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I am trying to determine why in mathematical expressions we can have a variable x which does not ‘refer’ to a specific number but can still 'denote' an unspecified object. In linguistic theory, is it possible to have something 'denote' but not 'refer'? (I understand you do not use variables in linguistics.)

Some sources suggest they are the same, others that they are different, even that to 'denote' is to 'signal' something so it does not need to be an entirely concrete reference.

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  • Are you speaking of "dummy variables" like the dx in integrals? Quantifiers (some, all, many) are usually represented in semantic logic as binding a variable, which is a term in the sentence. Both the quantifier and the variable are marked with a dummy variable (typically, of course, x). Similar remarks for negatives and modals, which all have scopes and foci.
    – jlawler
    Dec 7, 2022 at 21:11
  • @jlawler that's definitely part of it, I'm more confused that 'denote' seems to suggest a specific object that the symbol/string/phrase 'denotes' yet free variables do not necessarily refer to anything, I think I might me trying to mix 'mathematical' with real language where there aren;'t really 'variables', so in real languages phrases will usually have a concrete object it refers to. Sorry if that is a poor expanation, I will look further in my logic books for more explanation.
    – Confused
    Dec 7, 2022 at 21:30
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    "Denote" is a term from popular linguistics that's usually contrasted with "connote", also appearing in "denotation/connotation". That's roughly (and only roughly) the distinction now made between aspects of "meaning" that are semantic (logical) and those that are pragmatic (supplied by context).
    – jlawler
    Dec 7, 2022 at 22:02
  • @jlawler In this context is it a correct lingusitic usage to say 'x' denotes an unnamed object and that 'John "denotes" the person who lives across the road' (assume that there is a person called john across the road from me)? In one way 'x' does denote it, in that it represents it but wthout explictly naming anything as I have done in the latter example
    – Confused
    Dec 8, 2022 at 10:53
  • X is a variable in a formula; it has no meaning except what you assign it explicitly. John, on the other hand, is a name that can refer to (or denote) a human. In a situation like that, whether one uses denotes or refers to is largely a matter of individual taste, and linguistic theories, to the extent they are different. I would say that refer falutes higher, especially if you're arguing with philosophers or logicians, but there's not a whole lot of difference. It depends on your theory of reference and sociolinguistics.
    – jlawler
    Dec 8, 2022 at 19:11

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I was actually under the impression when you wrote a mathematical formula without stating that x has a particular (but omitted) value, it usually more formally is a stand-in for when you do decide to supply a value.

In other words, I do not know if a “free variable” is a concept in more axiomatic set theory. It’s more of a notational convention that’s saying, “for any x in this domain, this function associates it with the following expression”. So actually, the “free variable” is not conceptually different from the “bound” variable. They are only symbolic placeholders to be substituted with a value.

That said, we can think more philosophically about what you are saying. I am pretty sure it could be the pretty well-established idea of “sign” vs. “referent”. In your example, x is a “sign”. Once you state what it means, now the sign has a referent. This is a general characteristic of language, in that we use conventions and associate them with meanings.

Maybe another way of thinking about it is that it is possible to talk about a thing not merely without specifying which thing it is, but far more abstractly, that it is not even asserted what kind of thing it is, yet some proposition with regards to it is still coherent. I think this could be like saying “one of the marbles in the bowl is green”, vs. “Things which are heavy sink fast in water.” This may not be a perfect explanation, but the idea is indicating very clearly what set of things something is in and making it clear that you have decided mentally which one you are referring to, vs. talking about some hypothetical thing that technically has not been claimed to be real, yet. And, honestly, this I believe is part and parcel of semantics. I believe it’s called bound vs. unbound variables and has to do with the field of pragmatics, concerned with how people infer meaning from context, such as what an anaphoric word like “it” is referring to.

(P.s. linguists use variables.)

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