I don't know if the name "strict" is best, but the name "validate" is not good too. Current to_date does some validations too.
Obviously not enough, because it allows invalid dates. I'd say that the current to_date() merely validates the input format for string parsing, and that the date is in range. But there is not much validation on the date itself.
So the name can't be "strict" because of the conflict with "NULL" handling, and you don't like "valid" - what other options do you offer?
I have not - so third option looks best for me - it can be long name "only_correct_date", "only_valid_date", "only_valid_date_on_input" ...
Pavel
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