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

[Jon Awbrey]

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QUIPS.  Discussion Note 50

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JW = Jim Willgoose

Re: QUIPS-DIS 47.  http://stderr.org/pipermail/inquiry/2005-May/002740.html
In: QUIPS-DIS.     http://stderr.org/pipermail/inquiry/2005-May/thread.html#2602

JW: Where to begin? I think most of us begin with the definition
    of a sign (df.=S).  Assuming that we can (which is a lot since
    you could construct a sign without defining it first), I take the
    expression "f <g, o, i, > = S" as primitive.  This says that a sign
    is a function on <g, o, i >.  Notice that I have chosen NLC as my
    guide where "g" means ground.

Jim,

The ground of a relation tells us in what sense the things in it are related.
The reason that we have to specify a ground is that our vernacular words for
qualities and relations are, like all our words, used in many different ways.
For example, let's take "teacher" as a relative term.  The word by itself is
already equivocal, since we have at least the following options:  the 2-adic
senses exemplified by "Plato teaches Aristotle" and "Plato teaches Rhetoric",
and the 3-adic sense illustrated by "Plato teaches Aristotle Rhetoric".  The
gist of this is that setting the ground is really tantamount to defining the
relation itself.  This definition can be extensional or intensional, or both.
A k-adic relation is defined in extension as a subset of a cartesian product,
say, L c X_1 x ... x X_k.  Consequently, the information that determines the
ground g is equivalent in information to the information that determines the
relation L.  Up to information equivalence, they are one and the same thing.

JW: More importantly, others might make the sign an element of the n-tuple!
    It is difficult to tell whether differing approaches may have equivalent
    outcomes.  In any case, what sort of relation is the basic sign-relation
    (suggested by Peirce's various definitions)?  What are its properties?
    There is no apriori way for determining this.

JW: Semiotics is a "quasi-necessary" science.

More accurately, logic is formal, normative, or quasi-necessary semiotics.
Thus, there is room in semiotics simpliciter for a descriptive semiotics.

JW: This makes the task of determining properties of relations
    hypothetical since counterexample and argument would force
    some revision of our intuitions.  If so, the definition of
    a sign is more nearly a postulate of an empirical theory.
    Are there only a finite number of relations to consider
    initially?  It seems so.  The definition tells us as much.
    It is one triadic relation with a small number of elements.

A triadic relation is a set of ordered triples.
Thus the relation can have many, many elements.
It is only the number of components in a triple
that is limited to three.

JW: Thus, we can check all the combinations of at least the basic
    sign relation to assess the properties.  My conjecture is that
    none of the individual elements (<g, o, i >) are reflexive (this
    alone would come under heavy interpretive assault in the case of
    [g/o--->g] when "g/o" is interpreted as an icon  and "--->" as
    entailment) I will skip far forward.  The n-tuple has these
    properties:  irreflexive, symmetric, and transitive though
    to what degree of probability is disputable.

Properties like the ones that you mention are
not defined at the level of individual tuples,
but definable as properties of sets of tuples,
since these are what constitutes the relation.

Jon Awbrey

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