Pragmatic Semiotic Information • Discussion 12

Peter Smith, on his Logic Matters blog, queried readers about the type/token distinction and its possible origins before Peirce.  I shared the following observations:

It doesn’t help with the question of semiogenesis, which is no doubt lost to the mists of history, but Peirce being Peirce naturally discerned three shades of signs in this respect:  Tone, Token, Type.  I collected a few excerpts here:

The distinction between Original and Copy figures frequently in Plato, with echoes of still more ancient voices.  Aristotle on Categories gives an example where a word meaning both a live animal and its true-to-life image must be shorn of ambiguity prior to appearing in a court of logic.  Aristotle on Interpretation distinguishes objects from their copies, images, likenesses in the mind:

Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche);  written words are the signs of words spoken.  As writing, so also is speech not the same for all races of men.  But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata).  (Aristotle, De Interp. i. 16a4).

From a Peircean semiotic perspective we can distinguish an object domain and a semiotic plane, so we can have three types of type/token relations:  (1) within the object domain, (2) between objects and signs, (3) within the semiotic plane.  We could subtilize further but this much is enough for a start.

Type/token relations of type (1) are very common in mathematics and go back to the origins of mathematical thought.  These days computer science is rife with them.  I’ve seen a lot of confusion about this in Peircean circles as it’s not always grasped that type/token relations are not always all about signs.  It can help to speak of types versus instances or instantiations instead.

Aristotle covers type/token relations of types (2) and (3) in De Interp., the latter since he recognizes signs of signs in the clause, “written words are the signs of words spoken”.

Resources

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  ArchiveJournalOnline.
  • Limited Mark Universes • Peirce’s Note “On a Limited Universe of Marks”

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Abduction, Deduction, Induction, Analogy, Inquiry • 25

Re: Artem KaznatcheevReductionism : To Computer Science From Philosophy

The sense of reduction operative in complexity theory has its roots in Aristotle’s απαγωγη, variously translated as abduction or reduction and sometimes glossed as retroduction by C.S. Peirce.  See my project report on Inquiry and Analogy for a discussion, especially the following section:

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 8

Re: Systems ScienceLen Troncale

It’s been a while since I started this thread, with many sidetrips and tangents, so let me go back to the top and expand on the motivations I expressed there, addressing a few issues that have arisen in the meantime.

People interested in category theory as applied to systems may wish to check out the following article, reporting work I carried out while engaged in a systems engineering program at Oakland University.

Differential Logic and Dynamic Systems

This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.  The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

Category theory, as working mathematicians understand it, is one of the chief conceptual frameworks for the development of mathematics today, the other being set theory.  Every graduate math course I ever took began with a two- or three-week review of set theory and category theory before launching into the main subject matter.  From a logical point of view, however, category theory has a history stretching back to Aristotle.

I once started writing a sketch on the “Precursors of Category Theory”, collecting a sample of historical landmarks through the centuries, from Aristotle to category theory’s modern mathematical avatars.  Here’s a link to a survey page on my blog.

To be continued …

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 7

Re: Systems ScienceJS

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 …

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 6

Re: Systems ScienceJS

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 …

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 5

Re: Systems ScienceJS

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 …

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 4

Re: Systems ScienceLT

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.

Resources

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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.

Resources

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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

  • Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), 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 more complete 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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