Given that there are still attempts to prove borns rule in other ways I get the impression that there must be something other than the mere structure of the logic which requires that probabilities in quantum mechanics are given by the born rule (with respect to gleasons theorem).
For instance i know that the crucial assumption it makes is the non contextuality of probabilities. I presume it also assumes that there is some probability function and that is a function of the quantum state. I presume therefore presume that there must be some probabilistic assumption akin to the principle of indifference lurking behind it (unless my misconstrual of maksynzkis analysis of A implies B iff B more probable than A in all states, means all contexts, not just all valuations (if it just mean all logical/physical possible worlds, or all logical/physically possible valuations/probability values, regardless of what the actual probability value is, then this analysis would be just as true in classical logic as in quantum logic. But if it means for all ways of measuring that same outcome given and keeping fixed the amplitudes that nature says it does have in those contexts that is more informative)
Is the crucial assumption something to the effect that probability must be (1) a probability function (which the born rule satisfies), (2) a function of the amplitude/state, and (3) most importantly a constant function of the amplitude so that the same amplitude receives the same probability- or that the function of the amplitude is indifferent to either the context or the outcome it is attached to (that is the same function) -- and (4) the probability is the same in all contexts,
Whilst it can be shown that born rule naturally satisfies the requirements of a probability function i presume the idea is that, given that the function must be a constant function of the state, and must give out the same probabilities in the same contexts yet the amplitudes change, the amplitudes change in such a way that other constant amplitude/state functions would give out results consistent with the probability calclulus they wont ascribe the same probability to the same outcome in each context (because it is a function of the amplitude and the amplitude changes in these contexts). However, in these contexts it does not change in any old way such that the quadratic function still outputs the same result in each context
where the quadratic nature of the function arises as a result of the fact that different states and amplitudes have the same inner product or norm squared value (but have different signs) in all of these contexts in which they also, as it happens are supposed to have the same probabilities
Presumably if if the state is different in these contexts the only way to ensure uniqueness is by a quadratic function (insofar that the squared norms of such states in such contexts are the same) . BY context here what is meant? Do they mean different yes no questions or when the observable and outcome is measured in combination with distinct pairs of other measurements of observables (ie where we measure spin x and some other observable A, and when we measure spin x some other observable B, where A and B, or rather their projectors, are non commuting quantities)
The basic question is this. If there spin up x has the same amplitude norm squared as spin down x what forces us to assign these events the same probability according to gleason theorem. Surely one could always find a way to assign them different probabilities if one were allowed to alter the probability function (of the amplitude) in different contexts and still preserve probability non-contextuality.(that the events have the --- i presume the issue is that (1) must be the same function (of the state/amplitude) that is used for the same outcome in all contexts or worse, (2) that this function of the amplitude must be the same for all events or, or at least must be the same for any two events with the same amplitude. Is (2) assumed
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