if the is indeed a typo, please correct it at the top level post and link to this comment. The broader point is that the interpretation of P( H | X2, M) is probability of heads conditioned on Monday and X2, and P (H |X2) is probability of heads conditioned on X2. In the later paragraphs, you seem to use the second interpretation. In fact, It seems your whole post’s argument and “solution” rests on this typo.
Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb’s because one defines “CDT” as rational. The point of probability theory is to be helpful in constructing rational agents. If the agents that your probability theory leads to are not winning bets with the information given to them by said theory, the theory has questionable usefulness.
Just to clarify, I have read Probability, the Logic of science, Bostrom’s and Armstrong’s papers on this. I have also read https://meaningness.com/probability-and-logic. The question of the relationship of probability and logic is not clear cut. And as Armstrong has pointed out, decisions can be more easily determined than probabilities, which means it’s possible the ideal relationship between decision theory and probability theory is not clear cut, but that’s a broader philosophical point that needs a top level post.
No, P(H | X2, M) is Pr(H∣X2,M), and notPr(H∣X2,Monday). Recall that M is the proposed model. If you thought it meant “today is Monday,” I question how closely you read the post you are criticizing.
I find it ironic that you write “Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb’s”—in an earlier version of this blog post I brought up Newcomb myself as an example of why I am skeptical of standard betting arguments (not sure why or how that got dropped.) The point was that standard betting arguments can get the wrong answer in some problems involving unusual circumstances where a more comprehensive decision theory is required (perhaps FDT).
Re constructing rational agents: this is one use of probability theory; it is not “the point”. We can discuss logic from a purely analytical viewpoint without ever bringing decisions and agents into the discussion. Logic and epistemology are legitimate subjects of their own quite apart from decision theory. And probability theory is the unique extension of classical propositional logic to handle intermediate degrees of plausibility.
You say you have read PTLOS and others. Have you read Cox’s actual paper, or any or detailed discussions of it such as Paris’s discussion in The Uncertain Reasoner’s Companion, or my own “Constructing a Logic of Plausible Inference: A Guide to Cox’s Theorem”? If you think that Cox’s Theorem has too many arguable technical requirements, then I invite you to read my paper, “From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem” (preprint here). That proof assumes only that certain existing properties of classical propositional logic be retained when extending the logic to handle degrees of plausibility. It does not assume any particular functional decomposition of plausibilities, nor does it even assume that plausibilities must be real numbers. As with Cox, we end up with the result that the logic must be isomorphic to probability theory. In addition, the theorem gives the required numeric value for a probability Pr(A∣X) when X contains, in propositional form, all of the background information we are using to assess the probability of A. How much more “clear cut” do you demand the relationship between logic and probability be?
Regardless, for my argument about indexicals all that is necessary is that probability theory deals with classical propositions.
if the is indeed a typo, please correct it at the top level post and link to this comment. The broader point is that the interpretation of P( H | X2, M) is probability of heads conditioned on Monday and X2, and P (H |X2) is probability of heads conditioned on X2. In the later paragraphs, you seem to use the second interpretation. In fact, It seems your whole post’s argument and “solution” rests on this typo.
Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb’s because one defines “CDT” as rational. The point of probability theory is to be helpful in constructing rational agents. If the agents that your probability theory leads to are not winning bets with the information given to them by said theory, the theory has questionable usefulness.
Just to clarify, I have read Probability, the Logic of science, Bostrom’s and Armstrong’s papers on this. I have also read https://meaningness.com/probability-and-logic. The question of the relationship of probability and logic is not clear cut. And as Armstrong has pointed out, decisions can be more easily determined than probabilities, which means it’s possible the ideal relationship between decision theory and probability theory is not clear cut, but that’s a broader philosophical point that needs a top level post.
In the meantime, Fix Your Math!
No, P(H | X2, M) is Pr(H∣X2,M), and not Pr(H∣X2,Monday). Recall that M is the proposed model. If you thought it meant “today is Monday,” I question how closely you read the post you are criticizing.
I find it ironic that you write “Dismissing betting arguments is very reminiscent of dismissing one-boxing in Newcomb’s”—in an earlier version of this blog post I brought up Newcomb myself as an example of why I am skeptical of standard betting arguments (not sure why or how that got dropped.) The point was that standard betting arguments can get the wrong answer in some problems involving unusual circumstances where a more comprehensive decision theory is required (perhaps FDT).
Re constructing rational agents: this is one use of probability theory; it is not “the point”. We can discuss logic from a purely analytical viewpoint without ever bringing decisions and agents into the discussion. Logic and epistemology are legitimate subjects of their own quite apart from decision theory. And probability theory is the unique extension of classical propositional logic to handle intermediate degrees of plausibility.
You say you have read PTLOS and others. Have you read Cox’s actual paper, or any or detailed discussions of it such as Paris’s discussion in The Uncertain Reasoner’s Companion, or my own “Constructing a Logic of Plausible Inference: A Guide to Cox’s Theorem”? If you think that Cox’s Theorem has too many arguable technical requirements, then I invite you to read my paper, “From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem” (preprint here). That proof assumes only that certain existing properties of classical propositional logic be retained when extending the logic to handle degrees of plausibility. It does not assume any particular functional decomposition of plausibilities, nor does it even assume that plausibilities must be real numbers. As with Cox, we end up with the result that the logic must be isomorphic to probability theory. In addition, the theorem gives the required numeric value for a probability Pr(A∣X) when X contains, in propositional form, all of the background information we are using to assess the probability of A. How much more “clear cut” do you demand the relationship between logic and probability be?
Regardless, for my argument about indexicals all that is necessary is that probability theory deals with classical propositions.
I responded to David Chapman’s essay (https://meaningness.com/probability-and-logic) a couple of years ago here.