Facts refer to real objects. They are true because there are something of which such sentences are true of, following Aristotle. What would be the reference of a fact of the form "not P"? Or there wouldn't be any reference and "not" is a purely linguistic phenomenon?
To quote a bit from SEP on this... the level of ontological commitment needed for truth-makers for negative truths has been a matter of debate:
Russell reluctantly chose to acknowledge negative facts as truth-makers for negative truths. He just couldn’t see any way of living without them. But negative facts are an unruly bunch. Try to think of all the ways you are. Contrast that with the even harder task of thinking of all the ways you aren’t! If negative truths are acknowledged as truth-makers they will have to be indefinitely numerous, unbounded in their variety; choosing to live with them is a heavy commitment to make (Armstrong 2004: 55). What’s worse, if negative facts are akin to positive facts—as their name suggests—then they must be made up out of things, properties and relations arranged together. But, prima facie, many of these things, properties and relations aren’t existing elements of reality. So unless, like Meinong, we believe in the non-existent, we’ll have to admit that negative facts aren’t configurations of their constituents and so an entirely different kind of entity from positive facts altogether (Molnar 2000: 77; Dodd 2007: 388). [...]
For example, we recognise that what makes it true that there is no oil in this engine is different from what makes it true that there are no dodos left. What makes claims like these true are absences, lacks, limits, holes and voids, where these are conceived not as things but as “localised states of the world”, robustly first-order and “causally relevant” to what goes on (Martin 1996: 58, 65–6; Taylor 1952: 443–5). But, as many philosophers have argued, when we talk about an absence having causal effects what we’re really saying can be understood without reifying negative states and appealing instead to the actual effects, or the counterfactual effects, of a positive state (Molnar 2000: 77–80; Armstrong 2004: 64–7; Lewis 2004; Beebee 2004).
Although I don't see the SEP page touching on this angle, some ["causalist"] authors do note that a problem of demarcation is related to this issue in a way, because natural language can encode opposite states of fact in experssions that don't seem prima facie negating anything:
Now take [...] the claim that everything that exists is positive. This is undoubtedly the most dubious of [such maximalist] claims. What does it mean? [...] Any such attempt would raise, needlessly, difficult questions concerning general principles of demarcation: does not 〈Socrates is dead〉 negatively represent it not being the case that Socrates is alive and 〈Socrates is not dead〉 positively represent it being the case that Socrates is alive?
What is it then for some existent entity to be positive? The only sense I can think of is that existent entities are positive in that they satisfy the Eleatic Principle that causal powers are the mark of being (or rather of positive being). Anything that can properly be said to have ‘negative existence’, such as lacks, absences and negative states of affairs, is negative in virtue of not having causal powers.
I don't want to get into the details on the latter here, but it's been noted that assuming causalism as the solution to truthmakers immediately opens up the (larger) discussion of what entailment actually means... basically what kind of proofs would one accept (in particular for non-existence claims). Milne noted that a version of the liar paradox can basically be stated as:
This sentence has no truthmaker.
You are asking, if there is a truthmaker for P, what is the truthmaker for not-P? Say P is, "John has a dog." The truthmaker for P would be a specific dog that John has. Then not-P is, "it's not the case that John has a dog." Not-P is made true by the state of affairs in which John does not have a dog. This is not a physical object in the usual sense, but it is a physical condition.