There is also Deliberation in Latent Space via Differentiable Cache Augmentation by Liu et al. and Efficient Reasoning with Hidden Thinking by Shen et al.
cubefox
Karma: 1,619
I think picking axioms is not necessary here and in any case inconsequential.
By picking your axioms you logically pinpoint what you are talking in the first place. Have you read Highly Advanced Epistemology 101 for Beginners? I’m noticing that our inferential distance is larger than it should be otherwise.
I have read it a while ago, but he overstates the importance of axiom systems. E.g. he wrote:
You need axioms to pin down a mathematical universe before you can talk about it in the first place. The axioms are pinning down what the heck this ‘NUM-burz’ sound means in the first place—that your mouth is talking about 0, 1, 2, 3, and so on.
That’s evidently not true. Mathematicians studied arithmetic for two thousand years before it was axiomatized by Dedekind and Peano. Likewise, mathematical statisticians have studied probability theory long before it was axiomatized by Kolmogorov in the 1930s. Advanced theorems preceded these axiomatizations. Mathematicians rarely use axiom systems in their work even if they are theoretically available. That’s why it is hard to translate proofs into Lean code. Mathematicians just use well-known mathematical facts (that are considered obvious or already sufficiently established by others) as assumptions for their proofs.
No, you are missing the point. I’m not saying that this phrase has to be axiom itself. I’m saying that you need to somehow axiomatically define your individual words, assign them meaning and only then, in regards to these language axioms the phrase “Bachelors are unmarried” is valid.
That’s obviously not necessary. We neither do nor need to “somehow axiomatically define” our individual words for “Bachelors are unmarried” to be true. What would these axioms even be? Clearly the sentence has meaning and is true without any axiomatization.
I wouldn’t generally dismiss an “embarassing & confusing public meltdown” when it comes from a genius. Because I’m not a genius while he or she is. So it’s probably me who is wrong rather than him. Well, except the majority of comparable geniuses agrees with me rather than with him. Though geniuses are rare, and majorities are hard to come by. I still remember an (at the time) “embarrassing and confusing meltdown” by some genius.
My point is that if your picking of particular axioms is entangled with reality, then you are already using a map to describe some territory. And then you can just as well describe this territory more accurately.
I think picking axioms is not necessary here and in any case inconsequential. “Bachelors are unmarried” is true whether or not I regard it as some kind of axiom or not. I seems the same holds for tautologies and probabilistic laws. Moreover, I think neither of them is really “entangled” with reality, in the sense that they are compatible with any possible reality. They merely describe what’s possible in the first place. That bachelors can’t be married is not a fact about reality but a fact about the concept of a bachelor and the concept of marriage.
Rationality is about systematic ways to arrive to correct map-territory correspondence. Even if in your particular situation no one is exploiting you, the fact that you are exploitable in principle is bad. But to know about what is exploitable in principle we generalize from all the individual acts of exploatation. It all has to be grounded in reality in the end.
Suppose you are not instrumentally exploitable “in principle”, whatever that means. Then it arguably would still be epistemically irrational to believe that “Linda is a feminist and a bank teller” is more likely than “Linda is a bank teller”. Moreover, it is theoretically possible that there are cases where it is instrumentally rational to be epistemically irrational. Maybe someone rewards people with (epistemically) irrational beliefs. Maybe theism has favorable psychological consequences. Maybe Pascal’s Wager is instrumentally rational. So epistemic irrationality can’t in general be explained with instrumental irrationality as the latter may not even be present.
You’ve said yourself, meaning is downstream of experience. So in the end you have to appeal to reality while trying to justify it.
I don’t think we have to appeal to reality. Suppose the concept of bachelorhood and marriage had never emerged. Or suppose humans had never come up with logic and probability theory, and not even with language at all. Or humans had never existed in the first place. Then it would still be true that all bachelors are necessarily unmarried, and that tautologies are true. Moreover, it’s clear that long before the actual emergence of humanity and arithmetic, two dinosaurs plus three dinosaurs already were five dinosaurs. Or suppose the causal history had only been a little bit different, such that “blue” means “green” and “green” means “blue”. Would it then be the case that grass is blue and the sky is green? Of course not. It would only mean that we say “grass is blue” when we mean that it is green.
Somewhat related: A critique of “bad faith”.
Do you really have access to the GPT-4 base (foundation) model? Why? It’s not publicly available.
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:
E —> M —> TIf this graph is faithful, it follows that E and T are conditionally independent given 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:
E —> M —> T |_________^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 and are equally analytic, even though only the latter is used as an axiom.
Being Dutch-bookable is considered irrational because you systematically lose your bets.
I think the instrumental justification (like Dutch book arguments) for laws of epistemic rationality (like logic and probability) is too weak. 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.
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”.
And 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?
Hedonic and desire theories are perfectly standard, we had plenty of people talking about them here, including myself. Jeffrey’s utility theory is explicitly meant to model (beliefs and) desires. Both are also often discussed in ethics, including over at the EA Forum. Daniel Kahneman has written about hedonic utility. To equate money with utility is a common simplification in many economic contexts, where expected utility is actually calculated, e.g. when talking about bets and gambles. Even though it isn’t held to be perfectly accurate. I didn’t encounter the reproduction and energy interpretations before, but they do make some sense.
A more ambitious task would be to come up with a model that is more sophisticated than decision theory, one which tries to formalize your previous comment about intent and prediction/belief.
Interesting. This reminds me of a related thought I had: Why do models with differential equations work so often in physics but so rarely in other empirical sciences? Perhaps physics simply is “the differential equation science”.
Which is also related to the frequently expressed opinion that philosophy makes little progress because everything that gets developed enough to make significant progress splits off from philosophy. Because philosophy is “the study of ill-defined and intractable problems”.
Not saying that I think these views are accurate, though they do have some plausibility.
(This is off-topic but I’m not keen on calling LLMs “he” or “she”. Grok is not a man, nor a woman. We shouldn’t anthropomorphize language models. We already have an appropriate pronoun for those: “it”)