Changes

|}

|}

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.

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 present 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"

{| align="center" cellpadding="8" style="text-align:center"

|}

|}

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="10" style="text-align:center; width:90%"

{| align="center" cellpadding="8" style="text-align:center"

|

| [[Image:Factorization Function Example 2.jpg|500px]]

|-

o---------------------------------------o

| <math>\text{Figure 2.  Factorization}~ f = g \circ h</math>

|                                       |

|  Source X  =  {1, 2, 3, 4,    5}     |

|      g   |      \ | /    \  /      |

|          v        \|/      \ /        |

|      h   |        |        |        |

</pre>

|}

|}

X & = & \{ 1, 2, 3, 4, 5 \}

X & = & \{ 1, 2, 3, 4, 5 \}

\\[4pt]

\\[4pt]

M & = & \{ Q, T \}

M & = & \{ m, n \}

\\[4pt]

\\[4pt]

Y & = & \{ p, q, r, s, t, u \}

Y & = & \{ p, q, r, s, t, u \}

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?

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.

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 &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''.

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.

{| align="center" cellpadding="8" style="text-align:center"

{| align="center" cellpadding="8" style="text-align:center"

| [[Image:Factorization Sign Relation L.jpg|500px]]

| [[Image:Factorization Sign Relation 1.jpg|500px]]

|-

|-

| <math>\text{Figure 3.  Denotative Component of Sign Relation}~ L</math>

| <math>\text{Figure 3.  Denotative Component of Sign Relation}~ L</math>

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''.

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 <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.

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.

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

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

{| align="center" cellpadding="10" style="text-align:center; width:90%"

{| 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>

|  Source O  :>  o_1 o_2 o_3          |

|         |      o  o  o            |

|         |        \ |  /            |

|          v      ... o ...           |

|  Target S :>      s                |

</pre>

|}

|}

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:

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="10" style="text-align:center; width:90%"

{| 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>

|  Source O  :>  o_1 o_2 o_3          |

|         |      o  o  o            |

|       g  |        \ |  /            |

|  Medium M  :>  ... o ...            |

|      h |          |                |

</pre>

|}

|}

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.

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 &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.

 

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>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 &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.

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:

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:

{| align="center" cellpadding="8" style="text-align:center"

{| align="center" cellpadding="8" style="text-align:center"

| [[Image:Factorization Sign Relation L'.jpg|500px]]

| [[Image:Factorization Sign Relation 2.jpg|500px]]

|-

|-

| <math>\text{Figure 6.  Denotative Component of Sign Relation}~ L^\prime</math>

| <math>\text{Figure 6.  Denotative Component of Sign Relation}~ L^\prime</math>

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.

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.

{| align="center" cellspacing="10" style="text-align:center; width:90%"

{| 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>

| Denotative Component of L  |

|   Objects    |    Signs    |

o--------------o--------------o

|                             |

|  o_1  o------>             |

|              \             |

 

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:

 

| Denotative Component of L   |

</pre>

|}

|}

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.

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" cellspacing="10" style="text-align:center; width:90%"

{| align="center" cellpadding="8" style="text-align:center"

|

| [[Image:Factorization Sign Relation 3.jpg|500px]]

|-

o-----------------------------o

| <math>\text{Figure 10.  Denotative Component of Sign Relation}~ L^\prime</math>

| Denotative Component of L'  |

|  /  |  \           *        |

</pre>

|}

|}

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.

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" cellspacing="10" style="text-align:center; width:90%"

{| 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>

| Denotative Component of L'' |

|   Objects    |    Signs    |

o--------------o--------------o

|                             |

| o  o  o >>>>>>>>>>>> s    |

|    .  .  .            '    |

</pre>

|}

|}

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.

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.

{| align="center" cellspacing="10" style="text-align:center; width:90%"

{| align="center" cellpadding="8" style="text-align:center"

|

| [[Image:Factorization Sign Relation 5.jpg|500px]]

|-

o-----------------------------o

| <math>\text{Figure 12.  Denotative Component of Sign Relation}~ L^{\prime\prime\prime}</math>

| Denotative Component of L'''|

|    /|\   *                  |

|  / | \       *            |

|  /  |  \           *        |

</pre>

|}

|}

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>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} 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.

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.

==Document History==

==Document History==