Translation:The judge has a cat but not the lawyer.
Not so much. First of all, yargıcı can't be director, it's "judge." For the rest of it, I think it's a trick of English more than anything.
In English, "There is" can mean something exists ("var" in Turkish). Like, there is a cat in the house - Evde bir kedi var.
Or, it can mean you're showing somebody something ("o" or "orada" or something like that). Like, look, there is a cat!
Var can never mean you are SHOWING something directly in the second sense. It just means you're talking about something's existing in the first sense, or somebody HAS something. So you could be super pedantic in your translation and say "The judge's cat exists but the lawyer's does not". (That's a dreadful translation. Don't do that.) What you're REALLY saying is, the judge has a cat but the lawyer does not.
But when you say "there is the director's cat" it sounds like you're SHOWING where something specific is, and var just doesn't do that. That would be more like "yöneticinin kedisi orada!" (the director's cat is there!) or just "o yöneticinin kedisi!" or just "yöneticinin kedisi!"
Impossible :) "The lawyer has not" doesn't make sense (or at least hasn't since probably the late 19th century). You must include an object for the verb "to have" when it means "to possess" and you must include either "do" (American English) or use the form "have got" (British English) when you negate it.
In ny opinion, the translation should be "The judge has a cat but the lawyer doesn't." That would be clear and it would test the intended objective: whether you understand the sentence, rather than letting you (me :-) get tricked into blindly following the Turkish sentence structure and forgetting proper English :-D
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