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

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

QUIPS.  Discussion Note 3

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

JA = Jon Awbrey
JP = Jim Piat

Re: QUIPS-DIS 1.  http://stderr.org/pipermail/inquiry/2005-May/002602.html
In: QUIPS-DIS.    http://stderr.org/pipermail/inquiry/2005-May/thread.html#2602

In part:

JA: These requisite conditions for the necessary involvement of
    icons and indices are simply not compelled by any of the
    logically adequate definitions of sign relations that
    Peirce gives, and indeed, inference to the contrary
    would be inconsistent with the critical fact that
    genuine 3-adic relations are irreducibly 3-adic.

JP remarks:

JP: I take the irreducibility of triadic relations to refer to the fact that the
    three parts are not additive but interactive.  Such that a 3-adic relationship
    is not the sum of three dyads but something more.  It is more than the sum of
    its parts -- but this does not mean it has no parts.  The mystery (and why the
    separate existence of triads as a distinct category has so long eluded many)
    is in just how the whole can exceed the sum of its parts.  We gloss over it
    with terms like interactions but these are just names.  I think part of the
    difficulty is that we have no higher order relations from which to view and
    contrast triads.  We look at triads through the prism of a triad and its
    hard to tell which is which. Much easier to see the workings of a dyad
    from the vantage of a triad.  Something like that.

Jim,

When I speak of a 3-adic relation I am speaking of a set of 3-tuples of
the form <x, y, z>, so a 3-adic relation has parts that amount to its
individual members and each 3-tuple of the form <x, y, z> has parts
that amount to its ordered components x, y, z.  There may indeed
be 3-adic relations that consist of a single 3-tuple but these
are degenerate cases and many of the properties of interest
to us in the case of more generic 3-adic relations may
not apply in any interesting sense.  Irreducibility
is one of the properties that does not have much
useful application in the singleton case.

That is by the way, why we won't make much progress on this score
so long as we stay stuck on sign relations of a single 3-tuple.

The two kinds of "irreducibility" that we almost always find ourselves
discussing in this context can be called "compositional irreducibility"
and "projective irreducibility".  All 3-adic relation are irreducible
in the compositional sense, since the relational composition of 2-adic
relations always produces a 2-adic relation.  So we tend to put the
compositional type of irreducibility to one side -- at least, so
long as it's understood, which is hardly always the case -- and
turn our attention to projective irreducibility.  This is what
we tend to mean when we say that a sign relation is genuinely
3-adic, intuitively speaking, tthat it cannot be defined in
terms of 2-adic relations even if we can "cheat" by using
a specified 3-adic relation, like that denoted by the
logical symbol "and", as a part of the combination.

One of the reasons that it is so important to understand
just what sorts of formal objects and formal operations
are denoted by symbols like "and", "not", "of", is just
because clarifying the issue of 3-adic irreducibility
depneds on it.

Jon Awbrey

KM: Anyway, all participants alike use Peirce's definitions
    to substantiate there views.  And it just does not seem
    to be so that one could tell from the definitions, as
    Jon claims, whether there is a logical necessity of
    symbols as symbols involving icons and indices.
    Or else the discussion would have been a short
    and simple one, leading to an agreement.

JP: Ha -- seems to me both sides make the claim
    that Kirsti attributes to Jon alone.

JP: In any case I like the way you have put the matter below, Jon.  I think
    you've hit upon a truth that includes both sides.  Including whatever side
    I'm on and whatever side disagrees with the one whose spell has captured me.

JP: And in my case add the fact that I seldom understand fully what is being 
    said by the side I don't agree with.  That's usually what my not agreeing 
    is -- not understanding or misunderstanding.   Like I have these pockets
    of atheism in my pantheism.  Black holes of disunity.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o