Changes

There is a one piece of unfinished business concerning the presentation of this example that deserves further comment.

There is a one piece of unfinished business concerning the presentation of this example that deserves further comment.

Since the objects of reference, <math>\text{A}\!</math> and <math>\text{B},\!</math> are imagined to be interpretive agents, it is convenient to use their names to denote the corresponding sign relations.  Thus, the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> are self-referent and mutually referent to the extent that they have names for themselves and each other.  However, their discussion as a whole fails to contain any term for itself, and it even lacks a full set of grammatical cases for the objects in it.  Whether these recursions and omissions cause any problems for my discussion will depend on the level of interpretive sophistication, not of <math>\text{A}\!</math> and <math>\text{B},\!</math> but of the external systems of interpretation that are brought to bear on it.

Since the objects of reference <math>\text{A}\!</math> and <math>\text{B}\!</math> are imagined to be interpretive agents, it is convenient to use their names to denote the corresponding sign relations.  Thus, the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> are self-referent and mutually referent to the extent that they have names for themselves and each other.  However, their discussion as a whole fails to contain any term for itself, and it even lacks a full set of grammatical cases for the objects in it.  Whether these recursions and omissions cause any problems for my discussion will depend on the level of interpretive sophistication, not of <math>\text{A}\!</math> and <math>\text{B},\!</math> but of the external systems of interpretation that are brought to bear on it.

In defining the activities of interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> as sign relations, I have implicitly specified set-theoretic equations of the following form:

In defining the activities of interpreters A and B as sign relations, I have implicitly specified set theoretic equations of the following form:

A = { <A, "A", "A">, ... , <A, "i", "i">,

  <B, "B", "B">, ... , <B, "u", "u"> },

\text{A}

& = & \{ &

(\text{A}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}),

& \ldots, &

(\text{A}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}),

(\text{B}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}),

& \ldots, &

(\text{B}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime})

& \},

\text{B}

(\text{A}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime}),

\end{array}</math>

|}

The way I read these equations, they do not attempt to define the entities <math>\text{A}\!</math> and <math>\text{B}\!</math> in terms of themselves and each other.  Instead, they define the whole instrumental activity of each interpreter, a highly complex duty, in terms of the interpreters' more perfunctory roles as objects of reference, and in terms of their associated actions on signs as mere tokens of each other's existence.  In other words, the recursion does not have recourse to the full fledged faculties of <math>\text{A}\!</math> and <math>\text{B}\!</math> but only to their more inert excipients, their rote performances as inactive objects and passive images whose properly reduced complexities provide grounds for permitting the recursion to fly.

 

The way I read these equations, they do not attempt to define the entities A and B in terms of themselves and each other.  Instead, they define the whole instrumental activity of each interpreter, a highly complex duty, in terms of the interpreters' more perfunctory roles as objects of reference, and in terms of their associated actions on signs as mere tokens of each other's existence.  In other words, the recursion does not have recourse to the full fledged faculties of A and B, but only to their more inert excipients, their rote performances as inactive objects and passive images whose properly reduced complexities provide grounds for permitting the recursion to fly.

===6.17. Patterns of Self-Reference===

===6.17. Patterns of Self-Reference===