Because the article would make the sentence more definite and that's not the case here. I mean it would be like you were talking about a specific object, while what the sentence implies is that something is just not ours, not that this one is not ours while there's something else out there that is ours.. I can't think of an example in english, but if you speak french you can understand the difference when comparing "ça n'est pas notre" with "ça n'est pas le notre". Pffff it's really confusing trying to explain this in english.
I think I understand, and I can also understand your confusion because it got me too. For us English speakers, it seems there's no subject in the sentence. It looks like "not" "is" "own" "our" to us. But the point I think Hitoritana was making is that the subject is indefinite, and while in English we have a vague notion that "it" corresponds to some hypothetical noun in the world somewhere and that our sentences must have a subject, in Greek it's just left out.
Correct me if I'm wrong guys, bit of a n00b here
Yes, this is actually the case. Someone could say "Αυτό δεν είναι δικό μας" (And αυτό would indeed correspond to it), but it's redundant, and omitted, unless the speaker wants to be emphatic in some way. ^.^
Αυτό δεν είναι δικό μας - This is not ours (The speaker probably wants to emphasize on the fact that this specific object is not theirs.)
Δεν είναι δικό μας - It is not ours (There is no emphasis needed. The object is just not theirs.)
I wonder if ImogenJacobs asked this question because Italian often uses the definite article here, Non è il nostro (or la nostra depending on the grammatical gender of the antecedent). In Italian the definite article is optional in this sentence but my current learner's understanding is that Gk does not use the definite article in this case. But maybe it's optional in Gk?