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?
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?
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4$\begingroup$ Those are not propositions, as there is no truth value to them. Nor are they even open sentences, which are statements whose truth or falsity depends on what is input to it. I don't think what you've written even has a name - they're just fragments of what could be proper statements $\endgroup$– H. sapiens rexMay 30 at 1:54
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1$\begingroup$ en.m.wikipedia.org/wiki/Well-formed_formula $\endgroup$– JonathanZMay 30 at 2:42
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4$\begingroup$ Only if we adopt the "unusual" convention of reading $\forall x \in \mathbb R$ as $\forall x (x \in \mathbb R)$ $\endgroup$– Mauro ALLEGRANZAMay 30 at 5:39
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- 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,..."
- 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).