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Message: <blockquote data-quote="Hawkman" data-source="post: 444853" data-attributes="member: 23715"><h3>Hempel’s Ravens Paradox</h3>The philosopher Carl G. Hempel, in his 1965 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 as follows: 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 also 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.<img src="https://platonicrealms.com/sites/default/files/images/prime/portraits/raven1.jpg" alt="" class="fr-fic fr-dii fr-draggable " style="" />Black RavenThis 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 equivalent to the statement “if I do not live in Colorado then I do not live in Denver.” This rule of logic is incontrovertible.<img src="https://platonicrealms.com/sites/default/files/images/prime/portraits/blue_shirt.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />Non-Black Non-RavenOur 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.<h4>Addendum</h4>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-white non-raven….<h3>Resolving Hempel’s Raven Paradox</h3><h4><a href="https://philosophynow.org/authors/Fred_Leavitt" target="_blank">Fred Leavitt</a> reveals how the whiteness of swans proves the blackness of ravens.</h4>Many scientific theories and laws are of the form “All A is B.” Two examples are “Water at sea level boils at 100 degrees centigrade” and “Schizophrenia is associated with an excess of dopamine in the limbic system.” But philosopher and logician Carl Hempel pointed out a seeming paradox (Hempel, 1945). As virtually all logicians agree, the propositions “All ravens are black” and “All nonblack things are nonravens” are equivalent. To test the former, a scientist would look for ravens and check their colour. A black raven would provide supporting evidence. To test the latter, the scientist would look for nonblack things and check to see if they are nonravens. A white handkerchief would provide supporting evidence.But if a white handkerchief supports the proposition that all nonblack things are nonravens; and if “All nonblack things are nonravens” is equivalent to “All ravens are black;” a white handkerchief would appear to support the proposition that all ravens are black. The paradox has two aspects: first, that a white handkerchief should be as informative as a black raven; second, that a white handkerchief should have any bearing at all on the proposition “All ravens are black” (or green or red). The paradox seems to have important implications for testing scientific theories.What follows is my resolution.There are two propositions (P1 and P2) and two pieces of evidence (E1 and E2).P1: All ravens are black.P2: All nonblack things are nonravens.E1: This raven is black.E2: This white thing is not a raven.P1 and P2 are logically equivalent, i.e., any evidence that supports P1 supports P2 to the same extent. But E1 and E2 are not equivalent. E1 provides much stronger support for both propositions. To see why this is so, consider an aviary in which there are exactly two ravens among 100 birds.P3: All ravens in the aviary are black.P4: All nonblack birds in the aviary are nonravens.E3: This raven is black.E4: This white bird is not a raven.E3 represents 50% of the evidence needed to prove P3. Less obviously but equally true, it represents 50% of the evidence needed to prove P4. (If the only other raven is found to be black, P4 must be true.)By checking all 98 nonblack birds and verifying that they are not ravens, an investigator could prove P3. The proof would apply just as convincingly to P4. But a single nonblack nonraven would be less useful than a single black raven, because the former would represent 1/98th of the necessary evidence and the latter 50%.E1 and E3 are more significant than E2 and E4 because they account for a greater proportion of the total number of cases under consideration. Nevertheless, E2 to a trivial extent and E4 to a much greater extent, account for something – a nonblack nonraven eliminates one potential falsifier of the proposition “All ravens are black.”© Prof. F. Leavitt 1997</blockquote>

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