ONT Re: Peirce's three kinds of representations
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DW = David Whitten
DW: this is quote is interesting, to speak of Peirce's views,
but it doesn't even begin to address interesting logical
phenomenon like variables. unless a variable like P is
intended to be a symbol. But the way I read this, it is
the VALUE of P that is intended to be the symbol instead
of the variable itself.
This is from the Lowell Institute Lectures of 1866, almost pre-historical
in Peircean terms. Here, Peirce is in the process of changing over from
the 3-fold terminology of <copy-likeness, conventional sign, symbol> to
<icon, index, symbol>. Part of the problem is that if you don't delimit
the notion of a "conventional sign" very carefully, then the common sense
understanding of the term will tend to make its meaning overlap with the
sorts of qualities that one expects to find in symbols.
Demonstratives, pronouns, and variables all fall under
the heading of "indices", or "indexes", as you will.
DW: if an entire book is a proposition, as I read Pierce below,
then presumably each copy of the book is a variable whose
value is the proposition.
No, he said that a book should be a symbol -- which is to say
something about its artistic integrity in appealing to a reader.
CSP: | A proposition, an argument, even a whole book may be,
| and should be, a single symbol.
The mention of propositions and arguments (here, the sign-like expressions)
is meant to continue the series: term, premiss, argument, as in syllogism.
DW: But modern predicate logic would have problems with
saying X, Y, and Z are all 'constants' whose value
is the same propositional value, since they really
aren't variables in that case.
Not sure, I think that the reading may be too far off track at this point.
DW: Propositional logic probably would have problems with it too,
as I understand any given proposition is only one variable.
It goes against the rules as I understand them to have
P = some proposition, and Q = the same proposition.
because P and Q are expected to be independent,
and this situation makes them two names for
the same proposition.
DW: Or am I just confused about this?
Well, I can't make sense of what you are saying here.
When Peirce gets down to doing propositional logic,
say, as it later develops in the zeroth layers of
his entitative and existential graphs, and most of
the abstract systems that he develops before that
time, the calculus itself will be what we call an
"uninterpreted formal system", and so all of the
p's and q's will be just so many formal tokens
that are meant to be manipulated according to
a fixed set of transformation rules. When it
comes to interpretations, these can be various.
You can then allow yourself to think of these
expressions with variables as forming forms of
"algebraic expressions", relative to a basis of
"arithmetic expressions" that have no variables.
You can then think of the things that you are
now calling "variables" p, q as symbols that
stand for something, namely, the contemplated
presence or absence of various "truth values",
and you can rationalize what you mean by this
thing called a "truth value" by saying that
it stands for a transformational cluster of
arithmetic expressions, usually regarded
as a "logical equivalence class" (LEC),
if everything works out right as rain
for being able to call it that.
But it is commonly understood in mathematical calculi of every
brand and description that a symbol that stands for a value can
always stand for a function whose value it is. So this makes it
easy to adopt a functional reading of logical calculi, whereby the
"proposition letter" q comes to denote a function q : X -> %B%, and
this is one of the best interpretations to take up for the sake of
computational implementations in ordinary programming languages.
Which Is Where I Came In ...
Jon Awbrey
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