Logical Structure in Deductive and Inductive Reasoning

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Logical Structure in Deductive and Inductive Reasoning

Every good murder mystery, all good experiments in science, and every reflective lifestyle depend upon deductive and inductive reasoning. These two types of reasoning come so naturally to you that explaining them is, initially at least, more a matter of reminding you about something you already do than telling you something new. Of course, because they are so natural, you may never have paid close attention to them the way philosophers have.

Now is your chance. If you want to develop that ability to “think in stereo” that we discussed in Chapter One, you will need to be able to listen and read for deductive and inductive reasoning. Each of these types of reasoning has specific structures, values, and pitfalls. In this chapter, then, we introduce you to two of your most prized reflective practices: deductive and inductive reasoning. They are not only “Sherlock’s logic,” but the forms of thought that almost define what it means to be rational.

As you know from the last chapter, all reasoning involves saying one thing (the premises) as support for another (the conclusion). That support can be understood as argumentative, as when we are supporting a belief in the truth of the conclusion, or explanatory, when our premises claim to support an explanation of the conclusion.

Taking argumentative rationales for the moment, we can distinguish two “levels” of support for conclusions. Look at the following two arguments and see if you can distinguish the different degrees of support for the conclusion in each case:

Argument One:

1. A bachelor is an unmarried male.

2. John is an unmarried male.

C. John is a bachelor.

Argument Two:

1. A bachelor is an unmarried male.

2. John doesn’t wear a ring on his fourth finger.

3. Most people who are married wear a wedding ring.

C. John is a bachelor.

Aside from the fact that the second argument has one more premise than the first, what’s the difference? They both argue for the same conclusion.

Assume for the moment that the premises of both arguments are true. Are you more certain of the truth of first argument than the second? You should be. It is impossible for the premises of the first argument to be true while the conclusion is false. In the second argument, you might say that the conclusion is “likely” if the premises are true, but you could not say it was certain. The first argument is deductive and the second is inductive.

The goal of a deductive argument is to demonstrate the truth of a conclusion with absolute certainty.

Inductive arguments attempt to show that there is a high probability that the conclusion is true.

Recognizing deductive and inductive arguments in everyday speech and writing is usually not that difficult. There are situations in which you have an interpretive choice to make about whether to construe someone’s words as part of a deductive or inductive reasoning process. Suppose, for example, you ran into the argument above in the following natural language prose: “I’ll bet John’s a bachelor. He doesn’t wear a ring.” Of course, any reconstruction of this rationale as an argument will have to fill in premises. It isn’t obvious whether you should interpret the argument as deductive or inductive. You will have to decide based on your view of the speaker’s intent. These days it would be a naïve speaker who would think that everyone who is married wears a ring, so it would probably be unfair to fill in the argument with the premise, “All married people wear wedding rings.” A fairer interpretation would be an inductive interpretation.

Let’s get back to our basic distinction: Deductive arguments attempt to show conclusions with absolute certainty, while inductive arguments attempt to show that a conclusion is probale. Probability is often represented as a percentage or decimal, as when the weather report predicts a 60% chance of rain. We could also say that the probability of rain is .6. In these terms, deductive arguments have a probability of 1.0 while inductive arguments attempt to show conclusions with a probability of less than 1. To practice reading for deductive and inductive intent, consider the arguments in the following dialogue:

Lost in the Fog

Kara: Don’t you think we should stop and ask for directions? It’s already 6:30 and the concert starts at 7:00pm. We haven’t had any luck finding The Gorge so far. I don’t think we’ll find it without help.

Nick: Well, it has taken longer than I thought. But your friend Chris told me it was 10 miles from the highway exit and we’ve only gone 8, so let’s keep going.

Jennifer: That’s assuming we’re going in the right direction. The Gorge is down by the Columbia, but since we left the highway we’ve been going uphill.

Ryo: That’s true but I remember being here a long time ago and I think you had to go up a hill and then down. So maybe this is ok.

Kara: Don’t you think we would have seen more signs for it if we were only 2 miles away? I think we’re lost.

This passage is full of inductive and deductive arguments, so let’s dig in and sort them out. The first argument sounds inductive, because Kara is saying that their current failure to find the concert makes it seem unlikely that they will succeed without help. The second argument seems deductive because if the premises are true (1. The Gorge is 10 miles from the exit. 2. They have only gone 8.), then it follows with certainty that they have two miles to go. Of course, Jennifer points out the assumption in that argument and makes another deductive argument: 1. The Gorge is downhill from the highway. 2. If the Gorge is downhill from the highway, then anyone driving uphill from the highway is going in the wrong direction. 3. We are driving uphill from the highway. Therefore, we are going in the wrong direction. Ryo introduces a new wrinkle with his vague memory about going up a hill first. This should lead us to construe his argument as inductive because he only assigns a probability to the accuracy of his memory. Finally, Kara offers inductive evidence (evidence that creates a probability of truth in the conclusion) that they are lost. After all, it’s unlikely that there wouldn’t be signs along the road to the Gorge. Fortunately, one of them called the Gorge and they made it to their concert.

As you can see, when we adduce evidence to make a point or make predictions in which we have only a degree of confidence, we are usually making an inductive inference. When you feel that you just can’t be wrong because the facts just won’t admit of exceptions, and they are logically linked, you have a deductive inference. This happens most noticeably when we are dealing with definitions, abstract knowledge or “essences.” The example we gave about bachelors involved definitions. Arguments in geometry are usually deductive because abstract objects like triangles have properties that are essential to them and are in necessary relationships. For example, it can be shown from the postulates of geometry that the interior angles of a triangle must equal 180 degrees and that the sides opposite the larger angles are larger than the sides opposite the smaller angles.

When philosophy in the West was a younger discipline, philosophers were very excited about the possibility of knowing the world in the way that geometry (then also a new science) proceded. It was a required course as far as Plato was concerned. It seemed that if you could know the essence of something, like an abstract object (a triangle or, as Plato may have behaved, “the good”), with absolute certainty, then you could deduce the properties that it necessarily had to have. If essences (everything that is real) were arranged in a big hierarchy, then it might be possible to draw all knowledge out in a deductive fashion. That was one of the dreams of philosophy, and it continues to captivate some philosophers today.

In studying the law like behavior of the universe, we also seem to find essential truths that we can “unpack” by drawing them out from things already known. We often do this with deduction. The logical structure of many experimental investigations in science has a deductive form, even though most of the actual results of scientific investigation are expressed in probabilities and inductive causal inferences. The broader theoretical reasoning in a natural science is also largely inductive.

Induction is not the special property of the sciences, of course. There is also something that might be called an “inductive temperament” and while yyou may see it most prominently in scientific thinkers, it is also part of most people’s basic approach to knowing the world. To get a feeling for this. Consider the following description of the problem of knowing the world: “Imagine that we are living on an intricately patterned carpet. It may or may not extend to infinity in all directions. Some parts of the pattern appear to be random, like an abstract expressionist painting; other parts are rigidly geometrical. A portion of the carpet may seem totally irregular, but when the same portion is viewed in a larger context, it becomes part of a subtle symmetry… . Everywhere there is a mysterious mixing of order and disorder. (From Martin Gardner, “Mathematical Games,” Scientific American (March 1976), 119. Copyright © 1976 by Scientific American, Inc., New York. )

Kathleen Dean Moore, author of a book on induction, adds to Gardner’s image, which she quotes, by suggesting that we imagine that the carpet is only partially revealed to us and then often through a distorting filter. This is a good way to think about the problem of knowledge for many contemporary researchers and thinkers. But how does this “mixing of order and disorder” lead to knowledge, inductive or otherwise? The key is the assumption we make while studying the carpet. By making one important assumption we seem to have found a way to know, or at least operate reliably in, the world. That assumption is called the Principle of Induction. While it has many forms, it basically asserts that “Nature is uniform.” You could also say that the principle of induction is the belief that the future will be like the past, that events observed together will continue to be observed together, or, one of my favorite ways of putting it, “Patterns are pervasive and repeating.”

The Principle of Induction is not something that is easy to argue for. If you argue for it on the basis of past experience, you would seem to be assuming that the past is like the present, but that is the very thing you are trying to argue for! Arguments that depend upon believing what is being argued for (which have their conclusions somewhere in the premises) are called “circular” and most circular arguments are bad arguments (Philosophers sometimes distinguish between “vicious circularity” and plain old circularity distinction we will discuss later). But no one denies that the Principle of Induction, even if it can’t be argued for, is very useful. It seems completely natural to our minds and motivates the search for patterns.

Now that you have an introduction to the distinction between induction and deduction, you may want to read about “Deductive Argument Forms” and “Inductive Inference”