(1) What is a question? (2) Are there sentences that have the grammatical form of a legitimate question, yet nevertheless fail to be legitimate questions? (3) Does this sentence (i.e., (3)) have the grammatical form of a legitimate question, yet fail to be a real question?

The word 'question' has several senses. In one sense it is a grammatical term referring to sentences of interrogative form. In another sense it is a semantic term, referring to the sort of thing that could be the content of an interrogative sentence - as in "The question that 'What is the meaning of life' is really asking". In the latter sense there can be sentences that have the form of a legitimate question yet nevertheless fail to pose legitimate questions - e.g. "Why has no dog ever barked?" or "Why have more people been to Berlin?". The answer to (3) is 'no', so the answer to (3) is 'no'.

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