SUO: Can Arity Be Taut?
apease wrote:
>
> Jon,
>
> I think we're agreed that relations of higher arity
> are not strictly required. The issue is whether they
> are advantageous for reasons of clarity or convenience
> and whether that facility would have any downside.
> I believe they are advantageous for convenience and
> clarity and I'm unaware of any negatives that may
> result from their being employed.
>
> Adam
>
<...>
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Adam,
Here is a summary of my current understanding:
As commonly used to discuss relations, the notion of "reducibility"
is ambiguous, ill-defined, indefinite, vague, whatever, and requires
to be refined by more specific adjectives before there is any chance
of people arriving at any kind of agreement as to the nature and the
properties of the subject matter that they may desire to talk about.
My acquaintance with the nature of human interpreters probably makes
me less sanguine than most about the chances that this will actually
happen, and I think that we will eventually be forced to take these
characters of the human into account in a fully pragmatic treatment
of sign use, but that is a separate matter.
As commonly used in logic and mathematics, the notion of "composition"
is less ambiguous, considered to be fairly well-defined, and treated
by those particular communities of interpretation as something nigh
unto what we might call a "reserved word". For my part, I will try
to avoid treading on their reservations.
Taken out of context, as I fear that it might be, sooner or later,
the following lead-in, and I dare say "leading", statement of yours
harbors a whole host of ambiguities:
> I think we're agreed that relations of
> higher arity are not strictly required.
Let me enumerate:
Ambiguity 1. Regarding "Higher Arity" (HA)
Meaning 1a. "HA" = "> 2".
Meaning 1b. "HA" = "> 3".
Ambiguity 2. Regarding "X of type T is required."
Meaning 2a. "Required in Reality":
"Objects of type T are required,
in the sense that they exist,
and so need to be recognized."
Meaning 2b. "Required in Representation":
"Signs of type T are required,
in the sense that there exist
no signs of simpler types that
will denote the same objects."
What I understand is this:
"Relations of higher arity (> 2) are required"
is true in the sense of 2a (reality), and
is true in the sense of 2b (representation).
"Relations of higher arity (> 3) are required."
is true in the sense of 2a (reality), but
is false in the sense of 2b (representation).
I think that a large part of the confusion here
is coming from a failure to clearly distingusish
the role of an object from the role of a sign.
This is why I tried to exploit the analogy with
numbers (objects) and numerals (signs), but if
we confound these two, too, then the point of
the analogy is completely lost.
I am not the sort of person who argues for prescriptive
restrictions on our representations -- I do my best to
push their envelopes in all of the available dimensions.
So "12" and "1100" and "(2^2)(3^1)", and no doubt ten-thousand
others, are each of them beautiful in its own way as signs
for the number twelve, but none of them changes the fact
that 5 + 7 = 12, however you say it.
Another way to approach this is to think about the
analogy with formal languages and formal grammars.
Suppose that you have a formal language with a finite number
of sentences. Then that language is "finite state", because
it has a finite state grammar, namely, list all the sentences
as terminal symbols -- your grammar is a simple look-up table.
But nobody would do this if the number of sentences is large --
one would naturally try to exploit the redundancies in the set
of sentences, in effect, start trying to reduce the number of
states, maybe even get a handle on the phrase structure, just
as if one were faced with a language of a non-degenerate type.
The subtext of what I am trying to say here --
as all of us Peirceans appear to be doomed
to wander the face of the earth crying out
our triadic message in the dyadic wilderness --
is a suggestion that goes a bit like this:
| If we could just try to accustom ourselves
| to thinking in terms of the underlying
| triadic relations, we just might find,
| if we 'tri' sometime, that many of the
| classical problems, about "context",
| "intention", "purpose", "reference",
| "sense", and their protean effects
| on interpretation, would appear in
| a new light, one that just might
| lead us out of the loop-de-loop
| that we commonly fall into over
| these questions.
And Dat's Da Truth!
Jon
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