I don't see what practical difference there is between saying "all universes exist and we need to think with logical consistency about the subject" and "all logically possible universes exist."
Whatever the case, a related point might be worth considering: whether logic is the only such "invariant" involved. Level IV is supposed to involve variations of mathematical structure. But would there be various universes for various structures in deductive mathematical theories of optimization, probability, information, or logic? If probability theory supplied various different structures for different universes, it could certainly complicate the measure problem -- one could be contending with various theories of measure itself as sources of differences between various universes. If, on the other hand, all these mathematics (and they amount together to a great deal of mathematics, maths often regarded as applied, but also often regarded as mathematically deep) place no additional constraints, nor broaden the variety of universes, then they seem correlated not to Level IV, but perhaps to Level III, since Level III (if I understand and phrase it correctly) neither adds constraints nor broadens the variety of universes on top of what one already has at Level II, and also since they seem to be mathematics of the structures of the kinds of alternatives which MWI says are all actualized. Best, Ben Udell Hal Finney writes: >>Brent Meeker writes: >>I doubt that the concept of "logically possible" has any absolute meaning. >>It is relative to which axioms and predicates are assumed. Not long ago the >>quantum weirdness of Bell's theorem, or special relativity would have been >>declared "logically impossible". Is it logically possible that Hamlet >>doesn't kill Polonius? Is it logically possible that a surface be both red >>and green? > >I agree. We went around on this "logically possible" stuff a few weeks ago. >A universe is not constrained by logical possibility. Our understanding of >what is or is not a possible universe is constrained by our mental abilities, >which include logic as one of their components. > >If I say a universe exists where glirp glorp glurp, that is not meaningful. >But it doesn't constrain or limit any universe, it is simply a non-meaningful >description. It is a problem in my mind and my understanding, not a problem >in the nature of the multiverse. > >If I say a universe exists where p and not p, that has similar problems. It it >is not a meaningful description. > >Similarly if I say a universe exists where pi = 3. Saying this demonstrates >an inconsistency in my mathematical logic. It doesn't limit any universes. > >More complex descriptions, like whether green can be red, come down to our >definitions and what we mean. Maybe we are inconsistent in our minds and >failing to describe a meaningful universe; maybe not. But again it does not >limit what universes exist. > >To summarize, logic is not a property of universes. It is a tool that our >minds use to understand the world, including possible universes. We may fail >to think clearly or consistently or logically about what can and cannot exist, >but that doesn't change the world out there. > >Rather than expressing the AUH as the theory that all "logically possible" >universes exist, I would just say that all universes exist. And of course as >we try to understand the nature of such a multiverse, we will attempt to be >logically consistent in our reasoning. That's where logic comes in. > >Hal Finney