Later Chapman wrote the more technical Probability theory does not extend logic in which Chapman who has a MIT AI PHD shows how the core claim that probability theory is an extension of logic that’s made in the sequences is wrong.
As far as I can tell, his piece is mistaken. I’m going to copypaste what I’ve written about it elsewhere:
So I looked at Chapman’s “Probability theory does not extend logic” and some things aren’t making sense. He claims that probability theory does extend propositional logic, but not predicate logic.
But if we assume a countable universe, probability will work just as well with universals and existentials as it will with conjunctions and disjunctions. Even without that assumption, well, a universal is essentially an infinite conjunction, and an existential statement is essentially an infinite disjunction. It would be strange that this case should fail.
His more specific example is: Say, for some x, we gain evidence for “There exist distinct y and y’ with R(x,y)”, and update its probability accordingly; how should we update our probability for “For all x, there exists a unique y with R(x,y)”? Probability theory doesn’t say, he says. But OK — let’s take this to a finite universe with known elements. Now all those universals and existentials can be rewritten as finite conjunctions and disjunctions. And probability theory does handle this case?
I mean… I don’t think it does. If you have events A and B and you learn C, well, you update P(A) to P(A|C), and you update P(A∩B) to P(A∩B|C)… but the magnitude of the first update doesn’t determine the magnitude in the second. Why should it when the conjunction becomes infinite? I think that Chapman’s claim about a way in which probability theory does not extend predicate logic, is equally a claim about a way in which it does not extend propositional logic. As best I can tell, it extends both equally well.
(Also here is a link to a place where I posted this and got into an argument with Chapman about this that people might find helpful?)
But if we assume a countable universe, probability will work just as well with universals and existentials as it will with conjunctions and disjunctions.
If you regard probability as a tool for thinking , which is pretty reasonable, it’s not going to work, in the sense of being usable, if it contains countable infinities or very large finite numbers.
Also, it is not a good idea to build assumptions about how the world works into the tools you are using to figure out how the world works.
But the question wasn’t about whether it’s usable. The question was about whether there is some sense in which probability extends propositional logic but not predicate logic.
But OK — let’s take this to a finite universe with known elements.
If everything is known you don’t need probability theory in the first place. You just know what happens. See Probability is in the Mind.
Most of the factors that we encounter are not known and good decision making is about dealing with the unknown and part of the promise of Bayesianism is that it helps you dealing with the unknown.
So, I must point out that a finite universe with known elements isn’t actually one where everything is known, although it certainly is one where we know way more than we ever do in the real world. But this is irrelevant. I don’t see how anything you’re saying relates to the claim is that probability theory extends propositional logic but not predicate logic.
Why is it irrelevant when you assume a world where the agent who has to make the decision knows more than they actually know?
Decision theory is about making decisions based on certain information that known.
I don’t see how anything you’re saying relates to the claim is that probability theory extends propositional logic but not predicate logic.
I haven’t studied the surrounding math but as far as I understand according to Cox’s Theorem probability theory does extend propositional calculus without having to make additional assumptions about finite universe or certain things being known.
Why is it irrelevant when you assume a world where the agent who has to make the decision knows more than they actually know? Decision theory is about making decisions based on certain information that known.
I think you’ve lost the chain a bit here. We’re just discussing to what extent probability theory does or does not extend various forms of logic. The actual conditions in the real world do not affect that. Now obviously if it only extends it in conditions that do not hold in the real world, then that is important to know; but if that were the case then “probability theory extends logic” would be a way too general statement anyhow and I hope nobody would be claiming that!
(And actually if you read the argument with Chapman that I linked, I agree that “probability theory extends logic” is a misleading claim, and that it indeed mostly does not extend logic. The question isn’t whether it extends logic, the question is whether propositional and predicate logic behave differently here.)
But again all of this is irrelevant because nobody is claiming anything like that! I mentioned a finite universe, where predicate logic essentially becomes propositional logic, to illustrate a particular point—that probability theory does not extend propositional logic in the sense Chapman claims it does. I didn’t bring it up to say “Oho well in a finite universe it does extend predicate logic, therefore it’s correct to say that probability theory extends predicate logic”; I did the opposite of that! At no point did I make any actual-rather-than-illustrative assumption to the effect that that the real world is or is like a finite universe. So objecting that it isn’t has no relevance.
I haven’t studied the surrounding math but as far as I understand according to Cox’s Theorem probability theory does extend propositional calculus without having to make additional assumptions about finite universe or certain things being known.
Cox’s theorem actually requires a “big world” assumption, which IINM is incompatible with a finite universe!
I think this is getting off-track a little. To review: Chapman claimed that, in a certain sense, probability theory extends propositional but not predicate logic. I claimed that, in that particular sense, it actually extends both of them equally well. (Which is not to say that it truly does extend both of them, to be clear—if you read the argument with Chapman that I linked, I actually agree that “probability theory extends logic” is a misleading claim, and that it mostly doesn’t.)
So now the question here is, what are you arguing for? If you’re arguing for Chapman’s original claim, the relevance of your statement of Cox’s theorem is unclear, as it’s not clear that this relates to the particular sense he was talking about.
If you’re arguing for a broader version of Chapman’s claim—broadening the scope to allow any sense rather than the particular one he claimed—then you need to exhibit a sense in which probability theory extends propositional logic but not predicate logic. I can buy the claim that Cox’s theorem provides a certain sense in which probability theory extends propositional logic. And, though you haven’t argued for it, I can even buy the claim that this is a sense in which it does not extend predicate logic [edit: at least, in an uncountable universe]. But, well, the problem is that regardless if it’s true, this broader claim—or this particular version of it, anyway—just doesn’t seem to have much to do with his original one.
As far as I can tell, his piece is mistaken. I’m going to copypaste what I’ve written about it elsewhere:
So I looked at Chapman’s “Probability theory does not extend logic” and some things aren’t making sense. He claims that probability theory does extend propositional logic, but not predicate logic.
But if we assume a countable universe, probability will work just as well with universals and existentials as it will with conjunctions and disjunctions. Even without that assumption, well, a universal is essentially an infinite conjunction, and an existential statement is essentially an infinite disjunction. It would be strange that this case should fail.
His more specific example is: Say, for some x, we gain evidence for “There exist distinct y and y’ with R(x,y)”, and update its probability accordingly; how should we update our probability for “For all x, there exists a unique y with R(x,y)”? Probability theory doesn’t say, he says. But OK — let’s take this to a finite universe with known elements. Now all those universals and existentials can be rewritten as finite conjunctions and disjunctions. And probability theory does handle this case?
I mean… I don’t think it does. If you have events A and B and you learn C, well, you update P(A) to P(A|C), and you update P(A∩B) to P(A∩B|C)… but the magnitude of the first update doesn’t determine the magnitude in the second. Why should it when the conjunction becomes infinite? I think that Chapman’s claim about a way in which probability theory does not extend predicate logic, is equally a claim about a way in which it does not extend propositional logic. As best I can tell, it extends both equally well.
(Also here is a link to a place where I posted this and got into an argument with Chapman about this that people might find helpful?)
If you regard probability as a tool for thinking , which is pretty reasonable, it’s not going to work, in the sense of being usable, if it contains countable infinities or very large finite numbers.
Also, it is not a good idea to build assumptions about how the world works into the tools you are using to figure out how the world works.
But the question wasn’t about whether it’s usable. The question was about whether there is some sense in which probability extends propositional logic but not predicate logic.
If everything is known you don’t need probability theory in the first place. You just know what happens. See Probability is in the Mind.
Most of the factors that we encounter are not known and good decision making is about dealing with the unknown and part of the promise of Bayesianism is that it helps you dealing with the unknown.
So, I must point out that a finite universe with known elements isn’t actually one where everything is known, although it certainly is one where we know way more than we ever do in the real world. But this is irrelevant. I don’t see how anything you’re saying relates to the claim is that probability theory extends propositional logic but not predicate logic.
Edit: oops, wrote “point” instead of “world”
Why is it irrelevant when you assume a world where the agent who has to make the decision knows more than they actually know?
Decision theory is about making decisions based on certain information that known.
I haven’t studied the surrounding math but as far as I understand according to Cox’s Theorem probability theory does extend propositional calculus without having to make additional assumptions about finite universe or certain things being known.
I think you’ve lost the chain a bit here. We’re just discussing to what extent probability theory does or does not extend various forms of logic. The actual conditions in the real world do not affect that. Now obviously if it only extends it in conditions that do not hold in the real world, then that is important to know; but if that were the case then “probability theory extends logic” would be a way too general statement anyhow and I hope nobody would be claiming that!
(And actually if you read the argument with Chapman that I linked, I agree that “probability theory extends logic” is a misleading claim, and that it indeed mostly does not extend logic. The question isn’t whether it extends logic, the question is whether propositional and predicate logic behave differently here.)
But again all of this is irrelevant because nobody is claiming anything like that! I mentioned a finite universe, where predicate logic essentially becomes propositional logic, to illustrate a particular point—that probability theory does not extend propositional logic in the sense Chapman claims it does. I didn’t bring it up to say “Oho well in a finite universe it does extend predicate logic, therefore it’s correct to say that probability theory extends predicate logic”; I did the opposite of that! At no point did I make any actual-rather-than-illustrative assumption to the effect that that the real world is or is like a finite universe. So objecting that it isn’t has no relevance.
Cox’s theorem actually requires a “big world” assumption, which IINM is incompatible with a finite universe!
I think this is getting off-track a little. To review: Chapman claimed that, in a certain sense, probability theory extends propositional but not predicate logic. I claimed that, in that particular sense, it actually extends both of them equally well. (Which is not to say that it truly does extend both of them, to be clear—if you read the argument with Chapman that I linked, I actually agree that “probability theory extends logic” is a misleading claim, and that it mostly doesn’t.)
So now the question here is, what are you arguing for? If you’re arguing for Chapman’s original claim, the relevance of your statement of Cox’s theorem is unclear, as it’s not clear that this relates to the particular sense he was talking about.
If you’re arguing for a broader version of Chapman’s claim—broadening the scope to allow any sense rather than the particular one he claimed—then you need to exhibit a sense in which probability theory extends propositional logic but not predicate logic. I can buy the claim that Cox’s theorem provides a certain sense in which probability theory extends propositional logic. And, though you haven’t argued for it, I can even buy the claim that this is a sense in which it does not extend predicate logic [edit: at least, in an uncountable universe]. But, well, the problem is that regardless if it’s true, this broader claim—or this particular version of it, anyway—just doesn’t seem to have much to do with his original one.