## Differential Logic, Dynamic Systems, Tangent Functors • Discussion 7

Let’s stand back from the picture and see how the dimensions of syntax, semantics, and pragmatics look from a pragmatic semiotic or sign relational perspective.

$O$ is an object domain, a set of elements under view in a given discussion.  Depending on the application we might be calling it a universe of discourse, a population, a sample space, a state space, or any number of other things.

$S$ and $I$ are sets of signs related to $O$ by means of a triadic relation, $L \subseteq O \times S \times I.$  If the triadic relation $L$ satisfies a set of conditions set down in a definition of a sign relation then we say $L$ is a sign relation.

Peirce’s best definitions of a sign relation are pretty minimal in what they demand and cover a wide range of cases from barely formed to highly structured.

Let’s move on to the more structured types of sign relations forming our ultimate practical interest.

In a typical case like that, $S$ is a formal language defined by a formal grammar.

Generally speaking, we might think of $I$ as being more loosely defined in its own right but when it comes to formal investigations the so-called interpretant sign domain $I$ will also be a formal language.  Here the cases divide into two broad sorts.

• $(S = I).$  We use this case to discuss transitions in time from one sign to the next.
• $(S \ne I).$  We use this case to discuss translations from one language to another.

To be continued …

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