Are there possible worlds that include unimaginable suffering? On behalf of the Anselmian theist, Tom Morris denies that there are such worlds.
Such an [Anselmian] God is a delimiter of possibilities. If there is a being who exists necessarily, and is necessarily omnipotent, omniscient, and good, then many states of affairs which otherwise would represent genuine possibilities, and which by all non-theistic tests of logic and semantics do represent genuine possibilities, are strictly impossible in the strongest sense. In particular, worlds containing certain sorts or amounts of disvalue or evil are metaphysically ruled out by the nature of God, divinely precluded from the realm of real possibility. (‘The Necessity of God’s Goodness’ in his Anselmian Explorations 48 ff.)
But I think there’s an interesting argument that there are such worlds. I call it the Property Argument.
The property argument claims first that, for every coherent property P, there is some possible world in which P is instantiated. But a weaker claim will do: for every (non-world indexed and non-essential) property P instantiated by some sentient being in the actual world, there are worlds in which two or more sentient beings instantiate P. Consider the property of suffering intensely. The property is coherent. Lots of actual things instantiate that property. The property argument then goes this way.
1. w0 includes an instantiation of the property of suffering intensely. Assumption
2. If wn includes n instantiations the property of suffering intensely, then wn+1 includes n+1 instantiations of that property. Hypothesis
3. ∴ For any number n, there is some world that includes n sentient beings suffering intensely.
We haven’t quite reached the conclusion in (4), but we’re close.
4. ∴ There are worlds that are on balance extremely bad.
How can we avoid (4)? Here are some metaphysical hypotheses inconsistent with (4).
H1. For some finite number n, it is not possible that n beings suffer intensely.
H2. For some finite number n, no world with n+1 beings suffering intensely is overall worse than a world with n beings suffering intensely.
According to H1, for some n, it is metaphysically impossible that n+1 beings suffer intensely. But can it be true that the logic of suffering intensely includes this theorem?
P1. ☐(Pa1 & Pa2 & . . . & Pan ⟶ ~Pan+1 & ~Pan+2 & . . . & ~Pan+j)
(P1) states that necessarily, if a1 – an+1 all have the property of suffering intensely, then it is false that anyone else has that property.
H2 is true only if (a) H1 is true or (b) for every additional suffering being in a world, God causes someone else to cease suffering or (c) for every additional suffering being in a world, God creates another being that enjoys a pleasurable life. That yields another odd theorem in the logic of suffering intensely. I let x and y quantify over sentient beings distinct from a1 – an.
P2. ☐((Pa1 & Pa2 & Pa3 &. . .& Pan) ⟶ (∀x)(∃y)(Px ⟶ Hy))
(P2) states that necessarily, if a1 – an all have the property of suffering intensely, then if any distinct being x has the property of suffering intensely, then yet another distinct being has the property of being happy.
I can’t see any reason to believe that the logic of suffering intensely includes the theorems in (P1) or (P2). I conclude that Morris is mistaken, There are worlds that are on balance very bad.
 It is natural to think immediately of the property of being the set of all sets that are not -self-membered. That seems like a coherent property, but no set could instantiate the property. But the antinomy simply moves us to realize that we were thinking of the class of all sets that are not self-membered, and there is such a class. Similarly, it is a perfectly coherent have the property of not being a property. I have that property; so what if no property has it.