## Icon Index Symbol • 16

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • JASHR

Having lost my train of thought due to a week on the road, I would like to go back and pick up the thread at the following exchange:

JAS:
To be honest, given that the Sign relation is genuinely triadic, I have never fully understood why Peirce initially classified Signs on the basis of one correlate and two dyadic relations.
HR:
I have a guess about that:  I remember from a thread with Jon Awbrey about relation reduction something like the following:  A triadic relation is called irreducible, because it cannot compositionally be reduced to three dyadic relations.  Compositional reduction is the real kind of reduction.  But there is another kind of reduction, called projective (or projectional?) reduction, which is a kind of consolation prize for people who want to reduce.  It is possible for some triadic relations.

The course of discussion after that point left a great many of the original questions about icons, indices, and symbols unanswered, so I’d like to make another try at addressing them.  The relevant facts about triadic relations and relational reduction can be found at the following locations:

I introduced two examples of triadic relations from mathematics, two examples of sign relations from semiotics, and used them to illustrate the question of projective reducibility, in another way of putting it, whether the structure of a triadic relation can be reconstructed from the structures of three dyadic relations derived or “projected” from it.

• In the geometric picture of triadic relations, a dyadic projection is the shadow that a 3-dimensional body casts on one of the three coordinate planes.
• In terms of relational data tables, a dyadic projection is the result of deleting one of the three columns of the table and merging any duplicate rows.
• In Peircean terms, a projection is a type of “prescision” operation, abstracting a portion the structure from the original relation and ignoring the rest.

The question then is whether we keep or lose information in passing from a triadic relation to the collection of its dyadic projections.  If there is no loss of information then the triadic relation is said to be reducible to and reconstructible from its dyadic projections.  Otherwise it is said to be irreducible and irreconstructible in the same vein.