## Icon Index Symbol • 17

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Our object being to clarify the relationships among icons, indices, and symbols, I believe the maximum benefit possible at this point is to be gained from studying the simple examples of triadic relations and sign relations discussed in the following places:

Once we get used to dealing with small examples like that we can move on to tackling more complex examples on the order of those we might encounter in realistic applications.

The sort of sign relation we normally encounter in practice will be a subset $L$ of a cartesian product $O \times S \times I,$ where the object, sign, and interpretant-sign domains all have infinitely many members in principle, though of course we tend to get by with finite samples at any given moment and it may even be possible to start small and build capacity over time.

All the objects we need to reference in a given application will go into the object domain $O$ and all the signs and interpretant-signs we need to denote these objects will go into the sign domain $S$ and the interpretant-sign domain $I.$

It may be useful to note at this point that there are such things as monadic projections:

$\begin{array}{lll} \mathrm{proj}_O & : & O \times S \times I \to O \\[4pt] \mathrm{proj}_S & : & O \times S \times I \to S \\[4pt] \mathrm{proj}_I & : & O \times S \times I \to I \end{array}$

For example, $\mathrm{proj}_O (L)$ gives the set of all elements in $O$ that actually occur as first correlates in $L,$ sometimes called the $O$-range.

There are interesting classes of relations that take place internal to the various domains.  For example, there are the syntactic relations or parsing relations that operate within the sign domain, relating complex signs to their component signs.

But that’s a topic a little ways down the road …