Relations & Their Relatives : 14

I constructed the “Ann and Bob” examples of sign relations back when I was enrolled in a Systems Engineering program and had to explain how triadic sign relations would naturally come up in building intelligent systems possessed of a capacity for inquiry.  My advisor asked me for a simple, concrete, but not too trivial example of a sign relation and after cudgeling my wits for a while this is what fell out.  Up till then I had never much considered finite examples before as the cases that arise in logic almost always have formal languages with infinite numbers of elements as their syntactic domains if not also infinite numbers of elements in their object domains.

The illustration at hand involves two sign relations:

• $L_\text{A}$ is the sign relation that captures how Ann interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$
• $L_\text{B}$ is the sign relation that captures how Bob interprets the signs in the set $S = I = \{ {}^{\backprime\backprime}\text{Ann}{}^{\prime\prime}, {}^{\backprime\backprime}\text{Bob}{}^{\prime\prime}, {}^{\backprime\backprime}\text{I}{}^{\prime\prime}, {}^{\backprime\backprime}\text{you}{}^{\prime\prime} \}$ to denote the objects in $O = \{ \text{Ann}, \text{Bob} \}.$

Each of the sign relations, $L_\text{A}$ and $L_\text{B},$ contains eight triples of the form $(o, s, i)$ where $o$ is an object in the object domain $O,$ $s$ is a sign in the sign domain $S,$ and $i$ is an interpretant sign in the interpretant domain $I.$  These triples are called elementary or individual sign relations, as distinguished from the general sign relations that generally contain many sign relational triples.

If this much is clear we can move on next time to discuss the two types of reducibility and irreducibility that arise in semiotics.

To be continued …

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