A. What is Logic?
Logic is the study of the principles of reasoning, especially valid reasoning, as distinguished from invalid or irrational argumentation.
B. Inductive arguments
a. Scientific method and induction
We accept the truth of a statement usually on the basis of the evidence of observation. These are called inductive arguments.
Many of the conclusions or hypotheses we consider it reasonable to accept are supported by inductive argument alone. The theoretical statements of empirical science -- for example, statements about unobserved submicroscopic particles -- are neither empirically falsifiable nor empirically verifiable. Consider any statement of empirical science which concerns the behavior or properties of such particles as neutrinos. Nothing we could observe inside or outside the laboratory would entail that the neutrino has the properties ascribed to it. It is at least logically possible that the meter readings and other observable phenomena should occur and that there should not exist any neutrinos at all. It is logically possible, if scientifically implausible, to suppose that the correct explanation of the phenomena we observe inside the laboratory is one that does not depend on the hypothesis that neutrinos exist but rather on some as yet unconceived of and perhaps undreamt of theory which will be proposed many years hence. On the other hand, should the neutrino hypothesis come to be rejected in science, this will not result from our observing something that entails the falsity of the neutrino hypothesis. It will remain at least logically possible that our present theory is correct, that the neutrino really does exist, no matter what we observe. The neutrino hypothesis is neither conclusively verifiable nor falsifiable by observation.
Of course, these remarks are not intended to suggest that the results of scientific inquiry are a mere chimera, mere guessing. On the contrary, the theories and hypotheses scientists accept are in many cases well supported and justified by the evidence of observation. But the point is that the evidence is inductive, as in the inference made from it. Furthermore, the a posteriori statements that are neither conclusively verifiable nor falsifiable are not the only class of statements we accept on the basis of inductive evidence. On the contrary, most statements that are conclusively verifiable or falsifiable by observation are also accepted on the basis of inductive evidence.
The reason for this is quite simple.
There are many statements that could be falsified or verified by observation but are such that we are not in a position to observe the things in question. Consider some statement about the past, for example, that a certain man was born on January 10, 1936. the fact that he was born on that date is something that could be observed, but obviously, he is in no position to observe that hallowed event. If he accepts the statement, then his acceptance of it must be based on inductive evidence of the usual sort: his parents' word, the information on his birth certificate, and so forth. Indeed all statements about things that happen at other times and places are statements which, if we accept them at all, are accepted on the basis of evidence that does not entail their truth.
Universal statements, if accepted, must also be accepted on the basis of inductive evidence, because they are not conclusively verifiable by observation; and particular statements, if held to be false, must be so held on the basis of inductive evidence, because they are not conclusively falsifiable by observation. When we consider the vast number of things that we believe, we will soon discover that induction is the warrant of most of them. It is rare to elicit premises from observation from which one can validly deduce the truth of those a posteriori statements one believes. The deduction almost always fails, but the powers of human reason refuse to be restrained by the limits of deductive reasoning. When a deductive argument is not forthcoming to defend our beliefs but the evidence seems strong nonetheless, then induction is called upon to meet our needs. Hence it is essential that we obtain some understanding of this variety of argumentation.
In an inductive argument, the premises are evidence for the conclusion or hypothesis. Unlike a sound deductive argument in which the premises entail the conclusion, the evidence of sound inductive argument does not entail the hypothesis inferred from it. What then, is a sound inductive argument? One condition of soundness is that the evidence must consist of true statements.
A sound inductive argument is one in which the statements of evidence are true, and in which if the premises are true, then it is reasonable to accept the hypothesis as true. So the second condition of soundness of an inductive argument, which we shall call inductive cogency, may be put as follows: if the evidence is true, it is reasonable to accept the hypothesis as true also.
FORMS OF INDUCTIVE ARGUMENTS
One familiar variety of inductive argument is a statistical argument in which the evidence or the hypothesis is a statistical statement concerning the percentage of things of one sort that are another. One example of a statistical statement is the statement that 67 percent of the cats of Dibar are rabid. This statement may be a hypothesis of an argument inferred from the evidence of observation. It may also be used as evidence for some conclusion about the health of a cat whose health is undetermined. Two forms of argument that could be employed are induction by enumeration and statistical syllogism.
b. Induction by enumeration
X per cent of the examined members of A are B
X percent of the members of A are B
c. statistical syllogism
X percent of the members of A are B (X being greater than 50).
O is an unexamined member of A
O is a member of B
The following two arguments instantiate these forms:
67 percent of the examined cats of Dibar are rabid.
67 percent of the cats of Dibar are rabid.
The cat that bit me is an unexamined cat of Dibar.
The cat that bit me is rabid.
These two arguments illustrate very familiar forms of inductive statistical argument. It is apparent that the hypotheses inferred from the evidence are not validly deducible from them. It is logically possible that what we have observed to be true of a certain percentage of the cats in a sample is not characteristic of the same percentage of cats in the total population of Dibar, and it is logically possible that what is characteristic of a certain percentage of cats of Dibar is not characteristic of a particular unexamined cat. There is one exception that should be noted. If we have a statistical syllogism in which the evidence is that 100 percent of the members of A are B, and ) is a member of A (unexamined or not), then, of course, it follows deductively that O is B. However, except for this extreme case, we must add other restrictions to render plausible the claim that arguments of these forms are inductively cogent.
The foregoing argument illustrates a typical problem confronting the attempt to provide argument forms for inductive logic. There is an underlying difficulty that generates the problem. It is natural to assume that just as a valid deductive argument is one in which if the premises are true, then the conclusion must be true, so a cogent inductive argument is one in which, if the evidence statements are true, then the hypothesis is probable.
Probability, even high probability, will not suffice for inductive cogency. In both induction by enumeration and statistical syllogism we may suppose that the inferred hypothesis is probable, even highly probable, on the basis of the evidence. Thus one inclined toward the idea that the argument form is cogent. But this natural line of reasoning leads directly to inconsistency.
A more general argument is available to show that probability, even very high probability, of a hypothesis on the basis of evidence does NOT suffice for inductive cogency. It depends on considering fair lotteries which enable us to specify probabilities with precision. Suppose, for example, that we think any hypothesis having a probability of 99/100 or greater on the basis of evidence may be cogently inferred from the evidence of induction. Imagine we have a lottery containing 100 tickets numbered consecutively from 1 to 100. Imagine a ticket has been drawn and the lottery is fair. All this is our evidence. Now consider ticket number 100. The probability on the evidence that it was drawn is 1/100. There is one chance in a hundred that it was drawn. This means that the probability that some other ticket was drawn is 99/100. Assuming that this is a high enough probability for cogent inductive inference, we may cogently infer from the evidence that some ticket other than the 100 ticket was drawn. Beginning from the same evidence we could use an argument of the same form to infer that some ticket other than 99 was drawn, that some ticket other than the 98 ticket was drawn, and so forth. In each case the hypothesis would have a probability of 99/100 on the evidence. So for each ticket we could cogently infer that some other ticket was drawn. But then the set of conclusions would be inconsistent with our original evidence. For the set of conclusion would tell us, for each ticket, that it was not drawn, and this is inconsistent with our evidence which tells us that one ticket WAS drawn. In short, the set of hypotheses inductively inferred entails each of the tickets numbered from 1 to 100 is not drawn whereas our evidence tells us that one of them is.
The preceding argument shows that inductive arguments of the following form are NOT cogent:
d. induction by probability
It is highly probable that P
They are not cogent because such argument forms lead from true evidence statements to inconsistent statements. We have said that a cogent inductive argument is one in which, if the statements of evidence are true, then it is reasonable to accept the hypothesis as true for the purpose of accepting true hypotheses and avoiding error. By accepting an inconsistent set of statements we insure that some statement we accept is erroneous. Therefore, inductive argument forms are not cogent when they warrant the inference of an inconsistent set of statements from true evidence statements.
The preceding argument illustrates the difficulty of attempting to specify any form of argument that is inductively cogent. We may obtain an improved account of inductive cogency by noting the importance of the concept of competition among hypotheses as a feature of induction.
Whether it is reasonable to accept a statement as true depends on what other statements it competes with as well as on the probability of the statement as evidence.
To understand this, consider once again the conclusion of the induction by enumeration concerning cats. The hypothesis inductively inferred was that 67 percent of the cats of Dibar are rabid. Is it reasonable to accept that hypothesis on the basis of the evidence? The answer to the question depends on what statements you take that hypothesis to compete with. If the competition consists of other statements specifying the exact percentage of rabidity of the cats of Dibar, then tit would be more reasonable to accept that hypotheses than any of the others because it is more probable than any others. On the other hand, if the competition includes not only hypotheses concerning exact percentages but also less exact hypotheses -- for example, the statement that the percentage lies somewhere between 60 and 80 percent -- then the issue has been radically changed. For the hypothesis that the percentage lies within that interval is much more probable than the more exact hypothesis specifying the percentage at a single point within that interval.
We conclude that inductive cogency depends in an essential way on the evidential and conceptual context of reasoning. We can give a definition of inductive cogency in terms of the notion of competition as follows:
An inductive argument from evidence to hypothesis is inductively cogent if and only if the hypothesis is that hypothesis which, of all the competing hypotheses, has the greatest probability of being true on the basis of the evidence.
Thus, whether it is reasonable to accept a hypothesis as true if the statements of evidence are true is determined by whether that hypothesis is the most probable on the evidence of those with which it competes.
The conclusion we have reached supplies us with a methodology for checking the cogency of an inductive argument. Confronted with an inductive argument, one should pose two critical questions:
1. What statements does the hypothesis of the argument compete with?
2. Is the hypothesis more probable than those hypotheses with which it competes?
Only if the answer to the second question is affirmative may we consider the argument cogent.
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