My current understanding of logical counterfactuals is something like this: if the inconsistent formal theory PA+”the trillionth digit of pi is odd” has a short proof that the agent will take some action, which is much shorter than the proof in PA that the trillionth digit of pi is in fact even, then I say that the agent takes that action in that logical counterfactual.
Note that this definition leads to only one possible counterfactual action, because two different counterfactual actions with short proofs would lead to a short proof by contradiction that the digit of pi is odd, which by assumption doesn’t exist. Also note that the logical counterfactual affects all calculator-like things automatically, whether they are inside or outside the agent.
That’s an approximate definition that falls apart in edge cases, the post tries to make it slightly more exact.
(Btw, I think it should be mentioned that a central piece of motivation for this “logical counterfactuals” thing is that it’s probably the same construction that’s needed to evaluate possible actions in normal cases, without any contrived coins, for an agent that knows its own program. So for example although a counterfactual scenario can’t easily “lead” to two different actions, two different actions in that scenario can still be considered as possibly even more (easily shown to be) contradictory “logical counterfactuals” that include additional assumptions about what the action is.)
Try as I might, I cannot find any reference to what’s canonical way of building such counterfactual scenarios. Closest I could get was in http://lesswrong.com/lw/179/counterfactual_mugging_and_logical_uncertainty/ , where Vladimir Nesov seems to simply reduce logical uncertainty to ordinary uncertainty, but this does not seem to have anything to do with building formal theories and proving actions or any such thing.
To me, it seems largely arbitrary how agent should do when faced with such a dilemma, all dependent on actually specifying what it means to test a logical counterfactual. If you don’t specify what it means, whatever could happen as a result.
I am not sure there is a clean story yet on logical counterfactuals. Speaking for myself only, I am not yet convinced logical counterfactuals are “the right approach.”
I am not yet convinced logical counterfactuals are “the right approach.”
Me neither. Have you seen my post about common mistakes? To me it seems more productive and more fun to explore the implications of an idea without worrying if it’s the right approach.
My current understanding of logical counterfactuals is something like this: if the inconsistent formal theory PA+”the trillionth digit of pi is odd” has a short proof that the agent will take some action, which is much shorter than the proof in PA that the trillionth digit of pi is in fact even, then I say that the agent takes that action in that logical counterfactual.
Note that this definition leads to only one possible counterfactual action, because two different counterfactual actions with short proofs would lead to a short proof by contradiction that the digit of pi is odd, which by assumption doesn’t exist. Also note that the logical counterfactual affects all calculator-like things automatically, whether they are inside or outside the agent.
That’s an approximate definition that falls apart in edge cases, the post tries to make it slightly more exact.
(Btw, I think it should be mentioned that a central piece of motivation for this “logical counterfactuals” thing is that it’s probably the same construction that’s needed to evaluate possible actions in normal cases, without any contrived coins, for an agent that knows its own program. So for example although a counterfactual scenario can’t easily “lead” to two different actions, two different actions in that scenario can still be considered as possibly even more (easily shown to be) contradictory “logical counterfactuals” that include additional assumptions about what the action is.)
Try as I might, I cannot find any reference to what’s canonical way of building such counterfactual scenarios. Closest I could get was in http://lesswrong.com/lw/179/counterfactual_mugging_and_logical_uncertainty/ , where Vladimir Nesov seems to simply reduce logical uncertainty to ordinary uncertainty, but this does not seem to have anything to do with building formal theories and proving actions or any such thing.
To me, it seems largely arbitrary how agent should do when faced with such a dilemma, all dependent on actually specifying what it means to test a logical counterfactual. If you don’t specify what it means, whatever could happen as a result.
I am not sure there is a clean story yet on logical counterfactuals. Speaking for myself only, I am not yet convinced logical counterfactuals are “the right approach.”
Hi Ilya,
Me neither. Have you seen my post about common mistakes? To me it seems more productive and more fun to explore the implications of an idea without worrying if it’s the right approach.
I like “breadth first search” or more precisely “iterative deepening” better than “depth first search.”
(DFS is not guaranteed to find the optimal solution, after all!)