The phrase is used metaphorically to express a lack of evidence or support in an argument. Without witnesses, your claims about seeing a UFO won't have a leg to stand on.
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Legal Contexts
Commonly used in legal discussions where evidence is crucial for a valid argument. In court, without concrete evidence, the defense doesn't have a leg to stand on.
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Colloquial Expression
It's a colloquial saying, so it's best used in casual or less formal conversations. He tried to get a refund without a receipt, but he didn't have a leg to stand on.