[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 46

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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
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
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,
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
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 Y.

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

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,
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.

Jon Awbrey

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