This paper is reprinted from Enrichment Study 12, in Ron Yezzi, Practical Logic (Mankato: G. Bruno & Co., 1992). Copyright 1992 by Ron Yezzi

# Defining Deduction and Induction

## Problems

The distinction between deduction and induction based upon whether conclusions follow necessarily or probably from their premises has achieved nearly universal acceptance among writers on logic. It has become the standard distinction in introductory textbooks. Still I find the distinction unsatisfactory for several reasons.

First of all, a disturbing lack of parallelism exists between deduction and induction by which the standards for establishing inductive conclusions are more demanding than those for deductive ones. In deduction we only need to consider the logical form of the argument; whereas, in induction, we must also add information about the world to show that the conclusion follows with some degree of probability. For example, in induction by simple enumeration, we need to assess the fairness of the sample in terms of our knowledge of the actual world in order to show that the conclusion probably follows from the premises. In a modus ponens argument, on the other hand, we need only consider the logical form present. This particular lack of parallelism leads to the "second class" status of inductive logic, as shown by the degree to which logicians, and usually the logic courses they teach, concentrate on deductive logic.

Secondly, instructors in introductory logic courses ordinarily stress the need to evaluate arguments first in terms of the strength of the conclusion relative to the premises. Accordingly, students may be told to assume that premises are true. This procedure has the beneficial effect that it enables them to study the basic structure of arguments without needing to possess expert knowledge about the world. It seems however that we can seldom assess the strength of inductive arguments without possessing such knowledge. On the one hand, we seem to be saying that expert knowledge of the world should not be relevant to assessing the relation between premises and conclusion; on the other hand, we seem to be saying that such knowledge is relevant, in inductive logic. If we respond by saying that we can study the structure of inductive arguments without assessing the probabilities of premises relative to conclusion, we are left with the possibility that there exists some underlying understanding of inductive arguments aside from the definition in terms of probability.

And thirdly, the current distinction between deduction and induction leads to cases where the same argument should be considered both deductively and inductively. After all, an invalid deductive argument may still be a good inductive one, if the premises establish a high degree of probability for the conclusion. Consider the following argument that commits the fallacy of an undistributed middle term:

All individuals with normal or superior IQs are human beings
John Doe is a human being
_______________________________________________________________
John Doe is an individual with normal or superior IQ

This fallacious deductive argument would have to be regarded as a good inductive one, when we assess the degree of probability attaching to the conclusion, given the knowledge that individuals with normal or superior IQs constitute 75% of the class of human beings. Or consider

All individuals with normal or superior IQs are human beings
All college students are human beings
__________________________________________________________________________
All college students are individuals with normal or superior IQs

Now this fallacious deductive argument becomes an even better inductive one with a probability approaching closer to certainty, given the knowledge that nearly all college academic standards are sufficiently demanding to require normal or superior IQs. Although no inconsistency arises in asserting that the same argument is deductively invalid but still an excellent inductive argument, the situation is troubling when we consider the degree of stress we place on students' being able to detect the flaw in syllogistic arguments possessing an undistributed middle term. We can even go a step further and produce an apparent inconsistency. Consider

All human beings are mammals
All Greeks are mammals
____________________________________
All Greeks are human beings

What happens when we assess the inductive probability of this invalid deductive argument? If we set up a Venn diagram, then there are three possibilities for the relation between Greeks and human beings--namely, the Greeks can be in region 1, 2, or both. That is why the argument is deductively invalid. But we now want to proceed inductively. So we want to assess the probabilities attaching to these various possibilities. Now if we think about the meaning of "Greeks" and "human beings," we see that all the Greeks are inside the class of human beings. That is, "All Greeks are human beings" is an analytic statement. Thus, inductively, the conclusion follows necessarily from the premises. But if the conclusion follows necessarily, we can label it a valid deductive argument and then face an inconsistency. Note carefully that the conclusion follows necessarily here through the process of inductive assessment, not simply because the statement in the conclusion is analytic. (In other words, if the conclusion here were "All triangles are three-sided figures," then the conclusion would not follow.) It is tempting to deny the inconsistency by saying that additional information is being introduced that alters the original argument. But how can the degree of probability in an inductive argument be determined without introducing additional information? To eliminate appeals to additional information is equivalent to eliminating any inductive argument not based upon some calculable probability set up among the premises themselves. What all this shows are the difficulties encountered when we define induction in such a way that standards for assessing the argument depend upon something other than the logical form of the argument itself.

Likewise, for the fallacy of affirming the consequent, we may end up with a strong inductive argument. Consider this example where the fallacy occurs:

If she won the New York State lottery sweepstakes, then Jane Doe--the law-abiding, Utica, N.Y. welfare mother with two small children, a limp, a lisp, and no other relatives or wealthy friends--has become a multi-millionairess
Jane Doe--the law-abiding, Utica, N.Y. welfare mother with two small children, a limp, a lisp, and no other relatives or wealthy friends--has become a multi-millionairess
__________________________________________________________________
She won the New York State lottery sweepstakes

Arguments of this form are fallacious because the same consequent can have more than one antecedent. Suppose however that we consider the argument inductively by trying to assess the probability that the conclusion follows. We should recognize that the properties of Jane Doe very likely exclude the most common, plausible antecedents other than the one stated in the first premise. For example, given those properties, it is unlikely that she gained the money through illegal activities or the signing of a professional contract as an athlete or entertainer. Consequently, this argument seems to be inductively strong. Although no inconsistency arises, we can question whether logic instructors are prepared to stress the need for detecting the fallacy of affirming the consequent only to point out, however, that the argument can still be quite strong. And we can question whether they are prepared to show the ways in which all the standard, fallacious deductive arguments are to be evaluated inductively. Given the current definition of induction in terms of probability, it is not at all clear why logic texts treat, say, argument by simple enumeration and argument through agreement in a discussion of induction rather than selecting the standard fallacious deductive arguments instead. One may appeal to tradition; but the current definition of induction is not traditional. So far as I know, these considerations do not appear in any current introductory logic text. If they did appear, there would be a significant risk that the distinction between correct and incorrect reasoning, a basic purpose for logic, would become quite confusing. At the very least, there would be serious doubts raised about the importance of deductive logic.

## Solutions

I think that there is a way out of these difficulties. We can establish a parallelism between deduction and induction so that both kinds of argument can be studied strictly in terms of their logical form. We can apply the important distinction between validity and soundness to induction as well as deduction. We can eliminate any need to consider fallacious deductive arguments inductively because of the current definition of induction. And we can redefine deduction and induction in a way more closely resembling the traditional definitions that go at least as far back as Aristotle.

We can accomplish all this by taking a simple step in induction, equal in kind with what we do for deduction. We can make an appropriate assumption that is equal in kind to two assumptions we regard as legitimate in deduction, when testing for validity, namely, (1) assuming the truth of the premises and (2) assuming uniformity of meaning for component terms or symbols.

The first assumption requires no explanation here (although I will add some comments later in a discussion of the definition of validity). The second assumption, however, (because of the particular way in which I want to discuss it) does require some explanation, because it is so obvious that we never state it. The same term or symbol need not retain the same meaning throughout an argument. For example, the author of an argument can insert a rule that, for every successive, even-numbered instance of a term or symbol, its meaning becomes the negation of its immediately preceding meaning. According to the rule,

A implies B
B
____________
A

becomes a modus tollens argument. Although nothing forbids insertion of such a rule, it is so silly, confusing, and unnecessary that we safely assume that the authors of arguments do not insert it. Accordingly we assume uniformity of meaning.

What, then, are the appropriate assumptions that assure the validity of inductive arguments? (a) In enumerative induction, we should assume that the premises constitute a fair sample. (b) In eliminative induction, we should assume that the premises contains all the conditions possibly relevant to the predicate in the conclusion. (c) In analogical argument, we should assume a perfect analogy in the first premise. Once we make these assumptions and reduce the three kinds of arguments to standard form, the conclusions follow necessarily from their premises. Moreover, in making these assumptions, we are not really adding external premises since the assumptions are internal to the meaning of the argument. Thus, in simple enumeration, assumption of a fair sample is a precondition for the conclusion generated. For example, scientists would not conclude that "Water boils at 100 degrees C., under standard conditions" unless they presume that the instances of boiling water described in the premises constitute a fair sample, that is, one that is representative of the whole of which it is a sample.

Making these assumptions, we will assert the validity of some arguments quite at odds with our knowledge of the world. But we should not confuse validity with soundness. Just as there can be silly, but valid, deductive arguments, there can be silly, but valid, inductive arguments.

Making these assumptions, we may seem to be sweeping the problems in establishing inductive conclusions under the rug. But such is not really the case. We are merely setting aside problems unrelated to the logical form of the arguments, just as we set aside the issue of the actual truth or falsity of premises in testing the validity of deductive arguments. After we settle the validity issue, we can assess the degree of approximation to a fair sample, the completeness of the statement of possibly relevant conditions, or the degree of approximation to a perfect analogy.

We can make this new treatment of induction correspond even more closely to the validity-soundness distinction by broadening the concept of soundness so that it encompasses the practical reliability of a valid argument, not just the truth or falsity of its premises. Accordingly, determining soundness for the three kinds of inductive argument mentioned above would include an assessment of (a) the representativeness of a sample, (b) the completeness of the stated, possibly relevant conditions, or (c) the strength of the analogy--in addition to testing for validity and establishing the truth of the premises. A section dealing with this broadened concept of soundness--that is, with general ways of establishing soundness--would be a valuable addition to an introductory logic text or course.

## Redefinition

This new treatment of induction creates a problem of definition however since the distinction between deduction and induction in current use no longer applies. Moreover, we cannot simply reinstitute the traditional distinction--whereby deduction goes from general to particular, and induction from particular to general--because of well-known counter examples. Still, there is a way out, I think. We can retain definitions based on the direction of arguments, but without injecting the terms general and particular. There is a difference between deductio, a leading down, and inductio, a bringing in. In deduction, we seem to begin with statements that function as principles and we argue from them. In induction, we seem to begin with statements that do not function as principles but rather lead to them in conclusions. We are dealing here with a fairly subtle distinction regarding the significance we assign to various statements in arguments. Within the present context, a principle is a basic assertion for purposes of argument and explanation. (We can easily see how general statements can function as principles.) In deduction, the premises fit the concept of a principle quite well. They are the originating points for the argument and the argument proceeds from them. In induction, however, a strange reversal occurs. Although the premises are the originating points of the argument and the argument proceeds from them, we attach greater significance to the conclusion. The premises seem to function merely as tools to produce a conclusion that we can then use as an originating point for further argument and explanation. That is, deduction is argument leading from principles; and induction is argument leading toward principles. These definitions avoid the obvious counter examples that create problems. They can accommodate the standard deductive and inductive arguments--although, admittedly, some troubling cases will probably arise. Also, according to these definitions, mathematical induction and complete, or perfect, induction would once again be classified as inductive arguments--after having been labeled as deductive, based on the necessary-probable distinction.

Enumerative induction fits nicely within the new definition of induction. And eliminative induction does, too, once we recognize that the argument leads from a state of less definite premises toward a more definite conclusion. Analogical argument however does not fit well--since the first premise establishes a definite analogy from which the conclusion follows. Accordingly, analogical argument would now be classified as deductive.

This new treatment of deduction and induction would also have further, more far-reaching effects. (1) A unified symbolic logic, making use of both deductive and inductive arguments, becomes possible. And (2), the traditional problem of induction dissolves or, at least, becomes a misnomer, once validity is attributable to inductive arguments. Just as there is no traditional problem of deduction, there is not one of induction either.

These suggestions would require some substantive, but not especially intrusive, revision of introductory logic texts. There is, I think, nothing to be lost and a great deal to be gained.

## Objections

There are several problems arising due to ways of discussing induction and validity in other logic texts, which warrant some comment.

Objection 1: Do we really assume premises to be true in deductive arguments?

According to a number of introductory logic texts, an argument is valid if and only if it is impossible that the premises are true and the conclusion false. So an argument can be valid even though it has false premises. Therefore, apparently, we do not assume the truth of premises in deduction. This definition brings the general meaning of validity into accord with the truth-functional definition of validity in symbolic logic.

Reply: This definition of validity however is not universally adhered to. Indeed, when text writers introduce the concept of validity, their primary concern is not so much this definition in itself as it is the importance of logical form in arguments and the need for a necessary connection between premises and conclusions. They just want to render the actual truth or falsity of the premises irrelevant in determining the validity or invalidity of the argument.

Although the truth-functional definition of validity serves a useful function in symbolic logic, we need not regard it as expressing the real meaning of validity--just as we need not regard the definition of material implication as expressing the real meaning of implication. Put differently, I am just saying that validity need not be defined so that false premises can produce valid arguments.

In traditional logic at least, we can render the actual truth or falsity of premises irrelevant in considering the validity of arguments quite well by assuming that the premises are true. This procedure conforms easily to the basic nature of an argument, where the premises are regarded to serve an assertive function, which then warrants a conclusion.

Objection 2: If it is a characteristic mark of inductive arguments that the conclusion goes beyond what the premises assert, how can a conclusion ever follow necessarily?

It is often pointed out that conclusions are implicit in their premises for deductive arguments, so that the arguments produce no new knowledge. The ampliative character of induction, by contrast, provides a way to extend our knowledge. This ampliative character is a primary virtue of inductive arguments.

Reply: If we make the appropriate assumptions for induction mentioned earlier in the paper, however, we have a way of getting conclusions that follow necessarily. If we thereby deprive induction of its ampliative character, we should not take this result as a demerit against inductive arguments--given the well known success and importance of deductive arguments. Even the concept of ampliative argument is suspect, when we remind ourselves that it suggests getting something from nothing or, more precisely, that its conclusion claims more than its premises are capable of producing. When we want to extend our knowledge, it is sufficient to combine the techniques for establishing soundness (as treated in this Enrichment Study) with the techniques for arriving at valid arguments.

Objection 3: If it is a characteristic distinction between deduction and induction that additional premises added to a valid deductive argument cannot affect its validity whereas added premises can affect the strength of inductive arguments, then how can one say that an inductive conclusion follows necessarily?

In induction by simple enumeration we can see how an enlarging of the sample in the premises can affect the probability that the conclusion follows. But this cannot happen in valid deductive arguments because, regardless of additional premises, the conclusion always remains implicit in the initial premises.

Reply: Given our new assumptions however, any problem here disappears. If we assume a fair sample in simple enumeration, then no additional premise is going to deny the necessity with which the conclusion follows from the initial premises. Any added premise will concur with the initial ones. If someone asserts that an added premise can establish that the assumption of a fair sample was incorrect, then we need to note that, for deductive arguments likewise, an additional premise can establish that the assumption of true premises was incorrect. For example, suppose that we add the premise, Three human beings are immortal, to this argument:

All human beings are mortal
All college students are human beings
_____________________________________________
All college students are mortal

So either both deduction and induction have a problem here or else neither does. I favor the latter alternative, given the ways logicians treat validity.