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The question is: What sort of thing is a connotation? Is it a sign? That is to say, is it yet another term? Or is it something like an abstract attribute, namely, a character, an intension, a property, or a quality? And while we're asking, does it really even matter?
The question is: What sort of thing is a connotation? Is it a sign? — that is to say, yet another term? Or is it something like an abstract attribute — a character, intension, property, or quality? And while we're asking, does it really even matter?
"No" is one answer worth considering. But then: Why does it not matter? What reason might be given that would excuse the indifference?
"No" is one answer worth considering. But then: Why does it not matter? What reason could be given to excuse the indifference?
This is a question that has exercised me since my earliest studies of Peirce. I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other, but, alas, I haven't the foggiest notion what exactly my revelation amounted to. Perhaps that is all for the best, as the vagrancy of memory is frequently better than the vapidity of one's banalytic anamnesia.
This is a question that has exercised me since my earliest studies of Peirce. I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other — alas, I haven't the foggiest notion what exactly my revelation amounted to.
These days, I usually try to finesse the trick under the trumped up rubric of a factorization. So let me excavate my last attempts to explain this business and see if I can improve on them.
These days, I usually try to finesse the trick under the trumped up rubric of a factorization. So let me excavate my last attempts to explain this business and see if I can improve on them.
==Factoring Functions==
==Factoring Functions==
{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
|
<p>Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different. For instance, while a man and a portrait can properly both be called "animals" [since the Greek ''zõon'' applies to both], these are equivocally named. For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different. For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone. (''Categories'', p. 13).</p>
<p>Aristotle, "The Categories", in ''Aristotle, Volume 1'', H.P. Cooke and H. Tredennick (trans.), Loeb Classics, William Heinemann Ltd, London, UK, 1938.</p>
|}
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 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.
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'' or ''domain'' to a ''target'' or ''codomain''. Here is a picture of such function, <math>f : X \to Y,</math> as generic as it needs to be for our prsent purposes:
Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''. Here is a picture of such a function, <math>f : X \to Y,</math> as generic as it needs to be for our present purposes:
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Function Example 1.jpg|500px]]
|-
| <math>\text{Figure 1. Function}~ f : X \to Y</math>
|}
</pre>
It is a fact that any old function that you might pick "factors" into a functional composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example pictured below:
It is a fact that any old function that you might pick factors into a composition of two other functions, a surjective ("onto") function and an injective ("one-to-one") function, in the present example as pictured below:
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Function Example 2.jpg|500px]]
|-
| <math>\text{Figure 2. Factorization}~ f = g \circ h</math>
|}
</pre>
Writing functional compositions <math>f = g \circ h</math> "on the right", we have the following data about the situation:
Writing functional compositions <math>f = g \circ h</math> "on the right", we have the following data about the situation:
<pre>
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{ccl}
X & = & \{ 1, 2, 3, 4, 5 \}
\\[4pt]
M & = & \{ m, n \}
\\[4pt]
Y & = & \{ p, q, r, s, t, u \}
\\[8pt]
f & : & X \to Y, ~\text{arbitrary}
</pre>
\\[4pt]
g & : & X \to M, ~\text{surjective}
\\[4pt]
h & : & M \to Y, ~\text{injective}
\\[8pt]
f & = & g \circ h
\end{array}</math>
|}
What does all of this have to do with reification and so on?
What does all of this have to do with reification and so on?
To begin answering that question, suppose that the source domain <math>X\!</math> is a set of ''objects'', that the target domain <math>Y\!</math> is a set of ''signs'', and suppose that the function <math>f : X \to Y</math> indicates the effect of a classification, conceptualization, discrimination, perception, or some other type of sorting operation, distributing the elements of the set <math>X\!</math> of objects and into a set of sorting bins that are labeled with the elements of the set <math>Y,\!</math> 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 codomain <math>Y\!</math> to reference the objects in the source domain <math>X,\!</math> 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'' or ''equivocal''.
In general, if we use the signs in the target domain <math>Y\!</math> to denote or describe the objects in the source domain <math>X,\!</math> then we are engaged in a form of ''general denotation'' or ''plural reference'' with regard to those objects, that is, each sign may refer to each of many objects, in a way that would normally be called ''ambiguous'' or ''equivocal''.
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.
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.
==Factoring Sign Relations==
==Factoring Sign Relations==
Let us now apply the concepts of factorization and reification, as they are developed above, to the analysis of sign relations.
Let us now apply the concepts of factorization and reification,
as they are developed above, to the analysis of sign relations.
Suppose that we have a sign relation L c O x S x I, where the sets
Suppose we have a sign relation <math>L \subseteq O \times S \times I,</math> where <math>O\!</math> is the object domain, <math>S\!</math> is the sign domain, and <math>I\!</math> is the interpretant domain of the sign relation <math>L.\!</math>
Now suppose that the situation with respect to
Now suppose that the situation with respect to the ''denotative component'' of <math>L,\!</math> in other words, the projection of <math>L\!</math> on the subspace <math>O \times S,</math> can be pictured in the following manner:
the "denotative component" of L, in other words,
the "projection" of L on the subspace O x S, can
be pictured in the following manner, where equal
{| align="center" cellpadding="8" style="text-align:center"
| Denotative Component of L |
| [[Image:Factorization Sign Relation 1.jpg|500px]]
|-
| <math>\text{Figure 3. Denotative Component of Sign Relation}~ L</math>
|}
| / / / |
| / / / |
| x_2 o-/--------o y_2 |
| |
The Figure depicts a situation where each of the three objects, <math>o_1, o_2, o_3,\!</math> has a ''proper name'' that denotes it alone, namely, the three proper names <math>s_1, s_2, s_3,\!</math> respectively. Over and above the objects denoted by their proper names, there is the general sign <math>s,\!</math> which denotes any and all of the objects <math>o_1, o_2, o_3.\!</math> 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''.
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".
If you now ask, ''Is the object of the sign <math>s\!</math> one or many?'', the answer has to be ''many''. That is, there is not one <math>o\!</math> that <math>s\!</math> denotes, but only the three <math>o\!</math>'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 <math>s\!</math> to the abnominal fact that a unit <math>o\!</math> exists.
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
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 <math>O\!</math> to <math>S\!</math> or from <math>S\!</math> to <math>O,\!</math> 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.
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
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''.
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
For example, the assignment of the general term <math>s\!</math> to each of the objects <math>o_1, o_2, o_3\!</math> 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.
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
In our example of a sign relation, we had a functional subset of the following shape:
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation Piece 1.jpg|500px]]
|-
| <math>\text{Figure 4. One Part of Sign Relation}~ L</math>
|}
into the composition g o h, where g : O -> M, and h : M -> S
The function <math>f : O \to S</math> factors into a composition <math>g \circ h,\!</math> where <math>g : O \to M,</math> and <math>h : M \to S,</math> as shown here:
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation Piece 2.jpg|500px]]
|-
| <math>\text{Figure 5. Factored Part of Sign Relation}~ L</math>
|}
It may be difficult at first to see how anything of significance could follow from an observation so facile as the fact that an arbitrary function factors into a surjective function followed by an injective function. What it means is that there is no loss of generality in assuming that there is a domain of intermediate entities under which the objects of a general denotation or plural reference can be marshaled, just as if they had something more essential and more substantial in common than the shared attachment to a coincidental name. So the problematic status of a hypostatic entity like <math>o\!</math> is reduced from a question of its nominal existence to a matter of its local habitation. Is it more like an object or more like a sign? 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.
into a surjective ("onto") function followed
by an injective ("one-one") function is such
a deceptively trivial observation that I had
If we force the factored annotation function, initially extracted from the sign relation <math>L,\!</math> back into the frame from whence it came, we get the augmented sign relation <math>L^\prime,\!</math> shown in the next Figure:
we might as well assume that there is a domain of intermediate entities
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation 2.jpg|500px]]
|-
| <math>\text{Figure 6. Denotative Component of Sign Relation}~ L^\prime</math>
|}
This amounts to the creation of a hypostatic object <math>o,\!</math> which affords us a singular denotation for the sign <math>s.\!</math>
By way of terminology, it would be convenient to have a general name for the transformation that converts a bare, ''nominal'' sign relation like <math>L\!</math> into a new, improved ''hypostatically augmented or extended'' sign relation like <math>L^\prime.</math> Let us call this kind of transformation an ''objective extension'' or an ''outward extension'' 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'' or an ''inward extension'' of the underlying sign relation, and this is the topic that I will take up next.
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.
==Nominalism and Realism==
==Nominalism and Realism==
Let me now 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 occurs when a sign <math>s\!</math> is taken to denote each object <math>o_j\!</math> in a collection of objects <math>\{ o_1, \ldots, o_k, \ldots \},</math> a situation whose general pattern is suggested by a sign-relational table of the following form:
Let me now 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
the following device to symbolize the same configuration:
<br>
{| align="center" style="text-align:center; width:60%"
|
{| align="center" style="text-align:center; width:100%"
| <math>\text{Table 7. Plural Denotation}\!</math>
|}
|-
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%"
|- style="background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
</pre>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
|
<math>\begin{matrix}
o_1 \\ o_2 \\ o_3 \\ \ldots \\ o_k \\ \ldots
\end{matrix}</math>
|
<math>\begin{matrix}
s \\ s \\ s \\ \ldots \\ s \\ \ldots
\end{matrix}</math>
|
<math>\begin{matrix}
\ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots
\end{matrix}</math>
|}
|}
<br>
<pre>
For example, consider the sign relation <math>L\!</math> whose sign relational triples are precisely as shown in Table 8.
<br>
{| align="center" style="text-align:center; width:60%"
|
{| align="center" style="text-align:center; width:100%"
| <math>\text{Table 8. Sign Relation}~ L</math>
|}
|-
|
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:100%"
|- style="background:#f0f0ff"
| width="33%" | <math>\text{Object}\!</math>
| width="33%" | <math>\text{Sign}\!</math>
| width="33%" | <math>\text{Interpretant}\!</math>
|-
|
<math>\begin{matrix}
o_1 \\ o_2 \\ o_3
\end{matrix}</math>
|
<math>\begin{matrix}
s \\ s \\ s
\end{matrix}</math>
|
<math>\begin{matrix}
\ldots \\ \ldots \\ \ldots
\end{matrix}</math>
|}
|}
<br>
For the moment, it does not matter what the interpretants are.
For the moment, it does not matter what the interpretants are.
I would like to diagram this somewhat after the following fashion,
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 Sign columns of the table.
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.
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation Piece 1.jpg|500px]]
|-
| <math>\text{Figure 9. Denotative Component of Sign Relation}~ L</math>
|}
Notice the subtle distinction between these two cases:
Notice the subtle distinction between these two cases:
# A sign denotes each object in a set of objects.
# 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.
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 <math>f : O \to S,</math> such that <math>f(o_1) =\!</math> <math>f(o_2) =\!</math> <math>f(o_3) = s,\!</math> 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 <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to the intermediate entity between the objects <math>o_j\!</math> and the sign <math>s.\!</math>
Now, should you annex <math>i\!</math> to the object domain <math>O\!</math> 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.
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation 3.jpg|500px]]
|-
| <math>\text{Figure 10. Denotative Component of Sign Relation}~ L^\prime</math>
|}
But if you assimilate <math>i\!</math> to the realm of signs <math>S,\!</math> you will be showing your inclination to remain within the straight and narrow of ''conceptualist'' or even ''nominalist'' dogmas, and you may regard the intermediate entity <math>i\!</math> as an intelligible concept, or an ''idea'' of the safely decapitalized, mental impression variety.
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation 4.jpg|500px]]
|-
| <math>\text{Figure 11. Denotative Component of Sign Relation}~ L^{\prime\prime}</math>
|}
But if you dare to be truly liberal, you might just find that you can easily afford to accommodate both intellectual inclinations, and after a while you begin to wonder how all that mental or ontological downsizing got started in the first place.
==Discussion==
{| align="center" cellpadding="8" style="text-align:center"
| [[Image:Factorization Sign Relation 5.jpg|500px]]
|-
| <math>\text{Figure 12. Denotative Component of Sign Relation}~ L^{\prime\prime\prime}</math>
|}
<pre>
To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i_o,\!</math> that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects <math>o_j\!</math> that are plurally denoted by the sign <math>s.\!</math> Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} s \, {}^{\prime\prime}</math> to denote this shared intension <math>i_o.\!</math> Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the 2-adic components of any relation.
the fact that any function can be factored into a surjective part followed
==Document History==
==Document History==
===Nov 2000 — Factorization Issues===
===Nov 2000 — Factorization Issues===
'''Standard Upper Ontology'''
'''Standard Upper Ontology'''
# http://suo.ieee.org/email/msg02430.html
# http://suo.ieee.org/email/msg02430.html
# http://suo.ieee.org/email/msg02448.html
# http://suo.ieee.org/email/msg02448.html
'''Ontology List'''
* http://suo.ieee.org/ontology/thrd111.html#00007
# http://suo.ieee.org/ontology/msg00007.html
# http://suo.ieee.org/ontology/msg00025.html
# http://suo.ieee.org/ontology/msg00032.html
===Mar 2001 — Factorization Flip-Flop===
===Mar 2001 — Factorization Flip-Flop===