Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.4
http://inquiryintoinquiry.com/2014/05/01/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-4/
Posted : May 1, 2014 at 3:30 pm
Author : Jon Awbrey
Peircers,
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.
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 immediate use in this discussion,
let's take the “number of” function that Peirce denotes by means of square brackets and re-formulate
it as a dyadic relative term v as follows:
v(t) := [t] = the number of the term t.
To set the dyadic 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 a set of ''terms''. Then the dyadic relation V is evidently a function
from S to R. We may think to use the plain letter “v” to denote this function, v : S → R, but I
worry this may be a chaos waiting to happen. Also, I think 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 dyadic 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 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 change
the notation for partial functions, or pre-functions, to one that is less likely to be confounded.
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, at least, I will pretend that v is a function in R of S, written v : R ← S, amounting to
the functional alias of the dyadic relation V ⊆ R × S, and associated with the dyadic relative term
v whose relate lies in the set R of real numbers and whose correlate lies in the set S of syntactic
terms.
Note
====
The definitions of ''functions'' and ''pre-functions'' used in the above discussion can be found in
the following article.
Relation Theory
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
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