Which statement is false?
$A=$ a premise is a statement
$B=$ a statement is a sentence or a proposition
$C=$ a sentence and a proposition is not an open formula
$D=$ therefore a premise is not an open formula
If none of these sentences is false, then what is $P(x)$ in this first-order argument, if not a premise?
Universal Generalization.
$P(x)$
$\therefore \forall x P(x)$
Is one of these statements false?
$E=$ $x$ in $P(x)$ must be a free variable
$F=$ an open formula is a formula with a free variable
$G=$ therefore $P(x)$ is an open formula
$H=$ $P(x)$ is a premise to Universal Generalization
$I=$ $P(x)$ is a premise
$J=$ an open formula is a premise
$K=$ a premise can be an open formula\
NOTE: contradiction
$D=$ therefore a premise is not an open formula
$K=$ a premise can be an open formula\
I am not an expert with first-order logic so please excuse my lack of symbolizing everything. I just want to understand this here.
Such issues should be settled with the complete specification of the logical system they refer to; otherwise, wrong-headed judgements and fallacies would be inevitable.
I shall assume that the given the universal generalisation rule correctly stated as an unconditional rule:
$P(x)$
$\therefore \forall x P(x)$
In such systems, the open formula $P(x)$ is accepted as stating that $P$ is true of any $x$ (whatever $x$ is) and thus generalising to its universal closure is a legitimate step: If $P(x)$ is true of any $x$ arbitrarily taken, then it is true of all $x$'s. Such a system may not have inference rules of instantiation; these rules may be given in the associated tableaux method (I have boldfaced, because this is a frequent point of misunderstanding, since many systems include existential and universal instantiation rules).
Let us go over the rest stepwise giving priority to the firmly established terminology. Hence:
$P(x)$ is a premiss for the consequence $\forall x P(x)$: If one accepts $P(x)$ as a premiss in the above interpretation (i.e., $E$ is true), one can draw the consequence $\forall xP(x)$ from it. Since "open formula" is a firmly established term of logic, each statement in the sequence $F, \ldots, K$ is true.
We turn to $A, B, C, D$. Now, these are well-established:
Therefore, the question hinges on the definitions of proposition and statement as terms of logic. Unfortunately, there is still much divergence on these. I favour the usages roughly described as follows:
Proposition is a statement that is opaque with respect to internal structure and a truth-value can be assigned to. Then, strictly speaking, being open or closed is not applicable to a proposition. If an equivalence to a structured statement is forced for some reason, the conception I would prefer is to take proposition as a closed formula without quantifiers (i.e., constituted by constants); that brings it more in line with its philosophical usages.
Statement has the sense of "simply and meaningfully that what is stated", whether expressed in natural language or in a formal language, whether looked at with a syntactic or semantic focus. It functions as a metatheoretical covering term.
But these usages are in no way standard. Just recall that some authors call sentential logic what others call propositional logic. We cannot delve into a survey of usages and their combinations to employ in an answer to the question.
So, as an illustration, I take statement as a well-formed formula. Thus, $A$ is true. I have noted above that sentence is a closed formula. Let us admit that proposition is a closed formula as well. Then, $B$ turns out to be deficient, for a statement has the option of being be an open formula.
I should stress that the distinctions between terms should be resolved within the text in which they occur.