Is it valid to quantify two variables over the same universe using the same quantifier (like in $\forall a,b \in \mathbb{R} \:\: P(a,b)$)?

Question

I am wondering if we can quantify two variables over the same universe using the same quantifier, such as in $\forall a,b \in \mathbb{R}\:\: P(a,b).$

Are statements like this found in mathematical expressions? If so, do we regard this as quantifying a tuple of elements or two individual elements?

These are ubiquitous in practice. – Trebor Jul 30 '23 at 02:35 — Trebor, Jul 30 '23 at 02:35

ryang · Accepted Answer · 2023-08-02T00:54:12.410

4

These are equivalent to one another

$\forall a\,\forall b \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$
$\forall a\,\forall b \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big)$
$\forall (a,b) \; \big(a\in \mathbb{R} \land b\in \mathbb{R}\implies P(a,b)\big)$
$\forall (a,b) \; \big((a,b)\in \mathbb{R}^2\implies P(a,b)\big),$

while these are interchangeable abbreviations of the above

Addendum

The StackExchange Question

is there a way to prove that $$∀,\;(∀\:(≥⟹≥)⟹≥)\;?$$

suggests two other abbreviations:

edited Aug 02 '23 at 00:54

answered Jul 30 '23 at 01:58

ryang

Would you like to cite actual textbook examples where the likes of $\forall(a, b)$ are used? I can’t recall ever seeing that sort of usage out in the wild … – Peter Smith Jul 30 '23 at 17:24
@PeterSmith: the notation $\forall(a,b)\dots$ is commonly supported in proof assistants like those in the HOL family. For the logical quantifiers, this doesn't add much value, but these systems also support the notation $\lambda(a, b)\dots$, which is really useful. – Rob Arthan Jul 30 '23 at 21:39
@PeterSmith Actually, me neither; I'd merely reverse-engineered it from that last bullet; thanks Rob for supplying the example. To be clear, all the above bullets are examples of informal writing (for example, mathematics) rather than formal logic. – ryang Jul 31 '23 at 04:22

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