[Inquiry] Re: Questions Involving Pure Symbols -- Discussion

[Jon Awbrey]

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QUIPS.  Discussion Note 15

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JW = Jim Willgoose

JW: I did not want to suggest that making sense of a pure symbol is easy, useful,
    taken for granted by Peirce, or anything more than a limit case (Stjernfelt).
    But it occurred to me that a sign's relation to itself makes no reference to
    a sign's relation to an object.  Thus, if a symbol is a sign, then a symbol's
    relation to itself need not either.  If the symbol "--" is a type of sign,
    then a replica or token of the symbol is not the denotation of the symbol.
    (anymore than the word "and" denotes token instances of the word.  The word
    "and" doesn't denote).  Suppose I say with Peirce that "--" is  an indice
    of a token".  What sort of icon articulates the predicative content "..
    an indice of a token?"  You could draft a list of letters (x,y,z, etc.)
    denoted by the symbol "--" and list the letters as the extension of the
    symbol.  But that would not be pure.   So, if you want to avoid iconic
    elements, leave it undefined. (or give it a supposition that makes
    reference only to its internal quality.  There is nothing in the
    symbol "--" itself other than an internal quality.)  The word
    "and" can then be made contextually iconic by "--and--."  But
    that is a different symbol than either "--" or "and."  The
    symbol "--and--" is the beginning of iconicity in so far
    as we observe that two placeholders or marks are separated
    (joined?) by a different mark.  Put this way, context free
    symbols are pure but "ill-formed."  In any case, introducing
    context and meaning through recursion may at least give us
    the illusion of purity.  We all like our pure symbols, right?

Jim,

Let me suggest what is frequently a useful heuristic for approaching
complicated and confusing problems:  Start with a simpler version of
what is roughly the same problem type, and gradually work our way up
to the problem that we came in with.

Instead of thinking about any and all symbols, then, let us think
about numerals, the symbols that denote numbers.  For simplicity
let us think about the natural numbers:  N = {0, 1, 2, 3, ...}.
Instead of the symbols "and", "or", "of", let us think about
the various symbols for multiplication, addition, involution
on the natural numbers N.  In this Ascii text, these would
be the symbols "*", "+", "^", respectively.  Surely if our
prospective theory of signs cannot groom this well-trodden
garden, then there is not much use taking it into the wild.
 
Normally, we think of the numbers 0, 1, 2, 3, ...
as formal objects or hypostatic abstractions, and
we think of the numerals "0", "1", "2", "3", ...
as symbols that denote these abstract objects.

In this setting we can ask what sorts of objects do the symbols
"*", "+", "^" denote, and we get the context-definite answers
that they denote particular formal objects that we can think
of as 3-adic relations.  These are sets of triples <x, y, z>,
subsets of N x N x N, that satisfy the following equations:
x = y*z, x = y+z, x = y^z, respectively.

In order to force this example into a form more analogous to our
questions about logic and semiotocs, think of the numbers in N
as addresses of cottages on the beach -- you can see where my
mind is really at -- and so it makes sense to say that the
numbers are signs, in this case, indices.  You probably
know that I could create a calculus where the numbers
are identified with propositions, or anything else,
by the trick of arithmetization, but let's keep
it simple for now.

If I ask how many instances of the number 1 are on this page --
well that's a trick question and you needn't bother counting,
since the number 1 is nowhere on this page, only some tokens
of the numeral "1" can be found on this page.  That is what
the "pure original"/"base replica" distinction is all about,
and it's never been what I was talking about in this matter,
nor is it what Peirce is talking about in CP 4.447.  So let
us try to look it with fresh eyes:

| QUIPS 1.  http://stderr.org/pipermail/inquiry/2005-April/002564.html
|
| Every word is a symbol.  Every sentence is a symbol.  Every book is
| a symbol.  Every representamen depending upon conventions is a symbol.
| Just as a photograph is an index having an icon incorporated into it,
| that is, excited in the mind by its force, so a symbol may have an icon
| or an index incorporated into it, that is, the active law that it is may
| require its interpretation to involve the calling up of an image, or a
| composite photograph of many images of past experiences, as ordinary
| common nouns and verbs do;  or it may require its interpretation to
| refer to the actual surrounding circumstances of the occasion of its
| embodiment, like such words as 'that', 'this', 'I', 'you', 'which',
| 'here', 'now', 'yonder', etc.  Or it may be pure symbol, neither
| 'iconic' nor 'indicative', like the words 'and', 'or', 'of', etc.
|
| C.S. Peirce, 'Collected Papers', CP 4.447
|
|"On Existential Graphs, Euler's Diagrams, and Logical Algebra",
|"Logical Tracts, No. 2" (c. 1903), in 'Collected Papers', CP 4.418-509.
| http://www.existentialgraphs.com/peirceoneg/existentialgraphs4.418-529.htm

The sense of the term "pure symbol" in this context is this:

| a symbol may have an icon or an index incorporated into it, that is,
| the active law that it is may require its interpretation to involve
| the calling up of an image [in which case it is iconic], ... or it
| may require its interpretation to refer to the actual surrounding
| circumstances of the occasion of its embodiment [in which case it
| is indexical or indicative].  Or it may be pure symbol, neither
| iconic nor indicative, like the words 'and', 'or', 'of', etc.

Cast on to the stage of our current example,
this sense of the term "pure symbol" is more
analogous to the concept of a "prime number",
as having no non-trivial incorporation or
involvements of other numbers as factors.

And here we can get a glimmer of why the original/replica dimension is
irrelevant to the discussion of the simplicity/complexity dimension,
since we are talking about a type of complexity that is present in
the originals and not a type of complexity that is necessarily
represented at all directly in the replicas, though the degree
of the latter can vary with the particular representation used.

Jon Awbrey

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