# Log­i­cal Uncertainty

TagLast edit: 6 Jun 2022 7:00 UTC by

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 to 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.

## 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.

# His­tory of the Devel­op­ment of Log­i­cal Induction

29 Aug 2018 3:15 UTC
100 points

# An Un­trol­lable Math­e­mat­i­cian Illustrated

20 Mar 2018 0:00 UTC
157 points

# [Question] Log­i­cal in­duc­tors in mul­ti­stable situ­a­tions.

3 Jan 2019 23:56 UTC
8 points

# Ques­tions of Rea­son­ing un­der Log­i­cal Uncertainty

9 Jan 2015 17:37 UTC
29 points

# A mea­sure-the­o­retic gen­er­al­iza­tion of log­i­cal induction

18 Jan 2017 13:56 UTC
6 points

26 Jun 2020 22:14 UTC
48 points

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

10 Jul 2018 23:47 UTC
80 points

# Log­i­cal In­duc­tors that trust their limits

20 Sep 2016 23:17 UTC
5 points

# Reflec­tion in Prob­a­bil­is­tic Logic

24 Mar 2013 16:37 UTC
112 points

# The set of Log­i­cal In­duc­tors is not Convex

27 Sep 2016 9:05 UTC
5 points

# Asymp­totic Log­i­cal Uncer­tainty: Con­crete Failure of the Solomonoff Approach

22 Jul 2015 19:27 UTC
13 points

# Ap­pli­ca­tions of log­i­cal uncertainty

18 Oct 2014 19:26 UTC
30 points

# What Would I Do? Self-pre­dic­tion in Sim­ple Algorithms

20 Jul 2020 4:27 UTC
64 points

# Re­lat­ing HCH and Log­i­cal Induction

16 Jun 2020 22:08 UTC
50 points

# Prob­a­bil­is­tic Löb theorem

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51 points

# Toward a New Tech­ni­cal Ex­pla­na­tion of Tech­ni­cal Explanation

16 Feb 2018 0:44 UTC
85 points

# You can’t be­lieve in Bayes

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15 points

# Metauncertainty

10 Apr 2009 23:41 UTC
26 points

# Draw­ing Two Aces

3 Jan 2010 10:33 UTC
19 points

# Coun­ter­fac­tual Mug­ging and Log­i­cal Uncertainty

5 Sep 2009 22:31 UTC
11 points

# Log­i­cal and In­dex­i­cal Uncertainty

29 Jan 2014 21:49 UTC
46 points

# The Ul­ti­mate New­comb’s Problem

10 Sep 2013 2:03 UTC
46 points

# Log­i­cal Uncer­tainty and Func­tional De­ci­sion Theory

10 Jul 2018 23:08 UTC
15 points

# Beliefs at differ­ent timescales

4 Nov 2018 20:10 UTC
25 points

# Fun­da­men­tal Uncer­tainty: Chap­ter 1 - How can we know what’s true?

13 Aug 2023 18:55 UTC
16 points

# Limited agents need ap­prox­i­mate induction

24 Apr 2015 7:42 UTC
16 points

# Top-Down and Bot­tom-Up Log­i­cal Probabilities

22 Jul 2014 8:53 UTC
6 points

# [Question] In­finite tower of meta-probability

19 Oct 2023 16:44 UTC
6 points

# In­fra-Bayesi­anism Unwrapped

20 Jan 2021 13:35 UTC
54 points

# Log­i­cal Prob­a­bil­ity of Gold­bach’s Con­jec­ture: Prov­able Rule or Coin­ci­dence?

29 Dec 2022 13:37 UTC
5 points

# Com­po­si­tional lan­guage for hy­pothe­ses about computations

11 Mar 2023 19:43 UTC
26 points

# Mar­kets are Univer­sal for Log­i­cal Induction

22 Aug 2019 6:44 UTC
75 points

# In Log­i­cal Time, All Games are Iter­ated Games

20 Sep 2018 2:01 UTC
93 points

# Com­put­er­phile dis­cusses MIRI’s “Log­i­cal In­duc­tion” paper

4 Oct 2018 16:00 UTC
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# Log­i­cal un­cer­tainty and Math­e­mat­i­cal uncertainty

26 Jun 2018 1:08 UTC
35 points

# Log­i­cal Uncer­tainty as Probability

29 Apr 2012 22:26 UTC
4 points

# Log­i­cal In­duc­tion with in­com­putable sequences

17 Aug 2017 0:39 UTC
4 points

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

23 Jul 2020 21:37 UTC
4 points

# Bounded Or­a­cle Induction

28 Nov 2018 8:11 UTC
25 points

# [In­com­plete] What is Com­pu­ta­tion Any­way?

14 Dec 2022 16:17 UTC
12 points
(arxiv.org)

# De­mons from the 5&10verse!

28 Mar 2023 2:41 UTC
−3 points
(voidgoddess.org)

# Re­boot­ing AI Gover­nance: An AI-Driven Ap­proach to AI Governance

6 Aug 2023 14:19 UTC
1 point
(forum.effectivealtruism.org)

# Knigh­tian Uncer­tainty and Am­bi­guity Aver­sion: Motivation

21 Jul 2014 20:32 UTC
44 points

# De­con­fus­ing Log­i­cal Counterfactuals

30 Jan 2019 15:13 UTC
27 points

# A Prob­lem About Bar­gain­ing and Log­i­cal Uncertainty

21 Mar 2012 21:03 UTC
47 points

# In­fer­ence from a Math­e­mat­i­cal De­scrip­tion of an Ex­ist­ing Align­ment Re­search: a pro­posal for an outer al­ign­ment re­search program

2 Jun 2023 21:54 UTC
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# Mo­ti­vat­ing a Se­man­tics of Log­i­cal Counterfactuals

22 Sep 2017 1:10 UTC
22 points

# Hert­ford, Sour­but (ra­tio­nal­ity les­sons from Univer­sity Challenge)

4 Sep 2023 18:44 UTC
28 points
(www.oliversourbut.net)

# Per­sua­sive­ness vs Soundness

13 Apr 2009 8:43 UTC
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# Be Log­i­cally Informative

15 May 2009 13:23 UTC
4 points

# Su­per­nat­u­ral Math

19 May 2009 11:31 UTC
5 points

# Us­ing the uni­ver­sal prior for log­i­cal un­cer­tainty (re­tracted)

28 Feb 2018 13:07 UTC
15 points

# Prob­a­bil­ity in­ter­pre­ta­tions: Examples

11 May 2019 20:32 UTC
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# Bet­ting on what is un-falsifi­able and un-verifiable

14 Nov 2023 21:11 UTC
9 points