I don’t have too much to add to this, honestly. But this is a super high-effort comment which was a joy to read, so I’ll give some commentary (mostly not arguments though).
I think the metaphysics of S5 are weird indeed, but I probably end up thinking it’s okay. But I could be convinced either way on this, very weakly-held. I think your points here are good considerations about this. I think my comment thread with @jessicata raised interesting questions about metaphysics of S5, and whether we should believe that perfect-essential necessity is a property realizable at all. Though I do disagree with her, but her arguments were fantastic.
I guess my main disagreement is that, well, I would say I am a more “mathematical reader” of this argument (I also authored this post, but in how I evaluate it). However, I do think it is sensible to think about what else we could interpret the second-order predicate P to mean, and so whether this argument “proves too much.” Especially if one doesn’t have a huge amount of logic-background to understand it, this can give you a first-order reason to argue against it. Although, again I probably agree with you that most such objections tend to be sort of subtly misguided, probably due to lacking mathematical background.
About Oppy’s objections I’m not as familiar. I think I agree with you that the entailment thing ends up being pretty strong here. Maybe strong enough to be a reason to reject that perfect properties exist (since they have to satisfy the implication rule), but in so-believing I also end up rejecting Oppy’s argument.
I think when we get to this level of reasoning it’s less about actual arguments and more about trying to tease out intuitions and develop intuition pumps. Actual arguments are usually the easy part, the hard part is finding an intuition on which to hang an argument!
While I might agree that the metaphysics of S5 is strange, I don’t think that’s really a function of S5 as much as it is a function of any coherent metaphysics.
For example consider again the accessibility relation over possible worlds. I suspect we all agree it is reflexive, but to deny S5 is to deny that it is transitive or reflexive. I think a possibility relation which is not transitive or reflexive also very weird!
Surely, my intuition says, if world A is possible from B, and B is possible from C, then A is possible from C. Surely, my intuition says, if A is possible from B, then B is possible to A.
I’d have some sympathy for the denial of symmetry if we were talking about a possible future, so maybe you can get to future A from here and future B from here, but you can’t get to here from B, or from A, and can’t get from A to B or vice versa. Ok, but we are talking about whole worlds rather than future, so I don’t think that’s the right logic to use here.
Which I think is really my point in the intuitions in my first comment—this is less about whether S5 is true or not, and more about whether it’s the right system for the type of objects we are dealing with. Since the type of objects we are dealing with a kind of “universal”, my intuition is we should use the most “universal” logic, which is S5.
I read that other thread you referenced and didn’t find the arguments particularly compelling, perhaps because I am coming from more of a platonist perspective where I think formal systems exist regardless of what concrete objects exist which might instantiate them. If I didn’t think that, I’d likely deny that necessary existence was a coherent concept at all, and so the argument falls apart much earlier!
I don’t have too much to add to this, honestly. But this is a super high-effort comment which was a joy to read, so I’ll give some commentary (mostly not arguments though).
I think the metaphysics of S5 are weird indeed, but I probably end up thinking it’s okay. But I could be convinced either way on this, very weakly-held. I think your points here are good considerations about this. I think my comment thread with @jessicata raised interesting questions about metaphysics of S5, and whether we should believe that perfect-essential necessity is a property realizable at all. Though I do disagree with her, but her arguments were fantastic.
I guess my main disagreement is that, well, I would say I am a more “mathematical reader” of this argument (I also authored this post, but in how I evaluate it). However, I do think it is sensible to think about what else we could interpret the second-order predicate P to mean, and so whether this argument “proves too much.” Especially if one doesn’t have a huge amount of logic-background to understand it, this can give you a first-order reason to argue against it. Although, again I probably agree with you that most such objections tend to be sort of subtly misguided, probably due to lacking mathematical background.
About Oppy’s objections I’m not as familiar. I think I agree with you that the entailment thing ends up being pretty strong here. Maybe strong enough to be a reason to reject that perfect properties exist (since they have to satisfy the implication rule), but in so-believing I also end up rejecting Oppy’s argument.
Thank you for your kind words.
I think when we get to this level of reasoning it’s less about actual arguments and more about trying to tease out intuitions and develop intuition pumps. Actual arguments are usually the easy part, the hard part is finding an intuition on which to hang an argument!
While I might agree that the metaphysics of S5 is strange, I don’t think that’s really a function of S5 as much as it is a function of any coherent metaphysics.
For example consider again the accessibility relation over possible worlds. I suspect we all agree it is reflexive, but to deny S5 is to deny that it is transitive or reflexive. I think a possibility relation which is not transitive or reflexive also very weird!
Surely, my intuition says, if world A is possible from B, and B is possible from C, then A is possible from C. Surely, my intuition says, if A is possible from B, then B is possible to A.
I’d have some sympathy for the denial of symmetry if we were talking about a possible future, so maybe you can get to future A from here and future B from here, but you can’t get to here from B, or from A, and can’t get from A to B or vice versa. Ok, but we are talking about whole worlds rather than future, so I don’t think that’s the right logic to use here.
Which I think is really my point in the intuitions in my first comment—this is less about whether S5 is true or not, and more about whether it’s the right system for the type of objects we are dealing with. Since the type of objects we are dealing with a kind of “universal”, my intuition is we should use the most “universal” logic, which is S5.
I read that other thread you referenced and didn’t find the arguments particularly compelling, perhaps because I am coming from more of a platonist perspective where I think formal systems exist regardless of what concrete objects exist which might instantiate them. If I didn’t think that, I’d likely deny that necessary existence was a coherent concept at all, and so the argument falls apart much earlier!