This is simply saying that, given we’ve randomly selected a truth table, the probability that every snark is a boojum is 0.4.
Maybe I misunderstand your quantifiers, but I don’t think it says that. It says that for every monster, if we randomly pick a truth table on which it’s a snark, the propability that it’s also a boojum on that table is 0.4. I think the formalism is right here and your description of it wrong, because thats just what I would expect P(boojum|snark) to mean.
I think the source of the problem here is that Chapman isn’t thinking about subjective propability. So he sees something like “Cats are black with 40% propability” and wonders how this could apply to one concrete cat. and he similarly thinks about the dummy variable of the quantifier that way, and so how could all these different cats have the same propability? But the statement actually means something like “If all I know about something is that it’s a cat, I give it a 40% propability of being black”.
You are right. Thank you for the correction, and I like your description which I hope you don’t mind me using (with credit) when I edit this post. My error was not realizing that P(boojum(x)|snark(x)) is the marginal probability for one particular row in the table. Even though the syntax is (hopefully) valid, this stuff is still confusing to think about!
and he similarly thinks about the dummy variable of the quantifier that way, and so how could all these different cats have the same propability?
I’m not quite sure how Chapman is interpreting these things, but what you are describing does sound like a reasonable objection for someone who interprets these probabilities to be physically “real” (whatever that means). Though Chapman is the one who chose to assert that all conditional probabilities are 0.4 in this example. I think he want’s to conclude that such a “strong” logical statement as a “for-all” is nonsensical in the way you are describing, whereas something like “for 90% of x, P(boojum(x)|snark(x)) is between 0.3 and 0.5″ would be more realistic.
But the statement actually means something like “If all I know about something is that it’s a cat, I give it a 40% propability of being black”.
Or you can just interpret this as being a statement about your model, i.e. without knowing anything about particular cats, you decided to model the probability any each cat is (independently) black as 40%. You can choose to make these probabilities different if you like.
Maybe I misunderstand your quantifiers, but I don’t think it says that. It says that for every monster, if we randomly pick a truth table on which it’s a snark, the propability that it’s also a boojum on that table is 0.4. I think the formalism is right here and your description of it wrong, because thats just what I would expect P(boojum|snark) to mean.
I think the source of the problem here is that Chapman isn’t thinking about subjective propability. So he sees something like “Cats are black with 40% propability” and wonders how this could apply to one concrete cat. and he similarly thinks about the dummy variable of the quantifier that way, and so how could all these different cats have the same propability? But the statement actually means something like “If all I know about something is that it’s a cat, I give it a 40% propability of being black”.
You are right. Thank you for the correction, and I like your description which I hope you don’t mind me using (with credit) when I edit this post. My error was not realizing that P(boojum(x)|snark(x)) is the marginal probability for one particular row in the table. Even though the syntax is (hopefully) valid, this stuff is still confusing to think about!
I’m not quite sure how Chapman is interpreting these things, but what you are describing does sound like a reasonable objection for someone who interprets these probabilities to be physically “real” (whatever that means). Though Chapman is the one who chose to assert that all conditional probabilities are 0.4 in this example. I think he want’s to conclude that such a “strong” logical statement as a “for-all” is nonsensical in the way you are describing, whereas something like “for 90% of x, P(boojum(x)|snark(x)) is between 0.3 and 0.5″ would be more realistic.
Or you can just interpret this as being a statement about your model, i.e. without knowing anything about particular cats, you decided to model the probability any each cat is (independently) black as 40%. You can choose to make these probabilities different if you like.