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	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.		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.
−
−	~~==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==		==Document History==

@@ Line 434: / Line 434: @@
 ===Nov 2000 &mdash; Factorization Issues===
 '''Standard Upper Ontology'''
@@ Line 456: / Line 449: @@
 # http://suo.ieee.org/email/msg02430.html
 # http://suo.ieee.org/email/msg02448.html
 ===Mar 2001 &mdash; Factorization Flip-Flop===

@@ Line 42: / Line 42: @@
 |      g   |       \ | /    \   /       |
 |          v        \|/      \ /        |
-|   Middle M  =  {   b   ,    e   }     |
+|   Medium M  =  {   b   ,    e   }     |
 |          |         |        |         |
 |      h   |         |        |         |

@@ Line 170: / Line 170: @@
 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.
-<pre>
+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.
-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:
 {| align="center" cellpadding="10" style="text-align:center; width:90%"
@@ Line 224: / Line 205: @@
 |}
-<pre>
+This amounts to the creation of a hypostatic object <math>x,\!</math> which affords us a singular denotation for the sign <math>y.\!</math>
-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.
-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.
 ==Nominalism and Realism==

@@ Line 120: / Line 120: @@
 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.
-<pre>
+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.
-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''.
-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.
-In this case, we factor the function f : O -> S
+In our example of a sign relation, we had a functional subset of the following shape:
 {| align="center" cellpadding="10" style="text-align:center; width:90%"
@@ Line 160: / Line 145: @@
 |}
-into the composition g o h, where g : O -> M, and h : M -> S
+The function <math>f : O \to S</math> factors into a composition <math>g \circ h,\!</math> where <math>g : O \to M,</math> and <math>h : M \to S,</math> as shown here:
 {| align="center" cellpadding="10" style="text-align:center; width:90%"
@@ Line 172: / Line 157: @@
 |          |         \ | /              |
 |          v          \|/               |
-|   Middle M  :>   ... x ...            |
+|   Medium M  :>   ... x ...            |
 |          |           |                |
 |       h  |           |                |
@@ Line 182: / Line 167: @@
 </pre>
 |}
 <pre>
 What it means is that, "without loss or gain of generality" (WOLOGOG),
 we might as well assume that there is a domain of intermediate entities

@@ Line 77: / Line 77: @@
 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.
-<pre>
+==Factoring Sign Relations==
-Suppose that we have a sign relation L c OxSxI,
-where the sets O, S, I are the domains of the
+Let us now apply the concepts of factorization and reification, as they are developed above, to the analysis of sign relations.
-Object, Sign, Interpretant domains, respectively.
-Now suppose that the situation with respect to
+Now suppose that the situation with respect to the ''denotative component'' of <math>L,\!</math> in other words, the projection of <math>L\!</math> on the subspace <math>O \times S,</math> can be pictured in the following manner, where equal signs written between ostensible nodes identify them into a single actual node.
-{| align="center" cellpadding="10" style="text-align:center; width:90%"
+{| align="center" cellspacing="10" style="text-align:center; width:90%"
+|
 <pre>
@@ Line 121: / Line 116: @@
 |}
-<pre>
+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''.
-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.
 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.
@@ Line 292: / Line 269: @@
 </pre>
-==Note 2==
+==Nominalism and Realism==
 <pre>

@@ Line 7: / Line 7: @@
 |}
-==Note 1==
+==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.
@@ Line 31: / Line 31: @@
 |}
-Now, it is a fact that any old function that you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example just like so:
+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:
 {| align="center" cellpadding="10" style="text-align:center; width:90%"
@@ Line 52: / Line 52: @@
 |}
-Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:
+Writing the functional compositions <math>f = g \circ h</math> on the right, we have the following data about the situation:
 <pre>
@@ Line 69: / Line 69: @@
 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.
-<pre>
+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.
-Notice what I did not say here, that one sign denotes a "set" of objects,
+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
+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,
+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.
 Suppose that we have a sign relation L c OxSxI,
 where the sets O, S, I are the domains of the

@@ Line 15: / Line 15: @@
 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:
 <pre>
 o---------------------------------------o
@@ Line 27: / Line 29: @@
 o---------------------------------------o
 </pre>
 Now, it is a fact that any old function that you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example just like so:
 <pre>
 o---------------------------------------o
@@ Line 45: / Line 50: @@
 o---------------------------------------o
 </pre>
 Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:
@@ Line 113: / Line 119: @@
 signs, like "=", written between ostensible nodes,
 like "o", identify them into a single real node.
 o-----------------------------o
 | Denotative Component of L   |
@@ Line 139: / Line 149: @@
 |                             |
 o-----------------------------o
 This depicts a situation where each of the three objects,
 x_1, x_2, x_3, has a "proper name" that denotes it alone,
@@ Line 182: / Line 195: @@
 In this case, we factor the function f : O -> S
 o---------------------------------------o
 |                                       |
@@ Line 194: / Line 211: @@
 |                                       |
 o---------------------------------------o
 into the composition g o h, where g : O -> M, and h : M -> S
 o---------------------------------------o
 |                                       |
@@ Line 212: / Line 234: @@
 |                                       |
 o---------------------------------------o
 The factorization of an arbitrary function
 into a surjective ("onto") function followed
@@ Line 240: / Line 265: @@
 we get the augmented sign relation L', shown
 in the next vignette:
 o-----------------------------o
 | Denotative Component of L'  |
@@ Line 266: / Line 295: @@
 |                             |
 o-----------------------------o
 This amounts to the creation of a hypostatic object x,
 which affords us a singular denotation for the sign y.

@@ Line 9: / Line 9: @@
 ==Note 1==
-<pre>
+I would like to introduce a concept that I find to be of use in discussing the problems of hypostatic abstraction, reification, the reality of universals, and the questions of choosing among nominalism, conceptualism, and realism, generally.
-I will take this up first in the simplest possible setting,
+I will take this up first in the simplest possible setting, where it has to do with the special sorts of relations that are commonly called ''functions'', and after the basic idea is made as clear as possible in this easiest case I will deal with the notion of ''factorization'' as it affects more generic types of relations.
-Picture an arbitrary function from a Source (Domain)
+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:
 o---------------------------------------o
 |                                       |
@@ Line 37: / Line 26: @@
 |                                       |
 o---------------------------------------o
-Now, it is a fact that any old function that you might
+Now, it is a fact that any old function that you might pick ''factors'' into a surjective ("onto") function and an injective ("one-to-one") function, in the present example just like so:
 o---------------------------------------o
 |                                       |
@@ Line 56: / Line 44: @@
 |                                       |
 o---------------------------------------o
-Writing the functional compositions f = g o h "on the right",
+Writing the functional compositions <math>f = g \circ h</math> "on the right", as they say, we have the following data about the situation:
     X  =  {1, 2, 3, 4, 5}
     M  =  {b, e}
@@ Line 69: / Line 58: @@
     f = g o h
-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
-Well, suppose that the Source domain X is a set of "objects",
+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 (Co-domain) Y
 to reference the objects in the Source (Domain) X, then we will be