Yeah. Or rather, I guess modal logic can describe the world—but only if you meet its very strict demands. For example, to say something is “possible”, one must prove the impossibility of finding a contradiction between the thing and all evidence known so far, to either the speaker or the listener. If that requirement is met, then modal logic will give the right answers, at least until new evidence comes along :-)
I’m not super confident in this conception… I’d have to think about it. If I’m honest your argument doesn’t totally convince me (because logically possible worlds do seem like a thing I should be able to talk about). Unless the “evidence” you’re talking about is what we know to be logically possible? Anyway, seems like this is probably the crux. Think it is quite reasonable to say modal logic is too strong (especially S5, which is on the stronger end).
I guess this time I spoke too soon! Indeed if we talk about logical possibility, then we “only” need to prove that the imagined world isn’t contradictory in itself. Which is also hard, but easier than what I said.
Yeah but who knows if that’s really “correct” in that the world-we-actually-exist-in logically behaves like this. Not to mention the precise details of the rules for necessity and possibility. I fear we’re in danger of switching positions.
It’s nice that we got to the notion of logical possibility though. It’s familiar ground to me.
Let’s talk for example about mathematical properties of musical intervals. When a major scale C D E F G A B is played on a just-intonation instrument, all pairwise ratios of frequencies are fairly simple: 2⁄3, 15⁄16, all that. All except the interval from D to F, which is an uglier 27⁄32, unpleasant both numerically and to the ear. This raises the tantalizing possibility of a perfect tuning: adjusting the frequencies a little bit so that all pairwise ratios are nice, not all except one. The property of a tuning being perfect can be described mathematically.
Unfortunately, it can also be shown mathematically that a perfect tuning can’t exist. What does that mean in light of your Axiom 3? Must there be a “possible world”, or “logically possible world”, where mathematics is different and a perfect tuning exists? Or is this property unworthy of being called perfect? But what if we weren’t as good at math, and hadn’t yet proved that perfect tuning is inachievable: would we call the property perfect then? What does your framework say about this example?
Well, I don’t want to take too much credit—it’s Gödel’s framework, not mine. I guess it would be the case that then it would not be a perfect property, since it is logically impossible to achieve. We just wouldn’t know that without doing a bunch of mathematics first, and might be confused to begin with. There’s many similar examples in this vein, for sure. But just because you don’t know something, doesn’t mean it’s not true or false. There either is a perfect property, or there is not.
I’m now going to address something that’s sort of irrelevant to your main point, but perhaps interesting:
It would seem strange to say that frequencies of music are inherently perfect. This seems to rely on a lot of contingent facts about how human ears work, why we like certain sounds and not other sounds. What is pleasant to us. So I wouldn’t call any property like this “perfect” in this sense. But I know it’s meant as an illustrative example, so this is not really to your point.
Wait, if we can be confused whether a property is perfect or imperfect—then why do we assert (in axioms 4 and 6) that some specific properties are perfect? What if they’re also impossible, like the perfect tuning?
For those ones we just say that they are, since we need those for the argument to hold logically. So we stipulate that for those properties there is no confusion, and they are perfect. You are totally allowed to say “hmm, seems suspicious,” but by the definition of these properties, there’s nothing contradictory about asserting that these properties are perfect.
But for a property like “is omniscient” or whatever, we could be more unsure. Since maybe “being omniscient” implies you also “know what it’s like to kick a baby.” And so it implies that you have a property that is not perfect, for example.
Yeah. Or rather, I guess modal logic can describe the world—but only if you meet its very strict demands. For example, to say something is “possible”, one must prove the impossibility of finding a contradiction between the thing and all evidence known so far, to either the speaker or the listener. If that requirement is met, then modal logic will give the right answers, at least until new evidence comes along :-)
I’m not super confident in this conception… I’d have to think about it. If I’m honest your argument doesn’t totally convince me (because logically possible worlds do seem like a thing I should be able to talk about). Unless the “evidence” you’re talking about is what we know to be logically possible? Anyway, seems like this is probably the crux. Think it is quite reasonable to say modal logic is too strong (especially S5, which is on the stronger end).
I guess this time I spoke too soon! Indeed if we talk about logical possibility, then we “only” need to prove that the imagined world isn’t contradictory in itself. Which is also hard, but easier than what I said.
Yeah but who knows if that’s really “correct” in that the world-we-actually-exist-in logically behaves like this. Not to mention the precise details of the rules for necessity and possibility. I fear we’re in danger of switching positions.
It’s nice that we got to the notion of logical possibility though. It’s familiar ground to me.
Let’s talk for example about mathematical properties of musical intervals. When a major scale C D E F G A B is played on a just-intonation instrument, all pairwise ratios of frequencies are fairly simple: 2⁄3, 15⁄16, all that. All except the interval from D to F, which is an uglier 27⁄32, unpleasant both numerically and to the ear. This raises the tantalizing possibility of a perfect tuning: adjusting the frequencies a little bit so that all pairwise ratios are nice, not all except one. The property of a tuning being perfect can be described mathematically.
Unfortunately, it can also be shown mathematically that a perfect tuning can’t exist. What does that mean in light of your Axiom 3? Must there be a “possible world”, or “logically possible world”, where mathematics is different and a perfect tuning exists? Or is this property unworthy of being called perfect? But what if we weren’t as good at math, and hadn’t yet proved that perfect tuning is inachievable: would we call the property perfect then? What does your framework say about this example?
Well, I don’t want to take too much credit—it’s Gödel’s framework, not mine. I guess it would be the case that then it would not be a perfect property, since it is logically impossible to achieve. We just wouldn’t know that without doing a bunch of mathematics first, and might be confused to begin with. There’s many similar examples in this vein, for sure. But just because you don’t know something, doesn’t mean it’s not true or false. There either is a perfect property, or there is not.
I’m now going to address something that’s sort of irrelevant to your main point, but perhaps interesting:
It would seem strange to say that frequencies of music are inherently perfect. This seems to rely on a lot of contingent facts about how human ears work, why we like certain sounds and not other sounds. What is pleasant to us. So I wouldn’t call any property like this “perfect” in this sense. But I know it’s meant as an illustrative example, so this is not really to your point.
Wait, if we can be confused whether a property is perfect or imperfect—then why do we assert (in axioms 4 and 6) that some specific properties are perfect? What if they’re also impossible, like the perfect tuning?
For those ones we just say that they are, since we need those for the argument to hold logically. So we stipulate that for those properties there is no confusion, and they are perfect. You are totally allowed to say “hmm, seems suspicious,” but by the definition of these properties, there’s nothing contradictory about asserting that these properties are perfect.
But for a property like “is omniscient” or whatever, we could be more unsure. Since maybe “being omniscient” implies you also “know what it’s like to kick a baby.” And so it implies that you have a property that is not perfect, for example.