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The question is:  What sort of thing is a connotation?  Is it a sign?  That is to say, is it yet another term?  Or is it something like an abstract attribute, namely, a character, an intension, a property, or a quality?  And while we're asking, does it really even matter?

The question is:  What sort of thing is a connotation?  Is it a sign?  — that is to say, yet another term?  Or is it something like an abstract attribute — a character, intension, property, or quality?  And while we're asking, does it really even matter?

"No" is one answer worth considering.  But then:  Why does it not matter?  What reason might be given that would excuse the indifference?

"No" is one answer worth considering.  But then:  Why does it not matter?  What reason could be given to excuse the indifference?

This is a question that has exercised me since my earliest studies of Peirce.  I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other, but, alas, I haven't the foggiest notion what exactly my revelation amounted to.  Perhaps that is all for the best, as the vagrancy of memory is frequently better than the vapidity of one's banalytic anamnesia.

This is a question that has exercised me since my earliest studies of Peirce.  I can remember discussing it with my philosophy mentor at the time and I distinctly recall having arrived at some conclusion or other — alas, I haven't the foggiest notion what exactly my revelation amounted to.

These days, I usually try to finesse the trick under the trumped up rubric of a factorization.  So let me excavate my last attempts to explain this business and see if I can improve on them.

These days, I usually try to finesse the trick under the trumped up rubric of a factorization.  So let me excavate my last attempts to explain this business and see if I can improve on them.

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 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.

To begin answering that question, 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 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 codomain <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''.

In general, if we use the signs in the target domain <math>Y\!</math> to denote or describe the objects in the source domain <math>X,\!</math> then we are engaged in a form of ''general denotation'' or ''plural reference'' with regard to those objects, that is, each sign may refer to 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, 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.

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.