I have just proved that observational support for for a universal statement is impossible. — tom
All unenlightened's pockets everywhere in the universe are empty. — unenlightened
If I have looked at 16 of my 17 pockets and found them empty, I have probable grounds for thinking that the last one will be also empty. — unenlightened
One more point - you also have to stipulate that this was true when the observations occurred. Even then, it is only strictly true if those observations were simultaneous; otherwise, something could have appeared in the first pocket that you checked by the time that you got to the last one. Furthermore, the fact that your pockets were empty then does not warrant the claim that they are still empty now and will remain empty in the future. This gets at my earlier comment about a universal proposition having to include all potential members in the class, not just its actual members. — aletheist
Not really. Why would you think that? The contents (or lack thereof) of the first 16 pockets have no bearing whatsoever on the contents (or lack thereof) of the 17th pocket. — aletheist
To the extent that one has explored the logical space and found it empty of non.black ravens or pocketed stuff or whatever, to that extent it is probable that the space is empty. — unenlightened
If there is one white one and sixteen black ones, would you bet on the white one being the last one out of the bag, or some other place? — unenlightened
The difference is that you actually observed all of your pockets. The OP is claiming that a single observation provides evidential support for a universal proposition. Tom's proof shows that this is not the case - but it no longer applies once you have observed all members of the class, at which point you know whether the universal proposition is true (p=1) or false (p=0). — aletheist
I don't think so, Tom. I'll let you quote one that rules that not to be a universal. I'll even modify it a bit for you: All unenlightened's pockets everywhere in the universe are empty. — unenlightened
By believing that the actual color of the eggs is somehow indeterminate until one opens the carton. It is not; it is a fact that either they are all white (p=1) or that at least one is non-white (p=0), unless we are going to treat this as a quantum physics scenario like Schroedinger's cat where each egg is neither white nor non-white until one observes it. — aletheist
A universal proposition does not assert the actual existence of anything in the subject class, so it must apply to all potential things in the subject class.
You used universal propositions, not singular propositions, in the OP. Now you are claiming that the two propositions of interest are both singular - "if a is a raven, then a is black," and its contrapositive, "if a is not black, then a is not a raven." In this example, a is a green apple, so it is trivial to say that a is not black and not a raven; both propositions are true (p=1). A second observation of a green apple, call it b, would go with a different pair of singular propositions - "if b is a raven, then b is black," and its contrapositive, "if b is not black, then b is not a raven"; again, both are true (p=1). By definition, you cannot say anything general in a singular proposition
If I say that all humans are shorter than 9 feet I'm not saying that all potential humans are shorter than 9 feet. — Michael
When I say that the probability that "if something is not black then it is not a raven" is true is 0.5 I mean that that for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven. — Michael
And the above interpretation is wrong. When I say that the probability that "if something is not black then it is not a raven" is true is 0.5 I mean that that for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven. — Michael
How many non-black things do you need to select to show that your assertion that the probability of selecting a non-raven is 0.5? — tom
I think you are, unless you qualify it somehow. You are saying that anything taller than 9 feet cannot (ever) be human. — aletheist
Then this is a universal proposition after all, rather than a singular proposition; and it is, in fact, logically equivalent to "all non-black things are non-ravens." Your use of probability in this case is unobjectionable to me; you are simply saying that exactly 50% of all non-black things are non-ravens. — aletheist
That is a different scenario. If I knew nothing about the contents of the bag, and had already drawn 16 black ones, I might very well be tempted to bet that the last one would also be black - and I would be dead wrong. — aletheist
If I say that nobody in my house is American I'm not saying that nobody in my house can ever be American. — Michael
Wait, so you're saying that it's unobjectionable to claim that a universal proposition has a greater than 0 but less than 1 probability of being true? — Michael
No, but I can see why you misunderstood me. The universal proposition is "if something is not black then it is not a raven"; i.e., "all non-black things are non-ravens." The proposition that I find unobjectionable is "for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven." This is not the same (universal) proposition; it is instead a particular proposition, "some non-black things are non-ravens," with the additional information that the proportion of non-black things that are non-ravens is 50%. — aletheist
No, but I can see why you misunderstood me. The universal proposition is "if something is not black then it is not a raven"; i.e., "all non-black things are non-ravens." The proposition that I find unobjectionable is "for any randomly selected non-black thing, the probability is 0.5 that it will not be a raven." This is not the same (universal) proposition; it is instead a particular proposition, "some non-black things are non-ravens," with the additional information that the proportion of non-black things that are non-ravens is 50%. — aletheist
You did not stipulate any knowledge of how the marbles got into the bag. All we knew was that the first 16 marbles that we took out were black. — aletheist
This information alone is insufficient to calculate a meaningful probability that the 17th marble will also be black. — aletheist
Most people would indeed be likely to bet on it being black in that scenario, but again, they would be wrong if it turned out to be white. — aletheist
Then you need to show where my admittedly incomplete calculation has gone wrong, because I think I have shown that the probability is greater than 0.5, and somewhere close to 0.9 — unenlightened
That's the nature of probability, that one can be wrong. — unenlightened
You have basically invented a clever mathematical way of measuring your level of confidence in your guess that the 17th marble is black, based solely on the fact that the first 16 were black. — aletheist
If one can be wrong, then one is really talking about (subjective) confidence or degree of belief, rather than (objective) probability. — aletheist
A universal statement is one in which no individual names occur. — tom
That is a pile of crap of biblical proportions that I am not going to even try and clear up. — unenlightened
So universality - in practice, in the real world - obtains only by a failure to find otherwise. The absence of not-A as a particular, is inductive confirmation of the presence of A as a generality. — apokrisis
Yes, this was pretty much exactly the point of my egg thought experiment. So, if each black raven observed in the absence of white ones decreases the potential population of non-black ravens, thereby increasing the probability that they are all black, can we then not say that successive observations of black ravens confirms the hypothesis "all ravens are black" (contra some claims on this thread that no such confirmation can be had for universally-quantified propositions)?Firstly, if there are a limited number of ravens, then there are some ravens. So we are not saying merely that there are no non-black ravens, but also that there are some black ravens. Then each black raven found in the absence of any white ones decreases the population of potential non-black ravens, and so increases the probability that they are all black. — unenlightened
I have some questions about this. I don't see how H (hypothesis) logically implies E (evidence). I understand the hypothetico-deductive mode of reasoning (which, in very general terms, science adheres to), i.e. posit a hypothesis, deduce observational consequences of said hypothesis, and perform a test to look for said consequences. However, in this case, I don't see how "all ravens are black" implies "the sighting of another raven." I'm not sure what the latter statement even means, exactly (H seems to imply only that, if one were to observe a raven, then said raven would be black).I beg to differ! If there is such a thing as probabilistic support for a universal statement, then green apples do indeed support "all ravens are black". I have given the solution to this paradox earlier in the thread, so now let me prove it:
A well known result from probability calculus is:
p(he|b) = p(h|eb)p(eb)
Let h = "all ravens are black" i.e. the hypothesis
Let b = background knowledge e.g. all the ravens previously encountered
Let e = new evidence - the sighting of another raven
h logically implies e, so "h and e" is equivalent to h, so
p(h|b) = p(h|eb)p(eb)
Thus
p(h|eb)=p(h|b)/p(eb)
Do this again with an alternative hypothesis:
k = "NOT all ravens are black"
And divide one expression by the other, you get:
p(h|eb)/p(k|eb) = p(h|b)/p(k|b)
Now notice that no matter how h and k generalize under new evidence e, the evidence is incapable of affecting the ratio of their probabilities! What you are left with is the ratio of the prior probabilities, which you can have done nothing except arbitrarily set.
Thus there is no such thing as probabilistic support for a universal statement! — tom
I don't understand thing idea of increasing probability either. The probability of two coins landing on heads is .25 because with two coins four things can happen. It's .5 with one coin because only two things can happen, whereas the probability of having gotten heads twice after the fact is 100%, as only that can happen, because it already happened. Did the probability change because I changed the variables?
If I keep moving things around it certainly is going to seem so, but it really isn't. — Wosret
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