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It is a proof.

Q: What is a convincing argument that logically shows why a conjecture is true?

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It is a proof.

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burden of proof

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It's called a proof

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There is no exactness when describing the relative burdens of proof, different courts have different articulations. Preponderance of the evidence means the evidence supports one conclusion over another no matter how slightly, clear and convincing means the evidence clearly supports one conclusion over another, and beyond a reasonable doubt means that the evidence comes at least close to supporting only one conclusion. Clear and convincing is often required to overcome a legal presumption, such as appealing a guilty plea, finding mental incompetency, or arguing an apparently valid contract is the product of fraud.

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The clause beyond reasonable doubt simply means that there is enough evidence to convince the judge that you have committed the act or in other word, the crime.
That is the standard which must be met for conviction of a defendant in a criminal trial. Notice that the standard is not, "beyond ALL doubt," - only beyond REASONABLE doubt.
REASONABLE DOUBT - The level of certainty a juror must have to find a defendant guilty of a crime. Proof beyond a reasonable doubt, therefore, is proof of such a convincing character that you would be willing to rely and act upon it without hesitation in the most important of your own affairs. However, it does not mean beyond ALL doubt.

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