Dear Luca
Thank you very much for your comments and kind words.
You note some potential typos or mistakes. Let me look at these first.
> Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.
You are correct. It should be:
Given: If A is true, then B is true.
Learn: B is false.
Deduce: A is false
> And similarly WS-2: Learn A false, then probably B false.
Again, you are correct. It should be:
Given: If A is true, then B is true.
Learn: A is false.
Infer: B is less plausible.
> Finally on page 8. If u is undecidable and u follows from x1 and x2 and x3, than at least one of these must be undecidable. But the other also could be true. If one is undecidable then the conjunction also must be undecidable. (They do not to be true).
I am not quite following what you are saying here. If u is undecidable and u is the disjunction (OR) of three atomic statements: u = x1 or x2 or x3, then
at least one of x1, x2, and x3 must be undecidable. That you seem to agree with based on your following comment. So let's say that it is x1 that is undecidable. Then x2 and x3 must be either undecidable or false. If one of x2 and x3 were true, then we could deduce that u was true, and u would not be undecidable.
Similar arguments apply to the compliment of u. And the result is that the existence of an undecidable statement u in the hypothesis space implies that at least two of the atomic statements are undecidable. Now it is not assumed that the atomic statements are true. The atomic statements are mutually exclusive and exhaustive so that one and only one of them is true. However, that true atomic statement must be one of the undecidable ones.
> But aren't the atomic statement usually the axioms of a formal language? But aren't the axioms by definition true?
The atomic statements are not necessarily the axioms of a formal language.
I look forward to reading your essay. And that will probably help me to better understand your question.
Later you ask:
> Under which physical conditions is measure theory applicable? And the sum rule?
It would be best for me to point you to one of our most recent papers on this topic:
https://onlinelibrary.wiley.com/doi/full/10.1002/andp.201800057
In short, one needs to have closure so that if you combine one set of pencils with another set of pencils, you get a set of pencils. The combination operation must be commutative and associative so that shuffling does not matter. And last, there can be no problem with continuing to combine things.
But now, looking at your question, it appears that you are interested in physical properties. In terms of closure, this would come down to how we choose to classify things. Combining a set of pens with a set of pencils does not result in a set of pens. But if I choose to think of them as writing implements, then I have closure. So part of the applicability has to do with the choices we make when we classify things. I hope that this helps. Although, I expect that I will understand your question better after I read your essay.
Thank you very much for pointing out my two typos. I really appreciate it.
And I hope that I have, at least begun, to answer your questions.
Sincerely,
Kevin Knuth