A prior is not a program that tells you what to do with the data. A prior is a set of hypotheses with a number assigned to each. When data comes in, we compute the likelihoods of the data given each hypothesis on the list, and use these numbers to obtain a posterior over the same hypotheses. There's no general way to have a "none of the above" (NOTA) hypothesis in your prior, because you can't compute the likelihood of the data given NOTA.

Another equivalent way to think about it: because of the marginalization step (dividing everything by the sum of all likelihoods), Bayesian updating doesn't use the total likelihood of the data given all current hypotheses - only the relative likelihoods of one hypothesis compared to another. This isn't easy to fix because "total likelihood" is a meaningless number that doesn't indicate anything - it could easily be 1000 in a setup with an incorrect prior or 0.001 in a setup with a correct prior.