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What does well-founded mean?

Looking for the meaning or definition of the word well-founded? Here's what it means.

Adjective
  1. based on reasoning, evidence and good judgement
  2. properly substantiated
  3. (mathematics) Of a binary relation: such that every non-empty subset of the relation's domain has a minimal element with respect to the relation.
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Examples
Either the judge read this carefully and thinks the order is well-founded, in which case he isn't a very smart judge.
And so he realized their skepticism was only natural, their suspicions well-founded.
Tales of ecological woe, no matter how well-founded in science, rarely galvanize people to action.
There are very real and probably well-founded reasons for not being so easily moved.
To what extent was it based on estimates, guesses and interpretations, however well-founded?
What do you mean by a well-founded suspicion if you do not simply mean a reasonable suspicion?

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