Changes

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 &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.

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,

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%"

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

|}

|}

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 x,

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'.

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.

 

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.

==Nominalism and Realism==

==Nominalism and Realism==