I agree with what you’re trying to do, but I don’t think this proposed construction does it. There are a lot of really complicated statements of propositional calculus which turn out to be either tautologically true or tautologically false, and I’d like to be able to speak of uncertainty of those statements as well.
Constructions like this (or like fuzzy logic) don’t capture the principle that I take to be self-evident when discussing bounded agents, that new deductions don’t instantly propagate globally: if I’ve deduced A and also deduced (A implies B), I may not yet have deduced B. (All the more so when we make complicated examples.)
I don’t think the construction actually requires instant propagation. It requires a certain calculation to be made when you wish to assign a probability to a particular statement, and this calculation is provably finite.
In your example, you are free to have X contain “A” and “A=>B”, and not contain “B”, as long as you don’t assign a probability to B. When you wish to do so, you have to do the calculation, which will find that B∈OLC(X), and so will assign P(B)=1. Assigning any other value would indeed be inconsistent for any reasonable definition of probability, because if you know that A=>B, then you have to know that P(A)≤P(B), and then if P(A)=1, then P(B) must also be 1.
I agree with what you’re trying to do, but I don’t think this proposed construction does it. There are a lot of really complicated statements of propositional calculus which turn out to be either tautologically true or tautologically false, and I’d like to be able to speak of uncertainty of those statements as well.
Constructions like this (or like fuzzy logic) don’t capture the principle that I take to be self-evident when discussing bounded agents, that new deductions don’t instantly propagate globally: if I’ve deduced A and also deduced (A implies B), I may not yet have deduced B. (All the more so when we make complicated examples.)
I don’t think the construction actually requires instant propagation. It requires a certain calculation to be made when you wish to assign a probability to a particular statement, and this calculation is provably finite.
In your example, you are free to have X contain “A” and “A=>B”, and not contain “B”, as long as you don’t assign a probability to B. When you wish to do so, you have to do the calculation, which will find that B∈OLC(X), and so will assign P(B)=1. Assigning any other value would indeed be inconsistent for any reasonable definition of probability, because if you know that A=>B, then you have to know that P(A)≤P(B), and then if P(A)=1, then P(B) must also be 1.