Difference between revisions of "Directory:Jon Awbrey/Notes/Factorization Issues"
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{{DISPLAYTITLE:Factorization Issues}} | {{DISPLAYTITLE:Factorization Issues}} | ||
− | + | <pre> | |
+ | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | ||
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+ | IDS -- FI | ||
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+ | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | ||
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+ | FI. Factorization Issues | ||
+ | |||
+ | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | ||
+ | |||
+ | FI. Note 1 | ||
− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
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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 | ||
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f : X -> Y, just about as generic as it needs to be: | f : X -> Y, just about as generic as it needs to be: | ||
− | | Source X = {1, 2, 3, 4, 5} | + | o---------------------------------------o |
− | | | o o o o o | + | | | |
− | | f | \ | / \ / | + | | Source X = {1, 2, 3, 4, 5} | |
− | | | \|/ \ / | + | | | o o o o o | |
− | | v o o o o o o | + | | f | \ | / \ / | |
− | | Target Y = {A, B, C, D, E, F} | + | | | \|/ \ / | |
+ | | v o o o o o o | | ||
+ | | Target Y = {A, B, C, D, E, F} | | ||
+ | | | | ||
+ | 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 | ||
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example just like so: | example just like so: | ||
− | | Source X = {1, 2, 3, 4, 5} | + | o---------------------------------------o |
− | | | o o o o o | + | | | |
− | | g | \ | / \ / | + | | Source X = {1, 2, 3, 4, 5} | |
− | | v \|/ \ / | + | | | o o o o o | |
− | | Middle M = { b , e } | + | | g | \ | / \ / | |
− | | | | | | + | | v \|/ \ / | |
− | | h | | | | + | | Middle M = { b , e } | |
− | | v o o o o o o | + | | | | | | |
− | | Target Y = {A, B, C, D, E, F} | + | | h | | | | |
+ | | v o o o o o o | | ||
+ | | Target Y = {A, B, C, D, E, F} | | ||
+ | | | | ||
+ | o---------------------------------------o | ||
Writing the functional compositions f = g o h "on the right", | Writing the functional compositions f = g o h "on the right", | ||
as they say, we have the following data about the situation: | as they say, we have the following data about the situation: | ||
− | X = {1, 2, 3, 4, 5} | + | X = {1, 2, 3, 4, 5} |
− | M = {b, e} | + | M = {b, e} |
− | Y = {A, B, C, D, E, F} | + | Y = {A, B, C, D, E, F} |
− | f : X -> Y, arbitrary. | + | f : X -> Y, arbitrary. |
− | g : X -> M, surjective. | + | g : X -> M, surjective. |
− | h : M -> Y, injective. | + | h : M -> Y, injective. |
− | f = g o h | + | 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? | ||
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and injective factors. | and injective factors. | ||
− | Observe, however, that here is where the battles | + | Observe, however, that here is where all the battles tend to break out, |
for not all factorizations are regarded with equal equanimity by folks | for not all factorizations are regarded with equal equanimity by folks | ||
who have divergent philosophical attitudes toward the creation of new | who have divergent philosophical attitudes toward the creation of new | ||
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In this case, we factor the function f : O -> S | In this case, we factor the function f : O -> S | ||
− | | Source O :> x_1 x_2 x_3 | + | o---------------------------------------o |
− | | | o o o | + | | | |
− | | | \ | / | + | | Source O :> x_1 x_2 x_3 | |
− | | f | \ | / | + | | | o o o | |
− | | | \|/ | + | | | \ | / | |
− | | v ... o ... | + | | f | \ | / | |
− | | Target S :> y | + | | | \|/ | |
+ | | v ... o ... | | ||
+ | | Target S :> y | | ||
+ | | | | ||
+ | o---------------------------------------o | ||
into the composition g o h, where g : O -> M, and h : M -> S | into the composition g o h, where g : O -> M, and h : M -> S | ||
− | | Source O :> x_1 x_2 x_3 | + | o---------------------------------------o |
− | | | o o o | + | | | |
− | | g | \ | / | + | | Source O :> x_1 x_2 x_3 | |
− | | | \ | / | + | | | o o o | |
− | | v \|/ | + | | g | \ | / | |
− | | Middle M :> ... x ... | + | | | \ | / | |
− | | | | | + | | v \|/ | |
− | | h | | | + | | Middle M :> ... x ... | |
− | | | | | + | | | | | |
− | | v ... o ... | + | | h | | | |
− | | Target S :> y | + | | | | | |
+ | | v ... o ... | | ||
+ | | Target S :> y | | ||
+ | | | | ||
+ | o---------------------------------------o | ||
The factorization of an arbitrary function | The factorization of an arbitrary function | ||
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and this is the topic that | and this is the topic that | ||
I will take up next. | I will take up next. | ||
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− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
+ | |||
+ | FI. Note 2 | ||
+ | |||
+ | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | ||
− | + | Let me illustrate what I think that a plethora of our controversies | |
− | Let me illustrate what I think that a | ||
about nominalism versus realism actually boil down to in practice. | about nominalism versus realism actually boil down to in practice. | ||
From a semiotic or a sign-theoretic point of view, it all begins | From a semiotic or a sign-theoretic point of view, it all begins | ||
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is quite literally taken to denote each object x_j in a whole | is quite literally taken to denote each object x_j in a whole | ||
collection of objects {x_1, ..., x_k, ...}, a situation that | collection of objects {x_1, ..., x_k, ...}, a situation that | ||
− | I | + | I'd normally represent in a sign-relational table like so: |
o---------o---------o---------o | o---------o---------o---------o | ||
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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. | + | 1. A sign denotes each object in a set of objects. |
− | 2. 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, | The first option uses the notion of a set in a casual, | ||
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of the dyadic components of any relation. | of the dyadic components of any relation. | ||
− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
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− | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | ||
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− | + | FI. Factorization Issues -- Discussion | |
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− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
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− | + | FI. Discussion Note 1 | |
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− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
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− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
− | + | FI. Work Area | |
− | o | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o |
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The word "intension" has recently come to be stressed in our discussions. | 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 | As I first learned this word from my reading of Leibniz, I shall take it | ||
Line 1,032: | Line 549: | ||
analogous to the issue of where to place those other i-words, namely, "idea", | analogous to the issue of where to place those other i-words, namely, "idea", | ||
capitalized or not, "impresssion", "intelligible concept", and "interpretant". | capitalized or not, "impresssion", "intelligible concept", and "interpretant". | ||
− | |||
− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
− | + | FI. Factorization Issues -- 2000-2001 | |
− | + | SUO List | |
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− | |||
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− | |||
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− | |||
− | + | 01. http://suo.ieee.org/email/msg02332.html | |
+ | 02. http://suo.ieee.org/email/msg02334.html | ||
+ | 03. http://suo.ieee.org/email/msg02338.html | ||
+ | 04. http://suo.ieee.org/email/msg02340.html | ||
+ | 05. http://suo.ieee.org/email/msg02345.html | ||
+ | 06. http://suo.ieee.org/email/msg02349.html | ||
+ | 07. http://suo.ieee.org/email/msg02355.html | ||
+ | 08. http://suo.ieee.org/email/msg02396.html | ||
+ | 09. http://suo.ieee.org/email/msg02400.html | ||
+ | 10. http://suo.ieee.org/email/msg02430.html | ||
+ | 11. http://suo.ieee.org/email/msg02448.html | ||
+ | 12. http://suo.ieee.org/email/msg04334.html | ||
+ | 13. http://suo.ieee.org/email/msg04416.html | ||
+ | 14. http://suo.ieee.org/email/msg07143.html | ||
+ | 15. http://suo.ieee.org/email/msg07166.html | ||
+ | 16. http://suo.ieee.org/email/msg07182.html | ||
+ | 17. http://suo.ieee.org/email/msg07185.html | ||
+ | 18. http://suo.ieee.org/email/msg07186.html | ||
− | + | Ontology List | |
− | |||
− | |||
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− | + | 01. http://suo.ieee.org/ontology/msg00007.html | |
+ | 02. http://suo.ieee.org/ontology/msg00025.html | ||
+ | 03. http://suo.ieee.org/ontology/msg00032.html | ||
+ | 04. http://suo.ieee.org/ontology/msg01926.html | ||
+ | 05. http://suo.ieee.org/ontology/msg02008.html | ||
+ | 06. http://suo.ieee.org/ontology/msg03285.html | ||
− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
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− | + | FI. Factorization Issues | |
− | + | 00. http://stderr.org/pipermail/inquiry/2005-March/thread.html#2495 | |
− | + | 01. http://stderr.org/pipermail/inquiry/2005-March/002495.html | |
− | + | 02. http://stderr.org/pipermail/inquiry/2005-March/002496.html | |
+ | 03. | ||
− | + | o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o | |
− | + | </pre> | |
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− |
Revision as of 20:18, 24 June 2009
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o IDS -- FI o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Factorization Issues o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o I would like to introduce a concept that I find to be of use in discussing the problems of hypostatic abstraction, reification, the reality of universals, and the questions of choosing among nominalism, conceptualism, and realism, generally. I will take this up first in the simplest possible setting, where it has to do with the special sorts of relations that are commonly called "functions", and after the basic idea is made as clear as possible in this easiest case I will deal with the notion of "factorization" as it affects more generic types of relations. Picture an arbitrary function from a Source (Domain) to a Target (Co-domain). Here is one picture of an f : X -> Y, just about as generic as it needs to be: 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 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 | | | Source X = {1, 2, 3, 4, 5} | | | o o o o o | | g | \ | / \ / | | v \|/ \ / | | Middle M = { b , e } | | | | | | | h | | | | | v o o o o o o | | Target Y = {A, B, C, D, E, F} | | | o---------------------------------------o Writing the functional compositions f = g o h "on the right", as they say, we have the following data about the situation: X = {1, 2, 3, 4, 5} M = {b, e} Y = {A, B, C, D, E, F} f : X -> Y, arbitrary. g : X -> M, surjective. h : M -> Y, injective. f = g o h What does all of this have to do with reification and so on? Well, suppose that the Source domain X is a set of "objects", that the Target domain Y is a set of "signs", and suppose that the function f : X -> Y indicates the effect of a classification, conceptualization, discrimination, perception, or some other type of "sorting" operation, distributing the elements of the set X of objects and into a set of "sorting bins" that are labeled with the elements of the set Y, regarded as a set of classifiers, concepts, descriptors, percepts, or just plain signs, whether these signs are regarded as being in the mind, as with concepts, or whether they happen to be inscribed more publicly in another medium. In general, if we try to use the signs in the Target (Co-domain) Y to reference the objects in the Source (Domain) X, then we will be invoking what used to be called -- since the Middle Ages, I think -- a manner of "general reference" or a mode of "plural denotation", that is to say, one sign will, in general, denote each of many objects, in a way that would normally be called "ambiguous", "equivocal", "indefinite", "indiscriminate", and so on. Notice what I did not say here, that one sign denotes a "set" of objects, because I am for the moment conducting myself as such a dyed-in-the-wool nominal thinker that I hesitate even to admit so much as the existence of this thing we call a "set" into the graces of my formal ontology, though, of course, my casual speech is rife with the use of the word "set", and in a way that the nominal thinker, true-blue to the end, would probably be inclined or duty-bound to insist is a purely dispensable convenience. In fact, the invocation of a new order of entities, whether you regard its typical enlistee as a class, a concept, a form, a general, an idea, an interpretant, a property, a set, a universal, or whatever you elect to call it, is tantamount exactly to taking this step that I just now called the "factoring" of the classification function into surjective and injective factors. Observe, however, that here is where 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 "sense in general" (SIG). Now, opting for the "Object" choice of habitation would usually be taken as symptomatic of "realist" leanings, while opting out of the factorization altogether, or weakly conceding the purely expedient convenience of the "Sign" choice for the status of the intermediate entities, would probably be taken as evidence of a "nominalist" persuasion. Suppose that we have a sign relation L c OxSxI, where the sets O, S, I are the domains of the Object, Sign, Interpretant domains, respectively. Now suppose that the situation with respect to the "denotative component" of L, in other words, the "projection" of L on the subspace OxS, can be pictured in the following manner, where equal signs, like "=", written between ostensible nodes, like "o", identify them into a single real node. o-----------------------------o | Denotative Component of L | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | o | | /= | | / o y | | / /= | | / / o | | / / / | | / / / | | / / / | | / / / | | / / / | | x_1 o-/-/-----o y_1 | | / / | | / / | | x_2 o-/--------o y_2 | | / | | / | | x_3 o-----------o y_3 | | | o-----------------------------o This depicts a situation where each of the three objects, x_1, x_2, x_3, has a "proper name" that denotes it alone, namely, the three proper names y_1, y_2, y_3, respectively. Over and above the objects denoted by their proper names, there is the general sign y, which denotes any and all of the objects x_1, x_2, x_3. This kind of sign is described as a "general name" or a "plural term", and its relation to its objects is a "general reference" or a "plural denotation". Now, at this stage of the game, if you ask: "Is the object of the sign y one or many?", the answer has to be: "Not one, but many". That is, there is not one x that y denotes, but only the three x's in the object space. Nominal thinkers would ask: "Granted this, what need do we have really of more excess?" The maxim of the nominal thinker is "never read a general name as a name of a general", meaning that we should never jump from the accidental circumstance of a plural sign y to the abnominal fact that a unit x exists. In actual practice this would be just one segment of a much larger sign relation, but let us continue to focus on just this one piece. The association of objects with signs is not in general a function, no matter which way, from O to S or from S to O, that we might try to read it, but very often one will choose to focus on a selection of links that do make up a function in one direction or the other. In general, but in this context especially, it is convenient to have a name for the converse of the denotation relation, or for any selection from it. I have been toying with the idea of calling this "annotation", or maybe "ennotation". For a not too impertinent instance, the assignment of the general term y to each of the objects x_1, x_2, x_3 is one such functional patch, piece, segment, or selection. So this patch can be pictured according to the pattern that was previously observed, and thus transformed by means of a canonical factorization. In this case, we factor the function f : O -> S o---------------------------------------o | | | Source O :> x_1 x_2 x_3 | | | o o o | | | \ | / | | f | \ | / | | | \|/ | | v ... o ... | | Target S :> y | | | o---------------------------------------o into the composition g o h, where g : O -> M, and h : M -> S o---------------------------------------o | | | Source O :> x_1 x_2 x_3 | | | o o o | | g | \ | / | | | \ | / | | v \|/ | | Middle 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" (WOLOGOG), we might as well assume that there is a domain of intermediate entities under which the objects of a general denotation can be marshalled, just as if they actually had something rather more essential and really more substantial in common than the shared attachment to a coincidental name. So the problematic status of a hypostatic entity like x is reduced from a question of its nominal existence to a matter of its local habitation. Is it very like a sign, or is it rather more like an object? One wonders why there has to be only these two categories, and why not just form up another, but that does not seem like playing the game to propose it. At any rate, I will defer for now one other obvious possibility -- obvious from the standpoint of the pragmatic theory of signs -- the option of assigning the new concept, or mental symbol, to the role of an interpretant sign. If we force the factored annotation function, initially extracted from the sign relation L, back into the frame from whence it once came, we get the augmented sign relation L', shown in the next vignette: o-----------------------------o | Denotative Component of L' | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | o | | /= | | x o=o-------/-o y | | ^^^ / /= | | ''' / / o | | ''' / / / | | ''' / / / | | ''' / / / | | ''' / / / | | '''/ / / | | x_1 ''o-/-/-----o y_1 | | '' / / | | ''/ / | | x_2 'o-/--------o y_2 | | ' / | | '/ | | x_3 o-----------o y_3 | | | o-----------------------------o This amounts to the creation of a hypostatic object x, which affords us a singular denotation for the sign y. By way of terminology, it would be convenient to have a general name for the transformation that converts a bare "nominal" sign relation like L into a new, improved "hypostatically augmented or extended" sign relation like L'. I call this kind of transformation an "objective extension" (OE) or an "outward extension" (OE) of the underlying sign relation. This naturally raises the question of whether there is also an augmentation of sign relations that might be called an "interpretive extension" (IE) or an "inward extension" (IE) of the underlying sign relation, and this is the topic that I will take up next. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Note 2 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o Let me illustrate what I think that a plethora 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'd normally represent in a sign-relational table like so: o---------o---------o---------o | Object | Sign | Interp | o---------o---------o---------o | x_1 | y | ... | | x_2 | y | ... | | x_3 | y | ... | | ... | y | ... | | x_k | y | ... | | ... | y | ... | o---------o---------o---------o For brevity, let us consider the sign relation L whose relational database table is precisely this: o-----------------------------o | Sign Relation L | o---------o---------o---------o | Object | Sign | Interp | o---------o---------o---------o | x_1 | y | ... | | x_2 | y | ... | | x_3 | y | ... | o---------o---------o---------o For the moment, it does not matter what the interpretants are. I would like to diagram this somewhat after the following fashion, here detailing just the denotative component of the sign relation, that is, the 2-adic relation that is obtained by "projecting out" the Object and the Sign columns of the table. o-----------------------------o | Denotative Component of L | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | x_1 o------> | | \ | | \ | | x_2 o------>--o y | | / | | / | | x_3 o------> | | | o-----------------------------o I would like to -- but my personal limitations in the Art of ASCII Hieroglyphics do not permit me to maintain this level of detail as the figures begin to ramify much beyond this level of complexity. Therefore, let me use the following device to symbolize the same configuration: o-----------------------------o | Denotative Component of L | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | o o o >>>>>>>>>>>> y | | | o-----------------------------o Notice the subtle distinction between these two cases: 1. A sign denotes each object in a set of objects. 2. A sign denotes a set of objects. The first option uses the notion of a set in a casual, informal, or metalinguistic way, and does not really commit us to the existence of sets in any formal way. This is the more razoresque choice, much less risky, ontologically speaking, and so we may adopt it as our "nominal" starting position. Now, in this "plural denotative" component of the sign relation, we are looking at what may be seen as a functional relationship, in the sense that we have a piece of some function f : O -> S, such that f(x_1) = f(x_2) = f(x_3) = y, for example. A function always admits of being factored into an "onto" (surjective) map followed by a "one-to-one" (injective) map, as discussed earlier. But where do the intermediate entities go? We could lodge them in a brand new space all their own, but Ockham the Innkeeper is right up there with Old Procrustes when it comes to the amenity of his accommodations, and so we feel compelled to at least try shoving them into one or another of the spaces already reserved. In the rest of this discussion, let us assign the label "i" to the intermediate entity between the objects x_j and the sign y. Now, should you annex i to the object domain O you will have instantly given yourself away as having "Realist" tendencies, and you might as well go ahead and call it an "Intension" or even an "Idea" of the grossly subtlest Platonic brand, since you are about to booted from Ockham's Establishment, and you might as well have the comforts of your Ideals in your exile. o-----------------------------o | Denotative Component of L' | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | i | | /|\ * | | / | \ * | | / | \ * | | o o o >>>>>>>>>>>> y | | | o-----------------------------o But if you assimilate i to the realm of signs S, you will be showing your inclination to remain within the straight and narrow of "Conceptualist" or even "Nominalist" dogmas, and you may read this "i" as standing for an intelligible concept, or an "idea" of the safely decapitalized, mental impression variety. o-----------------------------o | Denotative Component of L" | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | o o o >>>>>>>>>>>> y | | . . . ' | | . . . ' | | ... ' | | . ' | | "i" | | | o-----------------------------o But if you dare to be truly liberal, you might just find that you can easily afford to accommmodate the illusions of both of these types of intellectual inclinations, and after a while you begin to wonder how all of that mental or ontological downsizing got started in the first place. o-----------------------------o | Denotative Component of L'" | o--------------o--------------o | Objects | Signs | o--------------o--------------o | | | i | | /|\ * | | / | \ * | | / | \ * | | o o o >>>>>>>>>>>> y | | . . . ' | | . . . ' | | ... ' | | . ' | | "i" | | | o-----------------------------o To sum up, we have recognized the perfectly innocuous utility of admitting the abstract intermediate object i, that may be interpreted as an intension, a property, or a quality that is held in common by all of the initial objects x_j that are plurally denoted by the sign y. Further, it appears to be equally unexceptionable to allow the use of the sign "i" to denote this shared intension i. Finally, all of this flexibility arises from a universally available construction, a type of compositional factorization, common to the functional parts of the dyadic components of any relation. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Factorization Issues -- Discussion o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Discussion Note 1 o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Work Area o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o 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". o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Factorization Issues -- 2000-2001 SUO List 01. http://suo.ieee.org/email/msg02332.html 02. http://suo.ieee.org/email/msg02334.html 03. http://suo.ieee.org/email/msg02338.html 04. http://suo.ieee.org/email/msg02340.html 05. http://suo.ieee.org/email/msg02345.html 06. http://suo.ieee.org/email/msg02349.html 07. http://suo.ieee.org/email/msg02355.html 08. http://suo.ieee.org/email/msg02396.html 09. http://suo.ieee.org/email/msg02400.html 10. http://suo.ieee.org/email/msg02430.html 11. http://suo.ieee.org/email/msg02448.html 12. http://suo.ieee.org/email/msg04334.html 13. http://suo.ieee.org/email/msg04416.html 14. http://suo.ieee.org/email/msg07143.html 15. http://suo.ieee.org/email/msg07166.html 16. http://suo.ieee.org/email/msg07182.html 17. http://suo.ieee.org/email/msg07185.html 18. http://suo.ieee.org/email/msg07186.html Ontology List 01. http://suo.ieee.org/ontology/msg00007.html 02. http://suo.ieee.org/ontology/msg00025.html 03. http://suo.ieee.org/ontology/msg00032.html 04. http://suo.ieee.org/ontology/msg01926.html 05. http://suo.ieee.org/ontology/msg02008.html 06. http://suo.ieee.org/ontology/msg03285.html o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o FI. Factorization Issues 00. http://stderr.org/pipermail/inquiry/2005-March/thread.html#2495 01. http://stderr.org/pipermail/inquiry/2005-March/002495.html 02. http://stderr.org/pipermail/inquiry/2005-March/002496.html 03. o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o