Re: Cybernetics • Cliff Joslyn

CJ:

For a given arbitrary triadic relation (let’s say that and are all finite, non‑empty sets), I’m interested to understand what additional axioms you’re saying are necessary and sufficient to make a sign relation.  I checked Sign Relations • Definition, but it wasn’t obvious, or at least, not formalized.

Dear Cliff,

From a purely speculative point of view, any triadic relation on any set might be capable of capturing aspects of objective structure immanent in the conduct of logical reasoning.  At least I can think of no reason to exclude the possibility à priori.

When we turn to the task of developing computational adjuncts to inquiry there is still no harm in keeping arbitrary triadic relations in mind, as entire hosts of them will turn up on the universe side of many universes of discourse we happen to encounter, if nowhere else.

Peirce’s use of the word definition understandably leads us to anticipate a strictly apodictic development, say, along the lines of abstract group theory or axiomatic geometry.  In that light I often look to group theory for hints on how to go about tackling a category of triadic relations such as we find in semiotics.  The comparison makes for a very rough guide but the contrasts are also instructive.

More than that, the history of group theory, springing as it did as yet unnamed from the ground of pressing mathematical problems, from Newton’s use of symmetric functions and Galois’ application of permutation groups to the theory of equations among other sources, tells us what state of development we might reasonably expect from the current still early days of semiotics.

To be continued …

Reference

• Charles S. Peirce (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.

Sources

• C.S. Peirce • On the Definition of Logic
• C.S. Peirce • Logic as Semiotic
• C.S. Peirce • Objective Logic

cc: Conceptual Graphs • Cybernetics • Laws of Form • Ontolog Forum
cc: FB | Semeiotics • Structural Modeling • Systems Science