Is "x is yellow" a statement by itself, given that no definition of x was previously provided? Does a statement have to affirm a certain state of affairs in order to be a statement and claim something of the world, or not?
The short answer is: it really depends on what you mean by ‘statement’. To give the longer answer, let’s recall the distinction between syntax and semantics. Starting on the syntactic side, compare the following three:
All three expressions consist only of first-order symbols. Further, (i) and (ii) were formed according to the syntactic rules of first-order logic: they are well-formed formulae. By contrast, (iii) wasn’t thus formed and is, in that sense, gibberish. (But one could imagine a language that accepts ‘vxP’ as well-formed.) Finally, while x occurs freely in (ii), all variables in (i) are bound. Therefore, (i) is a sentence of first-order logic, while (ii) is not. Note that these differences were drawn without considering what the individual expressions ‘mean’ or ‘assert’, i.e. without appealing to semantics.
To give corresponding English expressions, consider:
(i*) Everything is fine.
(ii*) It isn’t green.
(iii*) Or it taller.
Turning now to semantics, (i) above is going to be either true or false under any given interpretation: it ‘says’ or ‘states’ that everything is P – where the individual interpretation needs to specify the value of P and the domain of the universal quantifier. Turning to (ii), let I be a particular interpretation. Because x occurs freely in (ii), I doesn’t assign a truth-value to (ii) directly. However, each variable assignment of I does assign a truth value to (ii): first, a value is assigned to x, and that value then either is or isn’t in the set assigned to P. So, if the free x in (ii) is like 'it' in (ii*), then each variable assignment gives an answer to the question: ‘What do you mean – it’?
Thus, it would be wrong to say that (i) is meaningful while (ii) is not: both express a content only relative to something – viz. relative to an interpretation in the case of (i), and relative to a variable assignment (within an interpretation) in the case of (ii).
So, are (ii) and ‘x is yellow’ statements? Well, it really does depend on what we mean by a statement. If we mean to include expressions like ‘It isn’t green’, then (ii) is a statement. If we want ‘statement’ to be synonymous with ‘sentence’, then (ii) isn’t a statement.
The question needs to be qualified. A statement in formal mathematical Philosophy of Language? According to who? Which authority? Which text book? Or, a statement for some genus of ancient philosophy? For Socrates? Or, for some other form, a non-mathematical form, of modern Philosophy of Language? The most common definition given in Analytic Philosophy departments to undergrads? Or, what they tell the PhD students? Or, for some medieval authority?
Ergo, the answer is, one must become better informed about the nature of such questions.
Of course, the passion for idiocy in this forum will first of all attempt to pass off any attempt to deviate from a fixed idea of a conventional, universally given answer, as "personal" answer. Never dreaming that the current most popular answers are in the tiniest minority, when compared to the verdict of philosophy of the ages. The forum itself is, now, so far, gravely and massively ill-informed about philosophy.