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The Limits Of Logic

Brian D. Rude, 1974

      It is widely accepted that we are a rational people, that civilization is marked by logical thinking. Often this is extended into the idea that primitive peoples are not so logical or rational, either because they lack the ability or they lack the training. Evidence of this is the large amount of magic and superstition that can be found in almost any primitive society.

      But superstition is also found in modern civilized societies, though of course it is usually harder to recognize simply because of its familiarity. Does this mean then that we ourselves do not act logically, or that logic can coexist with superstition, or that the superstitions and magic of primitive peoples are overemphasized? I think all three of these answers are true to one extent or another. However I believe the main mistake is in ascribing too much importance to logic, and that is the thesis of this article. Logic, though important, is very much limited. In fact it is so limited that it is often almost irrelevant in comparison with other factors. I will give a few illustrations.

      An old woman can’t figure out why her lights don’t work. Finally she finds an empty light socket and immediately understands - her electricity is leaking out of this socket. This is perfectly rational I think, a good example of adding up clues, of seeing relations, the kind of thinking that makes us human and able to understand and build. It is a good deduction - simply wrong, for electricity does not leak out of empty light sockets.

      Most conclusions can be put in the form of a syllogism. For example:

           All dogs bark.
           This animal is a dog.
          This animal barks.

      The light bulb example can be put in this syllogism form:

           All things that flow can leak.
          Electricity flows.
          Electricity can leak.

      Both syllogisms are of the form:

           All A are B.
          C is A.
          Therefore C is B.

      In diagram form this can be represented:

C is wholly contained in A, and A is wholly contained in B. Therefore C must be wholly contained in B.

      The logic is perfectly valid. The truth of the statements is open to question. It is simply not true that “All things that flow can leak.”

      Consider another example:

           All mothers love their children.
          This woman is a mother.
          Therefore this woman loves her children.

      Here again the logic is valid. The fault lies with the truth of the statements. We cannot say that all mothers love their children.

      Yet another example.

           All soft drinks contain sugar.
          This coke is a soft drink.
          Therefore this coke contains sugar.

      Once again the logic is perfectly valid. The statements are not entirely true. There are exceptions to soft drinks containing sugar. There are exceptions to mothers who love their children, exceptions to things that flow and leak. Therefore we have to alter the syllogisms a bit:

           Most dogs can bark.
          This animal is a dog.
          This animal may or may not be able to bark.

           Most things that flow can leak.
          Electricity can flow.
          Electricity can probably leak.

           Nearly all mothers love their children.
          This woman is a mother.
          Therefore this woman will probably love her children.

           Most soft drinks contain sugar.
          This coke is a soft drink.
          This coke might contain sugar.

      How does logic apply to these? We could use probability and make predictions based on odds. Or we could just not make any conclusions at all and live in doubt. Of these two alternatives the former is preferable to the latter. However probability is a whole other field. in itself. The point of this article is how people do think, not how they could or should think.

      It is not hard to see errors in logic. For example:

           If you’re mad at me you won’t want to go to the show.
          You don’t want to go to the show.
          Therefore you must be mad at me.

      Which can be put in the form:

           If A, then B.
          B exists.
          Therefore A must exist.

      This does not follow, for A might exist from some cause other than B. Perhaps you don’t want to go to the show for some reason entirely unrelated to being mad at me. The inverse of the statement, “A implies B” is the statement “not A implies not B”. That is if you’re not mad at me then you will not avoid the movie. The converse of the statement “A implies B” is the statement “B implies A”, which is the above example. The contrapositive of “A implies B” is the statement “not B implies not A”. That is, if you do want to go to the show then you’re not mad at me. A little thought will show that the contrapositive of a statement holds, also that if the inverse holds the converse holds, for the inverse is the contrapositive of the converse, and vice versa.

      Such mistakes are one source of error in our thinking to be sure. I suppose they are everyday occurrences even for well educated people, but what have such mistakes to do with every day life? If I suspect you’re mad at me I carefully weigh the evidence and decide, knowing full well that I could make mistakes, and knowing full well that logic learned in a college math course has very little to do with it. Formal logic won’t tell us what car to buy or what food to eat. It won’t tell us whether an old bridge will hold our weight or why the Russians choose a different political system than we do. It is one thing to say that lack of logic will lead us to wrong conclusions, which is perfectly true, but it is quite another thing to expect that logic will always lead us right. The truth of the statements is more commonly the weakest link in our arguments. When we say “all” we usually mean “most”, or even “some” . When we say “none” we so often mean “only a few”, or “not too many”.

      The next step is to show that superstition or magic can come about by quite logical thinking. It will be useful to differentiate between inductive and deductive reasoning.

      Suppose a person sees dark and windy clouds and shortly gets rained on. After a few repetitions of this he can form the conclusion, “Dark clouds mean it will rain”. It happened several times and. so it must happen all the time. Dark clouds always mean rain. This is inductive reasoning. Inductive reasoning is going from a number of particular cases to a general rule. Inductive reasoning then is not a hundred per cent sure. It is a useful form of reasoning but not entirely valid.

      Deductive reasoning is just the opposite.

           Dark clouds always mean rain.
          That cloud is dark.
          It will rain.

      Here we are using a general statement to get to a particu1ar case. The first statement, “all dark clouds always mean rain”, may or may not be true of course, but if we accept it then we can confidently say, “It will rain”. Deductive reasoning is always valid. To reason inductively is to form generalizations. Put “hasty” in front of generalization and we have a villain. How can we set a hasty generalization apart from a more legitimate generalization? Only experience will tell. The school of hard knocks will tell which generalizations are hasty. My point is that generalization is important, indispensable to any rational thought. To illustrate this let us suppose for a minute that we are not going to generalize. The paper that I am writing on has lines on it. So did the previous page, and the page before that and so on. However I said I would not generalize so how can I know if the next page will have lines on it? Generalization seems to work here. We said it is not true that “dark clouds always mean rain.” So if we refuse to generalize we can then plan a picnic on a dark days having no reason not to. Of course we might be charged with not having enough sense to come in out of the rain. Both of these generalizations are not a hundred per cent guaranteed but we can make these generalizations and then base our actions on them. A baby touches a stove and is burned. Should he generalize that stoves are hot? We hope he will, for who could follow after him all his days and warn him, “Don’t touch that, it’s hot”? Should the baby generalize that any big white thing in the kitchen is hot and will burn? Or is this a hasty generalization? Is there anything wrong with this generalization? Do we have a dumb kid who can’t tell a stove from a refrigerator or do we have a smart kid who can learn from only one mistake? Discrimination is the opposite of generalization. The child touches the stove and generalizes, “All big white objects will burn”, and scrupulously avoids the stove and the refrigerator. Next he must discriminate between “all big white objects”. He separates them into those that do burn and those that don’t burn. This pattern of generalization and discrimination is common in most kinds of learning. In chemistry we learn first that all elements have something in common, their being composed of electrons, protons, and neutrons, and then proceed to discriminate between them, to learn how they are different from each other. On a deeper level in the same subject we learn that the halogens are all active nonmetals with seven electrons in their outer shell. We then discriminate between them and find that they are certainly not identical. Often the discrimination may precede the generalization. Details may lead to a broad overview and when the generalization is discovered the details are already very familiar.

      Generalization is induction, but discrimination is not deduction. In deduction the generalization is accepted and used, while in discrimination the generalization is altered. Discrimination is the breaking down of broad generalizations into smaller generalizations.

      I have discussed discrimination in order to more clearly illustrate generalization. I have talked about generalization in order to illustrate the legitimacy of induction. Now I want to give an example of how this type of perfectly valid reasoning, induction and deduction, can lead to something more than simple mistakes.

      One form of superstition or magic is simply misinterpretation of cause and effect. Valid observations and valid induction and deduction can lead to faulty conclusions: <

           observation: When animals roar they are angry.
          observation: When men and women roar they are angry.
          induction: Whenever anything roars it is angry.
          observation: The heavens are roaring. (thunder)
          deduction: The heavens are angry.

      The deductive reasoning is perfectly valid. The induction was not just a generalization it was a hasty generalization. But consider for a moment if the generalization was not made. We could then hit a roaring tiger if we like for we have no reason to believe it is angry. The penalty for generalizing was a wrong conclusion. The penalty for not generalizing is a tiger by the tail. The point is than that generalization, and other forms of reasoning, are necessary even if they are not perfect. Since people have to live with the consequences of their reasoning, and these consequences can be very serious, it is not surprising that some conclusions are held to very tenaciously. Some of these conclusions acquire the label of superstition. This is inevitable, but it should not necessarily reflect badly on either the people who hold these beliefs or the reasoning by which the beliefs were acquired.

      What then is necessary for rational thinking and behavior? In a word I would answer knowledge. Pure logic cannot be applied to the problem of the leaking electricity. Without a knowledge of electricity it is just not possible to know whether an empty bulb socket is or is not at fault. Without knowledge of sound, weather, air, etc., it is hardly possible to explain thunder rationally. Today’s fact may be tomorrow’s superstition. Tomatoes are poisonous according to accepted opinion just a few years ago. Now we call that only superstition. Our knowledge has increased, not our logic.

      Misinterpretation of cause and effect offers a partial explanation for magic but certainly not a complete explanation. It’s not too hard to see that man wants an explanation of thunder, but why does he choose this particular explanation? Why does the simple statement, “ the gods are angry”, have such strength. Why does it appear in one form or another among peoples all over the world?

      I expect it is true that every superstition has its own history. To trace down these histories would be an interesting but never-ending task. I would use a different approach. My approach would be to try to relate various types of superstitions and magic to different instinctual bases. In talking about thunder and angry gods I have hinted at paranoia. Another strong base of superstition and magic is the concept of mana. However that is a different story. My purpose in this article is simply to illustrate that logic is very limited by itself. Various factors that distort our thinking must be left for another time.