May 24, 1997 Deduction and Induction
In Logic we are studying the ways of distinguishing correct from incorrect reasoning. We will be examining and focusing upon whether or not the premises justify the attempted conclusion of an argument. There are two basic kinds of argument: deductive and inductive.
Deductive argument asserts that the conclusion follows necessarily from the truth of the premises. For example:
All men are mortal. Joe is a man. Therefore Joe is mortal. If the first two statements are true, then the conclusion must be true.
Inductive argument asserts that the conclusion follows, not necessarily, but only probably from the truth of the premises. For example:
This cat is black. That cat is black A third cat is black. Therefore all cats are are black.
This marble from the bag is black. . That marble from the bag is black. A third marble from the bag is black. Therefore all the marbles in the bag black. .
Neither of the above examples has a conclusion that follows with necessity from the truth of the premisses. The conclusion can be false in each case, and the premises will still remain true. All we need is one exception to the general statement "All cats are black", all we need is one white cat, to show that the conclusion does not follow with necessity from the premises. However, inductive arguments are different than deductive arguments. Deductive arguments attempt to conclude with necessity, but inductive arguments do not attempt to do so. Inductive argument arguments only attempt to conclude with probability.
Evaluation of Deductive and Inductive Arguments
The basic principle used in evaluation of deductive arguments is the principle of contradiction: the same thing, the same truth, cannot both be affirmed and denied at the same time and in the same respect. This principle is so basic that there is no way to prove. We implicitly affirm it to be true whenever we say any sentence at all. If I say that the homework is due, I mean that the homework is due; I do not mean that the homework is not due. In order to communicate meaningfully with another person, I have to implicitly affirm the truth of the principle of contradiction in everything that I say.
Here is an example of a deductive argument which is valid, that is, to say, correct or in accord with the principle of contradiction.
Premises: If Joe has acute appendicitis, he is very sick. Joe does have acute appendicitis. Conclusion: Therefore he is very sick. This is a correct or valid argument (Modus Ponens) according to the principle of contradiction. The principle states that one cannot both affirm and deny the same thing in the same respect. Now, a close inspection of the premises shows us that they are saying practically the same thing which the conclusion says. For if it is true that (1) if Joe has a.a, then he is very sick; and (2) Joe has a.a., then Joe has to be very sick. If Joe were not sick, then one of those premises would have to be false. For if both premises are true, then Joe has to be very sick. If we were to affirm that the premises are true but deny that the conclusion is true, we would be violating the principle of contradiction; we would be affirming and denying the same thing in the same respect at the same time.
Here is an example of argument which is invalid as a deductive argument but acceptable as an inductive argument:
Premises: If Joe has acute appendicitis, he is very sick. But Joe is very sick. Conclusion: Therefore Joe has acute appendicitis.
This is an incorrect or invalid argument deductively understood (Fallacy of Affirming the Consequent). For a close inspection of the premises shows that they do not require us to affirm that the conclusion under penalty of contradicting ourselves. For the premises leave open the possibility that there may be some other reason why Joe is seriously sick. Thus, even if we accept the two premises as true, the conclusion does not necessarily follow from the premises. If we affirm that the premises are true but deny that the conclusion is true, we would not be violating the principle of contradiction since the premises do not require us to affirm that the conclusion is true under penalty of contradicting ourselves if we do otherwise.
The above argument, although it is invalid deductively, may be understood as an inductive argument which attempts to affirm a probable conclusion. The conclusion becomes more and more probable, the more that other possible reasons for Joe's being sick are eliminated. In fact, the structure of the fallacious argument in deductive logic, the Fallacy of Affirming the Consequent, is at the heart of scientific method. There are three steps to scientific method:
(1) There is observation of facts and the attempt to generalize the description of these facts into a general law of nature. For example, this body heavier than air falls to the earth when unsupported; that body heavier than air falls to the earth when unsupported; therefore, all bodies heavier than air fall to the earth when unsupported. In this step of scientific method we have carefully observed the fact of objects falling to the earth. Very often, the scientist will attempt to give a mathematical description of the observed facts. He will measure and tabulate. Inductively we argue with probability from the specific facts we have observed to a generalization that all unsupported bodies heavier than air will fall to the earth. Logic can examine some of the guidelines for making good inductive generalizations. (Newton has the best inductive generalization about gravity.)
(2) Given the inductive generalization of Newton that bodies attract each other directly proportionally to their masses and inversely proportionally to the square of the distance between them, in the second step of scientific method we now try to develop a hypothesis about the nature of bodies that would explain why bodies behave the way we have observed them. A good hypothesis will not just be ad hoc (to this) relevant to predicting the already observed facts, a good hypothesis will be able to predict new facts that we have not yet observed. The hypothesis and predictions would be stated as follows:
If the proposed hypothesis is true, then the already observed facts would be predicted to occur and certain new facts not yet observed would also be predicted to occur.
The predictions must follow deductively or mathematically from the proposed hypothesis. However, it is obvious that we do not have an hypothesis which explains why the law of gravitation is as Newton has mathematically described it. If we could come up with such an hypothesis, we must be able to deduce, strictly logically, strictly mathematically, why the bodies would attract each other directly proportionally to their masses and inversely proportionally to the square of the distance between them. And if the hypothesis is a really good one from the viewpoint of scientific method, it would also predict new facts that we have not yet observed.
(3) In the third step of the scientific method, we experiment. We try to verify the hypothesis. The step adds a second premise and a conclusion to the already stated hypothesis as follows, when the experiment is successful:
If the proposed hypothesis is true, then the old facts would be observed and new facts would be observed. The old facts are observed, and the new facts are observed in a controlled experiment. Therefore the hypothesis is verified.
The logical structure of this deductively interpreted is Fallacy of Affirming the Consequent. However, now we are in scientific method; we are interpreting the argument inductively. And inductively speaking, we can say that we have a good argument since we have predicted new observations which have been verified (and since no other hypothesis seems likely to explain what we have found.) If the experiment is not successful, then we have as follows: If the proposed hypothesis is true, then the old and new facts would be obtained. The new facts are not found in the controlled experiment. Therefore, the proposed hypothesis is falsified.
This is a valid deductive argument in the form of Modus Tollens. Of course, we mean that if our deduction of the predicted new facts is correct and if we have correctly performed the experiment, then the conclusion we have reached must be true.
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