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How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations A and B be combined or developed into a new SOI that represents what agents like A and B are sure to know about each other's language use?  In order to make it clear that this is a non trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.

How can the fragmentary system of interpretation (SOI) constituted by the juxtaposition of individual sign relations A and B be combined or developed into a new SOI that represents what agents like A and B are sure to know about each other's language use?  In order to make it clear that this is a non trivial question, and in the process to illustrate different ways of combining sign relations, I begin by considering a couple of obvious suggestions for their integration that immediate reflection will show to miss the mark.

The first thing to try is the set theoretic union of the sign relations.  This commonly leads to a "confused" or "confounded" combination of the component sign relations.  For example, the sign relation defined as C = A U B is shown in Table 83.  Interpreted as a transition digraph on the four points of the syntactic domain S = {"A", "B", "i", "u"}, the sign relation C specifies the following behavior for the conduct of its interpreter:

The first thing to try is the set theoretic union of the sign relations.  This commonly leads to a "confused" or "confounded" combination of the component sign relations.  For example, the sign relation defined as C = A U B is shown in Table 86.  Interpreted as a transition digraph on the four points of the syntactic domain S = {"A", "B", "i", "u"}, the sign relation C specifies the following behavior for the conduct of its interpreter:

1. AC has a sling at each point of {"A", "i", "u"} and two way arcs on the pairs {"A", "i"} and {"A", "u"}.

1. AC has a sling at each point of {"A", "i", "u"} and two way arcs on the pairs {"A", "i"} and {"A", "u"}.

These sub-relations do not form equivalence relations on the relevant sets of signs.  If closed up under transitive compositions, then {"A", "i", "u"} are all equivalent in the presence of object A, but {"B", "i", "u"} are all equivalent in the presence of object B.  This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.

These sub-relations do not form equivalence relations on the relevant sets of signs.  If closed up under transitive compositions, then {"A", "i", "u"} are all equivalent in the presence of object A, but {"B", "i", "u"} are all equivalent in the presence of object B.  This may accurately represent certain types of political thinking, but it does not constitute the kind of sign relation that is wanted here.

Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co product might be demanded.  Table 84 presents the results of taking the disjoint union D = A U B to constitute a new sign relation.

Reflecting on this disappointing experience with using simple unions to combine sign relations, it appears that some type of indexed union or categorical co product might be demanded.  Table 87 presents the results of taking the disjoint union D = A U B to constitute a new sign relation.

Table 83.  Confounded Sign Relation C

 

 

Table 86.  Confounded Sign Relation C

Object Sign Interpretant

Object Sign Interpretant

A "A" "A"

A "A" "A"

B "u" "B"

B "u" "B"

B "u" "u"

B "u" "u"

Table 84.  Disjointed Sign Relation D

Table 87.  Disjointed Sign Relation D

Object Sign Interpretant

Object Sign Interpretant

AA "A"A "A"A

AA "A"A "A"A