Difference between revisions of "Directory:Jon Awbrey/Notes/Factorization Issues"

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{{DISPLAYTITLE:Factorization Issues}}
 
{{DISPLAYTITLE:Factorization Issues}}
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{| align="center" cellspacing="6" width="90%" <!--QUOTE-->
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<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>
  
−
==Note 1==
+
<p>Aristotle, "The Categories", in ''Aristotle, Volume 1'', H.P. Cooke and H. Tredennick (trans.), Loeb Classics, William Heinemann Ltd, London, UK, 1938.</p>
 +
|}
  
−
<pre>
+
==Factoring Functions==
−
| 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" [Greek 'zõon' means 'living' or 'true to life'],
 
−
| 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).
 
−
|
 
−
| Aristotle, "The Categories", in 'Aristotle, Volume 1',
 
−
| Translated by H.P. Cooke & H. Tredennick, Loeb Classics,
 
−
| William Heinemann Ltd, London, UK, 1938.
 
  
−
I would like to introduce a concept that I find to be of
+
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.
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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,
+
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.
−
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)
+
Picture an arbitrary function from a ''source'' or ''domain'' to a ''target'' or ''codomain''.  Here is one picture of an <math>f : X \to Y,</math> just about as generic as it needs to be:
−
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}
+
{| align="center" cellpadding="10" style="text-align:center; width:90%"
−
|          |      o  o  o  o    o
+
|
−
|      f  |      \ | /    \  /
+
<pre>
−
|          |        \|/      \ /
+
o---------------------------------------o
−
|          v      o  o  o  o  o  o
+
|                                      |
−
|  Target Y  =  {A, B, C, D, E, F}
+
|  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}    |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
  
−
Now, it is a fact that any old function that you might
+
It is a fact that any function you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example taking the following shape:
−
pick "factors" into a surjective ("onto") function and
 
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an injective ("one-to-one") function, in the present
 
−
example just like so:
 
  
−
|  Source X  =  {1, 2, 3, 4,    5}
+
{| align="center" cellpadding="10" style="text-align:center; width:90%"
−
|          |      o  o  o  o    o
+
|
−
|      g  |      \ | /    \  /
+
<pre>
−
|          v        \|/      \ /
+
o---------------------------------------o
−
|  Middle M  =  {  b  ,    e  }
+
|                                      |
−
|          |        |        |
+
|  Source X  =  {1, 2, 3, 4,    5}     |
−
|      h  |        |        |
+
|          |      o  o  o  o    o     |
−
|          v      o  o  o  o  o  o
+
|      g  |      \ | /    \  /       |
−
|  Target Y  =  {A, B, C, D, E, F}
+
|          v        \|/      \ /       |
 +
|  Medium M  =  {  b  ,    e  }     |
 +
|          |        |        |        |
 +
|      h  |        |        |        |
 +
|          v      o  o  o  o  o  o     |
 +
|  Target Y  =  {A, B, C, D, E, F}     |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
 +
 
 +
Writing the functional compositions <math>f = g \circ h</math> on the right, we have the following data about the situation:
  
−
Writing the functional compositions f = g o h "on the right",
+
<pre>
−
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}
  
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X =  {1, 2, 3, 4, 5}
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  f : X -> Y, arbitrary.
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M =  {b, e}
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  g : X -> M, surjective.
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Y =  {A, B, C, D, E, F}
+
  h : M -> Y, injective.
  
−
f : X -> Y, arbitrary.
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  f = g o h
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g : X -> M, surjective.
+
</pre>
−
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 <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.
  
−
What does all of this have to do with reification and so on?
+
In general, if we try to use the signs in the target domain <math>Y\!</math> to reference the objects in the source domain <math>X,\!</math> then we will be invoking what used to be called &mdash; since the Middle Ages, I think &mdash; 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.
−
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
+
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.
−
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,
+
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.
−
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
+
Observe, however, that here is where all the battles tend 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 senses in general.  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.
−
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,
+
==Factoring Sign Relations==
−
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,
+
Let us now apply the concepts of factorization and reification, as they are developed above, to the analysis of sign relations.
−
where the sets O, S, I are the domains of the
 
−
Object, Sign, Interpretant domains, respectively.
 
  
−
Now suppose that the situation with respect to
+
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>
−
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.
 
  
 +
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, where equal signs written between ostensible nodes identify them into a single actual node.
 +
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L  |
 
| Denotative Component of L  |
Line 157: Line 113:
 
|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
  
−
This depicts a situation where each of the three objects,
+
The Figure depicts a situation where each of the three objects, <math>x_1, x_2, x_3,\!</math> has a ''proper name'' that denotes it alone, namely, the three proper names <math>y_1, y_2, y_3,\!</math> respectively. Over and above the objects denoted by their proper names, there is the general sign <math>y,\!</math> which denotes any and all of the objects <math>x_1, x_2, x_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''.
−
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:
+
Now, at this stage of the game, if you ask: ''Is the object of the sign <math>y\!</math> one or many?'', the answer has to be:  ''Not one, but many''. That is, there is not one <math>x\!</math> that <math>y\!</math> denotes, but only the three <math>x\!</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>y\!</math> to the abnominal fact that a unit <math>x\!</math> exists.
−
"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
+
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>y</math> to each of the objects <math>x_1, x_2, x_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:
  
−
|  Source O  :>  x_1 x_2 x_3
+
{| align="center" cellpadding="10" style="text-align:center; width:90%"
−
|          |      o  o  o
+
|
−
|          |        \  |  /
+
<pre>
−
|      f  |        \ | /
+
o---------------------------------------o
−
|          |          \|/
+
|                                      |
−
|          v      ... o ...
+
|  Source O  :>  x_1 x_2 x_3           |
−
|  Target S  :>      y  
+
|          |      o  o  o           |
 +
|          |        \  |  /             |
 +
|      f  |        \ | /             |
 +
|          |          \|/               |
 +
|          v      ... o ...           |
 +
|  Target S  :>      y               |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
  
−
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:
  
−
|  Source O  :>  x_1 x_2 x_3
+
{| align="center" cellpadding="10" style="text-align:center; width:90%"
−
|          |      o  o  o
+
|
−
|      g  |        \  |  /
+
<pre>
−
|          |        \ | /
+
o---------------------------------------o
−
|          v          \|/
+
|                                      |
−
|  Middle M  :>  ... x ...
+
|  Source O  :>  x_1 x_2 x_3           |
−
|          |          |  
+
|          |      o  o  o           |
−
|      h  |          |
+
|      g  |        \  |  /             |
−
|          |          |
+
|          |        \ | /             |
−
|          v      ... o ...
+
|          v          \|/               |
−
|  Target S  :>      y
+
|  Medium M  :>  ... x ...           |
 +
|          |          |                |
 +
|      h  |          |                |
 +
|          |          |                |
 +
|          v      ... o ...           |
 +
|  Target S  :>      y               |
 +
|                                      |
 +
o---------------------------------------o
 +
</pre>
 +
|}
  
−
The factorization of an arbitrary function
+
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.
−
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),
+
What it means is that &mdash; without loss or gain of generality &mdash; 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 <math>x\!</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 &mdash; obvious from the standpoint of the pragmatic theory of signs &mdash; the option of assigning the new concept, or mental symbol, to the role of an interpretant sign.
−
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,
+
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:
−
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:
 
  
 +
{| align="center" cellpadding="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L'  |
 
| Denotative Component of L'  |
Line 276: Line 202:
 
|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
  
−
This amounts to the creation of a hypostatic object x,
+
This amounts to the creation of a hypostatic object <math>x,\!</math> which affords us a singular denotation for the sign <math>y.\!</math>
−
which affords us a singular denotation for the sign y.
 
  
−
By way of terminology, it would be convenient to have
+
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.
−
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
+
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.
−
an "objective extension" (OE) or
 
−
an "outward extension" (OE) of
 
−
the underlying sign relation.
 
  
−
This naturally raises the question of
+
==Nominalism and Realism==
−
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==
+
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 <math>y\!</math> is quite literally taken to denote each object <math>x_j\!</math> in a whole collection of objects <math>\{ x_1, \ldots, x_k, \ldots \},</math> a situation that can be represented in a sign-relational table like this one:
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<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
 
o---------o---------o---------o
 
| Object  |  Sign  | Interp  |
 
| Object  |  Sign  | Interp  |
Line 322: Line 228:
 
|  ...  |    y    |  ...  |
 
|  ...  |    y    |  ...  |
 
o---------o---------o---------o
 
o---------o---------o---------o
 +
</pre>
 +
|}
  
−
For brevity, let us consider the sign relation L
+
For brevity, let us consider a sign relation <math>L\!</math> whose relational database table is precisely this:
−
whose relational database table is precisely this:
 
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
|      Sign Relation L      |
 
|      Sign Relation L      |
Line 335: Line 245:
 
|  x_3  |    y    |  ...  |
 
|  x_3  |    y    |  ...  |
 
o---------o---------o---------o
 
o---------o---------o---------o
 +
</pre>
 +
|}
  
 
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" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L  |
 
| Denotative Component of L  |
Line 358: Line 270:
 
|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
  
−
I would like to -- but my personal limitations in the
+
I would like to &mdash; 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:
−
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:
 
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L  |
 
| Denotative Component of L  |
Line 374: Line 287:
 
|                            |
 
|                            |
 
o-----------------------------o
 
o-----------------------------o
 +
</pre>
 +
|}
  
 
Notice the subtle distinction between these two cases:
 
Notice the subtle distinction between these two cases:
  
−
1.  A sign denotes each object in a set of objects.
+
# A sign denotes each object in a set of objects.
 +
# A sign denotes 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.
  
−
The first option uses the notion of a set in a casual,
+
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(x_1) =\!</math> <math>f(x_2) =\!</math> <math>f(x_3) = y,\!</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.
−
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,
+
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.
−
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 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>x_j\!</math> and the sign <math>y.\!</math>
−
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
+
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.
−
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.
 
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 +
<pre>
 
o-----------------------------o
 
o-----------------------------o
 
| Denotative Component of L'  |
 
| Denotative Component of L'  |
Line 424: Line 321:
 
|                            |
 
|                            |
 
o-----------------------------o
 
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>
 
</pre>
 +
|}
  
−
==Note 3==
+
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 read this <math>i\!</math> as standing for an intelligible concept, or an ''idea'' of the safely decapitalized, mental impression variety.
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<pre>
−
JA = Jon Awbrey
 
−
SR = Seth Russell
 
−
 
−
JA figured:
 
−
 
 
o-----------------------------o
 
o-----------------------------o
−
| Denotative Component of L'" |
+
| Denotative Component of L'' |
 
o--------------o--------------o
 
o--------------o--------------o
 
|  Objects    |    Signs    |
 
|  Objects    |    Signs    |
 
o--------------o--------------o
 
o--------------o--------------o
 
|                            |
 
|                            |
−
|    i                      |
 
−
|    /|\  *                  |
 
−
|  / | \      *            |
 
−
|  /  |  \          *        |
 
 
| o  o  o >>>>>>>>>>>> y    |
 
| o  o  o >>>>>>>>>>>> y    |
 
|    .  .  .            '    |
 
|    .  .  .            '    |
Line 514: Line 343:
 
|                            |
 
|                            |
 
o-----------------------------o
 
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>
 
</pre>
 +
|}
  
−
==Note 4==
+
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.
  
 +
{| align="center" cellspacing="10" style="text-align:center; width:90%"
 +
|
 
<pre>
 
<pre>
−
JA = Jon Awbrey
 
−
SR = Seth Russell
 
−
 
−
JA glyped:
 
−
 
 
o-----------------------------o
 
o-----------------------------o
−
| Denotative Component of L'" |
+
| Denotative Component of L'''|
 
o--------------o--------------o
 
o--------------o--------------o
 
|  Objects    |    Signs    |
 
|  Objects    |    Signs    |
Line 575: Line 369:
 
|                            |
 
|                            |
 
o-----------------------------o
 
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>
 
</pre>
 +
|}
  
−
==Note 5==
+
To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object <math>i,\!</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>x_j\!</math> that are plurally denoted by the sign <math>y.\!</math>  Further, it appears to be equally unexceptionable to allow the use of the sign <math>{}^{\backprime\backprime} i \, {}^{\prime\prime}</math> to denote this shared intension <math>i.\!</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.
  
−
<pre>
+
==Document History==
−
JA = Jon Awbrey
 
−
SR = Seth Russell
 
  
−
Seth,
+
===Nov 2000 &mdash; Factorization Issues===
  
−
Let me try to come up with a more concrete version
+
'''Standard Upper Ontology'''
−
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>
 
−
 
 
−
==Work Area==
 
−
 
 
−
<pre>
 
−
The word "intension" has recently come to be stressed in our discussions.
 
−
As I first learned this word from my reading of Leibniz, I shall take it
 
−
to be nothing more than a synonym for "property" or "quality", and shall
 
−
probably always associate it with the primes factorization of integers,
 
−
the analogy between having a factor and having a property being one of
 
−
the most striking, at least to my neo-pythagorean compleated mystical
 
−
sensitivities, that Leibniz ever posed, and of which certain facets
 
−
of Peirce's work can be taken as a further polishing up, if one is
 
−
of a mind to do so.
 
−
 
 
−
As I dare not presume this to constitute the common acceptation
 
−
of the term "intension", not without checking it out, at least,
 
−
I will need to try and understand how others here understand
 
−
the term and all of its various derivatives, thereby hoping
 
−
to anticipate, that is to say, to evade or to intercept,
 
−
a few of the brands of late-breaking misunderstandings
 
−
that are so easy to find ourselves being surprised by,
 
−
if one shies away from asking silly questions at the
 
−
very first introduction of one of these parvenu words.
 
−
I have been advised that it will probably be fruitless
 
−
to ask direct questions of my informants in such a regard,
 
−
but I do not see how else to catalyze the process of exposing
 
−
the presumption that "it's just understood" when in fact it may
 
−
be far from being so, and thus to clear the way for whatever real
 
−
clarification might possibly be forthcoming, in the goodness of time.
 
−
Just to be open, and patent, and completely above the metonymous board,
 
−
I will lay out the paradigm that I myself bear in mind when I think about
 
−
how I might place the locus and the sense of this term "intension", because
 
−
I see the matter of where to lodge it in our logical logistic as being quite
 
−
analogous to the issue of where to place those other i-words, namely, "idea",
 
−
capitalized or not, "impresssion", "intelligible concept", and "interpretant".
 
−
</pre>
 
−
 
 
−
==Document History==
 
−
 
 
−
===Factorization Issues (Standard Upper Ontology, Nov&ndash;Dec 2000)===
 
  
 
* http://suo.ieee.org/email/thrd224.html#02332
 
* http://suo.ieee.org/email/thrd224.html#02332
Line 1,042: Line 384:
 
# http://suo.ieee.org/email/msg02334.html
 
# http://suo.ieee.org/email/msg02334.html
 
# http://suo.ieee.org/email/msg02338.html
 
# http://suo.ieee.org/email/msg02338.html
 +
# http://suo.ieee.org/email/msg02340.html
 +
# http://suo.ieee.org/email/msg02345.html
 
# http://suo.ieee.org/email/msg02349.html
 
# http://suo.ieee.org/email/msg02349.html
 +
# http://suo.ieee.org/email/msg02355.html
 
# http://suo.ieee.org/email/msg02396.html
 
# http://suo.ieee.org/email/msg02396.html
 
# http://suo.ieee.org/email/msg02400.html
 
# http://suo.ieee.org/email/msg02400.html
Line 1,048: Line 393:
 
# http://suo.ieee.org/email/msg02448.html
 
# http://suo.ieee.org/email/msg02448.html
  
−
===Factorization Issues (Ontology, Nov&ndash;Dec 2000)===
+
'''Ontology List'''
  
 
* http://suo.ieee.org/ontology/thrd111.html#00007
 
* http://suo.ieee.org/ontology/thrd111.html#00007
Line 1,055: Line 400:
 
# http://suo.ieee.org/ontology/msg00032.html
 
# http://suo.ieee.org/ontology/msg00032.html
  
−
===Factorization Issues (Standard Upper Ontology, Nov 2001)===
+
===Mar 2001 &mdash; Factorization Flip-Flop===
 +
 
 +
'''Ontology List'''
 +
 
 +
* http://suo.ieee.org/ontology/thrd71.html#01926
 +
# http://suo.ieee.org/ontology/msg01926.html
 +
# http://suo.ieee.org/ontology/msg02008.html
 +
 
 +
'''Standard Upper Ontology'''
 +
 
 +
* http://suo.ieee.org/email/thrd184.html#04334
 +
# http://suo.ieee.org/email/msg04334.html
 +
# http://suo.ieee.org/email/msg04416.html
 +
 
 +
===Apr 2001 &mdash; Factorization Flip-Flop===
 +
 
 +
* http://stderr.org/pipermail/arisbe/2001-April/thread.html#408
 +
# http://stderr.org/pipermail/arisbe/2001-April/000408.html
 +
 
 +
===Sep 2001 &mdash; Descartes' Factorization===
 +
 
 +
'''Arisbe List'''
 +
 
 +
* http://stderr.org/pipermail/arisbe/2001-September/thread.html#1053
 +
# http://stderr.org/pipermail/arisbe/2001-September/001053.html
 +
 
 +
'''Ontology List'''
 +
 
 +
* http://suo.ieee.org/ontology/thrd44.html#03285
 +
# http://suo.ieee.org/ontology/msg03285.html
 +
 
 +
===Nov 2001 &mdash; Factorization Issues===
  
 
* http://suo.ieee.org/email/thrd128.html#07143
 
* http://suo.ieee.org/email/thrd128.html#07143
Line 1,064: Line 440:
 
# http://suo.ieee.org/email/msg07186.html
 
# http://suo.ieee.org/email/msg07186.html
  
−
===Factorization Issues (Inquiry, Mar 2005)===
+
===Mar 2005 &mdash; Factorization Issues===
  
 
* http://stderr.org/pipermail/inquiry/2005-March/thread.html#2495
 
* http://stderr.org/pipermail/inquiry/2005-March/thread.html#2495
Line 1,070: Line 446:
 
# http://stderr.org/pipermail/inquiry/2005-March/002496.html
 
# http://stderr.org/pipermail/inquiry/2005-March/002496.html
  
−
===Factorization And Reification (Inquiry, May 2005)===
+
===May 2005 &mdash; Factorization And Reification===
  
 
* http://stderr.org/pipermail/inquiry/2005-May/thread.html#2747
 
* http://stderr.org/pipermail/inquiry/2005-May/thread.html#2747
 +
# http://stderr.org/pipermail/inquiry/2005-May/002747.html
 +
# http://stderr.org/pipermail/inquiry/2005-May/002748.html
 +
# http://stderr.org/pipermail/inquiry/2005-May/002749.html
 +
# http://stderr.org/pipermail/inquiry/2005-May/002751.html
 +
 +
===May 2005 &mdash; Factorization And Reification : Discussion===
 +
 +
* http://stderr.org/pipermail/inquiry/2005-May/thread.html#2758
 +
# http://stderr.org/pipermail/inquiry/2005-May/002758.html

Latest revision as of 03:18, 25 June 2009

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).

Aristotle, "The Categories", in Aristotle, Volume 1, H.P. Cooke and H. Tredennick (trans.), Loeb Classics, William Heinemann Ltd, London, UK, 1938.

Factoring Functions

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 or domain to a target or codomain. Here is one picture of an \(f : X \to Y,\) just about as generic as it needs to be: 

o---------------------------------------o
|                                       |
|   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}     |
|                                       |
o---------------------------------------o

It is a fact that any function you might pick factors into a surjective ("onto") function and an injective ("one-to-one") function, in the present example taking the following shape: 

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

Writing the functional compositions \(f = g \circ h\) on the right, 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 \(f : X \to Y\) 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 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 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.

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 all the battles tend 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 senses in general. 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.

Factoring Sign Relations

Let us now apply the concepts of factorization and reification, as they are developed above, to the analysis of sign relations.

Suppose we have a sign relation \(L \subseteq O \times S \times I,\) where \(O\!\) is the object domain, \(S\!\) is the sign domain, and \(I\!\) is the interpretant domain of the sign relation \(L.\!\)

Now suppose that the situation with respect to the denotative component of \(L,\!\) in other words, the projection of \(L\!\) on the subspace \(O \times S,\) can be pictured in the following manner, where equal signs written between ostensible nodes identify them into a single actual 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

The Figure 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 example, 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 our example of a sign relation, we had a functional subset of the following shape: 

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

The function \(f : O \to S\) factors into a composition \(g \circ h,\!\) where \(g : O \to M,\) and \(h : M \to S,\) as shown here: 

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

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 — 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 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.

If we force the factored annotation function, initially extracted from the sign relation \(L,\!\) back into the frame from whence it came, we get the augmented sign relation \(L^\prime,\!\) shown in the next Figure: 

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 \(L^\prime.\) 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.

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 happens when a sign \(y\!\) is quite literally taken to denote each object \(x_j\!\) in a whole collection of objects \(\{ x_1, \ldots, x_k, \ldots \},\) a situation that can be represented in a sign-relational table like this one: 

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 a 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 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.

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 \(f : O \to S,\) 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 \({}^{\backprime\backprime} i \, {}^{\prime\prime}\) 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 \({}^{\backprime\backprime} i \, {}^{\prime\prime}\) 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 2-adic components of any relation.

Document History

Nov 2000 — Factorization Issues

Standard Upper Ontology

  1. http://suo.ieee.org/email/msg02332.html
  2. http://suo.ieee.org/email/msg02334.html
  3. http://suo.ieee.org/email/msg02338.html
  4. http://suo.ieee.org/email/msg02340.html
  5. http://suo.ieee.org/email/msg02345.html
  6. http://suo.ieee.org/email/msg02349.html
  7. http://suo.ieee.org/email/msg02355.html
  8. http://suo.ieee.org/email/msg02396.html
  9. http://suo.ieee.org/email/msg02400.html
  10. http://suo.ieee.org/email/msg02430.html
  11. http://suo.ieee.org/email/msg02448.html

Ontology List

  1. http://suo.ieee.org/ontology/msg00007.html
  2. http://suo.ieee.org/ontology/msg00025.html
  3. http://suo.ieee.org/ontology/msg00032.html

Mar 2001 — Factorization Flip-Flop

Ontology List

  1. http://suo.ieee.org/ontology/msg01926.html
  2. http://suo.ieee.org/ontology/msg02008.html

Standard Upper Ontology

  1. http://suo.ieee.org/email/msg04334.html
  2. http://suo.ieee.org/email/msg04416.html

Apr 2001 — Factorization Flip-Flop

  1. http://stderr.org/pipermail/arisbe/2001-April/000408.html

Sep 2001 — Descartes' Factorization

Arisbe List

  1. http://stderr.org/pipermail/arisbe/2001-September/001053.html

Ontology List

  1. http://suo.ieee.org/ontology/msg03285.html

Nov 2001 — Factorization Issues

  1. http://suo.ieee.org/email/msg07143.html
  2. http://suo.ieee.org/email/msg07166.html
  3. http://suo.ieee.org/email/msg07182.html
  4. http://suo.ieee.org/email/msg07185.html
  5. http://suo.ieee.org/email/msg07186.html

Mar 2005 — Factorization Issues

  1. http://stderr.org/pipermail/inquiry/2005-March/002495.html
  2. http://stderr.org/pipermail/inquiry/2005-March/002496.html

May 2005 — Factorization And Reification

  1. http://stderr.org/pipermail/inquiry/2005-May/002747.html
  2. http://stderr.org/pipermail/inquiry/2005-May/002748.html
  3. http://stderr.org/pipermail/inquiry/2005-May/002749.html
  4. http://stderr.org/pipermail/inquiry/2005-May/002751.html

May 2005 — Factorization And Reification : Discussion

  1. http://stderr.org/pipermail/inquiry/2005-May/002758.html
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