the Zeroth Law Of Semiotics and the scene of Joseph Harry’s remarks.

JH:
“Meaning is a privilege not a right” would seem to be a meaningless proposition, since ‘privilege’ and ‘right’ are third-order evaluative, symbolic terms, while ‘meaning’ is a neutral second-order term, implying only existential individualized dynamic activity or process.  Driving (a car) is a privilege not a right, but meaning is neither.

Dear Joseph,

That may be too literal a reading for Zero‑Aster’s poetic figure.  If I read the oracle right, the contrast between “privilege” and “right” serves merely to mark the distinction between meanings optional and obligatory.  Whether any hint of “private law” or “law unto itself” is intended or involved is something I would have to spend more time thinking about.

Regards,

Jon

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