Tim--- To resolve your disagreement: Induction is not purely about deduction, but it nevertheless can be completely modelled by a deductive system.
More specifically, I agree with your claim about induction (see point 4 above). However, in defense of Eliezer’s claim that induction is a special case of deduction, I think you can model it in a deductive system even though induction might require additional assumptions. For one thing, deduction in practice seems to me to require empirical assumptions as well (i.e., the “axioms” and “inference rules” are chosen based on how right they seem), so the fact that induction needs some axioms should not itself prevent deductive style proofs using an appropriately formalized version of it. So, once one decides on various axioms, such as the various desiderata I list above for a Solomonoff-like system, you CAN describe via a mathematical deduction system how the process of induction would proceed. So, induction can be formalized and proofs can be made about the best thing for an agent to do; the AIXI model is basically an example of this.
If that is a defense of induction being a special case of deduction, then it’s a defense of anything being a special case of deduction—since logic can model anything.
The golden gate bridge is a special case of deduction, in this sense.
I am not impressed by the idea that induction is a special case of deduction—I would describe it as being wrong. You need extra axioms for induction. It is not the same thing at all.
Tim--- To resolve your disagreement: Induction is not purely about deduction, but it nevertheless can be completely modelled by a deductive system.
More specifically, I agree with your claim about induction (see point 4 above). However, in defense of Eliezer’s claim that induction is a special case of deduction, I think you can model it in a deductive system even though induction might require additional assumptions. For one thing, deduction in practice seems to me to require empirical assumptions as well (i.e., the “axioms” and “inference rules” are chosen based on how right they seem), so the fact that induction needs some axioms should not itself prevent deductive style proofs using an appropriately formalized version of it. So, once one decides on various axioms, such as the various desiderata I list above for a Solomonoff-like system, you CAN describe via a mathematical deduction system how the process of induction would proceed. So, induction can be formalized and proofs can be made about the best thing for an agent to do; the AIXI model is basically an example of this.
If that is a defense of induction being a special case of deduction, then it’s a defense of anything being a special case of deduction—since logic can model anything.
The golden gate bridge is a special case of deduction, in this sense.
I am not impressed by the idea that induction is a special case of deduction—I would describe it as being wrong. You need extra axioms for induction. It is not the same thing at all.
Yes, the golden gate bridge is a special case of deduction in the sense meant here. I have no problem with anything in your comment, I think we agree.