Oh, no, no, no. That is not how it works!
I am not a lawyer, but just watching Judge Judy will teach you the basics.
If you are suing someone, you need evidence. You can't just waltz in with allegations and expect the judge to side with you. You have to make your case.
If you're suing a wedding caterer for doing an inadequate job, you have to bring in your contract and evidence such as witnesses. Your word alone isn't enough.
Think about the huge patent lawsuits between companies such as Apple and Samsung. Do you think the suing company just comes in alleging that the other company has infringed on their patents and the judge will assume the allegations are true??? Heavens no! The suing company will spend hundreds of hours gathering evidence that they have to present to the court. It's only after they presented their case that the defendant responds.
If the defendant feels that the suing party hasn't presented sufficient evidence, they really don't have to do much. Is someone claims you keyed their car, but they don't know you from Adam and they have no witnesses, all you have to do is say, No, I wasn't there. You would win. They haven't carried their burden.
So no, you can't just go to court and allege that Obama is a naturalized citizen and he would have to produce proof he isn't. You, as the suing party, have to provide sufficient evidence to convince the court of that fact. Allegations are not presumed to be true. Only evidence counts in a court.
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What your watching is mediation in the format of a trial. The plaintiff and defendant have signed an agreement to have their case adjudicated by a mediator, Judge Judy. Judge Judy is not bound by the rules of the Court.