Mackysinki , and others (see wilce in stanford encylopedia) define two events to A and B to be equivalent iff P(A), s)=P(B,s) for all states.it is callsed: "outcome-separating" (see Four and a Half Axioms for Finite-Dimensional Quantum systems' i think by Wilce)
Is this just a probabilistic semantics, for a logic, as popper has a similar condition in his probability calculus but it is not specialized to logic.
What is meant by state in this, s. I presume this never holds for orthogonal events; when we defined the events as (spin up y|prepared in x and measured y), and spin down y\prepared in x and measured in x) and we range over all state given this. I presume it just mean valuations consistent with quantum logic (not with the laws of quantum mechanics).
where two events to be equivalent or probabilistically equivalent if separable; if for all states, s, the two events have the same probability. What is meant by state here. Do they mean physically equivalent according to the laws of quantum mechanics (or merely the laws of quantum logic? or is just those nomological laws of quantum mechanics localised to strict nomological connections between these two events); and what is held fixed. Do they when considering two events measured along the x component, spin up in x, and spin down in x, for a particle prepared in y, are equi-probable; and this presumably holds for all contexts.
Quantum Probability); and what are meant by states here ( i presume it just means, consistent truth valuations or atoms, in the domain of the model, just relatative to those two events in questions, and thus disregarding any other nomological relations about other events entails about them that nomologically entail anything about them or their probabilities given the laws of quantum mechanics (which are outside this context); so it would range over all quantum logic worlds which is not just the quantum mechanics law worlds).
So one would not hold fixed. when considering the a spin 1/2 particleof event (spin up and spin down) prepared in x and measured in y, the probabilities that the corresponding eigenstate in x nomologically necessitates (which is 0.5)' at most one would hold fixed any strict nomological relations just between spin up x and spin y down (and perhaps not even that, only strict analytic or logical connection-quantum logical connections between the two events)
So are the states supposed to quantify (1) over preparations, so that one can always prepare the states differently so that the two outcome are not equi-probable
(2) the preparation states are kept fixed; and one quantifies over all possible contexts; probabilistic non-contextuality would still ensure they are equally probable though
(3) Does he mean consistent with the 'if then structure' ie ie all possible quantum measure even those not allowed by the laws of quantum mechanics, except for strict nomological connections; given the state and preparation; where the fact that one cannot make one event more probable then the other would indicate that they are logically equivalent so that increasing the probability of one, would increase that of the other, or physically equivalent.
(3.A)What counts as a strict nomological connection here; presumably not the if then relation, between the amplitudes and thus probabilities of spin up for any component which uniquely determine the other probabilities and the certain state (the state that if measured, realizes the outcome with probability one)
(4) Does it therefore just mean the strict nomological connection between A and B if these are what are compared are held; and if one cannot coherently give one of the two a greater measure, then this indicates that is a strict, nomological if then relation between the two events. This would hold though even in classical mechanics. And its a little ambiguous in some cases as to which ones connections we hold fixed if this supposed to identify nomological equivalence.
And if its supposed to indicate logical equivalence, then this is just a general probabilistic semantical/syntactical condition, just as would hold in classical probability and in classical logic (karl popper has this as an axiom) and would not indicate any special quantum ordering; except insofar as there may be slightly different rules of inference. The strict logics and the probability logics correspond to each other as von neumman suggested (i presumed this just relates to the fact that setting the deterministic state, the eigenstate, sets the probabilities for all the other angles, so I presume its not talking about that- its a meta-lingusitic notion, that holds disregarding this extra structure and relates only to the analytic entailment relations within quantum logic, or only the strict nomological relations that must hold between the two events; given quantum mechanics and not that one can literally derive the probabilities through strict ranked 'if then' nomological connectives
(5) I presume this is distinct from operational equivalence as in spekkens case; where this is just equal probabilities across contexts. Operational equivalence can presumably hold for orthogonal events; ie those s pin 1/2 systems prepared and measured along an opposite anglefrom which they are preprared are such that in all contexts, the born rule probability of spin up=spin down?.
I presume this is distinct from operational equivalence which is probabilistic non-contextuality for measurements and outcomes; is independent state contextuality then defined for operational equivalents; and if so what would contextuality amount to in an-indeterministic model
Please can you make your question clearer? You provide too much miscellaneous information. You only need to provide the first two sentences, as no-one will read the rest – we like to answer questions, not read an essay lol :-)
I am basically asking whether
(1) A>B, the ordering relation just denotes implication or 'a greater amplitude than' relation
(2) when they say A>B iff P(A,s) >P(B,s) in all states,s what do they mean by states here; likewise depending on what this means, is A>~A possible in quantum logic, or in some quantum logics
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