Differential Logic, Dynamic Systems, Tangent Functors • Discussion 7

Let’s stand back from the picture and see how the dimensions of syntax, semantics, and pragmatics look from a pragmatic semiotic or sign relational perspective.

$O$ is an object domain, a set of elements under view in a given discussion.  Depending on the application we might be calling it a universe of discourse, a population, a sample space, a state space, or any number of other things.

$S$ and $I$ are sets of signs related to $O$ by means of a triadic relation, $L \subseteq O \times S \times I.$  If the triadic relation $L$ satisfies a set of conditions set down in a definition of a sign relation then we say $L$ is a sign relation.

Peirce’s best definitions of a sign relation are pretty minimal in what they demand and cover a wide range of cases from barely formed to highly structured.

Let’s move on to the more structured types of sign relations forming our ultimate practical interest.

In a typical case like that, $S$ is a formal language defined by a formal grammar.

Generally speaking, we might think of $I$ as being more loosely defined in its own right but when it comes to formal investigations the so-called interpretant sign domain $I$ will also be a formal language.  Here the cases divide into two broad sorts.

• $(S = I).$  We use this case to discuss transitions in time from one sign to the next.
• $(S \ne I).$  We use this case to discuss translations from one language to another.

To be continued …

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 6

A few of my readers are racing well ahead of me, exploring a range of different roads, but I’ll be making a dogged effort to stick to my math-bio-graphical narrative this time around, and try to tell how I came to climb down from logical trees and learned to love logical cacti.

As far as the logical ballpark goes, this is all just classical propositional logic, what my old circle used to call “zeroth order logic”, alluding to its basemental status for every storey built on it.  (But I have since found that others use that term for other things, so usage varies as it usually does.)

When it comes to semantics, the class of formal or mathematical objects residing among the referents of our propositional signs, I’m content for most purposes to say they’re all the same, namely, Boolean functions of abstract type $f : \mathbb{B}^k \to \mathbb{B},$ where $\mathbb{B} = \{ 0, 1 \}$ and $k$ is a non-negative integer.  Although we’re likely to have other sorts of meanings in mind, this class of models suffices for a ready check on logical consistency and serves us well, especially in practical applications.

The upshot is — I’m aiming for innovation solely in the syntactic sphere, the end being only to discover/invent a better syntax for the same realm of logical objects.

To be continued …

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 5

Yes, all these strands are strongly entangled.

I had already spent a full decade wrestling with the works of Charles Sanders Peirce and George Spencer Brown before my need to figure out what they were talking about in the way of logical graphs and math in general drove me to the extremes of enrolling in a graduate math program.  I found much diversion and measures of enlightenment there but soon encountered questions I just had to know the answers to.  One of my office mates suggested I devote some effort to developing a theorem prover to assist with the task.  Theorem provers in those days were so primitive everyone we knew was hacking out their own, so that is what I set to.  Naturally I turned to my previous tillage of logical graphs as a seed bed for my system.

To be continued …

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 4

To clarify my previous remark about General Systems Theory, I wasn’t trying to define a whole field but merely to describe my experience in forums like these, where it took me a while to realize that when I use the word “system” a great many people are not thinking what I’m thinking when I use it.  The first thing in my mind is almost always a state space $X$ and the possible trajectories of a representative point through it.  But a lot of people will be thinking of a “system”, like the word says, as a collection of parts “standing together”.  Naturally I’d like to reach the point of discussing such things, it’s just that it takes me a while, and considerable analysis of $X,$ to get there.

It goes without saying this has to do with the boundaries of my own experience and the emphases of my teachers and other influencers in systems, the early ones taking their ground in Ashby, Wiener, and the MIT school, the later ones stressing optimal control and learning organizations, but mostly it has to do with my current objectives and the species of intelligent systems, Inquiry Driven Systems, I want to understand and help to build.

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 3

Re: Systems Science • (1)(2)(3)

Various discussions in various places bring back to mind this thread from early this fall, prompting me to make a try at continuing it.  Here’s a series of blog posts where I kept track of a few points along the way:

Another thing to keep in mind here is the difference between General Systems Theory, following on Bertalanffy et al., and what is known as Dynamical Systems Theory (DST) or Mathematical Systems Theory (MST).  GST spends a lot of time studying part-whole hierarchies while DST/MST deals with the state space of a system and the possible trajectories of the system through it.

Category theory is especially useful in the latter application, abstracting or generalizing as it does the concepts of mathematical objects, functions, and transformations.

For my part I have come to take the DST/MST approach as more fundamental since it starts with fewer assumptions about the anatomy or architecture of the as-yet hypothetical agent, making it one of the first and continuing tasks of the agent to discover its own boundaries, potentials, and structures.

Semiotics, Semiosis, Sign Relations • 3

For ease of reference, here are two variants of Peirce’s 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

Reference

• Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.
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Semiotics, Semiosis, Sign Relations • 2

Here are links to fuller discussions of semiotics.

The approach described here develops from what I regard as the core definition of triadic sign relations, one explicit enough to support a consequential theory of signs.  Peirce gives that definition in the process of defining logic itself, as detailed in the following texts.

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