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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: