THINKING WELL AND THINKING LOGICALLY by Mark Hodes I. The Thesis The literature of skeptic

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THINKING WELL AND THINKING LOGICALLY by Mark Hodes I. The Thesis The literature of skepticism has generated its share of cliches. Skeptics claim to use scientific reasoning, to think logically, to be guided by evidence in forming beliefs as though these are clear-cut descriptions of well-understood activities. Thinking logically, by implication, is taken as an explication of thinking well or clearly. Those who think logically are regarded as less likely to err by adopting false beliefs or rejecting true ones. I will argue that thinking logically is not all it's cracked up to be, and that thinking logically is not the same as thinking well. II. Contrapositives In the propositional calculus of elementary symbolic logic, p -> q, material implication, has the following truth table: p q p->q -------------- 1. T T T 2. T F F 3. F T T 4. F F T Exemplar: If most live birds fly, then 8 - 3 = 5 (true). If most live birds fly, then 8 - 3 = 4 (false). If pigs fly, then 8 - 3 = 5 (true). If pigs fly, then 8 - 3 = 4 (true). Line 3 is true because the sentence makes no claim if in fact pigs do not fly. If in line 4 it seems strange that F -> F is T, consider the sentence "If I am president of France, then I have a chauffeur." I assure my readers that both the hypothesis and conclusion are false, though intuitively (as well as logically) the sentence is true. The next step in my argument is to consider the CONTRAPOSITIVE of p -> q. This is the sentence (not q) -> (not p). For example, if an exemplar of p -> q is "If something is a dog, then it is an animal", then the exemplar of (not q) -> (not p) would be "If something is not an animal, then it is not a dog." A conditional sentence and its contrapositive are logically equivalent. Their truth tables are: 1 2 3 4 5 6 p q p->q (not q) (not p) (not q)->(not p) ---------------------------------------------- T T T F F T T F F T F F F T T F T T F F T T T T The entries in columns 4 and 5 are derived as the opposites of the entries in 2 and 1, respectively. The matching entries in columns 3 and 6 demonstrate the logical equivalence of the sentences p -> q and (not q) -> (not p). In case you still are not convinced of the logical equivalence of a sentence and its contrapositive, select a conditional sentence and try to imagine a world in which it is true, but its contrapositive is false (or vice versa). Your inability to do this should convince you of their synonymity. III. Induction A primary epistemological activity of science is the generation of probably true or well-confirmed generalizations from evidence. The more extensive and diverse the evidence, the more soundly based are the generalizations. A single apparently disconfirming instance can, however, jeopardize any general statement. The following hackneyed example is often given as a model for inductive generalization. You observe a crow and notice that it is black. You seek out many other crows and find (surprise) that they, too, are black. You conclude, tentatively, that all crows are black. As the years roll by, you have occasion to observe other black crows, and never observe a crow that is not black. Your observations of black crows occur under widely varying conditions. Each new observation in the absence of disconfirming instances strengthens your belief that all crows are black. Originally, of course, being black was not a defining characteristic of crows. After years of study, you reformulate the concept of crowness by including melanism amongst the theoretical baggage of being a crow. You are awarded the Nobel Prize for Avian Trivia and retire to Woodshole, where you spend your declining years wiring flowers to the grave of Burt Lancaster to commemorate his portrayal of the Bird Man of Alcatraz. IV. Is Science Logical? The general statement involved in our example of generalization is "If something is a crow, then it is black." We have seen that this statement is logically equivalent to its contrapositive, "If something is black, then it is not a crow", or, more simply, "All non-blacks are non-crows." Now, if science really proceeds logically, any evidence that tends to confirm a statement should tend to confirm any logically equivalent statement to exactly the same extent. Similarly, disconfirming evidence should be equipotent in respect to a statement and its contrapositive. We have arrived at the end of my garden path. So you see the paradox? V. The Paradox Our Nobel Laureate could have confirmed his hypothesis by observing non-blacks and noticing that they never turn out to be crows. He could, for example, have gone to Sears, inventoried all non-black items, noticed that none was a crow, and proclaimed his tentative conclusion, "All crows are black." Notice that if Sears does not sell parrots, the generalization "All parrots are black" is equally well confirmed. Receiving a grant from William Proxmire, our savant could then have toured the shopping malls of Europe, sampling non-black merchandise, finding no crows, and proclaiming, "All non-blacks are non-crows", or, equivalently, "All crows are black." Now, you may feel that I have cheated in some way, because non-blacks are so much more numerous than crows. I am not sure how this affects the prinicple involved, but it does create an apparent asymmetry between the original condition and its contrapositive. Consider, however, an astronomer attempting to support her thesis, "All type G stars have planetary systems." Lacking the funding for adequate telescope time, she peers from the roof of her observatory onto the used car lot below. She proceeds by examining all Chevrolets failing to have planetary systems, and notes that none of them is a type G star. Given the frequency-of-repair records of GM cars, there are surely more type G stars than Chevrolets. So, here the numerical asymmetry is in the opposite direction. There is some rationale for our stargazer's procedure. The more diverse the confirmatory evidence, the more effective the confirmation. Chevrolets are more diverse (remember the '59 Impala?), though less numerous, than type G stars. Nevertheless, we would be a bit uncomfortable accepting our astronomer's conclusion on the basis of such evidence. She would be laughed off the roof of the observatory. VI. The Lineus Bottomus Our conclusion is that sentences that are logically equivalent are not necessarily epistemologically equivalent. This would come as no surprise to serious students of science, for the philosophical and psychological literature is replete with such examples. Ours is merely a cautionary tale for those who believe that thinking logically is an adequate explication of thinking well. Sorry, Mr. Spock.

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