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$x=1$ is not a proposition because if $x$ is zero it is false and if $x$ is one it is true. Such a formula is called an open formula.

Then, are all closed formulas—formulas without free variables—propositions?

I think that $\forall x \in \mathbb{R} (x=1)$ is a proposition, because it has a truth value False. On the other hand, the closed formula $\forall x (x=1)$ contains a quantification without range: it is true on $\{1\}$ and false on $\{0\}.$

Therefore, not all closed formulas are propositions, right?

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  • $\begingroup$ A proposition is rather a statement something like a theorem than a formula that can hold or not depending on $x$. $\endgroup$
    – Peter
    Commented Aug 13 at 11:57
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    $\begingroup$ Do you have a technical definition of proposition in mind? In logic and philosophy different things can fall under the term ‘proposition’. $\endgroup$
    – JoD17
    Commented Aug 13 at 12:37
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    $\begingroup$ @BBB Closed formulae and propositions are synonyms. $\quad$ Separate point: truth is relative to interpretation, and in general, a proposition's truth value depends on its context. You have illustrated that the truth of ∀x(x=1) depends on the universe of discourse, which is an aspect of the interpretation. Another example: the proposition "the square of every number is nonnegative" is true in $\mathbb R$ and false in $\mathbb C.$ To sum $\endgroup$
    – ryang
    Commented Aug 13 at 13:27
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    $\begingroup$ up: a proposition can have multiple truth values, though not within a single interpretation. More at: Is "the dog is Batman" a proposition? $\endgroup$
    – ryang
    Commented Aug 13 at 13:27
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    $\begingroup$ Usually, a closed formula is called a sentence and it has a definite truth value in an interpretation. $\endgroup$
    – Mauro ALLEGRANZA
    Commented Aug 13 at 14:55

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What I understand from reading comments and the proposed duplicate is that (1) closed formulae and propositions are synonyms, (2) a proposition has a truth value when and only when it has a context, in other words, is "interpreted".

So, in conclusion, closed formulas and propositions are the same (synonyms). I misunderstood that propositions always have a definite truth value; actually, a proposition's truth value depends on the context.

Both $\forall x {\in} \mathbb{R} (x=1)$ and $\forall x (x=1)$ are propositions. The former has a truth value because of the context $x\in \mathbb{R}$. The latter has no truth value because there is no context.

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  • $\begingroup$ Just to point out that the two comments that you're paraphrasing contains a link, whose final sentence mentions, "the context/interpretation, even if not explicit, is always at least tacitly in the background." $\endgroup$
    – ryang
    Commented Aug 14 at 3:57