## Icon Index Symbol • 11

### Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

HR:
A simple triadic or n-adic relation, I think, belongs to secondness, and has only two modes:  the quality, e.g. function or caprice (intension), and the resulting set of tuples (extension).

There is a kind of Peircean “secondness” involved in the use of set theory, indeed, there are several kinds of dyadic relations in the mix, all intimately related.  Letting $X$ be the universe of discourse, there is the dyadic elementhood or membership relation $x \in X,$ there is the dyadic subset relation $A \subseteq X,$ and every subset $A \subseteq X$ has a characteristic or indicator function $f_{A} : X \to \{ 0, 1 \}$ with $f_{A}(x) = 1$ if $x \in A$ and $f_{A}(x) = 0$ if $x \notin A.$  So one could say there is secondness afoot in the extensions of whatever symbols one uses to demarcate or distinguish portions of the universe.  As it usually turns out, though, if you know enough to invoke secondness, you usually know enough to say something more specific about the dyadic relations you have in mind.

This is a very old theme.  The very word existence, whether by folk etymology or not, is said to mean standing out, the way a subset stands out against its ground.  It’s a nice image if nothing else.  In another connection, some take the prevalence of these dyadic relations in set theory, along with the assumption of set theory’s foundational status, as proving all structure is ultimately dyadic.

Well, I have my reasons to doubt that …

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