# Are closed formulas and propositions the same? [duplicate]

$$x=1$$ 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?

• A proposition is rather a statement something like a theorem than a formula that can hold or not depending on $x$. Commented Aug 13 at 11:57
• Do you have a technical definition of proposition in mind? In logic and philosophy different things can fall under the term ‘proposition’. Commented Aug 13 at 12:37
• @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 Commented Aug 13 at 13:27
• up: a proposition can have multiple truth values, though not within a single interpretation. More at: Is "the dog is Batman" a proposition? Commented Aug 13 at 13:27
• Usually, a closed formula is called a sentence and it has a definite truth value in an interpretation. Commented Aug 13 at 14:55

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.