Changes

The connotative components <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>.  Since <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> are SERs, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations.  In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs.  In the present case, the arcs are given as follows:

The connotative components <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>.  Since <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> are SERs, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations.  In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs.  In the present case, the arcs are given as follows:

# Con(''A'') has the structure of a SER on ''S'', with a sling at each of the points in ''S'', two-way arcs between the points of {"''A''",&nbsp;"''i''"}, and two-way arcs between the points of {"''B''",&nbsp;"''u''"}.

:: <math>\operatorname{Con}(\text{A})</math> has the structure of a SER on <math>S</math>, with a sling at each of the points in <math>S</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>.

# Con(''B'') has the structure of a SER on ''S'', with a sling at each of the points in ''S'', two-way arcs between the points of {"''A''",&nbsp;"''u''"}, and two-way arcs between the points of {"''B''",&nbsp;"''i''"}.

Taken as transition digraphs, Con(''A'') and Con(''B'') highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters ''A'' and ''B'', respectively.

 

Taken as transition digraphs, <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively.

The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel.

The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel.