I have to leave soon, but I'm going to try to address some of these points in the next couple of days. I was going to say I regret using the term 'rational', because I had intended the method as a method of practical rationality, but now I see that that is a limiting point of view to take.
The 'method' is mathematically sound, because it is consistent with the axioms of probability. This fact is not very interesting. What seems to be at issue is the philosophical basis for interpreting the probability axioms as I did (as Bayesians do). In writing that post, I was presenting something which I have used in practice in the past, so I had not even given much thought to the interpretation beyond noting that elements of the event space need not be regarded as 'events' in the frequentist sense. Therefore, I will try to modify the interpretation to suit the objections that bbarr has raised.
Originally posted by bbarrBut you didn't need an involved reductio to prove that. It follows immediately from your definition of rational agent.
The fact that propositions like P cannot exist if we suppose that S exists is the very point the reductio establishes!
You said that a defining component of rationality is having credence in all logical truths.
Thus, it cannot be the case that there exists a logical truth in which a rational agent does not place credence. Hence P cannot exist.
Further, if P cannot exist and you deduce a contradiction from the conjunction of "P exists" and other assumptions, and then conclude that one of those other assumptions - such as the e.p. in question - is in error, then you have not correctly carried out an abductio-style proof.
That would be like me assuming something false like "A and not A" and "The earth is round", then deducing a contradiction, which is trivial in this case, and then concluding that the earth must not be round.
A proper reductio may take only one falsehood as a suppostion (although that one supposition may itself be a conjunction, but which shall ultimately be rejected in toto and not piecemeal). If "P exists" is a falsehoood, you have exhausted your false assumptions and you are not at liberty to introduce a new one and maintain a valid reductio proof.
This thread is quite baffling to me. I am math impaired and am starting to think that there are good reasons why I can never win an argument.
If... and it's a big if -- truth is a reducible quantity in any measure, then why is the world so messed up? Wouldn't somebody have done the "truth charts" on every issue by now so that we wouldn't be stuck wondering what is "good" and what is "bad"?
I feel the same reading this as I did forty years ago trying to understand Russel's "Truth Charts" and sets and joins and domains and ... errr. Well, anyway, it did my heart good to hear Kip Thorne say that he is amused at the waste of time and effort that was that part of Russels life. Who knew that quantum physics demands we strike "measurable reality" from our list of certainties?
Consider this: Perhaps the mind exists outside of our measurable universe -- maybe in a quantum flux. Can we ever trust anything that arrives by way of that -- thing! -- to be reliable?
I think that "rational" only applies to numbers and math. When it comes to human beings, it is a myth. Hell. What am I saying? The very concept of "myth" is probably all wrong! It holds as good a chance of being "real" as "reality".
Which is not to say that I won't be interested in how this develops. It is interesting in a very abstract way. RC - When you get ready to actually have a piece to work on, I will offer my "Induced" H that "Man can go faster than light, but NEVER as fast as a Woman". I have a whole list of P that are prolly irrelavent as all get out.
Keep up the good work,
In the glory of the Lord,
Mike
Originally posted by DoctorScribblesI can't believe you still aren't getting this.
But you didn't need an involved reductio to prove that. It follows immediately from your definition of rational agent.
You said that a defining component of rationality is having credence in all logical truths.
Thus, it cannot be the case that there exists a logical truth in which a rational agent does not place credence. Hence P cannot exist and you are not at liberty to introduce a new one and maintain a valid reductio proof.
1) It is obviously true that there could be a theorem that nobody actually had great credence in.
2) When the e.p. criterion is applied to P, a contradiction results.
3) This allows us to either infer the negation of P, or to dismiss the e.p. criterion. Since P could be true, we ought to dismiss the e.p. criterion. This is just what presenting a counterexample to some criterion consists of; showing that it entails a contradiction for some logically possible case.
4) It is simply false that a reductio can't proceed from two false premises. Consider:
a) Nothing is made of Camembert.
b) Suppose the moon is made of Camembert.
c) If b, then something is made of Camembert.
d) So, it is not the case that nothing is made of Camembert
e) So, it is both the case that nothing is made of Camembert and it is not the case that nothing is made of Camembert.
Now, we have derived a contradiction that rests on both a and b as assumptions. The rules of first-order logic permit us to infer the negation of either of these propositions, because the rules of first-order logic do not depend for their valid application upon the actual truth values of particular propositions. Rather, they specify the relations between propositions that result from the syntax of those propositions.
In the case at hand, the reductio shows that the application of the e.p. criterion to some class of propositions (that are obviously logically possible) entails a contradiction. So, the e.p. criterion cannot be applied to all propositions.
Originally posted by bbarrI don't see how this is so obviously true.
I can't believe you still aren't getting this.
1) It is obviously true that there could be a theorem that nobody actually had great credence in.
Construct a logical theorem that you do not have great credence in. I dare say it's an impossible task.
Without construction, what other means can you employ to demonstrate the existence of such things? And can't I simply rebut them by saying, "I hereby declare that I am placing credence in all logical theorems."
@bbarr: Which assumption is both essential to this model and entails a contradiction?
Edit: I'm sorry, I didn't even see your P(E|E) = 1 post. I also don't see what the problem is, however. The point of the system is to reduce something difficult (determining the extent to which we should believe a hypothesis conditional on given evidence) to something generallay easier (determining how likely we are to observe the evidence that we did observe given that our hypothesis is true, all other things being equal). I don't see the problem with taking our hypothesis to be the same as the piece of evidence we observe; it's a triviality, but not an inconsistency in the method. The only possible problem would result from a hypothesis whose negation would make the evidence impossible (ie would require us to divide by or take the logarithm of 0), but in that case we simply switch the roles of H and ¬H.
I feel like I haven't addressed your point at all, but I'm tired nearly to the point of incoherence right now and will give a proper answer in the morning.
Originally posted by royalchickenSee the proof above.
@bbarr: Which assumption is both essential to this model and entails a contradiction?
Edit: I'm sorry, I didn't even see your P(E|E) = 1 post. I also don't see what the problem is, however. The point of the system is to reduce something difficult (determining the extent to which we should believe a hypothesis conditional on given evidence) to somet ...[text shortened]... early to the point of incoherence right now and will give a proper answer in the morning.
Since our evidence is rarely, if ever, certain, then how are we justified in treating our evidence as certain (i.e., by conditionalizing upon it, as though it had probability 1)?
Originally posted by bbarrSince when do we have to assume statements have probability one to condition on them? We only have to assume that they have nonzero probability:
See the proof above.
Since our evidence is rarely, if ever, certain, then how are we justified in treating our evidence as certain (i.e., by conditionalizing upon it, as though it had probability 1)?
P(H|E) = P(HE)/P(E).
In the calculations above, we never have to consider P(E) alone anyway, because it cancels in the application of Bayes' theorem to H and ¬H.
I would have thought you'd attack the much more questionable assumption, which is that the individual evidence-statements are independent.
Originally posted by royalchickenIf evidential probabilities are probabilites conditional upon one's evidence, then evidence itself has probability 1. If probability is constured subjectively, as credence or degree of belief, then we must have the greatest possible degree of belief in our evidence. If, for any piece of evidence e, we have the greatest possible degree of belief in evidence e, then anything that conflicts with e must be dismissed. This seems like a problem, since we often derive our evidence through experience, and it seems like experience should be able to make us re-evaluate our evidence.
Since when do we have to assume statements have probability one to condition on them? We only have to assume that they have nonzero probability:
P(H|E) = P(HE)/P(E).
In the calculations above, we never have to consider P(E) alone anyway, because it cancels in the application of Bayes' theorem to H and ¬H.
I would have thought you'd attack th ...[text shortened]... ore questionable assumption, which is that the individual evidence-statements are independent.
Originally posted by bbarrIf evidential probabilities are probabilites conditional upon one's evidence, then evidence itself has probability 1.
If by 'evidential probability' you are referring to what I have denoted P(H|E), then this is not true. See the above post; the probability of the evidence, independent of its context, (ie the quantity P(E)) is completely irrelevant to this discussion. The method is based on the question 'How likely is it that we would observe the evidence which we have observed if our hypothesis is true, compared to the same likelihood in the case that our evidence is false?' These likelihoods are never treated separately. Someone prone to exaggeration would say that this is the whole point of the method.
If probability is constured subjectively, as credence or degree of belief, then we must have the greatest possible degree of belief in our evidence.
This is not relevant. Having the 'greatest possible degree of belief in our evidence' is equivalent to subjectively assigning a value to P(E), which this method does not require, other than to assume it is nonzero.
Thus when we observe some piece of evidence, we don't have to believe it is true or even likely (even after observing it!) in order to use it -- we only have to believe it is not impossible.
If, for any piece of evidence e, we have the greatest possible degree of belief in evidence e, then anything that conflicts with e must be dismissed.
This is misconstrual of how the method works. We are not trying to dismiss H by observing some piece of evidence which is incompatible with H, we are merely using our evidence to revise our beliefs about P(H). The measure S(H) depends on E and accomplishes this. In a situation where H is incompatible with the evidence, we will get a very negative S(H) (a large positive S(¬H)), which gives us a basis for making a decision. The real problem is determining what sort of value of S(H) is sufficient for making a decision, which is what the Experiment (incompletely) addresses.
Originally posted by royalchickenI'm talking about P [e/e], not P[e]. If the evidential probability of a proposition is the probability of that proposition conditional upon it's evidence, then application of this to the evidence itself entails trivially that evidence has probability 1.
[b]If evidential probabilities are probabilites conditional upon one's evidence, then evidence itself has probability 1.
If by 'evidential probability' you are referring to what I have denoted P(H|E), then this is not true. See the above post; the probability of the evidence, independent of its context, (ie the quantity P(E)) is completely irre sufficient for making a decision, which is what the Experiment (incompletely) addresses.[/b]
Originally posted by bbarrTo clarify, your question is: "How are we justified in conditioning on E, since P(E|E) = 1?"
If evidential probabilities are probabilites conditional upon one's evidence, then evidence itself has probability 1. If probability is constured subjectively, as credence or degree of belief, then we must have the greatest possible degree of belief in our evidence. If, for any piece of evidence e, we have the greatest possible degree of belief in evidence e ...[text shortened]... gh experience, and it seems like experience should be able to make us re-evaluate our evidence.
Originally posted by bbarrThanks for clarifying; I was quite sure I'd misunderstood you, because asking the question I had asked would be tantamount to saying that we're not allowed to use conditional probability at all if we interpret probabilities as degrees of belief, which would be silly, since conditional probabilities comply with the axioms in exactly the same way as unconditional ones do.
I think we're missing each other here. Let's start again. Given some hypothesis, how do propositions enter our evidential set for that hypothesis?
Your clarification is a good question, though. I have to go to class, but when I get back, may I try to answer it by giving an example of the method in action, being careful to point out where and how propositions have entered the evidential set?