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It's difficult to assign probability to incoherent statements, because since we can't mean anything by them, we can't assert a referent to the statement -- in that sense, the probability is indeterminate (additionally, one could easily imagine a language in which a statement such as "the green is either" has a perfectly coherent meaning -- and we can't say that's not what we meant, since we didn't mean anything). Recall also that each probability zero statement implies a probability one statement by its denial and vice versa, so one is equally capable of imagining them, if in a contrived way.
Putting this in a slightly more coherent way. (I was having some trouble understanding the explanation, so I broke it down into layman's terms, might make it more easily understandable)
If I assign P(0) to "Green is either" Then I assign P(1) to the statement "Green is not either"
If you assign absolute certainty to any one statement you are, by definition assigning absolute impossibility to all other possibilities.
It's difficult to assign probability to incoherent statements, because since we can't mean anything by them, we can't assert a referent to the statement -- in that sense, the probability is indeterminate (additionally, one could easily imagine a language in which a statement such as "the green is either" has a perfectly coherent meaning -- and we can't say that's not what we meant, since we didn't mean anything). Recall also that each probability zero statement implies a probability one statement by its denial and vice versa, so one is equally capable of imagining them, if in a contrived way.
Putting this in a slightly more coherent way. (I was having some trouble understanding the explanation, so I broke it down into layman's terms, might make it more easily understandable)
If I assign P(0) to "Green is either" Then I assign P(1) to the statement "Green is not either"
If you assign absolute certainty to any one statement you are, by definition assigning absolute impossibility to all other possibilities.
That's a useful heuristic to combat our tendency to see patterns that aren't there. It's not strictly necessary.
Another way to solve the same problem is to look at the first 500 questionnaires first. The scientists then notice that there is a correlation between excessive sneezing and large ears. Now the scientists look at the last 500 questionnaires -- an independent experiment. If these questionnaires also show correlation, that is also evidence for the hypothesis, although it's necessarily weaker than if another 1000-person poll were conducted.
So this shows that a second experiment isn't necessary if we think ahead. Now the question is, if we've already foolishly looked at all 1000 results, is there any way to recover?
It turns out that what can save us is math. There's a bunch of standard tests for significance when lots of variables are compared. But the basic idea is the following: we can test if the correlation between sneezing and ears is high, by computing our prior for what sort of correlation the two most closely correlated variables would show.
Note that although our prior for two arbitrary variables might be centered at 0 correlation, our prior for two variables that are selected by choosing the highest correlation should be centered at some positive value. In other words: even if the questions were all about unrelated things, we expect a certain amount of correlation between some things to happen by chance. But we can figure out how much correlation to expect from this phenomenon! And by doing some math, we might be able to show that the correlation between sneezing and having ears is too high to be explained in this way.
Okay, that makes tons more sense, I apparently wasn't thinking too clearly when I wrote the first post. (plus I didn't know about the standard tests)
Thanks for setting me straight.
Something popped into my mind while I was reading about the example in the very beginning. What about research that goes out to prove one thing, but discovers something else?
Group of scientists want to see if there's a link between the consumption of Coca-Cola and stomach cancer. They put together a huge questionnaire full of dozens of questions and have 1000 people fill it out. Looking at the data they discover that there is no correlation between Coca-Cola drinking and stomach cancer, but there is a correlation between excessive sneezing and having large ears.
So now we have a group of scientists who set out to test correlation A, but found correlation B in the data instead. Should they publish a paper about correlation B?
Before they publish anything (other than a article on Coca-Cola not being related to stomach cancer) they should first use a different test group in order to determine that the first result wasn't a sampling fluke or otherwise biased, (Perhaps sneezing wasn't causing large ears after all, or large ears were correlated to something that also caused sneezing.)
What brought the probability to your attention in the first place shouldn't be what proves it.
If A then B is a separate experiment than If C then D and should require separate additional proof.
I would be willing to run a study like this for you–it would have to be long distance/online, since I'm in Canada, but I could get some of the local LW-Ottawa people to help set up an online survey format.
I'd love to participate in such a study, I've noticed my moods go all over the place and would like to see how such a survey would work.
Have you played any, or are you a fan of, interactive fiction? If so, and you haven't played this particular game before, I recommend you look at The Gostak. It's an entire story written using standard IF principles and conventions, only every noun, verb, adjective, and adverb has been changed to be semantically unrecognizable but syntactically familiar to an English speaker. It is based on a thought experiment from The Meaning of Meaning; in short, the meaningless sentence "The gostak distims the doshes" allows you to generate three interconnected floating beliefs, one about the gostak, one about distimming, and one about doshes.
The core of language and communication is common convention. If your intent is to create a visually-pleasing pattern of pixels on a screen or ink on paper, you can change any part of your writing you like. You can change every word to a made-up word that only uses half-height letters, use the capital X as your sentence-ending punctuation, and as long as the story is internally consistent, people could still conceivably generate meaning from it. If they try hard enough, they might even generate the meaning you intended, but you would by necessity have a smaller audience.
Is your primary goal to communicate your ideas to others using a common language, or is it to create visual artwork? No matter your personal feelings on the aesthetics of different parts of the language, if you violate conventions, all you're doing is harming the former in favor of the latter.
Inspired by maia's post:
“When life gives you lemons, don’t make lemonade. Make life take the lemons back! Get mad! I don’t want your damn lemons, what the hell am I supposed to do with these? Demand to see life’s manager! Make life rue the day it thought it could give Cave Johnson lemons! Do you know who I am? I’m the man who’s gonna burn your house down! With the lemons! I’m gonna get my engineers to invent a combustible lemon that burns your house down!”
---Cave Johnson, Portal 2
Huh, I scrolled past this and read Nisan's post first, by the time I got any further this was already running through my head.
Not so sure that this is a good rationalist quote though.
Inspired by maia's post:
“When life gives you lemons, don’t make lemonade. Make life take the lemons back! Get mad! I don’t want your damn lemons, what the hell am I supposed to do with these? Demand to see life’s manager! Make life rue the day it thought it could give Cave Johnson lemons! Do you know who I am? I’m the man who’s gonna burn your house down! With the lemons! I’m gonna get my engineers to invent a combustible lemon that burns your house down!”
---Cave Johnson, Portal 2
Huh, I scrolled past this and read Nisan's post first, by the time I got any further this was already running through my head.
Not so sure that this is a good rationalist quote though.
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Meditation:
So far we've talked about two kinds of meaningfulness and two ways that sentences can refer; a way of comparing to physical things found by following pinned-down causal links, and logical reference by comparison to models pinned-down by axioms. Is there anything else that can be meaningfully talked about? Where would you find justice, or mercy?
Right, response to the meditation:
It gets rather difficult talking about human mental constructs, let's begin by asking myself where would I find justice/mercy; almost immediately (which means that I need to do some more thinking) I find that I think of human emotional constructs as a side effect of society and it's group mindset,
I would find that by grinding down the universe to it's component molecules would completely fail to find any number of things that humanity finds important; Humanity for one, to me rationalism is, above all the study of the universe and what it contains. And yet when it comes to most psychological phenomenon the models start to break down, does this mean that a more refined model would be equally unable to describe the phenomenon, not necessarily. Because as rationalists one of our key teachings is that we can observe something by studying it's causes and effects; justice and mercy exists insofar as we as humans can comprehend their nature. They exist because we can determine the differences between a universe where they exist and the ones where they don't exist.
-reposted in the right section