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Thanks, Ken, this is super helpful. I’ll definitely add the unconditional probabilities, as you suggest. And you’re right that I shouldn’t say that there’s NO sense in which preparation and measurement are the time-reverse of each other – I should say that they’re not entirely the time-reverse of each other.
Now on to your main point about intermediate measurements. First, I should clarify footnote 5 (and the sentence it attaches to). I shouldn’t say that the particle has spin properties for every direction in which its spin COULD be measured at R. Rather, what I meant to say is that it has spin properties for the direction in which its spin is ACTUALLY measured at R, whatever that direction might be (and footnote 5 provides the formula for assigning probabilities). Given that, I don’t feel so concerned that if you stick an intermediate measurement between S and R you change those probabilities. After all, you’ve changed the measurements that flank the source.
But you point out that the intermediate measurement can be “non-invasive” – it can simply divert the particle according to its spin in some direction, without any entropy-increasing action (like running it into a screen). You would rather not change the beables for such “measurements” (and, I assume reserve the term MEASUREMENT without the scare quotes for entropy-increasing interactions of a certain kind). It’s always a challenge to explain the role of measurement in retrocausal accounts without introducing a new measurement problem. I guess I’m committing myself to a particular view on measurement. Perhaps: a measurement of property p for a particle is any interaction in which the value of p becomes correlated with the position of the particle. This casts a pretty broad net (and has a special role for position, like Bohm does). So when you introduce a new SG device (even if there’s no entropy increase) then you introduce a new (genuine) measurement and you need to re-figure the probabilities. Are there problems lurking?