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The SEP article on deontic logic, in sec. 2.3, reads:

We assume that we have a set of possible worlds, W, and a relation, A, relating worlds to worlds, with the intention that Aij iff j is a world where everything that holds at j is acceptable from the standpoint of i, so that everything obligatory at i holds in j. For brevity, we will call all worlds so related to i, “i-acceptable” worlds and denote them by Ai.[31] We then add that the acceptability relation is “serial”: for every world, i, there is at least one i-acceptable world. [emphasis added]

Although we mostly tend to think of the ensemble of possible worlds as infinite, and this infinity can be variously derived from conditions like "possible recombinations of countably many base sentences" or "permutations of spacetime continua," is this necessary for possible-worlds talk modulo i-acceptability? If there is the actual world (as a possible world to boot) and one other possible world, yet the actual world is not i-acceptable, is having i be self-acceptable enough? The above-quoted article does go on to talk about something relevant to this consideration: "... we need the further requirement of 'secondary seriality': that any i-acceptable world, j, must be in turn acceptable to itself." And then: "Essentially, the ordering relation coupled with the limit assumption just gives us a way to generate the set of i-acceptable worlds instead of taking them as primitive in the semantics: j is i-acceptable iff j is i-best."

Is there a system-dependent lower bound on this "generation"? Or can deontic logics, particularly SDL, have arbitrarily few-or-many possible worlds, just as modal logic in general seem to be able to?

2 Answers 2

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As far as I can tell, SDL’s axiomatization is just that of KD, but with OB as the strong operator instead of the usual box operator. As such, it seems that only one self-related world is enough to guarantee seriality. To avoid transitivity, reflexivity, symmetry, etc. three worlds {0,1,2} are required such that 0R1, 1R2, 2R0. If we have just one or two worlds, then at least one of the worlds is going to violate seriality or entail more than just seriality for the accessibility relation.

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    Well, apparently I have to wait 23 hours but once those have passed, I will give you the points for the restarted bounty! This was an important question for me, not just for theoretical reasons but I was envisioning a multiversal fantasy story in which the question would play some kind of role. Commented Dec 7, 2023 at 19:18
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Since our world is a possible distribution, when we define all permutations, the more possible worlds we add when interpreting them, the more static we begin to be in practice. Because when we want to compile the existing laws of physics that cause possible situations to occur, the common points of all the laws of physics will begin to decrease. That is, until the ontology remains. We don't need to, since our current situation is already more likely.

The only reason parallel universes exist -if they exist- is because they are parallels. This is more difficult than metaphysically describing the connection of another reality to our reality.


Proof

s(W)<s(U)
i∈W∈U
s(Aim)=(s(i, α)-(1-1))×...×(s(i, α)-(m-1))
fserie(i)≠fserie(Aim, r)
fweight(Aim, r)=1-mα

According to such an infinite universe model, I specify the disciplines that would describe the laws of the universe as F:

F0: Determinism
F1: Neo-determinism/Negentropy
F2: Physicalism
F3: Relative System Laws
F4: Relative Game Laws/Entropy
F5: Degenerating
F6: Derealism
F7: Ignorance
F8: Ontology

In this sampling, the order of the laws F0 and F8 is clear.

fserie(i, F0)≠fserie(Aim, r, F0)
fserie(i, F1)≄fserie(Aim, r, F1)
fserie(i, F7)≃fserie(Aim, r, F7)
fserie(i, F8)=fserie(Aim, r, F8)

At least one of an equal number of possible causes is needed to update asymptotically equal laws. In this case, the less similarity of the universes to each other, the greater the probability that their asymptotic equation will be broken by insufficient language use or law updates. A very limited universe is needed for the laws of F1 in SDL. Neo-determinism/Negentropy, due to the asymptotic equality (if m = 1), shows that some of the very similar universes are abstract, so there are no parallel universes in its laws, since the possible futures are more limited, there are exceptions.

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  • So do you mean to say that (S)DL requires infinitely many worlds? That that is what the calculation of the given parameters necessarily comes out to? Or is a small finite number of worlds sufficient? Commented Dec 2, 2023 at 22:38
  • Even if there are an infinite number of worlds in SDL, a finite number of worlds must be needed. Imagine that a space stone behaves differently in two different worlds, and when we want to compile these laws, an exception may arise that will require us to redefine them. Except by interpreting the dynamics of possible i-acceptable worlds, one cannot induct that the existing laws derived from their intersection/combination will change absolutely.
    – fkybrd
    Commented Dec 3, 2023 at 3:49
  • Apologies in advance but I am having a hard time following your comments/answer... Do you have a precise calculation of the lower bound of possible worlds in SDL? Like a set of mathematical expressions, functions, etc. that can be evaluated to a specific number? Commented Dec 3, 2023 at 17:22

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