The context is *all* applications of probability theory. Look, when I tell you that A or not A is a rule of classical propositional logic, we don’t argue about the context or what assumptions we are relying on. That’s just a universal rule of classical logic. Ditto with conditioning on all the information you have. That’s just one of the rules of epistemic probability theory that *always* applies. The only time you are allowed to NOT condition on some piece of known information is if you would get the same answer whether or not you conditioned on it. When we leave known information Y out and say it is “irrelevant”, what that means is that Pr(A | Y and X) = Pr(A | X), where X is the rest of the information we’re using. If I can show that these probabilities are NOT the same, then I have proven that Y is, in fact, relevant.
“Look, when I tell you that A or not A is a rule of classical propositional logic, we don’t argue about the context or what assumptions we are relying on”—Actually, you get questions like, “This sentence is false”, which fall outside out classical propositional logic. This is why it is important to understand the limits which apply.
But within what context? You can’t just take a formula or rule and apply it without understanding the assumptions it is reliant upon.
The context is *all* applications of probability theory. Look, when I tell you that A or not A is a rule of classical propositional logic, we don’t argue about the context or what assumptions we are relying on. That’s just a universal rule of classical logic. Ditto with conditioning on all the information you have. That’s just one of the rules of epistemic probability theory that *always* applies. The only time you are allowed to NOT condition on some piece of known information is if you would get the same answer whether or not you conditioned on it. When we leave known information Y out and say it is “irrelevant”, what that means is that Pr(A | Y and X) = Pr(A | X), where X is the rest of the information we’re using. If I can show that these probabilities are NOT the same, then I have proven that Y is, in fact, relevant.
“Look, when I tell you that A or not A is a rule of classical propositional logic, we don’t argue about the context or what assumptions we are relying on”—Actually, you get questions like, “This sentence is false”, which fall outside out classical propositional logic. This is why it is important to understand the limits which apply.