Is saying that P is very unlikely, the same as saying that you believe ~P?
Does it follow rationally, that if P is very unlikely, one should believe ~P?
And would the evidence for P being very unlikely, be the same as the evidence for ~P?
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You might modify your question to make the term "unlikely" less ambiguous. Do you mean evidential/epistemic or metaphysical probability?– user461May 26, 2012 at 6:30
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@arbn both may be relevant to the question.. and I don't know what metaphysical probability is– barlopMay 26, 2012 at 6:39
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"Unlikely" may mean "undetermined given the set of available evidence". This is evidential probability, and we ought to assign degrees of belief to propositions proportionally with it. "Unlikely" can also mean "true in very few possible states of affairs". This just means that there are very few possible combinations which make the proposition true. The difference between these two concepts is pretty vital here.– user461May 26, 2012 at 6:42
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@arbn so what is metaphysical probability?– barlopMay 26, 2012 at 6:43
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1It's the notion of probability involving possible worlds. That is, it involves how many possible configurations of the universe would make the proposition true.– user461May 26, 2012 at 6:45
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3 Answers
If you take a standard assumption of fuzzy logic
P(X) + P(~X) = 1
Then let's examine your statements:
Is saying that P is very unlikely, the same as saying that you believe ~P?
Yes. Say "likely" is defined as > e. Then clearly P(X) < e ==> P(~X) >= e, hence X being unlikely proves ~X's likelihood.
Does it follow rationally, that if P is very unlikely, one should believe ~P?
I'm not certain how this differs from the first.
And would the evidence for P being very unlikely, be the same as the evidence for ~P?
Yes. You can reuse the first proof. P(X | E) < e ==> P(~X | E) >= e hence the evidence E suffices to both prove X's unlikelihood and ~X's likelihood.
Is saying that P is very unlikely, the same as saying that you believe ~P?
It can go either way, a person could believe P and ~P while knowing it is unlikely.
Does it follow rationally, that if P is very unlikely, one should believe ~P?
That is basically the probabilistic fallacy.
And would the evidence for P being very unlikely, be the same as the evidence for ~P?
If it is objective evidence it certainly points toward ~P without objectively proving it - but providing a clue for further inductive investigation. After that, more evidence might be found. Incomplete evidence can still be key to evidence.
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even with no proof of ~P. If objective evidence points towards ~P, then can't one believe ~P? I don't see how that is a probabilistic fallacy. So, if P is very unlikely, is it right to say that rationally, one should believe ~P? one could still change that belief if there is a change in evidence.– barlopMay 26, 2012 at 10:14
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You said "is it the same as saying you believe", not "can you believe". Of course you can believe anything you like. But "P is unlikely" and "I believe that P is not true" are different statements. Jan 2, 2015 at 17:38
Is saying that P is very unlikely, the same as saying that you believe ~P?
No. If P is very unlikely then there is a remaining small possibility that P is true, so I don't believe ~P. I may act as if ~P is true, if the "very unlikely" means a small risk. Which depends on how unlikely, and what the cost if I'm wrong. For example, if it is very unlikely that a cable is connected to lethal amounts of electrical voltage, then I'm not touching it.
Does it follow rationally, that if P is very unlikely, one should believe ~P? Answered.
And would the evidence for P being very unlikely, be the same as the evidence for ~P? No. I found one power cable, and one switch, and nothing else, and turned the switch off. It is now very likely that there is no power. But it doesn't prove there is no power. To prove it, I use some measuring device which will show that there is indeed no power.
Another example. It is very unlikely that an easy proof for Fermat's Last Theorem exists. Evidence: People have been looking for it for hundreds of years, and if there was an easy proof, then surely it would have been found by now. But this isn't evidence that such a proof doesn't exist.
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Re your second paragraph. Say there's a 95% chance that a cable has no power. Clearly a huge cost as you say.. so you don't touch it. But that doesn't mean you believe it has power. You are only acting as if it might and avoiding touching it, because of the high risk and easyness of not touching it. You may still believe that it has no power, if by belief I don't mean absolute certainty - since we don't have absolute certainty other than that there is a reality.– barlopJan 2, 2015 at 17:39
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Re your 4th paragraph, switch off, that is evidence of no power. Not a proof, but is evidence. If you had a few lightbulbs to try, it could be strong evidence. If you measure and see no power, that's strong evidence too(though not a proof, as the device may be broken or you may have mismeasured). I'm not talking proofs here, just evidence. Evidence of unlikely P = evidence of ~P– barlopJan 2, 2015 at 17:41
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Re your 5th/final paragraph, if we define evidence as convincing, then indeed it doesn't constitute evidence that a proof doesn't exist.. But if we define evidence as something that chances the probability, then sure the fact that no proof has been found is very very weak evidence or somewhat weak evidence in favour of a proof not existing.– barlopJan 2, 2015 at 17:43
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@barlop: An easy proof is very unlikely to exist. Easy proofs don't hide away from many clever mathematicians and a huge number of amateurs for 300 years. Jan 3, 2015 at 21:21
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gnasher, you wrote "An easy proof is very unlikely to exist.".<---- I never said otherwise. Please read my comments carefully– barlopJan 3, 2015 at 23:14