Are "∀x ∈ R" and "∃x ∈ R" propositions?

Question

1. Is the proposition which consists solely of "∀x ∈ R" considered true because x does not fail to satisfy any conditions we lay out? Or is it not a proposition?

2. Is the proposition which consists solely of "∃x ∈ R" considered true because we are asserting the existence of a real number? Or is it not considered a proposition?

Answer 1

1. Is the proposition which consists solely of "$∀x ∈ \mathbb R$" considered true because $x$ does not fail to satisfy any conditions we lay out?

I don't see that we have laid out any condition?

• The statement $$\text{every object $x$ is real}\\∀x\;\;x ∈ \mathbb R\tag1$$ can be read literally as "every object $x$ is such that it is real" or as "for every object $x,$ it is real".

• While formula $(1)$ has no standard abbreviation, the statement $$\text{every object $x$ that is real satisfies $P(x)$}\\∀x\;\big(x ∈ \mathbb R\to P(x)\big)\tag2$$ is typically abbreviated as $$∀x{\in}\mathbb R\,\;P(x).\tag2$$

• So, "$∀x{\in}\mathbb R$" is not a meaningful sentence; it is merely an abbreviation of a part of a full sentence. You could read it as "every $x$ that is real is such that..." or "for every $x$ that is real,..."

2. Is the proposition which consists solely of "∃x ∈ R" considered true because we are asserting the existence of a real number?

Similar to the above. Replace "every" with "some" and → (conditional) with ∧ (conjunction).