View source for Directory talk:Jon Awbrey/Notes/Factorization Issues

'''Note.'''  Older Version

==Note 1==

<pre>
I would like to introduce a concept that I find to be of
use in discussing the problems of hypostatic abstraction,
reification, the reality of universals, and the questions
of choosing among nominalism, conceptualism, and realism,
generally.

I will take this up first in the simplest possible setting,
where it has to do with the special sorts of relations that
are commonly called "functions", and after the basic idea
is made as clear as possible in this easiest case I will
deal with the notion of "factorization" as it affects
more generic types of relations.

Picture an arbitrary function from a Source (Domain)
to a Target (Co-domain).  Here is one picture of an
f : X -> Y, just about as generic as it needs to be:

|   Source X  =  {1, 2, 3, 4,    5}
|          |      o  o  o  o     o
|      f   |       \ | /    \   /
|          |        \|/      \ /
|          v      o  o  o  o  o  o
|   Target Y  =  {A, B, C, D, E, F}

Now, it is a fact that any old function that you might
pick "factors" into a surjective ("onto") function and
an injective ("one-to-one") function, in the present
example just like so:

|   Source X  =  {1, 2, 3, 4,    5}
|          |      o  o  o  o     o
|      g   |       \ | /    \   /
|          v        \|/      \ /
|   Middle M  =  {   b   ,    e   }
|          |         |        |
|      h   |         |        |
|          v      o  o  o  o  o  o
|   Target Y  =  {A, B, C, D, E, F}

Writing the functional compositions f = g o h "on the right",
as they say, we have the following data about the situation:

X  =  {1, 2, 3, 4, 5}
M  =  {b, e}
Y  =  {A, B, C, D, E, F}

f : X -> Y, arbitrary.
g : X -> M, surjective.
h : M -> Y, injective.

f = g o h

What does all of this have to do with reification and so on?
Well, suppose that the Source domain X is a set of "objects",
that the Target domain Y is a set of "signs", and suppose that
the function f : X -> Y indicates the effect of a classification,
conceptualization, discrimination, perception, or some other type
of "sorting" operation, distributing the elements of the set X of
objects and into a set of "sorting bins" that are labeled with the
elements of the set Y, regarded as a set of classifiers, concepts,
descriptors, percepts, or just plain signs, whether these signs
are regarded as being in the mind, as with concepts, or whether
they happen to be inscribed more publicly in another medium.

In general, if we try to use the signs in the Target (Co-domain) Y
to reference the objects in the Source (Domain) X, then we will be
invoking what used to be called -- since the Middle Ages, I think --
a manner of "general reference" or a mode of "plural denotation",
that is to say, one sign will, in general, denote each of many
objects, in a way that would normally be called "ambiguous",
"equivocal", "indefinite", "indiscriminate", and so on.

Notice what I did not say here, that one sign denotes a "set" of objects,
because I am for the moment conducting myself as such a dyed-in-the-wool
nominal thinker that I hesitate even to admit so much as the existence of
this thing we call a "set" into the graces of my formal ontology, though,
of course, my casual speech is rife with the use of the word "set", and
in a way that the nominal thinker, true-blue to the end, would probably
be inclined or duty-bound to insist is a purely dispensable convenience.

In fact, the invocation of a new order of entities, whether you regard
its typical enlistee as a class, a concept, a form, a general, an idea,
an interpretant, a property, a set, a universal, or whatever you elect
to call it, is tantamount exactly to taking this step that I just now
called the "factoring" of the classification function into surjective
and injective factors.

Observe, however, that here is where the battles begin to break out,
for not all factorizations are regarded with equal equanimity by folks
who have divergent philosophical attitudes toward the creation of new
entities, especially when they get around to asking:  "In what domain
or estate shall the multiplicity of newborn entities be lodged or yet
come to reside on a permanent basis?"  Some factorizations enfold new
orders of entities within the Object domain of a fundamental ontology,
and some factorizations invoke new orders of entities within the Sign
domains of concepts, data, interpretants, language, meaning, percepts,
and "sense in general" (SIG).  Now, opting for the "Object" choice of
habitation would usually be taken as symptomatic of "realist" leanings,
while opting out of the factorization altogether, or weakly conceding
the purely expedient convenience of the "Sign" choice for the status
of the intermediate entities, would probably be taken as evidence of
a "nominalist" persuasion.

Suppose that we have a sign relation L c OxSxI,
where the sets O, S, I are the domains of the
Object, Sign, Interpretant domains, respectively.

Now suppose that the situation with respect to
the "denotative component" of L, in other words,
the "projection" of L on the subspace OxS, can
be pictured in the following manner, where equal
signs, like "=", written between ostensible nodes,
like "o", identify them into a single real node.

o-----------------------------o
| Denotative Component of L   |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|                   o         |
|                  /=         |
|                 / o   y     |
|                / /=         |
|               / / o         |
|              / / /          |
|             / / /           |
|            / / /            |
|           / / /             |
|          / / /              |
|  x_1    o-/-/-----o  y_1    |
|          / /                |
|         / /                 |
|  x_2   o-/--------o  y_2    |
|         /                   |
|        /                    |
|  x_3  o-----------o  y_3    |
|                             |
o-----------------------------o

This depicts a situation where each of the three objects,
x_1, x_2, x_3, has a "proper name" that denotes it alone,
namely, the three proper names y_1, y_2, y_3, respectively.
Over and above the objects denoted by their proper names,
there is the general sign y, which denotes any and all of
the objects x_1, x_2, x_3.  This kind of sign is described
as a "general name" or a "plural term", and its relation to
its objects is a "general reference" or a "plural denotation".

Now, at this stage of the game, if you ask:
"Is the object of the sign y one or many?",
the answer has to be:  "Not one, but many".
That is, there is not one x that y denotes,
but only the three x's in the object space.
Nominal thinkers would ask:  "Granted this,
what need do we have really of more excess?"
The maxim of the nominal thinker is "never
read a general name as a name of a general",
meaning that we should never jump from the
accidental circumstance of a plural sign y
to the abnominal fact that a unit x exists.

In actual practice this would be just one segment of a much larger
sign relation, but let us continue to focus on just this one piece.
The association of objects with signs is not in general a function,
no matter which way, from O to S or from S to O, that we might try
to read it, but very often one will choose to focus on a selection
of links that do make up a function in one direction or the other.

In general, but in this context especially, it is convenient
to have a name for the converse of the denotation relation,
or for any selection from it.  I have been toying with the
idea of calling this "annotation", or maybe "ennotation".

For a not too impertinent instance, the assignment of the
general term y to each of the objects x_1, x_2, x_3 is
one such functional patch, piece, segment, or selection.
So this patch can be pictured according to the pattern
that was previously observed, and thus transformed by
means of a canonical factorization.

In this case, we factor the function f : O -> S

|   Source O  :>  x_1 x_2 x_3
|          |       o   o   o
|          |        \  |  /
|       f  |         \ | /
|          |          \|/
|          v       ... o ...
|   Target S  :>       y 

into the composition g o h, where g : O -> M, and h : M -> S

|   Source O  :>  x_1 x_2 x_3
|          |       o   o   o
|       g  |        \  |  /
|          |         \ | /
|          v          \|/
|   Middle M  :>   ... x ...
|          |           | 
|       h  |           |
|          |           |
|          v       ... o ...
|   Target S  :>       y

The factorization of an arbitrary function
into a surjective ("onto") function followed
by an injective ("one-one") function is such
a deceptively trivial observation that I had
guessed that you would all wonder what in the
heck, if anything, could possibly come of it.

What it means is that, "without loss or gain of generality" (WOLOGOG),
we might as well assume that there is a domain of intermediate entities
under which the objects of a general denotation can be marshalled, just
as if they actually had something rather more essential and really more
substantial in common than the shared attachment to a coincidental name.
So the problematic status of a hypostatic entity like x is reduced from
a question of its nominal existence to a matter of its local habitation.
Is it very like a sign, or is it rather more like an object?  One wonders
why there has to be only these two categories, and why not just form up
another, but that does not seem like playing the game to propose it.
At any rate, I will defer for now one other obvious possibility --
obvious from the standpoint of the pragmatic theory of signs --
the option of assigning the new concept, or mental symbol,
to the role of an interpretant sign.

If we force the factored annotation function,
initially extracted from the sign relation L,
back into the frame from whence it once came,
we get the augmented sign relation L', shown
in the next vignette:

o-----------------------------o
| Denotative Component of L'  |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|                   o         |
|                  /=         |
|   x   o=o-------/-o   y     |
|       ^^^      / /=         |
|       '''     / / o         |
|       '''    / / /          |
|       '''   / / /           |
|       '''  / / /            |
|       ''' / / /             |
|       '''/ / /              |
|  x_1  ''o-/-/-----o  y_1    |
|       '' / /                |
|       ''/ /                 |
|  x_2  'o-/--------o  y_2    |
|       ' /                   |
|       '/                    |
|  x_3  o-----------o  y_3    |
|                             |
o-----------------------------o

This amounts to the creation of a hypostatic object x,
which affords us a singular denotation for the sign y.

By way of terminology, it would be convenient to have
a general name for the transformation that converts
a bare "nominal" sign relation like L into a new,
improved "hypostatically augmented or extended"
sign relation like L'.

I call this kind of transformation
an "objective extension" (OE) or
an "outward extension" (OE) of
the underlying sign relation.

This naturally raises the question of
whether there is also an augmentation
of sign relations that might be called
an "interpretive extension" (IE) or
an "inward extension" (IE) of
the underlying sign relation,
and this is the topic that
I will take up next.
</pre>

==Note 2==

<pre>
Let me illustrate what I think that a lot of our controversies
about nominalism versus realism actually boil down to in practice.
From a semiotic or a sign-theoretic point of view, it all begins
with a case of "plural reference", which happens when a sign y
is quite literally taken to denote each object x_j in a whole
collection of objects {x_1, ..., x_k, ...}, a situation that
I would normally represent in a sign-relational table like so:

o---------o---------o---------o
| Object  |  Sign   | Interp  |
o---------o---------o---------o
|   x_1   |    y    |   ...   |
|   x_2   |    y    |   ...   |
|   x_3   |    y    |   ...   |
|   ...   |    y    |   ...   |
|   x_k   |    y    |   ...   |
|   ...   |    y    |   ...   |
o---------o---------o---------o

For brevity, let us consider the sign relation L
whose relational database table is precisely this:

o-----------------------------o
|       Sign Relation L       |
o---------o---------o---------o
| Object  |  Sign   | Interp  |
o---------o---------o---------o
|   x_1   |    y    |   ...   |
|   x_2   |    y    |   ...   |
|   x_3   |    y    |   ...   |
o---------o---------o---------o

For the moment, it does not matter what the interpretants are.

I would like to diagram this somewhat after the following fashion,
here detailing just the denotative component of the sign relation,
that is, the 2-adic relation that is obtained by "projecting out"
the Object and the Sign columns of the table.

o-----------------------------o
| Denotative Component of L   |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|  x_1  o------>              |
|               \             |
|                \            |
|  x_2  o------>--o  y        |
|                /            |
|               /             |
|  x_3  o------>              |
|                             |
o-----------------------------o

I would like to -- but my personal limitations in the
Art of ASCII Hieroglyphics do not permit me to maintain
this level of detail as the figures begin to ramify much
beyond this level of complexity.  Therefore, let me use
the following device to symbolize the same configuration:

o-----------------------------o
| Denotative Component of L   |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
| o   o   o >>>>>>>>>>>> y    |
|                             |
o-----------------------------o

Notice the subtle distinction between these two cases:

1.  A sign denotes each object in a set of objects.

2.  A sign denotes a set of objects.

The first option uses the notion of a set in a casual,
informal, or metalinguistic way, and does not really
commit us to the existence of sets in any formal way.
This is the more razoresque choice, much less risky,
ontologically speaking, and so we may adopt it as
our "nominal" starting position.

Now, in this "plural denotative" component of the sign relation,
we are looking at what may be seen as a functional relationship,
in the sense that we have a piece of some function f : O -> S,
such that f(x_1) = f(x_2) = f(x_3) = y, for example.  A function
always admits of being factored into an "onto" (surjective) map
followed by a "one-to-one" (injective) map, as discussed earlier.

But where do the intermediate entities go?  We could lodge them
in a brand new space all their own, but Ockham the Innkeeper is
right up there with Old Procrustes when it comes to the amenity
of his accommodations, and so we feel compelled to at least try
shoving them into one or another of the spaces already reserved.

In the rest of this discussion, let us assign the label "i" to
the intermediate entity between the objects x_j and the sign y.

Now, should you annex i to the object domain O you will have
instantly given yourself away as having "Realist" tendencies,
and you might as well go ahead and call it an "Intension" or
even an "Idea" of the grossly subtlest Platonic brand, since
you are about to booted from Ockham's Establishment, and you
might as well have the comforts of your Ideals in your exile.

o-----------------------------o
| Denotative Component of L'  |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|     i                       |
|    /|\   *                  |
|   / | \       *             |
|  /  |  \           *        |
| o   o   o >>>>>>>>>>>> y    |
|                             |
o-----------------------------o

But if you assimilate i to the realm of signs S, you will
be showing your inclination to remain within the straight
and narrow of "Conceptualist" or even "Nominalist" dogmas,
and you may read this "i" as standing for an intelligible
concept, or an "idea" of the safely decapitalized, mental
impression variety.

o-----------------------------o
| Denotative Component of L"  |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
| o   o   o >>>>>>>>>>>> y    |
|    .  .  .             '    |
|         . . .          '    |
|              ...       '    |
|                   .    '    |
|                       "i"   |
|                             |
o-----------------------------o

But if you dare to be truly liberal, you might just find
that you can easily afford to accommmodate the illusions
of both of these types of intellectual inclinations, and
after a while you begin to wonder how all of that mental
or ontological downsizing got started in the first place.

o-----------------------------o
| Denotative Component of L'" |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|     i                       |
|    /|\   *                  |
|   / | \       *             |
|  /  |  \           *        |
| o   o   o >>>>>>>>>>>> y    |
|    .  .  .             '    |
|         . . .          '    |
|              ...       '    |
|                   .    '    |
|                       "i"   |
|                             |
o-----------------------------o

To sum up, we have recognized the perfectly innocuous utility
of admitting the abstract intermediate object i, that may be
interpreted as an intension, a property, or a quality that
is held in common by all of the initial objects x_j that
are plurally denoted by the sign y.  Further, it appears
to be equally unexceptionable to allow the use of the
sign "i" to denote this shared intension i.  Finally,
all of this flexibility arises from a universally
available construction, a type of compositional
factorization, common to the functional parts
of the dyadic components of any relation.

Okay, there are a few pieces of this that I
will need to think over once or thrice more.
</pre>

==Note 3==

<pre>
JA = Jon Awbrey
SR = Seth Russell

JA figured:

o-----------------------------o
| Denotative Component of L'" |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|     i                       |
|    /|\   *                  |
|   / | \       *             |
|  /  |  \           *        |
| o   o   o >>>>>>>>>>>> y    |
|    .  .  .             '    |
|         . . .          '    |
|              ...       '    |
|                   .    '    |
|                       "i"   |
|                             |
o-----------------------------o

SR: Your diagrams dont tell the whole story.

JA: No diagram, no form of representation, tells the "whole" story.
    A representation becomes pretty useless if it tries to do that.

SR: .... here is the rest of the story all in one diagram.

SR: http://robustai.net/mentography/intensionExtension.gif

Seth,

Just off the bat, the arrows that are labeled "connotes",
"extension of", "intension of", and "isa" seem off base.

Just some random notes:

y and "i" are both signs.

x_1, x_2, x_3, and i are all objects
in the augmented sign relation L'''.

The intension (property, quality) i gets to be
an "object of conduct, discussion, or thought"
as soon as any agents (interpreters, observers)
start to act, to talk, or to think in some way
or another with regard to it.

Later, I will build separate hierarchies for the objects
and for the syntactic entities (signs, interpretants).

I forget now, but I don't remember saying anything yet
about interpretants in this example.  I will go check.
</pre>

==Note 4==

<pre>
JA = Jon Awbrey
SR = Seth Russell

JA glyped:

o-----------------------------o
| Denotative Component of L'" |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|     i                       |
|    /|\   *                  |
|   / | \       *             |
|  /  |  \           *        |
| o   o   o >>>>>>>>>>>> y    |
|    .  .  .             '    |
|         . . .          '    |
|              ...       '    |
|                   .    '    |
|                       "i"   |
|                             |
o-----------------------------o

SR giffed:

http://robustai.net/mentography/intensionExtension.gif

JA: No diagram, no form of representation, tells the "whole" story.
    A representation becomes pretty useless if it tries to do that.

SR: Point taken :)

JA: Just off the bat, the arrows that are labeled "connotes",
    "extension of", "intension of", and "isa" seem off base.

SR: Why?

JA: Just some random notes:
    y and "i" are both signs.

SR: You mean 'y' and 'i' , I presume.  And yes, I agree,
    and my mentograph shows both of those things in the
    context labeled signs.

No, let me explain ...

I'm trying to stay within what I'm able to say using
just one level of quotation marks, so bear with me.
To do any better in a truly systematic way requires
the explicit introduction of "higher order" (HO)
sign relations.  Maybe later.

I resort to analogy:

I am saying that y is a sign in S, much like the way I might say
that k is an integer in J = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

I am saying that "i" is a sign in S, much like the way I might say
that |j| is an integer in J, where the vertical bars indicate the
absolute value function -- this is just an example, it could have
been any other functional value f(j).

The point is that once we have a sign domain S, for example,
something like S = {"a", "b", "c", ... A", "B", "C", ...},
then we can use the elements listed to talk about signs in S,
or we can use other constant names and variable names to talk
about the elements of S.  For example, I can ask you to think
about a sign z such that z = "a", and so on.

JA: x_1, x_2, x_3, and i are all objects
    in the augmented sign relation L'".

> Yes I agree and have shown them as such in the context labeled objects in
> the mentograph.  I presume the 'sign relation L' to which you refer to is
> all the arcs labeled 'connotes', 'denotes', and 'represents' in my diagram.
> I may or may not have chosen correct words for these labels.  What words
> would you choose?

Just for clarity, here is the tabular version
of the twice augmented sign relation L'":

o-----------------------------o
|      Sign Relation L'"      |
o---------o---------o---------o
| Object  |  Sign   | Interp  |
o---------o---------o---------o
|    i    |   "i"   |   ...   |
|   x_1   |   "i"   |   ...   |
|   x_2   |   "i"   |   ...   |
|   x_3   |   "i"   |   ...   |
o---------o---------o---------o
|    i    |    y    |   ...   |
|   x_1   |    y    |   ...   |
|   x_2   |    y    |   ...   |
|   x_3   |    y    |   ...   |
o---------o---------o---------o

Okay, this has gotten way too abstract for me!
Let us back up and remember why we got into this
in the first place.  It had to do with some of the
hard cases of the ontology development process that
I commonly think of as "inquiry", and especially the
abductive generation of a new concept, hypothesis, or
term, or what is very similar, the semeiosis/semitosis
of some old such notion that has gotten too posh to be
useful without undergoing some further distinctions or
divisions in the over-extenuated mass of its extension.

Were you here when we were talking about metonymy?
There is something about this that reminds of that.

Here is one old note I found:

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

Subj:  Re: Meaning-Preserving Translations
Date:  Sat, 31 Mar 2001 23:00:31 -0500
From:  Jon Awbrey <jawbrey@oakland.edu>
  To:  Stand Up Ontology <standard-upper-ontology@ieee.org>,
       SemioCom <semiocom@listbot.com>
  CC:  John F. Sowa <sowa@bestweb.net>,
       Mary Keeler <mkeeler@u.washington.edu>

John F. Sowa wrote:
> 
> Jon,
> 
> Your quotation from Hugh T. is very helpful, because it
> illustrates a universal principle of natural languages:
> 
> > | It is worth noting in this connexion that the use of the words
> > | 'oros' (bound or limit), 'akron' (extreme), and 'meson' (middle) to
> > | describe the terms, and of 'diastema' (interval) as an alternative
> > | to 'protasis' or premiss, suggests that Aristotle was accustomed to
> > | employ some form of blackboard diagram, as it were, for the purpose
> > | of illustration.  A premiss was probably represented by a line joining
> > | the letters chosen to stand for the terms.   How quality and quantity
> > | were indicated can only be conjectured.
> > |
> > | Hugh Tredennick,
> > |"Introduction" to Aristotle's "Prior Analytics", page 184 in:
> > |'Aristotle, Volume 1', Translated by H.P. Cooke & H. Tredennick,
> > | Loeb Classical Library, William Heinemann, London, UK, 1938.
> 
> This example illustrates a kind of "metonomy", which refers to
> something by using a term (often more concrete or "diagrammatic")
> to refer to something abstract.  This usage is common not only in
> ordinary language, but also in the most formal of all sciences,
> mathematics.  We use terms like "limit", "boundary", or "interval"
> to refer to numbers, which are the entities denoted by numerals.
> In fact, it is very rare for mathematicians to mention the
> distinction between numbers and numerals explicitly, unless
> they are talking about the actual syntax of decimal, binary,
> or other representation.

Let me think.

Metonomy = "use of the name of one thing for that of another
of which it is the attribute or with which it is associated --
as in 'lands belonging to the crown'" (Webster's).

Accordingly, in this figure of metonymy, the term "crown" denotes
what the term "regent" denotes by virtue of the fact that a crown
is an associate or an attribute of a regent.

Apparently, we have a sign relation of the following form,
in which the figure of metonymy is embodied by the triples
of the form <o, s, i> in the lower four rows of the table:

¤~~~~~~~~~~¤~~~~~~~~~~¤~~~~~~~~~~¤
| Object   |  Sign    | Interp   |
¤~~~~~~~~~~¤~~~~~~~~~~¤~~~~~~~~~~¤
|          |          |          |
| Crown    | "Crown"  | "Crown"  |
|          |          |          |
| Regent   | "Crown"  | "Crown"  |
| Regent   | "Crown"  | "Regent" |
| Regent   | "Regent" | "Crown"  |
| Regent   | "Regent" | "Regent" |
¤~~~~~~~~~~¤~~~~~~~~~~¤~~~~~~~~~~¤

This may be diagrammed as follows, with denotative arcs
extending from signs to objects and with connotative arcs
extending from signs to interpretant signs:

  Crown  = o1 <----- s1 = "Crown"
                   / ^
                  /  |
                 /   |
                /    |
               v     v
  Regent = o2 <----- s2 = "Regent"

The projection of this sign relation on its SI-space forms an
equivalence relation, a "semiotic equivalence relation" (SER),
on the signs "Crown" and "Regent".  However, this SER does not
constitute a "referential equivalence relation" (RER), because
the parts of the associated partition of the syntactic domain,
the union of S & I, do not faithfully represent the structure
of the object domain O.

> I would interpret Aristotle's use of diagrammatic terms in
> the same way I would interpret the use of the word "top"
> to refer to the most general category of the ontology:
> it refers explicitly to the place where the mark occurs
> on the paper or blackboard, by metonomy to the word instance
> written in that place, by further metonomy to the word type,
> and by further metonomy to the concept expressed by that word.
> 
> In programming languages, a related term is "coercion", which
> refers to the automatic type conversion that takes place when
> necessary:
> 
>   - Integer to float:  In the expression, "2 + 3.75",
>     the integer value of the numeral "2" is automatically
>     converted (or "coerced") to float.
> 
>   - Character string to numeric:  In some languages,
>     arithmetic can be performed directly on numbers that
>     are represented by character strings.  In "2.6 + '55'"
>     the string '55' is coerced to the integer 55, which is
>     then coerced to the floating-point value.
> 
> Metonomy in natural language is extremely common and,
> I would say, extremely valuable in general.  And I admit
> that it can sometimes cause confusion.  But I would much
> rather take advantage of metonomy in what I read, write,
> and speak than to force myself and others to insert
> "conversion" operators for every change of type.
> 
> Bottom line:  I am willing to say "By 'top', I mean
> the concept expressed by the mark that occurs at the
> top of the type lattice."  But I'm only going to say
> that once.  From then on, I would just say "top".
> 
> > ...  The more pertinent question,
> > from the standpoint of a pragmatic theory of signs is:
> > "Exactly what roles does the given thing play within
> > a given moment (= elementary relation = triple) of
> > the relevant sign relation?"  So, of course, signs
> > can be objects -- no sooner do we talk about them
> > than they become objects of discussion, if others
> > would say "potential objects" (PO's), reserving
> > the honorific title "object" for the PO of some
> > consistent style of discussion and predication.
> 
> Yes, such analysis can be valuable.  But once the analysis
> has been done, I would go back to using language the way it
> has always been used:  take advantage of metonomy whenever
> it makes the expression more concise.

Sadly, until our computers get to understand the way we talk,
with all of these figures of speech, metaphor, metonymy, and
many more, somebody will have to do the dirty job of getting
them to grok it.

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

Okay, let's compare and contrast:

o---------o---------o---------o
| Object  |  Sign   | Interp  |
o---------o---------o---------o
|         |         |         |
| crown   | "crown" | "crown" |
|         |         |         |
| regent  | "crown" | "crown" |
| regent  | "crown" | "regent"|
| regent  | "regent"| "crown" |
| regent  | "regent"| "regent"|
o---------o---------o---------o

o-----------------------------o
|      Sign Relation L'"      |
o---------o---------o---------o
| Object  |  Sign   | Interp  |
o---------o---------o---------o
|    i    |   "i"   |   ...   |
|   x_1   |   "i"   |   ...   |
|   x_2   |   "i"   |   ...   |
|   x_3   |   "i"   |   ...   |
o---------o---------o---------o
|    i    |    y    |   ...   |
|   x_1   |    y    |   ...   |
|   x_2   |    y    |   ...   |
|   x_3   |    y    |   ...   |
o---------o---------o---------o

What's similar is this.  Signs are typically used in highly
ambiguous, equivocal, non-deterministic ways, and there is
just no substitute for intelligent interpreters, humane or
otherwise, when it gets down to the brass syntax of trying
to pin down the meaning of a text.  The way that metonymy
works is that when you hear the word "crown", not knowing
if it is capitalized or not, you have to decide whether
it literally means a crown, or whether it figuratively
means a regent.  In the literal case, you are taking
the word at its word and assigning it to a semantic
equivalence class with other words that are used
to denote physical crowns.  In the figurative
case, you are associating the word to a very
different sort of semantic equivalence class.

I need to break here and think about that a while.

Jon Awbrey

JA: The intension (property, quality) i gets to be
    an "object of conduct, discussion, or thought"
    as soon as any agents (interpreters, observers)
    start to act, to talk, or to think in some way
    or another with regard to it.

SR: Yes, absolutely ... this is not as yet in that graph.
    However I did make a stab in that direction in both
    of the mentographs:

    http://robustai.net/mentography/Tarskian3.gif and
    http://robustai.net/mentography/AnnBobYouI.gif

    ... which shows the perdicament broken
    into the contexts of different agents.

JA: Later, I will build separate hierarchies for the objects
    and for the syntactic entities (signs, interpretants).

SR: ... looking forward to it.

JA: I forget now, but I don't remember saying anything yet
    about interpretants in this example.  I will go check.

SR: You probably did not, yet I cannot in good conscience
    mentograph a sign relation leaving out one of the triads.

SR: ... thanks for the dialogue.
</pre>

==Note 5==

<pre>
JA = Jon Awbrey
SR = Seth Russell

Seth,

Let me try to come up with a more concrete version
that has the same structure as the present example.
Then I'll go back and try to answer your questions.

JA glyphed:

o-----------------------------o
| Denotative Component of L'" |
o--------------o--------------o
|   Objects    |    Signs     |
o--------------o--------------o
|                             |
|     i                       |
|    /|\   *                  |
|   / | \       *             |
|  /  |  \           *        |
| o   o   o >>>>>>>>>>>> y    |
|    .  .  .             '    |
|         . . .          '    |
|              ...       '    |
|                   .    '    |
|                       "i"   |
|                             |
o-----------------------------o

SR giffed:

http://robustai.net/mentography/intensionExtension.gif

The initial problem had to do with "nominal" thinking versus "real" thinking.

A.  Some maxims of nominal thinking are:
   
    1.  "Do not confuse a general name with the name of a general."  (Goodman, I think).
        In other words:  Just because we find it useful to employ general, plural, or
        universal terms, that does not mean that there is any such thing as a general
        property, a plurality such as a set, a universal form or a platonic idea that
        we are thus talking about, or thereby denoting by means of this general term.
        In the way that folks used to talk, the practice of really believing in such
        entities would have been criticized as "reifying an adjective" and so on.

    2.  Short versions:

        a.  "Generals are mere names."
 
        b.  "Universals are merely signs."

B.  The real thinker does not see the harm in supposing the existence of objects
    of thought like abstractions, categories, generalities, intensions, properties,
    qualities, universals, platonic ideas, and so on.

Where I came in, I was trying to explore the conditions under which
it really does appear to be perfectly harmless to talk as if we were
really talking about such things, and so I picked up the classical
notions of "general denotation" and "plural reference", examined
their analogy to function application, and then observed that
the canonical factorization of functions permits us to invoke
a realm of intermediate entities without having to wring our
hands in ontological anxiety about it.  That was Phase One.
Phase Two was more tentative and tenuous, trying to shove
these intermediate entities into one or the other or both
of the established domains, namely, objects and/or signs.
In mathematics, they usually do not bother with this,
but just refer to the equivalence classes explicitly.
Maybe that will turn out to be the best way after all.

Let's try this:

x_1  =  cat_1

x_2  =  cat_2 

x_3  =  cat_3

Options:

1.  y  =  "Cat", interpreted as denoting each item of a category.
           This is the nominal way of interpreting general terms,
           namely, as applying to each separate member of a group,
           but without having to posit the group as a whole or
           any of its qualities as separately existing entities.

           The nominal option is not to augment the sign relation,
           but just keep trying to get by with multiple referents.

2.  y  =  "Cat", interpreted as denoting a category of items.
           Here, one is asserting that a category is an object
           in its own right, over and above its items.

           Here, object i is a new entity like a class or a set.

3.  y  =  "Catitude", interpreted as denoting a quality that is
           possessed in common or shared by cat_1, cat_2, cat_3.

           Here, object i is a new entity like an intension or a property.

So, in general, it can happen that a use of the string of char "Cat"
may denote a particular cat, a category of cats, or a catitudiosity.
</pre>