ONT Re: Differential Logic A -- Discussion
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DLOG A. Discussion Note 5
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| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have. Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| Peirce, "Maxim of Pragmaticism",
| 'Collected Papers', CP 5.438.
Let me see what happens if I try to follow Peirce's instructions
in the sort of context where they were meant to be used, that is,
to clarify the meaning of a concept.
Interpreter J picks a concept y whose meaning J wants to clarify.
For the sake of the exercise, let us say that y has some object x.
We can safely suppose this without making any real commitments of
the ontological sort, since we can always abandon the supposition
if it leads to absurd consequences, whether logical or practical.
As far as what a concept is, Peirce follows a classical tradition
that regards a concept as a mental sign. More exactly, a concept
is a mental symbol. Now, a symbol is a type of sign that denotes
whatever objects it does just because some interpreter interprets
it as doing so. Let's put off saying what that means until later,
but it is bound to involve Peirce's definition of a sign relation,
and so knowing that much allows us to draw a picture of this sort:
y
o
/
/
x o--------@
\
\
o
z
This places the 3-ple <x, y, z> of the form <object, sign, sign'>
in the context of a suitable sign relation L c !O! x !S! x !I!,
where !O!, !S!, and !I! are the relational domains of objects,
signs, and interpretant signs available to the interpreter J.
Regarding y as a sign, J can ask whether y
has objects, and what its objects might be.
Regarding y as a symbol, what objects it has
depends on its interpreters, for instance, J.
That is, it depends on its interpreters in a
way that is "essential" and not eliminatable,
not without loss of generativity as a symbol.
For any interpreter K, let C_K (x) be K's concept of x.
For instance, in the present case, we have C_J (x) = y.
Relative to the object x of the concept y, the maxim advises J to consider
the set of effects, that might conceivably have practical bearings, that J
conceives the object x to have. Let E_J (x) be this collection of effects,
that might conceivably have practical bearings, that J conceives x to have.
The entities and relationships that we've seen so far
may be sketched in the form of the following diagram:
o y = C_J (x)
/
/
x o--------@
. \
. \
. o z
.
.
.
. o C_J (E_J (x))
. /
. /
o--------@
E_J (x) \
\
o
For the sake of a first approximation to the maxim,
I am overlooking the subtleties that may be lurking
in the proviso "conceivably have practical bearing".
Apart from the catches of that one remaining wrinkle,
the maxim apparently suggests some sort of descriptive
or normative equation between C_J (x) and C_J (E_J (x)).
Now, what possible use could such a formula have,
when it comes to revealing the meaning of C_J (x)?
One thing that comes to mind right off is the similarity of the
diagram that I drew above to the kinds that I commonly draw in
cases of functions defined by recursion. What I have in mind
here is the type of function whose value on "complicated"
arguments is arrived at by way of a specified relation
to its values on "simpler" arguments.
If this form of analogy is apt, then C_J would be analogous to
the recursive function in question, and E_J would be analogous
to the recursion relation between complex and simple arguments.
Of course, it has to be kept in mind that we are talking about
structures more general than functions, namely, sign relations.
But I think that I will sleep on it now
and explore this idea further tomorrow.
Jon Awbrey
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http://www.cs.bsu.edu/homepages/mighty/history.html
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