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It seems clear to me that statements expressing logical or probabilistic laws like or are "analytic". Similar to "Bachelors are unmarried".
The truth of a statement in general is determined by two things, it's meaning and what the world is like. But for some statements the latter part is irrelevant, and their meanings alone are sufficient to determine their truth or falsity.
Not to remove all limitations: I think the probability axioms are a sort of "logic of sets of beliefs". If the axioms are violated the belief set seems to be irrational. (Or at least the smallest incoherent subset that, if removed, would make the set coherent.) Conventional logic doesn't work as a logic for belief sets, as the preface and lottery paradox show, but subjective probability theory does work. As a justification for the axioms: that seems a similar problem to justifying the tautologies / inference rules of classical logic. Maybe an instrumental Dutch book argument works. But I do think it does come down to semantic content: If someone says "P(A and B)>P(A)" it isn't a sign of incoherence if he means with "and" what I mean with "or".
Regarding the map representing the territory: That's a more challenging thing to formalize than just logic or probability theory. It would amount to a theory of induction. We would need to formalize and philosophically justify at least something like Ockham's razor. There are some attempts, but I think no good solution.
Well, technically P(Ω)=1 is an axiom, so you do need a sample space if you want to adhere to the axioms.
For a propositional theory this axiom is replaced with , i.e. a tautology in classical propositional logic receives probability 1.
But sure, if you do not care about accurate beliefs and systematic ways to arrive to them at all, then the question is, indeed, not interesting. Of course then it's not clear what use is probability theory for you, in the first place.
Degrees of belief adhering to the probability calculus at any point in time rules out things like "Mary is a feminist and a bank teller" to simultaneously receive a higher degree of belief than "Mary is a bank teller". It also requires e.g. that if and then . That's called "probabilism" or "synchronic coherence".
Another assumption is typically that after "observing" . This is called "conditionalization" or sometimes "diachronic coherence".
1And how would you know which worlds are possible and which are not?
Yes, that's why I only said "less arbitrary".
Regarding "knowing": In subjective probability theory, the probability over the "event" space is just about what you believe, not about what you know. You could theoretically believe to degree 0 in the propositions "the die comes up 6" or "the die lands at an angle". Or that the die comes up as both 1 and 2 with some positive probability. There is no requirement that your degrees of belief are accurate relative to some external standard. It is only assumed that the beliefs we do have compose in a way that adheres to the axioms of probability theory. E.g. P(A)≥P(A and B). Otherwise we are, presumably, irrational.
A less arbitrary way to define a sample space is to take the set of all possible worlds. Each event, e.g. a die roll, corresponds to the disjunction of possible worlds where that event happens. The possible worlds can differ in a lot of tiny details, e.g. the exact position of a die on the table. Even just an atom being different at the other end of the galaxy would constitute a different possible world. A possible world is a maximally specific way the world could be. So two possible worlds are always mutually exclusive. And the set of all possible worlds includes every possible way reality could be. There are no excluded possibilities like a die falling on the floor.
But for subjective probability theory a "sample space" isn't even needed at all. A probability function can simply be defined over a Boolean algebra of propositions. Propositions ("events") are taken to be primary instead of being defined via primary outcomes of a sample space. We just have beliefs in some propositions, and there is nothing psychological corresponding to outcomes of a sample space. We only need outcomes if probabilities are defined to be ratios of frequencies of outcomes. Likewise, "random variables" or "partitions" don't make sense for subjective probability theory: there are just propositions.
I think the main problem from this evolutionary perspective is not so much entertainment and art, but low fertility. Not having children.
A drug that fixes akrasia without major side-effects would indeed be the Holy Grail. Unfortunately I don't think caffeine does anything of that sort. For me it increases focus, but it doesn't combat weakness of will, avoidance behavior, ugh fields. I don't know about other existing drugs.
I think the main reason is that until a few years ago, not much AI research came out of China. Gwern highlighted this repeatedly.
I agree with the downvoters that the thesis of this post seems crazy. But aren't entertainment and art superstimuli? Aren't they forms of wireheading?
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Yes, the meaning of a statement depends causally on empirical facts. But this doesn't imply that the truth value of "Bachelors are unmarried" depends less than completely on its meaning. Its meaning (M) screens off the empirical facts (E) and its truth value (T). The causal graph looks like this:
If this graph is faithful, it follows that E and T are conditionally independent given M. E⊥T∣M. So if you know M, E gives you no additional information about T.
And the same is the case for all "analytic" statements, where the truth value only depends on its meaning. They are distinguished from synthetic statements, where the graph looks like this:
That is, we have an additional direct influence of the empirical facts on the truth value. Here E and T are no longer conditionally independent given M.
I think that logical and probabilistic laws are analytic in the above sense, rather than synthetic. Including axioms. There are often alternative axiomatizations of the same laws. So P(A∨B)=P(A)+P(B)−P(A∧B) and P(⊤)=1 are equally analytic, even though only the latter is used as an axiom.
I think the instrumental justification (like Dutch book arguments) for laws of epistemic rationality (like logic and probability) is too wealk. Because in situations where there happens to be in fact no danger of being exploited by a Dutch book (because there is nobody who would do such an exploit) it is not instrumentally irrational to be epistemically irrational. But you continue to be epistemically irrational if you have e.g. incoherent beliefs. So epistemic rationality cannot be grounded in instrumental rationality. Epistemic rationality laws being true in virtue of their meaning alone (being analytic) therefore seems a more plausible justification for epistemic rationality.