The jury does not democratically decide if someone is guilty of something, they must unanimously agree that a crime was committed without a reasonable doubt. In fact, quite the opposite of democracy, a single jurist can disagree and change the outcome.
And even if a jury does find a person guilty, that person still has a right to appeal.
Trials are actually good examples of what democracy is not. Trials are lawful, rather than collective. Trials are designed to protect the rights of the individual against the democratic will of the majority.
Ron Paul's Convention Speech
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