Overwhelming exception

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An overwhelming exception is a logical fallacy similar to a hasty generalization. It is a generalization which is accurate, but comes with one or more qualifications which eliminate so many cases that what remains is much less impressive than the initial statement might have led one to assume.

Examples:

• "All right, but apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, a fresh water system, and public health, what have the Romans ever done for us?" (The attempted implication (fallaciously false in this case) is that the Romans did nothing for us). This is a quotation from Monty Python's Life of Brian.
• "Our foreign policy has always helped other countries, except of course when it is against our National Interest..." (The false implication is that our foreign policy always helps other countries).
• "All Americans are useless at foreign languages. Ok, I'll make an exception for those who live in multi-ethnic neighbourhoods, have parents who speak a foreign language, are naturally gifted in languages, have lived abroad or who went to a school with a good foreign language program, but the rest are absolutely useless at foreign languages."
• All dogs are black, except for those which are not black. (This is also a tautology, or rather a validity.)

See also faulty generalization for other fallacies involving generalization.