Rather than dwelling on the 'madnesss' of querying hinges, it seems much more natural to say that while they may not be propositions in the (sub-) game which they enable, they can be propositions in a game which interrogates the playability of the game that they enable.
This points, of course, at the questions which I think are important - to do with the hinges of any possible language game and the (fundamental) unintelligibility of asking whether we can talk - whether we can ask questions. Whether it is possible to do philosophy is not a philosophical question.
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