4. Induction and Deduction

“JM-FB3x”

I was driving my son to school one morning, and said to him, “That kid in front of us has been on the football team for three years. And his family are Yankees fans, probably moved here from New York.”

“How do you know?” he asked.

“I read his license plate, and the back of his car: Yankees logo on the hitch, Giants logo on the license plate holder, ‘NY Budget Sales’ painted on the rear.”

Sherlock Holmes might have said “Elementary deduction, my dear Dylan,” but Dylan’s lived with a logic professor long enough to know that that would be incorrect. This kind of reasoning is called “induction,” not “deduction.”

It is an interesting question why Conan Doyle gets this wrong. Virtually none of the reasoning Holmes engages in could properly be called “deduction.” It’s nearly all cause-effect reasoning, or generalization, or arguing by analogy. I don’t know the history of the word “induction” well enough to be confident that Conan Doyle would have learned this in his schooling, but the use of it we’re sticking with in this course, and in any Logic course, goes back at least one hundred years before Conan-Doyle, to the 18th century Scottish philosopher David Hume.

     

What typifies induction is the recognition that the reasoning engaged in is at best highly probable. When I announced that the kid in front of us was a football player, I was making quite a few assumptions which might well turn out to be false, such as: that the car he’s driving is his own or even his family’s. And even if it is theirs, it is possible that it’s his brother, not him, who was on the football team, or that the “JM-FB3x” doesn’t mean what I took it to mean (“James Monroe Football”). Maybe it meant “Jim lied three times.”

Deduction, on the other hand, is the kind of reasoning that claims to be free of degrees of probability. Deductive arguments are the kind in which the premises are intended to make a completely airtight case for the conclusion. Or is that a watertight case? Same difference I suppose, since in both metaphors the point is about nothing getting away or sneaking in. No “if’s,” “and’s” or “but’s.”

“I think, therefore I am,” for instance, is a deductive argument. If the premise (“I think”) is true, then the conclusion (“I am”) is inescapable. It’s what Humpty Dumpty might call “glory.”

Arguments come in two general categories: the kind that claim that their conclusions follow from their premises with necessity are called “deductive.” The kind that only claim their conclusions follow from their premises with some probability are called “inductive.”

Of course, as was true before, it remains important to appreciate that “claiming to follow from” and “following from” are not the same thing. Arguments claim that their conclusions follow from their premises, and in good ones they do—in bad ones they don’t. That will continue to be the case when we factor in whether they claim this as a matter of necessity or as a matter of probability. And since we’re distinguishing between two large families of arguments now, when we start talking about the goodness and badness of arguments, we’re going to be using enough different words that we can tell which kind we’re talking about right off. “Good” and “Bad” aren’t going to be all that helpful, i.e., all that good or bad. (Can you see that if we didn’t have the use/mention distinction to follow, the preceding sentence would be absurd?)

We’ll begin by looking at two sets of examples that typify the two families of Induction and Deduction. These are typical patterns, pretty easily recognized when encountered, that will help to make the distinction more clear and concrete.

Inductive Patterns

The example I began this section with counts as what we call an argument from signs. I inferred that the kid was a three-time football player at James Monroe High School because he was driving a car that had a license plate on it. I inferred his family was from New York from the signs on the car’s hitch and license-plate holder. We reason by appeal to signs all the time, choosing which restroom to use, for example, is a matter of recognizing the sign (and reasoning in an immediate way that no pranksters have switched them). It’s easy to appreciate that reasoning from what a sign says is far from drawing a conclusion that is inescapable.

Another example, something different, from Conan Doyle this time, The Adventure of the Dancing Men:

Holmes had been seated for some hours in silence with his long, thin back curved over a chemical vessel in which he was brewing a particularly malodorous product. His head was sunk upon his breast, and he looked from my point of view like a strange, lank bird, with dull gray plumage and a black top-knot.

“So, Watson,” said he, suddenly, “you do not propose to invest in South African securities?”

I gave a start of astonishment. Accustomed as I was to Holmes’s curious faculties, this sudden intrusion into my most intimate thoughts was utterly inexplicable.

“How on earth do you know that?” I asked.

He wheeled round upon his stool, with a steaming test-tube in his hand, and a gleam of amusement in his deep-set eyes.

“Now, Watson, confess yourself utterly taken aback,” said he.

“I am.”

“I ought to make you sign a paper to that effect.”

“Why?”

“Because in five minutes you will say that it is all so absurdly simple.”

“I am sure that I shall say nothing of the kind.”

“You see, my dear Watson”–he propped his test-tube in the rack, and began to lecture with the air of a professor addressing his class–“it is not really difficult to construct a series of inferences, each dependent upon its predecessor and each simple in itself. If, after doing so, one simply knocks out all the central inferences and presents one’s audience with the starting-point and the conclusion, one may produce a startling, though possibly a meretricious, effect. Now, it was not really difficult, by an inspection of the groove between your left forefinger and thumb, to feel sure that you did NOT propose to invest your small capital in the gold fields.”

“I see no connection.”

“Very likely not; but I can quickly show you a close connection. Here are the missing links of the very simple chain. 1. You had chalk between your left finger and thumb when you returned from the club last night. 2. You put chalk there when you play billiards, to steady the cue. 3. You never play billiards except with Thurston. 4. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him. 5. Your check book is locked in my drawer, and you have not asked for the key. 6. You do not propose to invest your money in this manner.”

“How absurdly simple!” I cried.

“Quite so!” said he, a little nettled.

The key element in this reasoning is cause-effect relations. Since there was chalk on Watson’s finger and thumb, Holmes infers that he’d been at the club playing billiards with Thurston, and so on: “the cause of the chalk on his fingers is that he applied it at the club.”But if Watson were a clever man, and a little mischievous, he might have simply applied the chalk in the stairway on his way home, to throw Holmes off, and make him run through a set of inferences. Reasoning from cause to effect, or from effect to cause (as in this case) is inductive reasoning, and at best it’s highly probable. Now, the inference in the middle, “You never play billiards except with Thurston” therefore you saw Thurston, fits our definition of deduction –that’s because of the “never.”

Reasoning from effect to cause may seem similar to reasoning from a sign, but the difference is more important than the similarity: one thing or event bringing about another is different from something signifying something else. If I see a string tied to a person’s finger, I might reason –causally—that she put it there to remind herself of something; when she sees it, however, she will reason—by virtue of its being a sign—that there is something she is supposed to remember to do. Now, if she is really forgetful, she might reason causally, and say “What the heck is this string doing on my finger? Someone must have put it there, I wonder if it was me?” But once she gets that cleared up, there’s still the inference from the sign to be drawn: “Oh right, I have to pick up that book at the library on my way home.”

There’s something more than just reasoning from effect to cause in this Holmes example. Holmes’s ultimate conclusion is not that Watson was at the club, but is actually a prediction that Watson will not make the investment. This prediction is based on Watson’s not having asked for his checkbook despite his having seen Thurston and the month running out. All those things add up to the very good likelihood that he is not planning to make the investment. Predictions have something to do with causal reasoning, but they are about future rather than past events (“pre-dict” means saying in advance).

For a fourth pattern of induction, let us consider the following passage from Hume’s Dialogues Concerning Natural Religion. It is the argument known today by the words “intelligent design”:

Look round the world: contemplate the whole and every part of it: you will find it to be nothing but one great machine, subdivided into an infinite number of lesser machines, which again admit of subdivisions to a degree beyond what human senses and faculties can trace and explain. All these various machines, and even their most minute parts, are adjusted to each other with an accuracy which ravishes into admiration all men who have ever contemplated them. The curious adapting of means to ends, throughout all nature, resembles exactly, though it much exceeds, the productions of human contrivance; of human designs, thought, wisdom, and intelligence. Since, therefore, the effects resemble each other, we are led to infer, by all the rules of analogy, that the causes also resemble; and that the Author of Nature is somewhat similar to the mind of man, though possessed of much larger faculties, proportioned to the grandeur of the work which he has executed. By this argument a posteriori, and by this argument alone, do we prove at once the existence of a Deity, and his similarity to human mind and intelligence.

The argument states that the fact that everything in the natural world has its purpose (the “curious adapting of means to ends,” e.g., teeth are for chewing, feet are for walking, wings are for flying, the seeds inside a piece of fruit are for reproduction, etc.) resembles something we find in all things made by humans: all human products have a purpose. Human-made products like shoes, carriages, boats, dinnerware, etc., don’t come about by chance, but by deliberate design and planning. Since, therefore, the effects resemble each other, we are led to infer, by all the rules of analogy, that the causes also resemble. The conclusion is that an intelligent designer is the cause behind all natural things and the world itself.

This is called reasoning by analogy. The more similar two different things are, the more likely they are to be similar in some further way. Human machines and the things that occupy the natural world are declared to be similar at the outset of this reasoning, so it is expected that they will have a similarity in their causes: the cause of the one is intelligence, so the cause of the other must be too.

Notice that even though inductive reasoning is the kind in which all that’s claimed is a degree of probability rather than necessity, it is not uncommon to find words like “must be” as part of the presentation of the argument. We don’t let words like that, which amount to a matter of rhetoric or style, rather than of substance or content, settle any of our questions in Logic. How good an argument from analogy is will depend on how relevant the similar features are to the further feature one is reasoning about. In the Dialogue, one person replies that we have actually witnessed the cause –the bringing about—of houses, dinnerware and shoes, but have never witnessed the creation of natural phenomena, so inferring that intelligence is behind the existence of this hammer is much better founded than inferring that intelligence is behind the existence of water or of planets. He’s pointing out that there are significant dissimilarities between machines and natural phenomena too.

We use the word “generalization” in Logic to name another kind of inductive argument. Often in daily life, we use this word to mean a general claim, like the generalization that all teenagers like skateboarding. In Logic, we use it a little differently, to name the reasoning that yields such a general statement or claim. If I point out that Sartre’s three best novels were each over 400 pages long, you might infer that all his novels are that long. That would be a generalization: moving from knowledge of some cases –a few or many—to a claim about more or about all.

French philosophers also write fiction: just look at Sartre, Beauvoir and Rousseau.

Depending on whether that occurence of “French philosophers” was meant to mean “All French philosophers” or “Some French philosophers,” this generalization will be worse or better.

The last pattern of induction we’re going to name for this course is the argument from authorityNone of Sherlock Holmes’ “elementary deductions” are really deductions at all, because my Logic teacher said so, is an argument from authority. Your Logic teacher knows more about the field of Logic than some other people, so it seems appropriate to draw on his expertise on such matters.You could stick “Lady Gaga” in there, in place of “my Logic teacher,” and it would still be an argument from authority. As with the other five patterns I’ve been introducing here, this is just a pattern of argument; there is nothing inherently good or bad about an argument in one of these patterns. It’s a matter of degree; it will vary with the context and the topic, whether the premises give good reason to believe the conclusion or whether they don’t. On a question of Logic, you have better reason to believe me than to believe Lady Gaga.

Why? Because I said so?

 

Now we can talk about the terminology we use for talking about how good or bad inductive arguments are. At the first level, this terminology is pretty intuitive. We call inductive arguments “strong” or “weak,” often using the “–er” suffix, in fact, depending on how likely it is that the conclusion will be true as a result of the premises. “Strong” and “weak” are only used when we are talking about induction; and most of the talk in this course about induction will be about one of these six patterns: argument from signs, causal reasoning, prediction, argument by analogy, generalization or argument from authority. It’s pretty clear that the appeal to Lady Gaga on the use of “deduction” is weak compared to the appeal to the authority of the Logic professor.

 

It’s one thing to evaluate whether or not a conclusion is likely, given the premises. Another issue can come up too, which is whether or not the premises themselves are true.

In the Holmes example above, the conclusion that Watson is not going to invest in that property followed just fine from the premises. But the first premise in Holmes’ reasoning was that Watson had been at the club last night. If, in reality, Watson had been at the opera with a friend he didn’t want Holmes to know about, and had applied the chalk to his hand on the way home to fool Holmes, then despite the strength of Holmes’ argument, we would have reason to doubt the conclusion was likely be true. After all, the reasoning would be beside the point, grounded on a false assumption.

Consider another example. Since every President so far has been a woman, probably the next one will be a woman too. On the basis of this premise, this conclusion is quite likely, so this argument is a strong one. But, as we all know, the premise is false, and so we have no reason to really believe the conclusion. So is this a good argument or not? In one sense it is, and in another sense it is not. Since there are two distinct senses in which we can talk about an argument being good or not, we just don’t use the word “good”; it’s not good enough! We talk about an inductive argument being either weak or strong, and we also talk about it as being either cogent or uncogentStrength is about whether the conclusion probably follows from the premises; cogency is about whether the premises—the starting point for the reasoning—are true.

We don’t call an argument “cogent” unless it meets both of these conditions: a) it has to be strong, and b) all its premises have to be true. That’s the important central point. It follows that: every weak argument is also uncogent, and that some uncogent arguments are strong (like the one about women Presidents).

And one more very important point, we don’t use “true” and “false” at all for talking about arguments. These words apply only to the statements that make up arguments, but not to arguments themselves. This may take some getting used to, and I will correct you mercilessly if you ever say that an argument sounds true to you.

 

Deductive Patterns

 

The most common association with deductive reasoning is mathematical reasoning, but that’s much too limiting. Also, we can reason about situations in which mathematics is involved, but still be reasoning inductively. For instance, were someone to argue that since the price of gas is over a dollar higher today than a year ago, this time next year it will probably be two dollars higher, this would just be a prediction, inductive. Or if we argued that since five coins drawn from a jar were pennies, they will all be pennies, we would be generalizing, and that reasoning would be inductive.

Still, the connection with mathematics is a legitimate, if limiting, one. An argument from mathematics is one in which the conclusion turns on a calculation. Bush is the 43rd president, so whoever gets elected in 2008 will be the 44th –that’s an argument from mathematics, that’s a deductive argument. Try a couple of variations on that:

a) Bush is the 50th President so whoever gets elected in 2008 will be the 51st.

b) Bush is the 43rd President, so whoever gets elected in 2008 will be the 42nd.

Something is wrong with both of these in comparison to the first one. In a) the conclusion definitely follows from the premise, but the premise is clearly false. In b) the conclusion is silly, and definitely doesn’t follow from the premise; and the premise is true.

We’re going to identify four more common patterns of deductive reasoning before we set out the terminology that corresponds, in the realm of deduction, to what “strength” and “cogency” refer to in the realm of induction.

One kind is fairly similar to the argument from math, except it is about meanings of words instead of about calculations. For instance, The call on WMD in Iraq was a slam dunk, so it was an easy one. What it means to say that something is a slam dunk is that you can’t get it wrong, so it follows from the claim that the call on WMD was a slam dunk, that it was impossible to be wrong about it.

Here are two more examples of arguments from definition:

c) The day after Christmas is always a Friday, since Christmas always falls on Thursday.

d) Rush Limbaugh says he’s pro-life, so he must be against the death penalty.

 

Most students have heard this example before: All men are mortal, and Socrates is a man, so he’s mortal. This is known as a Categorical Syllogism.

A syllogism is an argument with two premises, so it’s easy enough to see why it’s a syllogism. What makes it “categorical”?

A category is a set of things. Syllogisms like this one are made up of statements that make connections between categories, like the category of men and the category of things that are mortal, the category of things that are Socrates and the category of things that are men, etc. Here are two more:

e) All spiders have six legs, and all six-legged things drive Toyotas, so all spiders drive Toyotas.

f) No dentists wear braces, and Julie is a dentist, so she wears braces.

You can tell these are categorical syllogisms because they are made up of statements that each relate a category of things to another category of things. All categorical syllogisms are deductive, so you can tell these are deductive arguments, since they are categorical syllogisms. (That, of course, was another one.)

 

We can identify a couple of other kinds of syllogisms now. They differ from categorical syllogisms because the smallest things they are made up of are not category names, but simple statements.

For instance: Mom said we’d either go to Chipolte or to Bonefish Grill after the movie, but Bonefish Grill is closed, so we have to go to Chipolte. The key element of this pattern is that it starts off from an “either…or…” sentence, eliminates one of the alternatives, and concludes the other. In Logic, “either…or…” statements are called “disjunctions,” so this is called a disjunctive syllogism. Two more examples:

g) Either Freud was French or else his wife was. But his wife was Austrian, so he must have been French.

h) Either Sartre didn’t follow Heidegger’s ideas accurately or else he misunderstood them. But he couldn’t have misunderstood him, so he followed them accurately.

 

Most students are familiar with the notion of a hypothesis, a proposed statement or idea. A hypothetical syllogism is any syllogism in which “if…then…” is the key structural element. Later on we’ll make a finer distinction, but for now we can say that any argument that has at least one hypothetical or conditional statement in it counts as a form of hypothetical syllogism. For instance, If you’re a fool, then you and your money are soon parted. Jim has been very successful managing his money, so he’s no fool. We’ll talk more about how conditional statements work, but for now you might find it useful to note that the “if” clause of a conditional statement is not claimed to be true; it’s the whole “if…then…” that is claimed to be true. Here are two more to consider:

i) If George Washington was assassinated then he died, and he was assassinated, so he’s dead.

j) If Washington was assassinated, then he died, but he wasn’t assassinated, so he’s not dead.

 

Five patterns of deductive reasoning: Argument from mathematics, Argument from definition, Categorical syllogism, Disjunctive Syllogism, and Hypothetical Syllogism. As I presented these patterns I gave three examples of each. If you look back closely at them, you’ll note the following: the example in the paragraph in which the pattern is discussed is a better argument than either of the examples that follow, indicated by letters “a” through “j.” For most of them, you can probably tell this intuitively.

But what does “better than” mean here? The situation is analogous to the strength/ cogency distinction we brought up regarding induction. But deduction and induction are very different kinds of arguments. We can’t use a word like “strong” to talk about an argument that presents its conclusion as following inevitably or necessarily from its premises, if we are already using it to mean an argument that presents its conclusion as probably true. The admission of mere likelihood that typifies inductive reasoning is sharply contrasted by the claim of “watertight-ness” or “glory” that a deductive argument makes. For one thing, when you claim that your argument for a conclusion is a nice knockdown argument, you are claiming to be absolutely right; and in response to that claim, there are only two possibilities: either you are right that you are right, or you are wrong! No middle ground, no “pretty right” or “kind of right.” That’s one reason why it’s so silly of Humpty Dumpty to hesitate about the number of days on which one can get an unbirthday present. The reasoning is an argument from mathematics, and it’s valid.

Logicians use “valid” and “invalid” to name the relation between premises and conclusion in deduction. If a conclusion follows inescapably from the premises that are provided, the argument is called “valid.” In the examples above, the ones embedded in the paragraphs of text are all valid, so they are “good” at the first level. But they are also “good” at the second level, because they also all have true premises.

In the lettered examples, however, the first one in each case is valid, but does not have all true premises; there is at least one false premise in each.

a) Bush is the 50th President so whoever gets elected in 2008 will be the 51st.

 

c) The day after Christmas is always a Friday, since Christmas always falls on Thursday.

 

e) All spiders have six legs, and all six-legged things drive Toyotas, so all spiders drive Toyotas.

 

g) Either Freud was French or else his wife was. But his wife was Austrian, so he must have been French.

 

and i) If George Washington was assassinated then he died, and he was assassinated, so he’s dead.

 

There is something confusing about these examples. You don’t want arguments getting called “valid,” which connotes being good or trustworthy, to lead to such nonsense as “the day after Christmas is always a Friday,” “Spiders drive Toyotas,” or “Freud was French.” You want them to lead to true statements, like “Washington is dead.”

But this confusion can be cleared up. We need to focus on the central defining character trait of an argument: it claims that, given its premises, its conclusion follows. That claim does not address the question of whether or not the premises are true. Leave the question of truth aside for a moment, and you can see that in each of these, the conclusion is the inescapable consequence of the premises provided. For each of these, we can say “it’s valid, but the premises are not true.” Given that defect, it is no surprise that the conclusions cannot be trusted to be true.

But wait a minute, Washington is dead! The conclusion of i) is true! What are we to make of that? Just this: that we can only be confident that the conclusion of a deduction is true if the argument is both valid and has all true premises. If it is either not valid or does not have all true premises, we cannot be confident that its conclusion is true or that it is false.

 

You may be expecting that good reasoning is going to lead to true conclusions and that therefore poor reasoning will lead to false ones. But poor reasoning doesn’t reliably lead to false statements; poor reasoning is not reliable at all! Some bad arguments happen to have true conclusions, while others have false ones. You can’t judge the quality of the argument from knowing the truth-value of the conclusion. That’s what Logic is for.

 

Let’s take a look now at the other lettered examples from above:

 

b) Bush is the 43rd President, so whoever gets elected in 2008 will be the 42nd.

 

d) Rush Limbaugh says he’s pro-life, so he must be against the death penalty.

 

f) No dentists wear braces, and Julie is a dentist, so she wears braces.

 

h) Either Sartre didn’t follow Heidegger’s ideas accurately or else he misunderstood them. But he couldn’t have misunderstood him, so he followed them accurately.

 

j) If Washington was assassinated, then he died, but he wasn’t assassinated, so he’s not dead.

 

What these all have in common is that the conclusions do not follow from the premises.

To b) you want to say “Nonsense, that doesn’t follow, that’s not what happens when you add 1 to 43.”

To d) you want to say “Nonsense, that doesn’t follow, that’s not what ‘pro-life’ means.”

To f) you want to say, “Nonsense, that doesn’t follow, that conclusion contradicts the premise!”

To h) you want to say, “Huh?” But even if you don’t know who Heidegger is, you can spot that this says “Either A is not true or else B is true. But B is not, so A is.” Nonsense! That’s not what follows!

To j) you want to say, “Nonsense, that doesn’t follow, in fact we all know he is dead!”

 

When the conclusion does not follow from the premise or premises, the argument is called “invalid.”

In deduction, the conclusion either follows or it doesn’t. There is no in-between like there are degrees of strength or weakness in induction.

What is the language we use in deduction that corresponds to what “cogency” names in inductive contexts?

Soundness.” An argument is sound if it is valid and all its premises are true. If it isn’t valid, it isn’t sound; and if it doesn’t have all true premises, it also isn’t sound. Sound arguments are what we want most often, because they are completely reliable: if you provide true premises in a valid argument structure, you’re going to get a true conclusion as a result, 100% guaranteed, every time.

Here’s a brief audio recording on the basics:

 

* * * * * * *  This link will open a powerpoint that reviews this material: 4 Induction and Deduction

Now get yourself a sheet of paper, and make a list of the 6 inductive and 5 deductive patterns, and list the appropriate evaluative terminology for induction and deduction.(Why not also see if you can list the 7 non-arguments we named?) You need to commit all of that to memory, at least for the duration of this course.

Homework:  Generate two examples of each of these 11 patterns. Spotting them as you read or listen to people is good as well, but begin practicing building arguments. These are patterns to think in terms of.  Bring your examples to class/ tutoring session to share.

There are also examples on the following Exercise page that you can try out

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