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.
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.
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?
Classical logics make the assumption that all statements are either exactly true or exactly false, with no other possibility allowed. Hence classical logic will take shortcuts like admitting not(not(X)) as a proof of X, under the assumptions of consistency (we’ve proved not(not(X)) so there is no proof of not(X)), completeness (if there is no proof of not(X) then there must be a proof of X) and proof-irrelevance (all proofs of X are interchangable, so the existence of such a proof is acceptable as proof of X).
The flaw is, of course, the assumption of a complete and consistent system, which Goedel showed to be impossible for systems capable of modelling the Natural numbers.
Constructivist logics don’t assume the law of the excluded middle. This restricts classical ‘truth’ to ‘provably true’, classical ‘false’ to ‘provably false’ and allows a third possibility: ‘unproven’. An unproven statement might be provably true or provably false or it might be undecidable.
From a probability perspective, constructivism says that we shouldn’t assume that P(not(X)) = 1 - P(X), since doing so is assuming that we’re using a complete and consistent system of reasoning, which is impossible.
Note that constructivist systems are compatible with classical ones. We can add the law of the excluded middle to a constructive logic and get a classical one; all of the theorems will still hold and we won’t introduce any inconsistencies.
Another way of thinking about it is that the law of the excluded middle assumes that a halting oracle exists which allows us to take shortcuts in our proofs. The results will be consistent, since the oracle gives correct answers, but we can’t tell which results used the oracle as a shortcut (and hence don’t need it) and which would be impossible without the oracle’s existence (and hence don’t exist, since halting oracles don’t exist).
The only way to work out which ones are shortcuts is to take ‘the long way’ and produce a separate proof which doesn’t use an oracle; these are exactly the constructive proofs!
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.
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.
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.)
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.
Are there any particular arguments in constructive logic that you formerly believed, and now no longer believe?
Or is this just a thing where you are forever doomed to say “minus epsilon” every time you say “1″ but it doesn’t actually change what arguments you accept?
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.
Could you be so kind as to expand on that?
Classical logics make the assumption that all statements are either exactly true or exactly false, with no other possibility allowed. Hence classical logic will take shortcuts like admitting not(not(X)) as a proof of X, under the assumptions of consistency (we’ve proved not(not(X)) so there is no proof of not(X)), completeness (if there is no proof of not(X) then there must be a proof of X) and proof-irrelevance (all proofs of X are interchangable, so the existence of such a proof is acceptable as proof of X).
The flaw is, of course, the assumption of a complete and consistent system, which Goedel showed to be impossible for systems capable of modelling the Natural numbers.
Constructivist logics don’t assume the law of the excluded middle. This restricts classical ‘truth’ to ‘provably true’, classical ‘false’ to ‘provably false’ and allows a third possibility: ‘unproven’. An unproven statement might be provably true or provably false or it might be undecidable.
From a probability perspective, constructivism says that we shouldn’t assume that P(not(X)) = 1 - P(X), since doing so is assuming that we’re using a complete and consistent system of reasoning, which is impossible.
Note that constructivist systems are compatible with classical ones. We can add the law of the excluded middle to a constructive logic and get a classical one; all of the theorems will still hold and we won’t introduce any inconsistencies.
Another way of thinking about it is that the law of the excluded middle assumes that a halting oracle exists which allows us to take shortcuts in our proofs. The results will be consistent, since the oracle gives correct answers, but we can’t tell which results used the oracle as a shortcut (and hence don’t need it) and which would be impossible without the oracle’s existence (and hence don’t exist, since halting oracles don’t exist).
The only way to work out which ones are shortcuts is to take ‘the long way’ and produce a separate proof which doesn’t use an oracle; these are exactly the constructive proofs!
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.
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.
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.)
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.
Are there any particular arguments in constructive logic that you formerly believed, and now no longer believe?
Or is this just a thing where you are forever doomed to say “minus epsilon” every time you say “1″ but it doesn’t actually change what arguments you accept?