Changes

===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 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.

I am optimistic that the some of the tethering material that I spun

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.

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.

The first obstacle to get past is the order convention

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''.

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.

To set the 2-adic relative term 'v' within a suitable context of interpretation,

To set the 2-adic relative term 'v' within a suitable context of interpretation, let us suppose that 'v' corresponds to a relation ''V'' ⊆ ''R'' × ''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'' : ''S'' → ''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.

let us suppose that 'v' corresponds to a relation V c R x 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 : S -> 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.

First, I will adapt the functional arrow notation so that it allows us

First, I will adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page.  Second, I will need to change the notation for "pre-functions", or "partial functions", from one likely confound to a slightly less likely confound.  This gives the scheme:

to detach the functional orientation from the order in which the names

of domains are written on the page.  Second, I will need to change the

notation for "pre-functions", or "partial functions", from one likely

confound to a slightly less likely confound.  This gives the scheme:

  q : X -> Y means that q is functional at X.

: ''q'' : ''X'' → ''Y'' means that ''q'' is functional at ''X''.

  q : X <- Y means that q is functional at Y.

: ''q'' : ''X'' &larr; ''Y'' means that ''q'' is functional at ''Y''.

  q : X ~> Y means that q is pre-functional at X.

: ''q'' : ''X'' ~> ''Y'' means that ''q'' is pre-functional at ''X''.

  q : X <~ Y means that q is pre-functional at Y.

: ''q'' : ''X'' <~ ''Y'' means that ''q'' is pre-functional at ''Y''.

For now, I will pretend that v is a function in R of S, v : R <- S,

For now, I will pretend that ''v'' is a function in ''R'' of ''S'', ''v'' : ''R'' &larr; ''S'', amounting to the functional alias of the 2-adic relation ''V''&nbsp;&sube;&nbsp;''R''&nbsp;&times;&nbsp;''S'', and associated with the 2-adic relative term ''v'' whose relate lies in the set ''R'' of real numbers and whose correlate lies in the set ''S'' of syntactic terms.

amounting to the functional alias of the 2-adic relation V c R x S,

and associated with the 2-adic relative term 'v' whose relate lies

in the set R of real numbers and whose correlate lies in the set S

of syntactic terms.

===Commentary Note 11.5===

===Commentary Note 11.5===