Jeff, List,
In response to my post on relation theory in general, Helmut
asked a question about sign relations in particular which he
expressed as follows:
• "Are interpretants an own class?"
After some thought and discussion I reformulated and generalized
the question in the lead paragraph of the blog post copied below,
which Helmut said captured his sense. The rest of the blog post
is a slight rewrite of the subtraction example that I was hoping
would help to clarify the issue.
In discussing Peirce's concept of a triadic sign relation as existing among
objects, signs, and interpretant signs the question arises whether any of
the classes so related are classes by themselves, that is, whether there
is necessarily anything distinctive about the being of an object, the
being of a sign, or the being of an interpretant sign.
Maybe I can clear up a few points about the relational standpoint by
resorting to a familiar case of a triadic relation, one I'm guessing
we all mastered early in our schooling, namely, the one involved in
the operation of subtraction, x - y = z. When I was in school we
learned a set of quaint terms for the numbers x, y, z in the
relation and I wasn't sure they still taught such things so
I checked the web and found a page that described the terms
just as I remembered them:
☞ Maths Is Fun • Subtraction
( http://www.mathsisfun.com/numbers/subtraction.html )
• The number x is called the minuend.
• The number y is called the subtrahend.
• The number z is called the difference.
So we come to the questions:
• Are minuends a class by themselves?
• Are subtrahends a class by themselves?
• Are differences a class by themselves?
To answer these questions we need to observe the distinction
between relational roles and absolute essences (inherent qualities,
ontological substances, or permanent properties).
If our notion of number is generous enough to include negative
numbers then any number can appear in any one of the three places,
so minuend, subtrahend, and difference are relational roles and not
absolute essences. We can tell this because it follows from the
definition of the subtraction operation.
When it comes time to ask the same questions of objects, signs,
and interpretant signs then any hope of a definitive answer must
come from the definition of a sign relation we've chosen to fit
our subject matter.
Now, one of the most important qualifiers in my formulation of the
question was the word "necessarily" in the question "whether there
is necessarily anything distinctive about the being of an object,
the being of a sign, or the being of an interpretant sign."
The qualification of necessity implies that we are concerned with
the full range of possible sign relations, not just our favorite,
isolated, peculiar examples.
That is why I considered a moderately generic case of the
subtraction relation in my illustrative example, rather
than any number of degenerate examples that we learned
in one grade and learned to generalize in the next.
Regards,
Jon
On 6/20/2015 11:32 PM, Jon Awbrey wrote:
Jeff, List,
I have run completely out of time for today, and some of what people
have written I wrote is blurring my memory of what I actually wrote,
so let me just prepare for the morrow's continuance by copying out
the augmented blog edition of my subtraction example and take up
your comments in a context where all the qualifications I made
will be fresh in my and others' minds.
Regards,
Jon
On 6/20/2015 11:16 AM, Jeffrey Brian Downard wrote:
Jon, List,
The arithmetic example you offer in "Relations & Their Relatives: 9"
is quite clear. Having said that, what should we say about a number
system that only allows positive integers starting with 1? In this
number system, the number 1 can serve in the role of a subtrahend
or a difference. It cannot, however, serve the role of a minuend.
What does this show us about the number 1 in the system of positive
integers starting with 1?
How do things change when we work with a number system allowing only
positive integers and 0? In this case, the number 1 can now serve
in any of the three places. The same follows for the case of 0.
But 0 can be the minuend only where it is also the subtrahend
and the difference. What does this show us about the numbers
0 and 1 in the system of positive integers that include 0?
Off the top of my head, I would think that the unit has a special place
in all three number systems we are considering (all integers including 0
and negative, only positive integers starting with 1, and all positive
integers including 0) because the basic operations of addition and
subtraction are both understood in terms of the operation of adding
one more or taking one away. What is more, the number 0 has a very
special place in the number systems that allow such an expression.
How might we explain the special role that 0 and 1 play in such
systems?
--Jeff
Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Jon Awbrey [jawb...@att.net]
Sent: Friday, June 19, 2015 12:40 PM
To: Helmut Raulien
Cc: Peirce List
Subject: [PEIRCE-L] Re: Survey of Relation Theory • 1
Helmut, List,
I wasn't completely sure about the meaning of your question:
• "Are interpretants an own class?"
Is "own" a translation of "eigen" maybe?
At any rate I went with my best guess and took you to be
asking whether interpretants (and the other two classes)
were ontologically distinctive in some way. I rewrote my
last reply as a blog post with this interpretation in mind:
• Relations & Their Relatives : 9
( http://inquiryintoinquiry.com/2015/06/19/relations-their-relatives-9/ )
Please let me know if my reading of your sense is right or not.
Regards,
Jon
On 6/18/2015 6:51 PM, Helmut Raulien wrote:
Supplement: On the other hand, even if interpretants are not an own class (or is
the word "domain"?), their representations in a mind may well be, and certainly
are. So- triadicity is rescued for me, I now think.
Dear Jon, Peircers,
I am wondering whether, mathematically spoken, there really are 3-adic relations
in semiotics. An interpretant is a 2-adic relation (between representamen and
object). But are interpretants an own class? Or are they a common class with
representamens (syntactic domain)- or are some of them so, while others (the
final interpretants) re-enter into the domain of objects? And: to regard the
three sets objects, representamens, interpretants, doesnt this regarding (action
of a mind) mean that they are represented? And doesnt representation by a mind
mean, that these representations are all objects, other than the represented?
So: Is the triadic relation between representamen, object and interpretant
possibly a relation between three objects? In this case, it is not triadic: It
is a 2-adic relation between the set of objects, and the same set of objects- at
least reducible to this, I suspect. On the other hand one might say: The objects
of a mind are divided into three classes: Representations of representamens,
objects, and interpretants. Three classes mean 3-adicity. But then there is the
problem again that I have mentioned: Are interpretants an own class?
Best,
Helmut
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