## Icon Index Symbol • 5

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Given that sign relations are special cases of triadic relations, we can get significant insight into the structures of both cases by examining a few simple examples of triadic relations, without getting distracted by all the extra features that come into play with sign relations.

When I’m talking about a $k$-place relation $L$ I’ll always be thinking about a set of $k$-tuples.  Each $k$-tuple has the form:

$(x_1, x_2, \ldots, x_{k-1}, x_k),$

or, as Peirce often wrote them:

$x_1 : x_2 : \ldots : x_{k-1} : x_k.$

Of course, $L$ could be a set of one $k$-tuple but that would be counted a trivial case.

That sums up the extensional view of $k$-place relations, so far as we need it for now.

Using a single letter like $``L"$ to refer to a set of $k$-tuples is already the genesis of an intensional view, since we now think of the elements of $L$ as having some property in common, even if it’s only their membership in $L.$  When we turn to devising some sort of formalism for working with relations in general, whether it’s an algebra, logical calculus, or graph-theoretic notation, it’s in the nature of the task to “unify the manifold”, to represent a many as a one, to express a set of many tuples by means of a single sign.  That can be a great convenience, producing formalisms of significant power, but failing to discern the many in the one can lead to no end of confusion.

To be continued …

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