## Sign Relations • Discussion 12

CJ:
For a given arbitrary triadic relation $L \subseteq O \times S \times I$ (let’s say that $O,$ $S,$ and $I$ are all finite, non‑empty sets), I’m interested to understand what additional axioms you’re saying are necessary and sufficient to make $L$ 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 $L \subseteq X \times X \times X$ on any set $X$ 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

This entry was posted in C.S. Peirce, Logic, Mathematics, Peirce, Relation Theory, Semiosis, Semiotics, Sign Relations and tagged , , , , , , , . Bookmark the permalink.

### 1 Response to Sign Relations • Discussion 12

This site uses Akismet to reduce spam. Learn how your comment data is processed.