Changes

For example, consider the aspects of sign use that concern two people, say, Ann and Bob, in using their own proper names, "Ann" and "Bob", and in using the pronouns, "I" and "you".  For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  The abstract consideration of how A and B use this set of signs to refer to themselves and to each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by A and by B, respectively.

For example, consider the aspects of sign use that concern two people, say, Ann and Bob, in using their own proper names, "Ann" and "Bob", and in using the pronouns, "I" and "you".  For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  The abstract consideration of how A and B use this set of signs to refer to themselves and to each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by A and by B, respectively.

Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),</math> where the object <math>x\!</math> belongs to the object domain <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the sign <math>y\!</math> belongs to the sign domain <math>S\!,</math> where the interpretant sign <math>z\!</math> belongs to the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math>

Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math>

The set-up to this point can be summarized as follows:

The set-up to this point can be summarized as follows: