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===Commentary Note 11.4===

===Commentary Note 11.4===

The task before us now is to get very clear about the relationships among relative terms, relations, and the special cases of relations that are constituted by equivalence relations, functions, and so on.

The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations that are given by equivalence relations, functions, and so on.

I am optimistic that the some of the tethering material that I spun along the "Relations In General" (RIG) thread will help us to track the equivalential and functional properties of special relations in a way that will not weigh too heavy on the rather capricious lineal embedding of syntax in 1-dimensional strings on 2-dimensional pages.  But I cannot see far enough ahead to forsee all the consequences of trying this tack, and so I cannot help but to be a bit experimental.

I am optimistic that the some of the tethering material that I spun along the "Relations In General" thread will help us to track the equivalential and functional properties of special relations in a way that will not weigh too heavily on the embedding of syntax in 1-dimensional strings on 2-dimensional pages.  But I cannot see far enough ahead to foresee all the consequences of trying this tack, and so it must remain a bit experimental.

The first obstacle to get past is the order convention that Peirce's orientation to relative terms causes him to use for functions.  By way of making our discussion concrete, and directing our attentions to an immediate object example, let us say that we desire to represent the "number of" function, that Peirce denotes by means of square brackets, by means of a 2-adic relative term, say 'v', where 'v'(''t'') = [''t''] = the number of the term ''t''.

The first obstacle to get past is the order convention that Peirce's orientation to relative terms causes him to use for functions.  To focus on a concrete example of immedeiate use in this discussion, let's take the "number of" function that Peirce dneotes by means of square brackets and re-formulate it as a 2-adic relative term &mdash; say <math>~v~</math> &mdash; where:

 

| <math>v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.</math>

To set the 2-adic relative term 'v' within a suitable context of interpretation, let us suppose that 'v' corresponds to a relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', where ''R'' is the set of real numbers and ''S'' is a suitable syntactic domain, here described as "terms".  Then the 2-adic relation ''V'' is evidently a function from ''S'' to ''R''.  We might think to use the plain letter "''v''" to denote this function, as ''v''&nbsp;:&nbsp;''S''&nbsp;&rarr;&nbsp;''R'', but I worry this may be a chaos waiting to happen.  Also, I think that we should anticipate the very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain S that we choose, so it is probably better to account the 2-adic relation ''V'' as a partial function from ''S'' to ''R''.  All things considered, then, let me try out the following impedimentaria of strategies and compromises.

To set the 2-adic relative term 'v' within a suitable context of interpretation, let us suppose that 'v' corresponds to a relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', where ''R'' is the set of real numbers and ''S'' is a suitable syntactic domain, here described as "terms".  Then the 2-adic relation ''V'' is evidently a function from ''S'' to ''R''.  We might think to use the plain letter "''v''" to denote this function, as ''v''&nbsp;:&nbsp;''S''&nbsp;&rarr;&nbsp;''R'', but I worry this may be a chaos waiting to happen.  Also, I think that we should anticipate the very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain S that we choose, so it is probably better to account the 2-adic relation ''V'' as a partial function from ''S'' to ''R''.  All things considered, then, let me try out the following impedimentaria of strategies and compromises.