## Sign Relations, Triadic Relations, Relations • 11

In pursuing applications of pragmatic semiotics to scientific research the following distinctions are critical.

We have the relational roles known as Object, Sign, and Interpretant Sign.  These are places or roles a thing may occupy in a given moment in a given context.  They are not absolute essences or fixed ontological characters.

We can formalize the “moment” mentioned above as an ordered triple $(o, s, i),$ where $o$ is the object, $s$ is the sign, and $i$ is the interpretant sign in view.

We can formalize the “context“ mentioned above as a set of ordered triples, each one having the form $(o, s, i).$  This set is called a sign relation.

We can formalize a given sign relation $L$ as a subset of a cartesian product, $L \subseteq O \times S \times I,$ where $O$ is the set of objects under consideration in a given context, $S$ is the set of signs, and $I$ is the set of interpretant signs being considered in the same context.

It is critically important to distinguish the triples $(o, s, i),$ which may be called elementary sign relations, from the sign relation proper, $L \subseteq O \times S \times I.$  Among other things, this is important because sets have properties their elements do not and it amounts to a category mistake to confuse the 2 levels.  In particular, the properties of reducibility and irreducibility are defined at the level of whole sign relations, not their individual elements.

Another very important distinction we have to keep in mind is the difference between the formal objects we are discussing and the formal signs and syntax we use to discuss them.  I’ll speak to that point more next time.

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