In the setup of the question you caused my type checker to crash and so I’m not giving an answer to the math itself so much as talking about the choices I think you might need to make to get the question to type check for me...
Here is a the main offending bit:
So I… attach beliefs to statements of the form S(f)= “my initial degree of belief is represented with probability density function f.”
Well this is not quite possible since the set of all such f is uncountable. However something similar to the probability density trick we use for continuous variables should do the job here as well.
When you get down into the foundations of math and epistemology it is useful to notice when you’re leaping across the entire conceptual universe in question in single giant bounds.
(You can of course, do this, but then to ask “where would I be heading if I kept going like this?” means you leave the topic, or bounce off the walls of your field, or become necessarily interdisciplinary, or something like that.)
When you “attach beliefs to statements” you might be attaching them to string literals (where you might have logical uncertainty about whether they are even syntactically valid), or maybe you’re attaching to the semantic sense (Frege’s Sinn) that you currently impute to those string literals? Or maybe to the semantic sense that you WILL impute to those string literals eventually? Or to the sense that other people who are better at thinking will impute?
...or maybe are you really attaching beliefs to possible worlds (that is, various logically possible versions of the totality of what Frege’s Bedeutungare embedded within) that one or another of those “senses” points at (refers to) and either “rules in or rules out as true” under a correspondence theory of truth...
...or maybe something else? There’s lots of options here!
When I search for [bayesian foundations in event spaces] there’s an weird new paper struggling with fuzzy logic (which is known to cause bayesian logic to explode because fuzzy logic violates the law of the excluded middle) and Pedro Teran’s 2023 “Towards objective Bayesian foundations with fuzzy events” found some sort of (monstrous?) alternative to bayes that don’t work totally the same way?
Basically, there’s a lot of flexibility in how you ground axioms to things that seem like they could be realized in physics (or maybe mere “realized” in lower level intuitively accessible axioms).
Using my default assumptions, my type checker crashed on what you said because all of the ways I could think to ground some of what you said in a coherent way… lead to incoherence based on other things you said.
I was able to auto-correct your example S(f) to something like you having a subjective probability that could be formalized P(“As a skilled subjective Bayesian, fryolysis should represent fryolysis’s uncertainty about a single stable fair coin’s possible mechanical/structural biases that could affect fair tosses with the pdf f(x)=(nh)xh(1−x)n−h after observing h heads out of n tosses of the coin.”)
But then, for your example S(f), you claimed they were uncountable!?
But… you said statements, right?
And so each S(f) (at least if you actually say what the f is using symbols) can be turned into a gödel number, and gödel numbers are COUNTABLY finite, similarly to (and for very similar reasons as) the algebraic numbers.
One of the main ideas with algebraic numbers is that they don’t care if they point to a specific thing hiding in an uncountable infinity. Just because the real neighborhood of π (or “pi” for the search engines) is uncountable doesn’t necessarily make π itself uncountable. We can point to π in a closed and finite way, and since the pointing methods are countable, the pointing methods (tautologically)… arecountable!
You said (1) it was statements you were “attaching” probabilities to but then you said (2) there were uncountably many statements to handle.
I suspect you can only be in reflective equilibrium about at most one of these claims (and maybe neither claim will survive you thinking about this for an adequately long time).
This is being filed as an “Answer” instead of a “Comment” because I am pointing to some of the nearby literature, and maybe that’s all you wanted? <3
In the setup of the question you caused my type checker to crash and so I’m not giving an answer to the math itself so much as talking about the choices I think you might need to make to get the question to type check for me...
Here is a the main offending bit:
When you get down into the foundations of math and epistemology it is useful to notice when you’re leaping across the entire conceptual universe in question in single giant bounds.
(You can of course, do this, but then to ask “where would I be heading if I kept going like this?” means you leave the topic, or bounce off the walls of your field, or become necessarily interdisciplinary, or something like that.)
When you “attach beliefs to statements” you might be attaching them to string literals (where you might have logical uncertainty about whether they are even syntactically valid), or maybe you’re attaching to the semantic sense (Frege’s Sinn) that you currently impute to those string literals? Or maybe to the semantic sense that you WILL impute to those string literals eventually? Or to the sense that other people who are better at thinking will impute?
...or maybe are you really attaching beliefs to possible worlds (that is, various logically possible versions of the totality of what Frege’s Bedeutung are embedded within) that one or another of those “senses” points at (refers to) and either “rules in or rules out as true” under a correspondence theory of truth...
...or maybe something else? There’s lots of options here!
When I search for [possible worlds foundations bayes] the best of the first couple hits is to a team trying to deploy modal logics: The Modal Logic of Bayesian Belief Revision (2017).
When I search for [bayesian foundations in event spaces] there’s an weird new paper struggling with fuzzy logic (which is known to cause bayesian logic to explode because fuzzy logic violates the law of the excluded middle) and Pedro Teran’s 2023 “Towards objective Bayesian foundations with fuzzy events” found some sort of (monstrous?) alternative to bayes that don’t work totally the same way?
Basically, there’s a lot of flexibility in how you ground axioms to things that seem like they could be realized in physics (or maybe mere “realized” in lower level intuitively accessible axioms).
Using my default assumptions, my type checker crashed on what you said because all of the ways I could think to ground some of what you said in a coherent way… lead to incoherence based on other things you said.
I was able to auto-correct your example S(f) to something like you having a subjective probability that could be formalized P(“As a skilled subjective Bayesian, fryolysis should represent fryolysis’s uncertainty about a single stable fair coin’s possible mechanical/structural biases that could affect fair tosses with the pdf f(x)=(nh)xh(1−x)n−h after observing h heads out of n tosses of the coin.”)
But then, for your example S(f), you claimed they were uncountable!?
But… you said statements, right?
And so each S(f) (at least if you actually say what the f is using symbols) can be turned into a gödel number, and gödel numbers are COUNTABLY finite, similarly to (and for very similar reasons as) the algebraic numbers.
One of the main ideas with algebraic numbers is that they don’t care if they point to a specific thing hiding in an uncountable infinity. Just because the real neighborhood of π (or “pi” for the search engines) is uncountable doesn’t necessarily make π itself uncountable. We can point to π in a closed and finite way, and since the pointing methods are countable, the pointing methods (tautologically)… are countable!
You said (1) it was statements you were “attaching” probabilities to but then you said (2) there were uncountably many statements to handle.
I suspect you can only be in reflective equilibrium about at most one of these claims (and maybe neither claim will survive you thinking about this for an adequately long time).
This is being filed as an “Answer” instead of a “Comment” because I am pointing to some of the nearby literature, and maybe that’s all you wanted? <3