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Directory:Jon Awbrey/Notes/Factorization And Reification (view source)
Revision as of 13:42, 25 June 2009
, 13:42, 25 June 2009→Factoring Sign Relations: go back to "o" for objects and "s" for signs
| o |
| o |
| /= |
| /= |
| / o y |
| / o s |
| / /= |
| / /= |
| / / o |
| / / o |
| / / / |
| / / / |
| / / / |
| / / / |
| x_1 o-/-/-----o y_1 |
| o_1 o-/-/-----o s_1 |
| / / |
| / / |
| / / |
| / / |
| x_2 o-/--------o y_2 |
| o_2 o-/--------o s_2 |
| / |
| / |
| / |
| / |
| x_3 o-----------o y_3 |
| o_3 o-----------o s_3 |
| |
| |
o-----------------------------o
o-----------------------------o
|}
|}
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''.
The Figure depicts a situation where each of the three objects, <math>o_1, o_2, o_3,\!</math> has a ''proper name'' that denotes it alone, namely, the three proper names <math>s_1, s_2, s_3,\!</math> respectively. Over and above the objects denoted by their proper names, there is the general sign <math>s,\!</math> which denotes any and all of the objects <math>o_1, o_2, o_3.\!</math> This kind of sign is described as a ''general name'' or a ''plural term'', and its relation to its objects is a ''general reference'' or a ''plural denotation''.
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.
Now, at this stage of the game, if you ask: ''Is the object of the sign <math>s\!</math> one or many?'', the answer has to be: ''Not one, but many''. That is, there is not one <math>o\!</math> that <math>s\!</math> denotes, but only the three <math>o\!</math>'s in the object space. Nominal thinkers would ask: ''Granted this, what need do we have really of more excess?'' The maxim of the nominal thinker is ''never read a general name as a name of a general'', meaning that we should never jump from the accidental circumstance of a plural sign <math>s\!</math> to the abnominal fact that a unit <math>o\!</math> exists.
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 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 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>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.
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:
o---------------------------------------o
o---------------------------------------o
| |
| |
| Source O :> x_1 x_2 x_3 |
| Source O :> o_1 o_2 o_3 |
| | o o o |
| | o o o |
| | \ | / |
| | \ | / |
| | \|/ |
| | \|/ |
| v ... o ... |
| v ... o ... |
| Target S :> y |
| Target S :> s |
| |
| |
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
o---------------------------------------o
| |
| |
| Source O :> x_1 x_2 x_3 |
| Source O :> o_1 o_2 o_3 |
| | o o o |
| | o o o |
| g | \ | / |
| g | \ | / |
| | \ | / |
| | \ | / |
| v \|/ |
| v \|/ |
| Medium M :> ... x ... |
| Medium M :> ... o ... |
| | | |
| | | |
| h | | |
| h | | |
| | | |
| | | |
| v ... o ... |
| v ... o ... |
| Target S :> y |
| Target S :> s |
| |
| |
o---------------------------------------o
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.
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 <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 — 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.
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 <math>o\!</math> is reduced from a question of its nominal existence to a matter of its local habitation. Is it more like an object or more like a sign? One wonders why there has to be only these two categories, and why not just form up another, but that does not seem like playing the game to propose it. At any rate, I will defer for now one other obvious possibility — obvious from the standpoint of the pragmatic theory of signs — the option of assigning the new concept, or mental symbol, to the role of an interpretant sign.
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:
| o |
| o |
| /= |
| /= |
| x o=o-------/-o y |
| o o=o-------/-o s |
| ^^^ / /= |
| ^^^ / /= |
| ''' / / o |
| ''' / / o |
| ''' / / / |
| ''' / / / |
| '''/ / / |
| '''/ / / |
| x_1 ''o-/-/-----o y_1 |
| o_1 ''o-/-/-----o s_1 |
| '' / / |
| '' / / |
| ''/ / |
| ''/ / |
| x_2 'o-/--------o y_2 |
| o_2 'o-/--------o s_2 |
| ' / |
| ' / |
| '/ |
| '/ |
| x_3 o-----------o y_3 |
| o_3 o-----------o s_3 |
| |
| |
o-----------------------------o
o-----------------------------o
|}
|}
This amounts to the creation of a hypostatic object <math>x,\!</math> which affords us a singular denotation for the sign <math>y.\!</math>
This amounts to the creation of a hypostatic object <math>o,\!</math> which affords us a singular denotation for the sign <math>s.\!</math>
By way of terminology, it would be convenient to have a general name for the transformation that converts a bare, ''nominal'' sign relation like <math>L\!</math> into a new, improved ''hypostatically augmented or extended'' sign relation like <math>L^\prime.</math> Let us call this kind of transformation an ''objective extension'' or an ''outward extension'' of the underlying sign relation.
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.