2 Induction

generalize from particular instances

Inductive reasoning is a method for inferring a generalization from particular instances or premises, provided there is no decisive, overriding reason to the contrary. For example: when you’re around my dog, your nose stuffs up, so you might have an allergy to dog dander.

Goals

• Distinguish inductive methods from deductive form
• Identify the elements of a strong and weak analogy
• Apply the three tests for corresponding cause
• Assess the strength of a representative sample

Inductive Argument

Inductive arguments have methods of making a point.

• That which is true of a representative sample is true of the general population (arguing from a fair sample).
• That which is true of one case is true of another case that is similar in relevant ways (arguing by analogy).

Deductive arguments, by comparison, have form.

• If A, then B; A is true; therefore, B is true. If you have a ticket, then you may enter; you do have a ticket; so, you may enter.
• A or B; not A; therefore, B. The light is on or off; it isn’t on; therefore, the light is off.

In the methods of induction, consider the content of what is said.

• Are the premises relevant and sufficient for believing the conclusion?
• Is there is a decisive, overriding reason to believe something contrary?

Especially whether there are over-riding factors that contravene the conclusion.

• I like vanilla and I’m fairly typical, so most people prefer vanilla.
• Rain or shine, the train will always arrive on time. 

If the premises aren’t relevant or sufficient to believe the claim, the argument is weak.

• Many people saw the movie, therefore it must be good — but did most like it?
• The flashlight won’t shine, so the batteries must be dead — but is it switched on?

But if they are, then the inductive argument is said to be strong.

• Some of the egg salad is moldy, so the rest of the salad is probably not good to eat.
• Around here, people tend to stay indoors on rainy days and many watch movies, so rainy days promote movie rentals.

A strong induction with true premises is said to be cogent. 

• The construction company should not be given the job of designing the new bridge over the river. In recent years this same company designed three other bridges. All have collapsed. The company is not competent to complete the job safely.
• The operation of a camera is similar to the operation of an eye. To see anything in a dark room, the pupils of your eyes must first dilate. So too, to take a photo (without flash) in a dark room, the aperture of the camera lens must first be opened.

An analogy is a type of induction that argues ‘that which is true of one case is true of another case similar in relevant ways.’

• A pipe resists the flow of water. A hydraulic circuit and electrical circuit both have a source (pump/battery), connections (pipe/wire), and controls (valve/switch). Source, connection, and controls are relevant to the flow of the contained medium. No overriding dissimilarity: both vessels are tubular containers that carry a flow. Therefore, the wire likely resists the flow of electricity.
• Animals and humans are both experience sensations, desires, fears, pleasures, and pains. Humans have a right not to be subjected to needless pain. It follows that animals have a right not to be subjected to needless pain.

The less alike the cases are, the weaker the analogy.

• Weak: Puzzles and chores are both time-consuming and difficult. I like doing crossword puzzles, so I suppose that I like doing chores.
• Weak: Just as it was wrong to deny women the vote, it is also wrong to deny the vote to children.

And the cases must not be dissimilar in any way that over-rides their similarity. 

• The duck decoy may appear life-like, but only a living mallard can fly. 
• A pile of spilled pills look alike, but a laxative won’t do for an aspirin. 

Even then, a conclusion by even the strongest analogy is likely, not certain. 

• Adding fertilizer, my strawberry garden had about 25% higher yield of fruit over previous years. As neighbours, our gardens have similar soil and weather . We both grow strawberries. Soil, location, and weather are relevant to the yield of fruit. There is nothing different about the two gardens (e.g., one isn’t enclosed in a greenhouse and the other openly exposed to the elements). Therefore, if you use fertilizer in your garden, it will likely yield more strawberries as well. 
• The dogs at your kennel ate the kibble. You have yellow labs and I have golden retrievers. The dogs are similar in size, age, health, and breed – factors that can affect what they will eat. There is no major difference in the dogs nor in their kennels. We live in the same climate and the dogs get about the same exercise. So, when I give some of that kibble to my retrievers, I suspect they will eat it.

What do you think?

1. Formal deduction and methodical induction both sound like strategies or recipes for proof. So why does only deduction guarantee the truth of the conclusion? 
2. “No one, upon encountering a watch lying on a forest trail, would expect that it had simply appeared there without having been made by some force. For the same reason, no one should expect that the universe simply appeared without having been made by some force.” This is an adequate analogy, not a fallacy. But does it infer a supernatural force or being?

Corresponding Cause

Correspondence is reason to believe one event is the cause of the other.

• Two events have a cause-and-effect (a.k.a. causal) relationship if one brings about or influences the other.
• Corresponding cause tests whether events match by presence, absence, variation, or remainder.

Testing by Presence: whenever B occurs, A precedes. 

• If in all cases where an effect occurs, there is a single prior factor that is common to all those cases, then that factor is the cause of the effect.
• If cookies are stolen only when Bart is present in a group of children, we would suspect Bart as the thief. If another child were also always present we could use this method only to narrow the suspects down to those two.

Testing by Absence: whenever A is absent, B is absent. 

• Where one situation leads to an effect, another situation does not lead to that effect, and the only difference is the presence of a single factor in the first situation, then this factor is the cause of the effect.
• Cookies are always missing from the cookie jar except on days when Bart is not present. We suspect Bart is the thief because the cookies remain safe when Bart is gone. However, another cookie thief could implicate Bart by stealing cookies only when Bart is present, so a more thorough analysis might be needed to discover the real culprit.

Testing by Presence and Absence: whenever B occurs, A precedes and whenever A is absent, B is absent.

• Cookies are always missing from the cookie jar whenever Bart is in a group of children, and never when Bart is missing from one or more of those same groups. This does not apply to any other child. Therefore, we suspect Bart as the thief.
• When you want something, you’re polite; otherwise, you are rude. So acting nice is just your way of getting what you want.

There is a variation, however: If A varies, B varies. When one event varies, another event varies in proportion.

• The number of cookies missing from the jar in the morning is proportional to the amount of food Bart leaves on his plate at lunchtime. Similar corresponding variations are not observed in any other child. Therefore, we suspect Bart is the thief.
• We could not determine the cause at first. Then we noticed that there were more cases of the infection when more monkeys from a certain nation were present. 

In testing by Remainder: by eliminating known correspondences, the remaining events must correspond.

• New cookies appear in the jar one day shortly after Thomas, Richard, and Harold arrive for work. We know that Tom brought just the sandwiches and Richard brought only coffee. So, we figure that Harold has brought the cookies.
• Part of the damage to the aircraft could be attributed to its impact with the ground. Another part was definitely due to the wind shear that the plane experienced as it fell from the sky. However, some of the damage cannot be accounted for by either of these factors. Investigators are examining this evidence closely for evidence of explosives.

What do you think?

1. If nobody ever saw Adams jogging, Adams is probably not a jogger. Corresponding absence. If nobody ever saw you do something, is it probably true that you don’t do it — even if, in fact, you do it?
2. If I eat oysters, I get hives. I never get hives when I eat other food, so oysters probably cause my hives. Presence and absence. There is an old joke about a patient who tells his family physician: when I do this, it hurts. She replies: then don’t do that. How can corresponding presence or absence alone be sufficient proof?

Fair & Statistical

A generalization about a population is fair if based on a random (or at least not biased) sample whose composition is similar to that of the population.

That which is true of a representative sample is likely true of the general population.

• Purple bears are again the preferred prize at the carnival among pre-tweens. Pollsters stationed at midways across the region observed a thousand game-winners age 9 to 12, given a choice of prize. The result is the same as the Kewpie Prize Poll conducted last year.
• If one strand of spaghetti is cooked to al dente, then the pot of pasta (all the spaghetti strands) is just as done and firm when bitten.

In a statistical syllogism, that which is true in general is likely true in a particular instance.

• A statistical generalization is a statement that is usually true. 
• What is true of the group reasonably applies to an arbitrary member of that group. 

The closer the generalization is to 100%, the stronger the induction.

• Bob is a mechanic. Most mechanics have dirty fingernails, so Bob probably does too.
• The first card dealt from a well-shuffled deck is probably not going to be an Ace.

Statistical syllogism can be quantified, relatively or percentage.

• Most, usually, frequently, rarely, scarcely.
• 99% sure, 60% chance of rain, 0 chance of winning.

At times the quantifier is unstated, but implied.

• Lions are (usually) faster than zebras.
• Barley is (commonly) used in making beer.

What do you think?

1. Most birds can fly. The kiwi is a bird, so probably it likely can fly. Probably, but not actually — so is it still a statistical syllogism?
2. Are left-handers the victims of product design by statistical syllogism?