The question in the title; if there is no existent precluding factor (whatsoever) for the existence of some x, must such a x exist?
If there is no existent precluding factor (whatsoever) for the existence of some x, must such a x exist?
The way this is worded is interesting, because any case that quickly comes to mind to try and falsify it (say, a Uranium atom popping into existence) has a "precluding factor": the conservation of energy. So, I'll address this question from the perspective of the most relevant philosopher that comes to my mind: Spinoza. Note: this response of mine has been heavily edited, in response to NotThatGuy’s criticism in the comments below. I have flipped 180 on this issue, and you will see why.
In Spinoza's Ethics (1677) he makes a move in Proposition XI, in using what he established beforehand, in making an argument for God's existence, using logic tied to your question. You'll see this in one of the passages below Prop. XI, where he says:
Another proof.—The potentiality of non—existence is a negation of power, and contrariwise the potentiality of existence is a power, as is obvious. If, then, that which necessarily exists is nothing but finite beings, such finite beings are more powerful than a being absolutely infinite, which is obviously absurd; therefore, either nothing exists, or else a being absolutely infinite necessarily exists also. Now we exist either in ourselves, or in something else which necessarily exists (see Axiom. i. and Prop. vii.). Therefore a being absolutely infinite—in other words, God (Def. vi.)—necessarily exists. Q.E.D.
Here Spinoza invokes the principle at question, that, if something cannot prevent the existence of something, then it exists. Like my example earlier of the Uranium atom, there is something that can preclude it existing simpliciter: Uranium atoms must abide by certain conservation laws (e.g., of energy). However, something infinite, of which there can be no greater power, there is nothing in principle to preclude it's existence. Thus, it does.
So, Spinoza would answer your question in the affirmative. Note: although he speaks of God in Prop. XI, his reasoning applies to what he defines as a "substance" (he simply elaborates on what substances exist in the process of this Part of the Ethics, and concludes that the sole absolutely infinite substance is God).
Now onto the critique of this line of reasoning: as NotThatGuy mentioned to me (repeating here in case those comments get deleted), Spinoza’s argument in the quoted passage above seems horribly circular. That is, he begins by saying the potential to exist is a power, which presupposes the principle in question. While true for that passage, I want to point out that Spinoza’s deduction to Prop. XI is not circular. But, as we will see shortly, the above critique has force anyway.
Spinoza is implicitly presuming the Principle of Sufficient Reason (PSR) in his argument (which he states a form of it as Axioms I and III).
I. Everything which exists, exists either in itself or in something else.
III. From a given definite cause an effect necessarily follows; and, on the other hand, if no definite cause be granted, it is impossible that an effect can follow.
As you can see, Axioms I and III effectively support the thesis that “if nothing precludes X from existing, X exists.” The problem is, Spinoza asserts these as Axioms, that is, they do not require justification in his eyes. While these were commonly held to be self evident in the 1600s, since Hume and Kant, they are not anymore. Especially problematic is Axiom I, since with that Spinoza can easily deduce that infinite substances (those whose power is no greater than any other substance of their kind) exist necessarily. That is, Spinoza’s argument is rigged.
If one wants to employ the principle that “X exists if nothing precludes X’s existence” a priori, one has a steep hill to climb in justifying its use.
So, your question is an interesting one, and since Parmenides’ formula of “from nothing comes nothing” some 2500 years ago, the principle you ask about seemed to be defensible a priori. However, with advancements in physics in the early 20th century that show beyond a reasonable doubt that our world has counter-intuitive features, we can’t be making arguments like Spinoza did in the 1600s anymore. One must be careful when reading statements like these by physicists: this interesting answer by a high reputation user over at the Physics.SE. This has a context that cannot be overstated. This is in regard to well-verified probabilities through empirical research, and thus do not admit of a priori applications.
You ask a good question, but it must be answered in the negative in regard to any a priori or philosophical use, until someone makes a rigorous proof. Relying on its intuitive appeal is no longer acceptable in philosophy.
No. You are assuming that for a thing not to exist there must be a factor that prevents it from existing, which is an unnecessary assumption. There is nothing to prevent me from creating a passible likeness of Bertrand Russell by carving a turnip- that does not mean that I must create it. Indeed, were that not the case, my house would be full of carvings and other objet d'art nothing is preventing me from creating.
There are infinitely many things that don't seem to be specifically precluded from existing, yet do not exist, as best we can tell.
As such, the answer to whether such things "must" exist seems to trivially be "no".
You could perhaps say that those things are precluded from existing due to a lack of things that caused it to exist, or due to some vague appeal to the laws of nature. But that would lead us to somewhat of a tautology: something came into existence because it came into existence. This wouldn't be a meaningful conclusion, and I'd be suspicious of any argument that tries to make use of something like this.
Exists no Y such that if Y, not X, therefore X... is a valid inference.
Note that for us to move from internally consistent logic (connecting an arbitrary premise to a conclusion that need not be about reality) to statements about reality we need to go do experiments and collect evidence. That is: to know that there "exists no Y" we need evidence incompatible with Y. Our real argument then becomes:
Z implies exists no Y such that if Y, not X. Therefore X.
Contrast the similar argument from ignorance, a fallacy:
Exists no Y known to me such that if Y, not X, therefore X... is fallacious.
Consider a trial. Alice is accused of burning Bob's precious experimentation textbooks.
In one hypothetical, detectives have found phone messages in which Alice says, "Bob, there's nothing anybody could do to stop me from burning your precious experimentation textbooks!"
In another hypothetical, detectives have no evidence.
As usual, what do you mean by 'exist'?
If you mean 'material existence', as in, made of particles from the Standard Model and being part of physical reality as studied by Physics, then... you should ask physicists! Some answers/comments stated that quite possibly there is always something preventing...
If you mean 'exist' in the mathematical sense, for a mathematical object, then Mathematics has very precise criteria for existence (which depend on the specific theory). Perhaps yes, adding more sets and sets of sets? Maybe in some cases there are models for and against? Ask a mathematician!
If you mean something more nebulous, whereby 'ideas exist', then... I don't know... Here the problem is, if you think "there's nothing preventing idea X from existing, but does it exist?"... then by thinking about it X already came into existence (in this sense).