You write: “When you look at proposed alternatives to probability theory, however, the important point of departure seems to be the assumption, used in Cox’s Theorem and variants, that the credibility of a proposition can be represented as a *single* number, rather than requiring, say, a pair of numbers as in Dempster-Shafer belief-function theory.”
This hits the nail on the head. And it means that the way E. T. Jaynes wants to interpret Cox’s Theorem is just flat out wrong, just like David Chapman tries to explain. It might be easier to see this by not looking at predicate logic, but instead looking at the canonical non-classical propositional logic: intuitionistic logic. You might interpret it as the logic of open subsets of a topological space. If you interpret classical logic in that context as the logic of subsets of a given set (the universe set), it becomes pretty obvious why the restriction to a *single* number is basically a bad idea, if taken literally (like E. T. Jaynes does).
Maybe I want to say more and provide some links with examples where E. T. Jaynes has done some really nice work, examples where he has totally misunderstood some concepts from mathematically logic and said some unbelievably stupid things, and examples where the misunderstandings put into the world by E. T. Jaynes lead some other people into some quite questionable actions. But let me first see, how the comment above gets received.
kevin@ksvanhorn.comsays
There is a more recent result that derives probability theory as the unique extension of classical propositional logic (CPL) without assuming that the credibility of a proposition can be represented as a single real number. Instead, this is found to be a consequence of requiring that the extended logic retain various properties of CPL.
You write: “When you look at proposed alternatives to probability theory, however, the important point of departure seems to be the assumption, used in Cox’s Theorem and variants, that the credibility of a proposition can be represented as a *single* number, rather than requiring, say, a pair of numbers as in Dempster-Shafer belief-function theory.”
This hits the nail on the head. And it means that the way E. T. Jaynes wants to interpret Cox’s Theorem is just flat out wrong, just like David Chapman tries to explain. It might be easier to see this by not looking at predicate logic, but instead looking at the canonical non-classical propositional logic: intuitionistic logic. You might interpret it as the logic of open subsets of a topological space. If you interpret classical logic in that context as the logic of subsets of a given set (the universe set), it becomes pretty obvious why the restriction to a *single* number is basically a bad idea, if taken literally (like E. T. Jaynes does).
Maybe I want to say more and provide some links with examples where E. T. Jaynes has done some really nice work, examples where he has totally misunderstood some concepts from mathematically logic and said some unbelievably stupid things, and examples where the misunderstandings put into the world by E. T. Jaynes lead some other people into some quite questionable actions. But let me first see, how the comment above gets received.
There is a more recent result that derives probability theory as the unique extension of classical propositional logic (CPL) without assuming that the credibility of a proposition can be represented as a single real number. Instead, this is found to be a consequence of requiring that the extended logic retain various properties of CPL.
“From propositional logic to plausible reasoning: a uniqueness theorem.”