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I am reading the book Language, Proof and Logic by Jon Barwise and John Etchemendy. In section $9.1$ it says the following. "Variables are a kind of auxiliary symbol. In some ways they behave like individual constants, since they can appear in the list of arguments immediately following a predicate or function symbol. But in other ways they are very different from individual constants. In particular, their semantic function is not to refer to objects. Rather, they are placeholders that indicate relationships between quantifiers and the argument positions of various predicates."

My question is, why are they not refering to objects? When I write $\forall x [P(x)]$ I thought that the interpretation is: Given any object, call it $x$, we have $P(x)$. In this sentence, $x$ refers to an object. Another example is $\exists y [P(y)]$, which translates to: There exists an object, call it $y$, satisfying $P(y)$. Again, $y$ is refering to an object. Is my interpretation false or does the passage of the book mean something else?

user1578232
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    To what object does the "$x$" in unquantified "$P(x)$" refer? – Eric Towers Feb 11 '22 at 22:41
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    @EricTowers Well I thought that the statement $\forall x [P(x)]$ is meant, and not only the $P(x)$ part. In the first case, if my interpretation is correct, $x$ should refer to an arbitrary but fixed object satisfying $P(x)$. – user1578232 Feb 11 '22 at 22:49
  • Also, "relationships between quantifiers and the argument positions" suggests you should be thinking about the difference between $\forall x [P(x,C)]$ and $\forall x [P(C,x)]$ (where $C$ is a suitable constant). What other method would we use to distinguish between quantification of the first argument versus the second? Also, it is worthwhile to not conflate a variable with hypothetical objects to which it may or may not refer (especially later, with typed variables -- for instance, "for all integers, n", "for all continuous functions, f", "for all predicates, P", ...). – Eric Towers Feb 11 '22 at 22:55
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    @EricTowers What is the problem with conflating variables with hypothetical objects they may refer to? For all integers $n$ could be interpreted as "given any integer, call it $n$". I am not seeing where this would lead to a problem. As far as I can see it, this is the natural interpretation, please correct me if I am wrong. – user1578232 Feb 11 '22 at 23:00
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    Free variables are like pronouns (it...): thet refer in context. – Mauro ALLEGRANZA Feb 12 '22 at 08:19
  • Regarding the universal quantifier: "Given any object, call it x, we have P(x)", more or less. The basic inference rule regarding $\forall$ is Instantiation: from $\forall x Px$ to $Pa$ for an $a$ whatever. The intuition is obvious: if every object is P, then an object whatever is P. – Mauro ALLEGRANZA Feb 12 '22 at 11:53
  • But the sentence $\forall x Px$ does not refer to a single object; it refers to the domain: every object of the domain is P. – Mauro ALLEGRANZA Feb 12 '22 at 11:54
  • The given description is a good "smoothed out" explanation sufficient for almost all practical intents and purposes. Deeper, it is not that smooth; see my post. – Tankut Beygu Feb 13 '22 at 12:22
  • See also this post as well as this one – Mauro ALLEGRANZA Feb 14 '22 at 16:03
  • It's a bad way of phrasing it; you are right that variables do refer to objects. What they meant was more something like variables are not used to single out a particular object but rather as a kind of dummy to talk about arbitrary objects. – Natalie Clarius Feb 15 '22 at 17:15
  • @lemontree Well, but I thought that a variable can mean any object of the domain, but once it has been introduced it means a fixed but arbitrary object. "Given any object, call it $x$" means that I can pick any object from the domain of discourse and call it $x$. After that is done, it is a fixed object, but chosen arbitrarily - I don't know anything about it other than it being in the domain of discourse. Hence it does single out an object in this view, but again, an arbitrary one. Is this interpretation wrong then? – user1578232 Feb 15 '22 at 22:29
  • Your understanding is correct, guess my wording wasn't better. Perhaps it makes sense like this: Whenever a variable is used it will likely be in the context of a quantifier, so that, while during the recursive evaluation it will indeed be assigned a fixed object, the interpretation on the level of the whole formula will encompass a range of objects; a variable would typically not occur unquantified and be used to refer to one particular object globally. But don't overthink that somewhat philosophical paragraph; your technical understanding of how variables work is absolutely right. – Natalie Clarius Feb 17 '22 at 16:24

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