## Icon Index Symbol • 12

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

HR:
Example:  The triadic function $x_1 + x_2 = x_3",$ with the three sets $X_1, X_2, X_3$ not being classes of any kind, at least not of the special kind (whatever that is), that would allow representation, and make it having to do with the third category.

Mathematics is rife with examples of triadic relations having all three relational domains the same.  For instance, the binary operation indicated by ${+}"$ in the expression ${x_1 + x_2 = x_3}"$ is associated with a function $\mathrm{Fun}[+]$ of the form $\mathrm{Fun}[+] : X \times X \to X$ and also with a triadic relation $\mathrm{Rel}[+]$ of the form $\mathrm{Rel}[+] \subseteq X \times X \times X.$

Semiotics, by contrast, tends to deal with relational domains $O, S, I$ where the objects in $O$ are distinct in kind from the signs in $S$ and the interpretant signs in $I.$  As far as $S$ and $I$ go, it is usually convenient to lump them all into one big set $S = I,$ even if we have to partition that set into distinct kinds, say, mental concepts and verbal symbols, or signs from different languages.  But even if it’s how things tend to work out in practice, as we currently practice it, there does not seem to be anything in Peirce’s most general definition of a sign relation to prevent all the relational domains from being the same.  So I’ll leave that open for now.