Jason (jcreed) wrote,

So Rowan's take on generalizing the Weather Report to arbitrary accessibility relations allows a surprising modal "lookahead" behavior. An example of its consequences is that that under an accessibility relation characterized only by the "supertransitivity" axiom

for all a, b, c, d, if a > b > c > d, then a > d

you get the following things:
(1) ◇◇◇A |- ◇—‡A
(2) ◇◇0 |- ◇—‡0

I put (1) there because it's "what you expect", from supertransitivity. (2) is the surprise, and it follows from ◇—‡ allowing you to get access to the 0 on the left not because it's at a world that you can actually get the right-hand-side to be located at, but because hypothetically under a sequence of imaginary ◇—‡L invocations, you could. This baked-in transitivity-ish behavior I find weird.

But! Yesterday I think I more or less figured out how to incorporate it into my focus-preserving-encoding worldview, and so I can sleep again at night. This means that my earlier suspicion that it didn't cut eliminate was wrong, but it's also the case that the funny side condition on 0 can be permuted up to a side condition on the ◇—‡L rule, meaning that my general intuition about nonstandard 0 rules is still corroborated by this experience.
Tags: logic, modal
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