RSS

Log­i­cal Uncertainty

TagLast edit: 23 Sep 2020 18:53 UTC by Ruby

Logical Uncertainty is probabilistic uncertainty about the implications of beliefs. (Another way of thinking about it is: uncertainty about computations.) Probability theory typically assumes logical omniscience, IE, perfect knowledge of logic. The easiest way to see the importance of this assumption is co consider Bayesian reasoning: to evaluate the probability of evidence given a hypothesis, , it’s necessary to know what the implications of the hypothesis are. However, realistic agents cannot be logically omniscient.

See Also: Logical Induction

Motivation

Is the googolth digit of pi odd? The probability that it is odd is, intuitively, 0.5. Yet we know that this is definitely true or false by the rules of logic, even though we don’t know which. Formalizing this sort of probability is the primary goal of the field of logical uncertainty.

The problem with the 0.5 probability is that it gives non-zero probability to false statements. If I am asked to bet on whether the googolth digit of pi is odd, I can reason as follows: There is 0.5 chance that it is odd. Let P represent the actual, unknown, parity of the googolth digit (odd or even); and let Q represent the other parity. If Q, then anything follows. (By the Principle of Explosion, a false statement implies anything.) For example, Q implies that I will win $1 billion. Therefore the value of this bet is at least $500,000,000, which is 0.5 * $1,000,000, and I should be willing to pay that much to take the bet. This is an absurdity. Only expenditure of finite computational power stands between the uncertainty and 100% certainty.

Logical Uncertainty & Counterfactuals

Logical uncertainty is closely related to the problem of counterfactuals. Ordinary probability theory relies on counterfactuals. For example, I see a coin that came up heads, and I say that the probability of tails was 0.5, even though clearly, given all air currents and muscular movements involved in throwing that coin, the probability of tails was 0.0. Yet we can imagine this possible impossible world where the coin came up tails. In the case of logical uncertainly, it is hard to imagine a world in which mathematical facts are different.

References

An Un­trol­lable Math­e­mat­i­cian Illustrated

abramdemski20 Mar 2018 0:00 UTC
150 points
38 comments1 min readLW link

Ques­tions of Rea­son­ing un­der Log­i­cal Uncertainty

So8res9 Jan 2015 17:37 UTC
28 points
19 comments2 min readLW link

Rad­i­cal Prob­a­bil­ism [Tran­script]

26 Jun 2020 22:14 UTC
45 points
12 comments6 min readLW link

What Would I Do? Self-pre­dic­tion in Sim­ple Algorithms

Scott Garrabrant20 Jul 2020 4:27 UTC
51 points
13 comments5 min readLW link

Toward a New Tech­ni­cal Ex­pla­na­tion of Tech­ni­cal Explanation

abramdemski16 Feb 2018 0:44 UTC
82 points
36 comments18 min readLW link

Mar­kets are Univer­sal for Log­i­cal Induction

johnswentworth22 Aug 2019 6:44 UTC
62 points
0 comments4 min readLW link

His­tory of the Devel­op­ment of Log­i­cal Induction

Scott Garrabrant29 Aug 2018 3:15 UTC
87 points
4 comments5 min readLW link

In Log­i­cal Time, All Games are Iter­ated Games

abramdemski20 Sep 2018 2:01 UTC
77 points
8 comments5 min readLW link

Bayesian Prob­a­bil­ity is for things that are Space-like Separated from You

Scott Garrabrant10 Jul 2018 23:47 UTC
74 points
22 comments2 min readLW link

Re­lat­ing HCH and Log­i­cal Induction

abramdemski16 Jun 2020 22:08 UTC
49 points
4 comments5 min readLW link

Com­put­er­phile dis­cusses MIRI’s “Log­i­cal In­duc­tion” paper

Parth Athley4 Oct 2018 16:00 UTC
43 points
2 comments1 min readLW link
(www.youtube.com)

Log­i­cal un­cer­tainty and Math­e­mat­i­cal uncertainty

AlexMennen26 Jun 2018 1:08 UTC
35 points
6 comments4 min readLW link

Log­i­cal Uncer­tainty and Func­tional De­ci­sion Theory

swordsintoploughshares10 Jul 2018 23:08 UTC
15 points
4 comments2 min readLW link

[Question] Log­i­cal in­duc­tors in mul­ti­stable situ­a­tions.

Donald Hobson3 Jan 2019 23:56 UTC
8 points
4 comments1 min readLW link

Asymp­totic Log­i­cal Uncer­tainty: Con­crete Failure of the Solomonoff Approach

Scott Garrabrant22 Jul 2015 19:27 UTC
13 points
0 comments1 min readLW link

A mea­sure-the­o­retic gen­er­al­iza­tion of log­i­cal induction

Vanessa Kosoy18 Jan 2017 13:56 UTC
4 points
0 comments7 min readLW link

Log­i­cal Uncer­tainty as Probability

gRR29 Apr 2012 22:26 UTC
4 points
22 comments1 min readLW link

Log­i­cal In­duc­tors that trust their limits

Scott Garrabrant20 Sep 2016 23:17 UTC
4 points
0 comments1 min readLW link

The set of Log­i­cal In­duc­tors is not Convex

Scott Garrabrant27 Sep 2016 9:05 UTC
4 points
0 comments1 min readLW link

Log­i­cal In­duc­tion with in­com­putable sequences

AlexMennen17 Aug 2017 0:39 UTC
4 points
0 comments1 min readLW link

Op­ti­miz­ing ar­bi­trary ex­pres­sions with a lin­ear num­ber of queries to a Log­i­cal In­duc­tion Or­a­cle (Car­toon Guide)

Donald Hobson23 Jul 2020 21:37 UTC
3 points
2 comments2 min readLW link

Bounded Or­a­cle Induction

Diffractor28 Nov 2018 8:11 UTC
25 points
0 comments9 min readLW link

Beliefs at differ­ent timescales

Nisan4 Nov 2018 20:10 UTC
25 points
12 comments2 min readLW link

Limited agents need ap­prox­i­mate induction

Manfred24 Apr 2015 7:42 UTC
16 points
10 comments8 min readLW link

Top-Down and Bot­tom-Up Log­i­cal Probabilities

Manfred22 Jul 2014 8:53 UTC
6 points
4 comments4 min readLW link

Ap­pli­ca­tions of log­i­cal uncertainty

alex_zag_al18 Oct 2014 19:26 UTC
30 points
19 comments4 min readLW link

In­fra-Bayesi­anism Unwrapped

adamShimi20 Jan 2021 13:35 UTC
19 points
0 comments24 min readLW link

Prob­a­bil­ity in­ter­pre­ta­tions: Examples

So8res11 May 2019 20:32 UTC
38 points
23 comments5 min readLW link

Us­ing the uni­ver­sal prior for log­i­cal un­cer­tainty (re­tracted)

cousin_it28 Feb 2018 13:07 UTC
15 points
13 comments2 min readLW link

Prob­a­bil­is­tic Löb theorem

Stuart_Armstrong26 Apr 2013 18:45 UTC
39 points
40 comments3 min readLW link

A Prob­lem About Bar­gain­ing and Log­i­cal Uncertainty

Wei_Dai21 Mar 2012 21:03 UTC
45 points
49 comments1 min readLW link

De­con­fus­ing Log­i­cal Counterfactuals

Chris_Leong30 Jan 2019 15:13 UTC
25 points
16 comments11 min readLW link

Coun­ter­fac­tual Mug­ging and Log­i­cal Uncertainty

Vladimir_Nesov5 Sep 2009 22:31 UTC
10 points
21 comments3 min readLW link

The Ul­ti­mate New­comb’s Problem

Eliezer Yudkowsky10 Sep 2013 2:03 UTC
30 points
116 comments1 min readLW link

Draw­ing Two Aces

Eliezer Yudkowsky3 Jan 2010 10:33 UTC
19 points
93 comments1 min readLW link

Knigh­tian Uncer­tainty and Am­bi­guity Aver­sion: Motivation

So8res21 Jul 2014 20:32 UTC
35 points
44 comments13 min readLW link

Mo­ti­vat­ing a Se­man­tics of Log­i­cal Counterfactuals

Sam_A_Barnett22 Sep 2017 1:10 UTC
21 points
3 comments2 min readLW link

Metauncertainty

jimmy10 Apr 2009 23:41 UTC
26 points
4 comments2 min readLW link

Per­sua­sive­ness vs Soundness

Patrick13 Apr 2009 8:43 UTC
0 points
19 comments2 min readLW link

Be Log­i­cally Informative

steven046115 May 2009 13:23 UTC
4 points
0 comments3 min readLW link

Su­per­nat­u­ral Math

saturn19 May 2009 11:31 UTC
5 points
58 comments1 min readLW link

You can’t be­lieve in Bayes

PhilGoetz9 Jun 2009 18:03 UTC
13 points
60 comments1 min readLW link
No comments.