Originally posted by DoctorScribblesThat's certainly not my view...or at least extremely probably not.
This is just Bayes v. Skepticism. Under Pawnokeyhole's view, any event can be given any desired likelihood of occuring by skeptically constructing a sufficient number of possible but imaginary cases to offset by the desired amount those cases indicated by actual information and evidence. This reduces probabilistic statements to worthless nonsense.
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Originally posted by PawnokeyholeNo, that would be silly. Your situation is not coherent. How did you determine that the witness is only 60% reliable, if not based on information? If you have such a reliability figure, it can't be the case that the only relevant information you have is the testimony itself.
Suppose the only "information" I have about who committed a crime is the testimony of a witness who says X did it. Suppose I also have reason to believe that the witness is only 60% reliable. Should I now conclude that the probability that X did it is 100%, summarizing the available information?
Your statement "Regardless of how probably true that information is?" begs the question of what your assessment of this question would be based on, if not information. This is the root of skepticism.
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Originally posted by PawnokeyholeSure it is. What non-skeptical evidence do you have to suggest that frogs might in fact jump to Jupiter?
That's certainly not my view...or at least extremely probably not.
You are stating that the probability is slightly greater than 0, which is equivalent to saying that you have some information that indicates that they do and a lot of information that indicates that they don't.
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Originally posted by PawnokeyholeUnder your view, I can conclude that all coins land heads under 50% of the time. For example, any coin could warp mid-flight and become a double-tailed coin, since we can't be certain that the laws of physics preclude this. This is no different than your analysis that frogs jump to Jupiter with a positive probability since we can't be certain of the laws that indicate that they can't.
That's certainly not my view...or at least extremely probably not.
Originally posted by DoctorScribblesOf course I don't have any positive evidence that frogs might jump to Jupiter.
Sure it is. What non-skeptical evidence do you have to suggest that frogs might in fact jump to Jupiter?
You are stating that the probability is slightly greater than 0, which is equivalent to saying that you have some information that indicates that they do and a lot of information that indicates that they don't.
I nonetheless assert that the probability that this might occur is still slightly greater than zero, because the scientific laws that prohibit it are not known with certainty to be true. However, because those laws are empirically very well-supported, and because the scientific method works so well in general, those laws are obviously very likely to be true; so I'm not advocating wholesale skepticism about science and evidence. That is, nothing in my position permits anything but an infinitesimal, and for all practical purposes negligible, probability to be assigned to weird, anomalous events--contrary to what you assert.
I think that your definition of probability is too restrictive, suited only to particular situations in the empirical world, and not, for example, to metaphysical questions. For example, I believe that I can state quite coherently that it is improbable that I am a brain in a vat. Most people would agree with me. But on what information do I base the judgment that I am not a brain in a vat? What are the pros and cons that I summarize, given that everything would look exactly the same whether I was a brain in a vat or I wasn't? This is not a case for weighing evidence in a Bayesian fashion, as one might in court of law, or in scientific practice. But, I contend, it's not likely that I am a brain in a vat, and most people would understand what I mean by this.
Originally posted by DoctorScribblesYes. Ultimately, there is an utterly tiny non-zero probability that this weird event could could occur, even though there is no positive evidence to believe that it ever would.
Under your view, I can conclude that all coins land heads under 50% of the time. For example, any coin could warp mid-flight and become a double-tailed coin, since we can't be certain that the laws of physics preclude this. This is no different than your analysis that frogs jump to Jupiter with a positive probability since we can't be certain of the laws that indicate that they can't.
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Originally posted by PawnokeyholeBut your concession of this point leads to an immediate paradox.
Yes. Ultimately, there is an utterly tiny non-zero probability that this weird event could could occur, even though there is no positive evidence to believe that it ever would.
You concede that all coins land heads with a probability of less than 50% because of the possibility of double-tail warping.
But now allow me to introduce the skeptical possibility that all coins can warp to be double-headed, since we can't be certain of the physical laws that would prevent this.
By the same logic used to make your concession, you would have to conclude that all coins land tails with a probability of less than 50%, while you initially concluded that all coins land heads with a probability of less than 50%.
The only real way out of this paradox is to ignore all of the skeptical outcomes, either actually by never considering them as factors in the probability assessment, or effectively by concluding that they cancel each other's likelihood.
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Originally posted by DoctorScribblesEverything about probabilities follows from:
Under your view, I can conclude that all coins land heads under 50% of the time. For example, any coin could warp mid-flight and become a double-tailed coin, since we can't be certain that the laws of physics preclude this. This is no different than your analysis that frogs jump to Jupiter with a positive probability since we can't be certain of the laws that indicate that they can't.
(i) A probability is a surjection from an event space to the interval [0,1];
(ii) The probability of the union of every event in the event space is 1;
(iii) If two elements of the event space are disjoint, then the probability of their union is the sum of their probabilities.
Let's modify these, creating the PSPPF (Potentially Strawmanular Pawnokeyhole Probability Formalisation):
PSPPF(i) A probability is a surjection from an event space to the interval (0,1];
PSPPF(ii) The probability of the union of every event in the event space is 1;
PSPPF(iii) If two elements of the event space are disjoint, then the probability of their union is the sum of their probabilities.
These are obviously inconsistent or incomplete (either P({...}) is positive, violating (iii) or undefined, violating (i)).
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I'm convinced it's impossible to find a remotely similar formalisation of probability in which all events are uncertain that can, say, give a coherent account of continuous random variables, since (iii) implies continuity of the probability function on event spaces whose elements are subsets of an appropriate metric space and since continuous maps map compact sets to compact sets. For example, under these axioms, I could define the range of a random variable X to be [0, 1] and Pawnokeyhole would be forced to tell me that the probability that X takes a value anywhere in this interval is less than 1, because that probability would be the limit of the probabilities of any sequence of closed intervals converging to the whole interval, by continuity, and that is less than 1 by assumption.
This leads to a mess by (iii) in several ways. In short, any formalisation of probability has to define a probability function with a compact image. This is why we have probability measures!
The non-nonsensical philosophical questions about probability all arise from its interpretation, not from its formalisation. Hold the axioms sacred and then decide whether to be a Bayesian or a frequentist or a whateverelsian; that way, you might be overly longwinded or silly, but never flat-out absurd.
Originally posted by DoctorScribblesThis is old ground - we've been over it several times.
Sounds like another case of Simon Says. They teach it's true without "officially" teaching that it's true. Shameful that they'd run such a play on children.
The Church does not have a position on every verse in the Bible. It teaches that some events are historical fact (like the Incarnation, the Virgin Birth, the Crucifixion, the Resurrection) and some are meant to be read figuratively (like the Genesis creation account). For the rest, any interpretation that follows sound hermeneutical principles and is consistent with the teaching of the Church is okay.
Originally posted by DoctorScribblesThe latter solution to the "paradox" strikes me as perfectly viable. So my view entails no contradiction.
But your concession of this point leads to an immediate paradox.
You concede that all coins land heads with a probability of less than 50% because of the possibility of double-tail warping.
But now allow me to introduce the skeptical possibility that all coins can warp to be double-headed, since we can't be certain of the physical laws that w ...[text shortened]... probability assessment, or effectively by concluding that they cancel each other's likelihood.
Originally posted by DoctorScribblesOn what basis do you believe that they are not? Isn't what's good for the goose good for the gander?
How do you assess the likelihood of each speculative case? What basis do you have for believing that they are equilikely?
Also, I don't mean to be pedantic, but the probability that a typical coin lands heads, and the probability that that coin lands tails--even in the absence of weird stuff happening--doesn't actually sum to 1. Why?
Originally posted by PawnokeyholePresumably it could land on its side. How is that relevant?
On what basis do you believe that they are not? Isn't what's good for the goose good for the gander?
Also, I don't mean to be pedantic, but the probability that a typical coin lands heads, and the probability that that coin lands tails--even in the absence of weird stuff happening--doesn't actually sum to 1. Why?
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Originally posted by PawnokeyholeI don't believe that they are not equallly likely. I believe they are both 0% likely.
On what basis do you believe that they are not?
As all probabilities are assessed - on the basis of information. In this case, the information dictates that each is 0% likely. However, you are the one claiming that each is greater than 0% likely, based on speculation, a claim which has confusion at its root. Speculation does not constitute an informational basis for assessing probabilities. Further, you haven't stated why you think the two cases are equally likely.