We've been talking about indexicals a lot the last while. In Peircean terms these are dyadic relations (two variables) that are related. The example Peirce uses is a natural index like a weather vane where the direction of the vane indicates the wind. However also clearly most of our language includes an indexical relationship. Over at Maverick Philosopher they've been discussing indexes. I just wanted to add a few points.
First while language can be indexical it doesn't logically follow that I know the indexical relationship. Sure I may talk about my keyboard right now and be sure that the referential function is to that keyboard. But let's say I say, "I think it is sunny out today." There are two semantic indexes in that sentence. "I" and "today." However do I know those absolutely? No. While we know what "I" entails to a degree, who am I? The notion of self is hardly philosophically unproblematic. Likewise time indexes are problematic, not just because how we use them is loose (thus at best indicating a range of possible values). But consider if I utter this sentence today and ten years from now encounter this sentence in a journal I wrote today. In this case, the question of "who" for "I" is hardly much better open than it is to some stranger who reads it. I know there's some historical connection to the me of today, but who is the me of yesteryear?
Now I don't want to make this seem mysterious the way some do. It's just a feature of indexes that they aren't quite as clear in their indications as we think. As Peirce puts it, while we have the index in a sign, the object of this index is always indicated only by a hint.
Consider the weather vane example. That's a natural sign. However we never have access to the pure index relationship. We only have representations of the index which are not, themselves, pure indexes.
The other thing to keep in mind is the type-token relationship. We can consider sentences independent of utterance for instance. Indexical expressions in these "de-grounded" sentences are still indexes but are very general ones. The sentence "I am here" has for its index all possible expressions of the sentence. This is what in terms of signs we call a general sign. So the index is to a general, much like the sentence fragment "all generals" is a general index.
Concerning you're last point about indexical-types, I think a good place to start is Strawson's "On Referring":
"Consider another case of an expression which has a uniquely referring use, viz. the expression 'I'; and consider the sentence, 'I am hot'. Countless people may use this same sentence; but it is logically impossible for two different people to make the same use of this sentence: or, if this is preferred, to use it to express the same proposition. The expression 'I' may correctly be used by (and only by) any one of innumerable people to refer to himself. To say this is to say something about the expression 'I': it is, in a sense, to give its meaning. This is the sort of thing that can be said about expressions. But it makes no sense to say of the expression 'I' that it refers to a particular person. This is the sort of thing that can be said only of a particular use of the expression." (326-7)
Strawson is considering sentence- and expression-types. He argues that the expression-type 'I' in the sentence-type 'I am hot' has no reference because it makes no sense to ask whether the sentence is true or false.
It seems that you are claiming that the expression-type 'I' in the sentence-type 'I am hot' does have a reference insofar as it has an index. The reference of the expression-type is 'all I's', and the sentence is about all possible tokens of the type 'I am hot'. Or have I misread your comment?
How are you understanding the quantifier (e.g., do you take it to represent an infinite conjunction)? Could you elaborate on the index of the sentence?
It depends upon how you take vague propositions. Strawson is taking them as more or less "meaningless" (in a certain sense) independent of a pragmatic setting. As I understand Peirce he is taking them as expressing a general sign which is roughly the extension entailed by any possible use of the sentence.
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