23 Feb 06
Originally posted by DoctorScribblesPerhaps, but I'd bet there are propositions relating to rational beings in other universes that would lead to counterexamples.
Only if it's quick and easy. Otherwise I'll take your word - I just don't see it.
My point is, can your objection be addressed by something analagous to the fix applied to set theory to eliminate its classic paradox? Can the rational agent be defined as being restricted to analyzing things in a universe in which he does not exist, thereby eliminating your objection?
23 Feb 06
10 edits
Originally posted by bbarrThe rational agent only has great credence in P & H if he acknowledges that P is a logical truth. If he has no evidence regarding the truth of P, which is likely by H which you assert, he would have to have less credence in P&H than in H by virtue of his rationality.
Since P is a logical truth, H is logically equivalent to the conjunction
P & H.
Since rational beings would have the same credence in logically equivalent hypotheses, he would have great credence in P & H.
Rational beings place the same credence in logically equivalent hypotheses, but P&H is only logically equivalent to H once the truth value of P is known to the rational being. This is because P&H is not formally equivalent to H; it depends on the actual truth value of P. (You didn't mean that P is a formal tautology, did you?) For example, you can't tell me, for some Z that I formulate, whether Z&H is logically equivalent to H until you know that Z is true, so a being in another universe who knows that Z is true can't simply conclude that you will accept Z&H as logically equivalent to H. For all the rational being knows, P could be false, so you can't require him to accept the logical equivalence and put great credence in P&H.
EDIT: OK, I'm done editing. Feel free to respond without fear of more.
23 Feb 06
Originally posted by DoctorScribblesIf S is rational, the S knows that P is a logical truth, and thus places great credence in P. Are you assuming that RC is using the term 'rational' to refer to less than perfect theoretical rationality? Since P is a logical truth (a theorem of logic), it is logically equivalent to the conjunction of P & H, because P and P&H yield equivalent truth tables.
The rational agent only has great credence in P & H if he acknowledges that P is a logical truth. If he has no evidence regarding the truth of P, which is likely by H which you assert, he would have to have less credence in P&H than in H by virtue of his rationality.
Rational beings place the same credence in logically equivalent hypotheses, but ...[text shortened]... redence in P&H.
EDIT: OK, I'm done editing. Feel free to respond without fear of more.
23 Feb 06
10 edits
Originally posted by bbarrOK, you are saying that P is a tautology. Then my objection to your proof is different.
If S is rational, the S knows that P is a logical truth, and thus places great credence in P.
P is defined: Let P be some logical truth such that, in this world, it is very probable on our evidence that nobody has great credence in P.
If S exists, then P cannot exist, for if S exists, then it is certain on our evidence that somebody - namely S - has great credence in every logical truth.
So I don't see how your proof even gets started since P can't exist.
Or if you're going to stipulate that P exists, then it follows that S can't exist and your proof concludes nothing. That is, there is no entity holding the absurd state of beliefs.
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23 Feb 06
1 edit
Originally posted by DoctorScribblesThe proof is a reductio. It shows that there are some propositions such that a contradiction arises when we apply the epistemic probability criterion to it. So, not all propositions are such that their epistemic probability is determined by the credence a rational agent would lend to them. The fact that the proof results in a contradiction is the very point!
P is defined: Let P be some logical truth such that, in this world, it is very probable on our evidence that nobody has great credence in P.
If S exists, then P cannot exist, for if S exists, then it is certain, due to his rationality, that somebody - namely S - has great credenece in every rational truth.
So I don't see how your proof even ge r proof concludes nothing. That is, there is no entity holding the absurd state of beliefs.
23 Feb 06
1 edit
Originally posted by bbarrHmmmmm. I'll have to think it over. I didn't even realize it was a reductio. I'm not clear on what supposition you are claiming leads to a contradiction, and which you thus reject and in turn conclude its negation. I don't see it as part of your proof.
The proof is a reductio. It shows that there are some propositions such that a contradiction arises when we apply the epistemic probability criterion to it. So, not all propositions are such that their epistemic probability is determined by the credence a rational agent would lend to them.
At any rate, I have to go out of town for four days. I hope to rejoin this discussion when I return.
23 Feb 06
4 edits
Originally posted by bbarrBut I don't think it does yield a contradiction, since you haven't demonstrated or stipulated that both P and S exist.
The fact that the proof results in a contradiction is the very point!
If you stipulate that they both exist, then that is the supposition that yields a contradiction, and which you must reject. But I can do that rejection a priori without appealing to the proof but merely by pointing to their definitions.
23 Feb 06
1 edit
Originally posted by DoctorScribblesWell, it is certainly possible that there are logical theorems complex or opaque enough that no actual agent would lend a high degree of credence to them, even though their objective probability is (of course) 1. So I don't think it would be wise to adopt a theory of epistemic probability that entails that these sorts of theorems can't exist. It's probably prudent to reject the epistemic probability criterion...
Hmmmmm. I'll have to think it over. I didn't even realize it was a reductio. I'm not clear on what supposition you are claiming leads to a contradiction, and which you thus reject and in turn conclude its negation. I don't see it as part of your proof.
At any rate, I have to go out of town for four days. I hope to rejoin this discussion when I return.
Have fun on your trip, Herr Doctor. I'll talk to you when you return.
23 Feb 06
1 edit
Originally posted by DoctorScribblesYou don't need to demonstrate the truth of premises to engage in a reductio. Reductios show that if a premise is true, and if we assume something further, then a contradiction results. In virtue of deriving a contradiction, we are warranted by the rules of the propositional calculus, to infer the negation of any premise upon which the contradiction is based. In this case, which premise ouught we reject? Should we reject that it is possible that a theorem of the sort described exists? Obviously not, because it obviously is possible for there to be a logical truth that no actual person lends any credence to. So, we should reject the other assumption that led to the contradiction, and that was the application of the epistemic probability criterion to the hypothesized theorem.
But I don't think it does yield a contradiction, since you haven't demonstrated that both P and S exist.
23 Feb 06
1 edit
Originally posted by bbarrIf you're going to distinguish between real rational agents and theoretical rational agents, then I would suggest that your conclusion contains an equivocation and is not an absurdity at all.
Obviously not, because it obviously is possible for there to be a logical truth that no actual person lends any credence to.
"But this entails that he has great credence in the proposition 'P, and nobody has great credence that P', which is absurd."
Under your distinction, this really reads:
"But this entails that the theoretically rational agent has great credence in the proposition 'P, and no actual agent has great credence that P', which is absurd."
But the latter is actually not absurd.
23 Feb 06
Originally posted by DoctorScribblesThere's no equivoation in the proof. It is the theorist, that by invoking the e.p.criterion, invites the rational agent into the domain of discourse by way of an existentially quantified supposition (i.e., (Ex)(X is perfectly rational and has our evidence)) and then conditionalizes upon that supposition. But the antecedent of the conditional has contradictory entailments when the proposition evaluated is something like P.
If you're going to distinguish between real rational agents and theoretical rational agents, then I would suggest that your conclusion contains an equivocation and is not an absurdity at all.
"But this entails that he has great credence in the proposition 'P, and nobody has great credence that P', which is absurd."
Under your distinction, this ...[text shortened]... has great credence that P', which is absurd."
But the latter is actually not absurd.
23 Feb 06
4 edits
Originally posted by bbarrDoes the 'nobody' in P refer to actual or theoretically rational agents?
There's no equivoation in the proof.
If the former, then the alleged absurdity is not actually an absurdity for the reason I just cited.
If the latter, then "But the antecedent of the conditional has contradictory entailments when the proposition evaluated is something like P" is immaterial since propositions like P cannot exist (since no theoretical rational agents can fail to have credence in a logical truth, by definition).
If it refers to all agents, theoretical and actual, then the same objection as for the latter case holds.
23 Feb 06
3 edits
Originally posted by DoctorScribblesThe 'nobody' in P refers to actual agents. When the e.p.criterion is applied, another agent is brought into the domain of agents via an hypthetical assumption on the part of the theorist ("Suppose there is some rational agent that has our evidence, if so...." ). This agent is such that when we suppose that he has our evidence, a contradiction results.
Does the 'nobody' in P refer to actual or theoretically rational agents?
If the former, then the alleged absurdity is not actually an absurdity for the reason I just cited.
If the latter, then "But the antecedent of the conditional has contradictory entailments when the proposition evaluated is something like P" is immaterial since propositions ...[text shortened]... to all agents, theoretical and actual, then the same objection as for the latter case holds.
If the latter, then "But the antecedent of the conditional has contradictory entailments when the proposition evaluated is something like P" is immaterial since propositions like P cannot exist...
The fact that propositions like P cannot exist if we suppose that S exists is the very point the reductio establishes!