This question occurred to me while reviewing a skeptical argument from Kripke regarding semantics:
Suppose that I’ve never dealt with numbers larger than 57. (Given our finite nature and the infinitude of the natural number series, there will always in fact be such a number.)!!! I’m asked to perform the computation '68+57', and I arrive at the answer ‘125’, which I take to be right. However, a “bizarre skeptic” (Kripke 1982: 8) questions my certainty. She suggests that in the past I used ‘plus’ and ‘+’ to mean a different function, which she calls “quaddition”. Quaddition yields the same result as addition if the numbers are lower than 57, and 5 otherwise, so the correct result of the aforementioned computation is ‘5’, not ‘125’. I should answer ‘5’ if I intend to use ‘plus’ in the same way in which I have been using it in the past, or so the sceptic suggests.
... “there must be some fact about my past usage that can be cited to refute” that hypothesis (1982: 9). That is, there must be some fact about my past usage that determines that I meant addition by ‘plus’ in the past, and thus that (again, assuming that I intend to use the expression in the same way I have been using it so far) I should answer ‘125’ rather than ‘5’. Importantly, the sceptic does not question my memory concerning past use; indeed, she goes as far as to allow that the exercise of my cognitive powers is faultless, and that I have access to all the facts about my mind and behaviour that are potentially constitutive of my meaning one thing rather than another (1982: 14). Her thought is that if I am not able, even in such cognitively ideal conditions, to provide the fact in virtue of which I mean addition, a fact that properly singles out the function of addition rather than the function of quaddition, it is because there is no such fact. Furthermore, the focus is on past use because “if I use language at all, I cannot doubt coherently that ‘plus’, as I now use it, denotes plus” (1982: 13). But, if the sceptic’s challenge succeeds, it can be generalized, for “if there was no such thing as my meaning plus rather than quus in the past, neither can there be any such thing in the present” (1982: 21). (The fact that the sceptic grants the idealisation of our cognitive powers in the way that she does shows that her argument is not of a piece with the argument of the epistemological sceptic, who is concerned with whether our actual cognitive capacities can lead to knowledge. See Boghossian 1989 [2002: 150]. For dissent over this point, see Ginsborg 2018. Martin Kusch takes the argument to be metaphysical [2006: xiv], but, in contrast with Boghossian, he takes the dialogic setting to play an essential role in it.)
As we’ll see, the search for a fact fails, and the sceptic concludes that “the entire idea of meaning vanishes into thin air” (Kripke 1982: 22).
Sec. 4 of the article goes over a dispositionalist approach to the problem:
The most widely discussed attempt at a straight solution to the sceptical challenge is reductive dispositionalism. According to a simple version of reductive dispositionalism, the fact that Jones has the concept of addition rather than of quaddition is to be identified with (or is constituted by) his disposition to produce the result of adding (and not quadding) the numbers x and y in response to arithmetical queries of the form ‘x + y = what?’, and the fact that he means cat by ‘cat’ is to be identified with (or is constituted by) his disposition to apply ‘cat’ to cats. (See Horwich 1998, 2010, 2012 for a systematic development of dispositionalism; an answer to Kripke’s challenge is articulated in Horwich 2015.)
A rough-and-(un)ready gloss of this, on my end, is something sloganish like, "The meaning of a term 'X' is a disposition, in users of 'X,' to refer to X upon consciousness of their (or an interlocutor's) transcription/vocalization/semiosis of 'X'." The disposition is the intension, and the reference is the extension; but so the disposition is ours more than the word's, so to say. But might we construe the tendency of certain words to cause us (when we recognize the words) to refer to certain other things, as the intension? So "cats" means cats (on average, in English), because (for the linguistic demographic that covers those qualifiers) people who perceive the word "cats," in this perception, are caused to refer, even if sometimes only (further) in Mentalese (perhaps), to cats?
I admit I am asking this question in an (un)fairly tortuous way: my target conclusion, so to say, is the idea that no word, as such, refers to anything whatsoever (I thus recapitulate much of the skeptical argument with which we started), but that reference is a case-by-case function from our capacity to semiotically cover things; the only stable intensionality is in a causal regularity associated with something like the nominal "baptism" Kripke also speaks of, and then further facts by virtue of which the regularity is (willfully) sustained over time.
This would not be too out of line with at least English use of the word "reference" itself: consider e.g. "I wasn't talking about Sam but was referring to Dean." Here, whatever wording is in dispute is not that which does the referring; I am the one who refers, through the wording to be sure, but not, then, because the wording itself carries some "referentiality" in its own essential character.
!!!This claim is (I think) inessential to the argument, and false anyway: anyone who uses the transfinite numbers uses a number larger than every finite number, and for those who accept things like Fregean anti-zero or von Neumann V, there is use of a number larger than all other numbers whatsoever, even the relatively transfinite ones. For those who desire to avoid the alephs and the omegas, though, there are still algebraic facts like A + B = B + A, for all A, B ∈ ℕ, such that, if we refer to these facts, we might be styled as referring to all natural numbers, hence all that are as large as we please.