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# Neel_Krishnaswami comments on My Wild and Reckless Youth - Less Wrong

23 30 August 2007 01:52AM

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Comment author: 30 August 2007 10:51:57AM 2 points [-]

One of my mistakes was believing in Bayesian decision theory, and in constructive logic at the same time. This is because traditional probability theory is inherently classical, because of the axiom that P(A + not-A) = 1. This is an embarassingly simple inconsistency, of course, but it lead me to some interesting ideas.

Upon reflection, it turns out that the important idea is not Bayesianism proper, which is merely one of an entire menagerie of possible rationalities, but rather de Finetti's operationalization of subjective belief in terms of avoiding Dutch book bets. It turns out there are a lot of ways of doing that, because the only physically realizable bets are of finitely refutable propositions.

So you can have perfectly rational agents who never come to agreement, no matter how much evidence they see, because no finite amount of evidence can settle questions like whether the law of the excluded middle holds for propositions over the natural numbers.

Comment author: 08 October 2012 08:09:35PM 2 points [-]

One of my mistakes was believing in Bayesian decision theory, and in constructive logic at the same time. This is because traditional probability theory is inherently classical, because of the axiom that P(A + not-A) = 1.

Could you be so kind as to expand on that?

Comment author: 31 January 2013 01:33:59PM -1 points [-]

0 And 1 Are Not Probabilities - there is no finite amount of evidence that allows us to assign a probability of 0 or 1 to any event. Many important proofs in classical probability theory rely on marginalising to 1 - that is, saying that the total probability of mutually exclusive and collectively exhaustive events is exactly 1. This works just fine until you consider the possibilty that you are incapable of imagining one or more possible outcomes. Bayesian decision theory and constructive logic are both valid in their respective fields, but constructive logic is not applicable to real life, because we can't say with certainty that we are aware of all possible outcomes.

Constructive logic preserves truth values - it consists of taking a set of axioms, which are true by definition, and performing a series of truth-preserving operations to produce other true statements. A given logical system is a set of operations defined as truth-preserving - a syntax into which semantic statements (axioms) can be inserted. Axiomatic systems are never reliable in real life, because in real life there are no axioms (we cannot define anything to have probability 1) and no rules of syntax (we cannot be certain that our reasoning is valid). We cannot ever say what we know or how we know it; we can only ever say what we think we know and how we think we know it.

Comment author: 31 January 2013 01:51:47PM 1 point [-]

there is no finite amount of evidence that allows us to assign a probability of 0 or 1 to any event.

To be more precise, there is no such finite evidence unless there already exist events to which you assign probability 0 or 1. If such events do exist, then you may later receive evidence that allows them to propagate.

Comment author: 01 February 2013 06:11:15PM *  -1 points [-]

Even if we have infinite evidence (positive or negative) for some set of events, we cannot achieve infinite evidence for any other event. The point of a logical system is that everything in it can be proven syntactically, that is, without assigning meaning to any of the terms. For example, "Only Bs have the property X" and "A has the property X" imply "A is a B" for any A, B and X - the proof makes no use of semantics. It is sound if it is valid and its axioms are true, but it is also only valid if we have defined certain operations as truth preserving. There are an uncountably infinite number of logical systems under which the truth of the axioms will not ensure the truth of the conclusion - the reasoning won't be valid.

Non-probabilistic reasoning does not ever work in reality. We do not know the syntax with certainty, so we cannot be sure of any conclusion, no matter how certain we are about the semantic truth of the premises. The situation is like trying to speak a language you don't know using only a dictionary and a phrasebook - no matter how certain you are that certain sentences are correct, you cannot be certain that any new sentence is gramatically correct because you have no way to work out the grammar with absolute certainty. No matter how many statements we take as axioms, we cannot add any more axioms unless we know the rules of syntax, and there is no way at all to prove that our rules of syntax - the rules of our logical sytem - are the real ones. (We can't even prove that there are real ones - we're pretty darned certain about it, but there is no way to prove that we live in a causal universe.)

Comment author: 01 February 2013 07:37:51PM 0 points [-]

Well, yes. If we believe that A=>B with probability 1, it's not enough to assign probability 1 to A to conclude B with probability 1; you must also assign probability 1 to modus ponens.

And even then you can probably Carroll your way out of it.