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 —> T
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:
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 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.
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.
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.
I think we are in agreement here.
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 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.
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.
Epistemic rationality laws being true in virtue of their meaning alone (being analytic) therefore seems a more plausible justification for epistemic rationality.
You’ve said yourself, meaning is downstream of experience. So in the end you have to appeal to reality while trying to justify it.
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.
“Bachelors are unmarried” is true whether or not I regard it as some kind of axiom or not.
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.
Moreover, I think neither of them is really “entangled” with reality
You’ve drawn the graph yourself, how meaning is downstream of reality. This is the kind of entanglement we are talking about. The choice of axioms is motivated by our experience with stuff in the real world. Everything else is beside the point.
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”.
Yes. That’s, among other things, what not being instrumentally exploitable “in principle” means. Epistemic rationality is a generalisation of instrumental rationality the same way how arithmetics is a generalisation from the behaviour of individual objects in reality. The kind of beliefs that are not exploitable in any case other than literally adversarial cases such as a mindreader specifically rewarding people who do not have such beliefs.
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.
I think the problem is that you keep using the word Truth to mean both Validity and Soundness and therefore do not notice when you switch from one to another.
Validity depends only on the axioms. As long as you are talkin about some set of axioms in which P defined in such a way that P(A) ≥ P(A&B) is a valid theorem, no appeal to reality is needed.
Likewise, you can talk about a set of axioms where P(A) ≤ P(A&B). These two statements remain valid in regards to their axioms.
But the moment you claim that this has something to do with the way beliefs—a thing from reality—are supposed to behave you start talking about soundness, and therefore require a connection to reality. As soon as pure mathematical statements mean something you are in the domain of map-territory relations.
Moreover, it’s clear that long before the actual emergence of humanity and arithmetic, two dinosaurs plus three dinosaurs already were five dinosaurs.
Territory behaved the way that we now can describe in the map as 2+3=5. But no maps existed back then. If we are in agreement about it, there is nothing substabtial to argue about.
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.
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 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.
I think we are in agreement here.
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.
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.
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 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.
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.
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.
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.
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.
You’ve drawn the graph yourself, how meaning is downstream of reality. This is the kind of entanglement we are talking about. The choice of axioms is motivated by our experience with stuff in the real world. Everything else is beside the point.
Yes. That’s, among other things, what not being instrumentally exploitable “in principle” means. Epistemic rationality is a generalisation of instrumental rationality the same way how arithmetics is a generalisation from the behaviour of individual objects in reality. The kind of beliefs that are not exploitable in any case other than literally adversarial cases such as a mindreader specifically rewarding people who do not have such beliefs.
I think the problem is that you keep using the word Truth to mean both Validity and Soundness and therefore do not notice when you switch from one to another.
Validity depends only on the axioms. As long as you are talkin about some set of axioms in which P defined in such a way that P(A) ≥ P(A&B) is a valid theorem, no appeal to reality is needed.
Likewise, you can talk about a set of axioms where P(A) ≤ P(A&B). These two statements remain valid in regards to their axioms.
But the moment you claim that this has something to do with the way beliefs—a thing from reality—are supposed to behave you start talking about soundness, and therefore require a connection to reality. As soon as pure mathematical statements mean something you are in the domain of map-territory relations.
Territory behaved the way that we now can describe in the map as 2+3=5. But no maps existed back then. If we are in agreement about it, there is nothing substabtial to argue about.
I have read it a while ago, but he overstates the importance of axiom systems. E.g. he wrote:
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.
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.