Hempel’s Ravens Paradox
Suppose you see a raven, and you note that it is black. “Hmm,” you say, “that raven was black.” Sometime later you notice a couple more ravens, and they too are black. “What a coincidence,” you remark, “those ravens are black too.” Time goes by and you see many more ravens. And it happens that all the ravens you see are black. “This is beyond coincidence,” you might reasonably think, and with the instincts of a good and observant scientist you form a hypothesis: All ravens are black.
This is a deliberately simplistic example, but it lays bare what the first step in the scientific method, commonly understood, really amounts to: one makes observations, and forms an inductive hypothesis. The next step, of course, is experimentation to confirm or refute the hypothesis – and it is here that the trouble occurs. In a case like this, experimentation amounts to observing as many ravens as possible, and confirming that they are all black. Now it is impossible, even in principle, to observe every raven, for many no longer exist, many do not yet exist, and it is conceivable that there are creatures one would also wish to call ravens that exist in inaccessible places, such as other planets. There are always limits to an experimental apparatus, even if the apparatus is just a matter of observing as many ravens as possible to check their color. Nonetheless, we feel justified in saying that each new observation of a black raven tends to confirm the hypothesis, and in time, if no green or blue or otherwise non-black ravens are observed, our hypothesis will eventually come to have the status of a natural law.
But is this logical? Note that, logically put, our hypothesis “all ravens are black” has the form of a conditional, that is, a statement of the form “if A then B.” In short, we are saying that if a given object is a raven, then that object is black. According to the laws of logic, a conditional is equivalent to its contrapositive. That is, a statement of the form “if A then B” is equivalent to the statement “if not B then not A.” For example, the statement “if I live in Denver then I live in Colorado” is logically Our hypothesis “all ravens are black” therefore has the equivalent form “all non-black things are non-ravens,” or more precisely, “if an object isn't black then it is not a raven.” Consequently, if every sighting of a black raven confirms our hypothesis, then every sighting of a non-black non-raven equally confirms our hypothesis.
I look at my shirt. It's blue. And it is not a raven. Confirmation! My hypothesis that all ravens are black is strengthened! My coffee cup is red. More confirmation. The grass is green, the sky is blue, my computer is gray, my dog is white – all confirming the hypothesis “all ravens are black.”
Silly, isn't it? (Isn't it?) But by the laws of logic, if I accept inductive hypotheses and confirmation by experiment, then every observation except one that refutes my hypothesis – confirms it. Even if it is totally irrelevant.
Very well, you might say, but maybe every sighting of a non-black non-raven really does confirm, even if only to an infinitessimal degree, the hypothesis that all ravens are black. After all, if we could, somehow, check every non-black object in the universe, and if none of them were ravens, our statement that all ravens are black would be proved.
Just so. Maybe my blue shirt does reinforce, even if only to some tiny degree, the hypothesis that all ravens are black. But if so, then it must also reinforce – to the same degree – a completely contradictory statement, namely, the hypothesis that all ravens are white. After all, my shirt is a non-whitenon-raven . . . .
The philosopher Carl G. Hempel, in his1965 essay “Studies in the Logic of Confirmation,” brought to light a central paradox in the scientific method as it is commonly understood.
The problem is with inductive reasoning, and Hempel’s example was the one stated above.
(taken from
http://www.mathacademy.com/pr/prime/articles/paradox_raven/index.asp)
Probatio Diabolica
legal requirement to achieve an impossible proof. Where a legal system would appear to require an impossible proof, the remedies are reversing the burden of proof, or giving additional rights to the individual facing the probatio diabolica. For example, one party might patent a process for manufacturing an item. Another party might then make the item. The patent holder would normally have to show that the patented process had been improperly used; this is a probatio diabolica since on the face of it the patent holder has no information on what process was actually used, and this could render the patent useless. Two possible solutions exist:
* the burden of proof is reversed by presuming that the second manufacturer has improperly used the patented process, unless or until he demonstrates that he has used some other process; or
* the patent holder is given discovery rights, enabling him to get information from the second manufacturer on the process actually used. (www.wikipedia.com)
Zeno's Paradox
Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.
What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.
Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?" If you are giving the matter your full attention, it should begin to make you squirm a bit, for on its face the logic of the situation seems unassailable. You shouldn't be able to cross the room, and the Tortoise should win the race! Yet we know better. Hmm.
Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno's Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile!
At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't – in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn't so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)
Now the resolution to Zeno's Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long – only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.
Schrodinger's Cat
principle in quantum theory of superposition , proposed by Erwin Schrödinger in 1935. Schrödinger's cat serves to demonstrate the apparent conflict between what quantum theory tells us is true about the nature and behavior of matter on the microscopic level and what we observe to be true about the nature and behavior of matter on the macroscopic level.Here's Schrödinger's (theoretical) experiment: We place a living cat into a steel chamber, along with a device containing a vial of hydrocyanic acid. There is, in the chamber, a very small amount of a radioactive substance. If even a single atom of the substance decays during the test period, a relay mechanism will trip a hammer, which will, in turn, break the vial and kill the cat. The observer cannot know whether or not an atom of the substance has decayed, and consequently, cannot know whether the vial has been broken, the hydrocyanic acid released, and the cat killed. Since we cannot know, the cat is both dead and alive according to quantum law, in a superposition of states. It is only when we break open the box and learn the condition of the cat that the superposition is lost, and the cat becomes one or the other (dead or alive). This situation is sometimes called quantum indeterminacy or the observer's paradox : the observation or measurement itself affects an outcome, so that the outcome as such does not exist unless the measurement is made. (That is, there is no single outcome unless it is observed.)
We know that superposition actually occurs at the subatomic level, because there are observable effects of interference, in which a single particle is demonstrated to be in multiple locations simultaneously. What that fact implies about the nature of reality on the observable level (cats, for example, as opposed to electrons) is one of the stickiest areas of quantum physics. Schrödinger himself is rumored to have said, later in life, that he wished he had never met that cat.
(from
http://whatis.techtarget.com/)
Subject: Crazy Theories 
Sat May 09, 2009 7:33 am
