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Let $P$ be a predicate. The expression $\forall x:P(x)$ seems to have two different meanings, depending on the context:

  1. If the domain of discourse $D$ is clear from context, then the statement "$\forall x:P(x)$" should actually be read as an abbreviation of "$\forall x\in D:P(x)$". For instance, in real analysis the statement $\forall x:x^2\ge0$ would be understood as an abbreviation of $\forall x\in\mathbb R:x^2\ge0$.
  2. In set theory, it seems that $\forall x:P(x)$ means "for any set $x$, $P(x)$ is true". For instance, the axiom of extensionality is written as $\forall x:\forall y:\forall z:(z\in x\iff z\in y)\implies x=y$.

I don't fully understand the meaning of $\forall x:P(x)$ in the context of set theory. Naïvely, the statement $\forall x:P(x)$ seems to be an abbreviation of $\forall x\in \mathbf{U}:P(x)$, where $\mathbf{U}$ is the universal set. However, the "universal set" is not actually a meaningful concept in standard formulations of set theory, and so this interpretation is clearly wrong. So how should the statement $\forall x:P(x)$ actually be interpreted?

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    $\begingroup$ @Joe Actually, $\forall x \in D(P(x))$ is an abbreviation of $\forall x(x\in D\to P(x))$, so you have the same problem. $\endgroup$
    – Git Gud
    Commented Sep 19, 2021 at 18:19
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    $\begingroup$ The membership symbol, $\in$, has a formal definition. Definition under which, as you say, the "universal set" does not exist. Let's just dispense with that here. Your interpretation correct: $\forall x P(x)$ means "for any set $x$, $P(x)$" is true. In an informal setting, you just accept that as part of the metalanguage of the subject at hand. In a formal setting, like in mathematical logic, those symbols are part of the language and basically are devoided of meaning. An expression like $\forall x\left(P(x)\right)$ is legal by virtue of the rules of formation of expressions. $\endgroup$
    – Git Gud
    Commented Sep 19, 2021 at 18:19
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    $\begingroup$ @Joe That's right. You start with a bunch of symbols, among which the logical connectives, the quantifiers, variables (and others) and then some rules are defined that say "a well-formed formula is an expression built from the following set of rules": <set of rules>, and that's that. You, as a human, and because you know mathematics, and also because these formal systems were created to mimic mathematics in practice, can attribute meaning to it. $\endgroup$
    – Git Gud
    Commented Sep 19, 2021 at 18:30
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    $\begingroup$ @Joe Right. People who care for such details wouldn't even write such "sets". The axiom schema of specification basically tells you that you need something like $\{x \in A\colon P(x)\}$, you can't do without the $\in A$ part, for some set $A$. $\endgroup$
    – Git Gud
    Commented Sep 19, 2021 at 21:35
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    $\begingroup$ $\forall x Px$ is interpreted differently in different interpretations... In the domain of naturals with "x is Even" as meaning for Px we have..., while in the domain of humans with "x is a Philosopher" as meaning of Px we have... So the issue is: what is a "suitable" domain of an interpretation for axiomatic set theory? See e.g. Von Neumann universe. $\endgroup$ Commented Sep 20, 2021 at 6:41

2 Answers 2

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Actually, there is nothing wrong with understanding "$∀x ( Q(x) )$" in a foundational set theory to mean "$∀x{∈}U\ ( \ Q(x) \ )$" where $U$ is the type of all objects (i.e. $U$ denotes the entire intended domain). $U$ does not have to be a set. In pure ZFC, $U$ is not an object, because otherwise we get a contradiction. $U$ is commonly called a class. In extensions of ZFC that can reason about classes as objects, such as NBG or MK, we have two sorts, one for "sets" and one for "classes", and $U$ is still not a set, even though it is now an object.

Typically "$∀x{\in}S\ ( \ Q(x) \ )$" can be treated as a short-form for "$∀x\ ( \ x∈S ⇒ Q(x) \ )$", and "$x∈U$" simply reduces to "true". Note that the "$\in$" here does not require any set theory in a strict sense, because $S$ can just be a sort (in many-sorted FOL). However, it is convenient to use the same symbol as for set membership because there is no danger of ambiguity. After all, in NBG and MK we literally have $E ∈ \{ t : Q(t) \} ⇔ Q(E)$ where "$\{ t : Q(t) \}$" is the class notation.

In fact, this usage of quantifying over classes is extremely common in higher set theory, such as "$∀k{∈}ORD$" where $ORD$ is the class of ordinals, which is (as explained above) perfectly fine even in pure ZFC.

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  • $\begingroup$ Thanks for this answer. It helps to get a different perspective. $\endgroup$
    – Joe
    Commented Sep 25, 2021 at 8:33
  • $\begingroup$ @Joe: You're welcome! $\endgroup$
    – user21820
    Commented Sep 25, 2021 at 9:34
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The question in the title is. . .

How should logical quantifiers be understood if the domain of discourse is not specified?

And the answer is that you don't understand them unless they're specified. It's possible the domain of discourse is clear from context, but that has much more to do human communication than with the mathematical content.

In the context of set theory, the domain of discourse is that of all sets.

This brings to light the following apparent issue.

Naïvely, the statement $\forall x:P(x)$ seems to be an abbreviation of $\forall x\in \mathbf{U}:P(x)$, where $\mathbf{U}$ is the universal set. However, the "universal set" is not actually a meaningful concept in standard formulations of set theory, and so this interpretation is clearly wrong.

And the question. . .

So how should the statement $\forall x:P(x)$ actually be interpreted?

As mentioned in the comments, your interpretation "for any set $x$, $P(x)$ is true" is correct. And this fine (or at least as fine as it can be). In this setting, the quantifiers are not formal entities. They're part of the metalanguage, just like English. We just enrinch the natural language (in this case English) with some extra symbols to facilitate communication. The $\in$ symbol is different, however, because in set theory it has a formal meaning.

There are contexts (e.g. predicate calculus) in which the quantifiers are formal symbols, in which they're not part of the metalanguage. In these contexts, the quantifiers are part of formal language just as much as $\in$ is in set theory. There are also formation rules that determine what formulas are legal, and in such systems a formula like $\forall x P(x)$ (it can be a little bit different, it may require some more parentheses, but it's something like this) is legal and that's that.

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