# What is the difference between 'denotation' and 'reference' in linguistics?

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.

• 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. 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. Dec 7, 2022 at 21:30
• "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). 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 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. Dec 8, 2022 at 19:11