Translation:The judge does not have a relationship with his sister.
Yes and I'm not sure how to reflect that in English, so please tell me if either of them is wrong. I'd have suggested: Beziehung mit seiner Schwester = relationship with his sister (they're lovers), Beziehung zu seiner Schwester = relationship to his sister (they call or see and probably like each other).
Oh ok, that makes sense. In English the difference is something like this: "[what is his] relationship to Karen" they're siblings (she's his sister). "[How is his] relationship with his sister" (e.g. caring, loving, affectionate, detached). For a romantic relationship with a sibling you usually just include the adjective incestuous "he's in an incestuous relationship with his sister".
To be more precisely: Beziehung mit always means a love-relationship in German – I can't think of any counter-example. Beziehung zu can mean any kind of relationship, like sister to mother, line x to cube y, company to client, etc.:
How is the relationship with our latest customer developing? - Wie entwickelt sich die Beziehung zu...
How does line x relate to cube y? - Wie verhält sich Linie x zu Würfel y? This might also be translated, less literally, but sometimes clearer in German: In welcher Beziehung steht Linie x zu Würfel y?
People might look at you really puzzled if you said: Die Beziehung mit meinem Hund ist sehr liebevoll.
The same statements are indeed valid in both languages. Relationship to.. = Beziehung zu..; Relationship with.. = Beziehung mit.. It's rare for German and English to correspond so closely. The only problem then resides in the meaning of the German sentence. The sister is a relation of of the judge and thus, by definition, has a relationship to him. That is, the sentence is self-contradictory.
I believe 'zu' is always Dativ, not a 2-way preposition.
female version: ... Richterin
In my opinion that logic doesn't really work .. "Nein" is "No" in German, but it doesn't mean 9, does it? If that logic was correct German would be a pretty broken langauge. :P