Larry, you have not proven that 6 would be a prime number if PA proved 6 was a prime number, because PA does not prove that 6 is a prime number.
The theorem is only true for the phi that it's true for.
The claim that phi must be true because if it's true then it's true, and if it's false then "if PA |- phi then phi" has an officially true conclusion whenever PA does not imply phi, is bogus.
It's simply and obviously bogus, and I don't understand why there was any difficulty about seeing it.
Caledonian, it's possible to care deeply about choices that were made in a seemingly-arbitrary way. For example, a college graduate who takes a job in one of eight cities where he got job offers, might within the year care deeply about that city's baseball team. But if he had taken a different job it would be a completely different baseball team.
You might care about the result of arbitrary choices. I don't say you necessarily will.
It sounds like you're saying it's wrong to care about morals unless they're somehow provably correct? I'm not sure I get your objection. I want to point out that usually when we have a war, most of the people in each country choose sides based on which country they are in. Less than 50% of americans chose to oppose the current iraq fiasco before it happened. Imagine that russia had invaded iraq with the same pretexts we used, all of which would have worked as well for russia as well as they did for us. Russians had more reason than us to fear iraqi nukes, they didn't want iraq supporting chechen terrorists, they thought Saddam was bad man, etc. imagine the hell we would have raised about it.... But I contend that well over a hundred million americans supported the war for no better reason that they were born in america and so they supported invasions by the US military.
Whether or not there's some higher or plausible morality that says we should not choose our morals at random, still the fact is that most of us do choose our morals at random.
I haven't read Roko's blog, but from the reflection in Eliezer's opposition I find I somewhat agree.
To the extent that morality is about what you do, the more you can do the higher the stakes.
If you can drive a car, your driving amplifies your ability to do good. And it amplifies your ability to do bad. If you have a morality that leaves you doing more good than bad, and driving increases the good and the bad you do proportionately, then your driving is a good thing.
True human beings have an insatiable curiousity, and they naturally want to find out about things, and one of the things they like is to find out how to do things. Driving a car is a value in itself until you've done it enough that it gets boring.
But if you have a sense of good and bad apart from curiousity, then it will probably seem like a good thing for good smart people to get the power to do lots of things, while stupid or evil people should get only powers that are reasonably safe for them to have.
"You should care about the moral code you have arbitrarily chosen."
No, I shouldn't. Which seems to be the focal point of this endless 'debate'.
Well, you might choose to care about a moral code you have arbitrarily chosen. And it could be argued that if you don't care about it then you haven't "really" chosen it.
I agree with you that there needn't be any platonic absolute morality that says you ought choose a moral code arbitrarily and care about it, or that if you do happen to choose a moral code arbitrarily that you should then care about it.
We are born with some theorems of right (in analogy to PA).
Kenny, I'd be fascinated to learn more about that. I didn't notice it in my children, but then I wouldn't necessarily notice.
When I was a small child people claimed that babies are born with only a fear of falling and a startle reflex for loud noises. I was pretty sure that was wrong, but it wasn't clear to me what we're born with. It takes time to learn to see. I remember when I invented the inverse square law for vision, and understood why things get smaller when they go farther away. It takes time to notice that parents have their own desires that need to be taken into account.
What is it that we're born with? Do you have a quick link maybe?
Larry, one of them is counterfactual.
If you draw implications on a false asumption then the result is useful only to show that an assumption is false.
So if PA -> 1=2 then PA -> 1<>2. How is that useful?
If PA -> 6 is prime then PA also -> 6 is not prime.
Once you assume that PA implies something that PA actually implies is false, you get a logical contradiction. Either PA is inconsistent or PA does not imply the false thing.
How can it be useful to reason about what we could prove from false premises? What good is it to pretend that PA is inconsistent?
Honestly I do not understand how you can continue calling Eliezer a relativist when he has persistently claimed that what is right doesn't depend on who's asking and doesn't depend on what anyone thinks is right.
Before I say anything else I want you to know that I am not a Communist.
Marx was right about everything he wrote about, but he didn't know everything, I wouldn't say that Marx had all the answers. When the time is ripe the proletariat will inevitably rise up and create a government that will organize the people, it will put everybody to work according to his abilities and give out the results according to the needs, and that will be the best thing that ever happened to anybody. But don't call me a Communist, because I'm not one.
Oh well. Maybe Eliezer is saying something new and it's hard to understand. So we keep mistaking what he's saying for something old that we do understand.
To me he looks like a platonist. Our individual concepts of "right" are imperfect representations of the real true platonic "right" which exists independently of any or all of us.
I am more of a nominalist. I see our concepts as things that get continually re-created. We are born without any concept of "right" and we develop such concepts as we grow up, with the foundations in our families. The degree to which we develop similar concepts of "right" is a triumph for our societies. There's nothing inevitable about it, but there's a value to moral uniformity that goes beyond the particular beliefs.
So for example about "murder". Americans mostly believe that killing is sometimes proper and necessary. Killing in self defense. Policemen must sometimes kill dangerous criminals. It's vitally necessary to kill the enemy in wartime. Etc. We call it "murder" only when it is not justified, so of course we agree that murder is wrong.
We would be better off if we all agreed about when killing is "right". Is it right to kill adulterous spouses? The people they have sex with? Is it right to kill IRS agents? Blasphemers? Four years ago a man I met in a public park threatened to kill me to keep me from voting for Kerry. Was he right? Whatever the rules are about killing, if we all agreed and we knew where we stood, we'd be better off than when we disagree and don't know who to expect will try to kill us.
And that is why in the new society children will be taken from their parents and raised in common dormitories. Because individual families are too diverse, and they don't all raise their children to understand that "from each according to his abilities, and to each according to his needs" is the most basic and important part of morality.
But don't call me a Communist, I already explained that I wasn't a Communist in my first sentence above.
But Larry, PA does not actually say that 6 is prime, and 6 is not prime.
You could say that if PA proved that every theorem is false then every theorem would be false.
Or what would it mean if PA proved that Lob's theorem was false?
It's customary to say that any conclusion from a false premise is true. If 6 is prime then God's in his heaven, everything's right with the world and we are all muppets. Also God's in hell, everything's wrong with the world, and we are all mutant ninja turtles. It doesn't really matter what conclusions you draw from a false premise because the premnise is false.
Your argument about what conclusion we could draw if PA said that 6 is prime is entirely based on a false premise. PA does not say that 6 is prime.
Let me try to say that clearer.
Suppose that A is false.
How the hell are you going to show that if PA proves A true then A will be true, when A is actually false?
If you can't prove what would happen if PA proved A true when A is actually false, then if you can prove that if PA proves A is true then A has to be true, it must be that A is true in the first place.
If this reasoning is correct then there isn't much mystery involved here.
One more time. If PA proves you are a werewolf, then you're really-and-truly a werewolf. PA never proves anything that isn't actually true. OK, say you aren't a werewolf. And I say I have a proof that if PA proves you're a werewolf then you're actually a werewolf. But in reality you aren't a werewolf and PA does not prove you are. How do I prove that PA would do that, when PA does not do that?
Once more through the mill. If PA proves that 6 is a prime number, then 6 is really a prime number. Can you prove that if PA proved 6 was a prime number then 6 would be a prime number? How would you do it?
If you definitely can't prove that, then what does it mean if I show you a proof that if PA proves 7 is a prime number then 7 must actually be prime? If you can't make that proof unless 7 is prime, and you have the proof, then 7 is actually prime.
The problem is with trying to apply material implication when it does not work.
I thought of a simpler way to say it.
If Hillary Clinton was a man, she wouldn't be Bill Clinton's wife. She'd be his husband.
Similarly, if PA proved that 6 was prime, it wouldn't be PA. It would be Bill Clinton's husband. And so ZF would not imply that 6 is actually prime.