Inductive Arguments

1. Identifying Inductive Arguments

So far, we have established some rules and some language for discussing arguments. For example, we have suggested that one measure of an argument is validity, which means that the conclusion follows from the premises, and another measure is soundness, which means the argument is valid and the premises are true. These measures of quality only work on certain types of arguments.

But consider the following arguments. Are they valid?

1. No human so far has lived past the age of 122.
2. Therefore, no human will ever live to be 123.

1. The oldest any person has been known to live is 122.
2. There are more people over 100 years old now than at any time in history.
3. It is likely somebody in that group of people over 100 years old will set a new record for longevity.

You may remember that if you can come up with a counterexample that is consistent with the premises, but where the conclusion is false, the argument is not valid. In the first example above, you may simply say that somebody who is still alive will reach the age of 123 (or even older!); this is as yet unproven, but it is enough to make the argument invalid.

The second example asks us to draw a conclusion from a large sample, but it is (like the first one) not based on known facts; we can propose a scenario where nobody out of the approximately 300,000 global centenarians lives to see their 123rd birthday.

However, these arguments represent a different kind of argument. So far, we have only described deductive reasoning, where a conclusion must follow logically from the premises. Deductive arguments are measured by validity (the relationship between the premises and the conclusion) and soundness (whether the premises are true, and the argument is valid).

But there is another kind of argument that is called inductive. In inductive reasoning, we do not expect to have absolute assurance that the conclusion is never wrong; we instead seek sufficient evidence to support the conclusion as “likely,” or “probable.” Instead of valid or invalid, we describe these arguments as “strong” or “weak,” depending on how well the premises support the conclusion. While validity is an either-or value, strength of an inductive argument is a matter of degrees, and perhaps perception.

did you know

Inductive arguments are the keystone of much of hard science. The scientific method tells us what conclusions are highly probable. We are never guaranteed the truth of the conclusions of our scientific experiments in the same way we are guaranteed the truth of the conclusion of a mathematical proof.

EXAMPLE

Say that you go to a diner one Friday for lunch, and the lunch special is a tuna melt. You return the following Friday, and the special is again a tuna melt. After four or five Fridays, you might determine that the restaurant simply always has tuna melts on Friday, perhaps to accommodate people who don’t eat red meat or poultry on Fridays for religious reasons. With each successive week, that conclusion becomes stronger, or more supported by the evidence. This means if one Friday you take a friend, they may ask you what the special is, and you would say it is likely to be a tuna melt. You know it’s not a logical certainty, but you have good reason to believe this.

In fact, if you show up and the special that day is egg salad, your argument is not shown to be invalid. Unlike deductive arguments, a single exception does not make the whole argument invalid. However, the exception does weaken your conclusion. You may revise it in light of new information.

Let's look at inductive arguments in standard form. Suppose that there are two opaque glass jars with different color marbles in them.

1. All the marbles in jar #1 are blue.
2. This marble I drew from one of the jars is blue.
3. This marble probably came from jar #1.

1. This marble came from jar #2.
2. This marble is red.
3. All the marbles in jar #2 are probably red.

Both of these arguments may seem hasty given the limited information, but what if we drew 100 marbles from jar #2 and found that they were all red? Let’s add that as a new premise to the first argument:

1. All the marbles in jar #1 are blue.
2. Of 100 marbles from jar #2, 100 are red.
3. This marble is blue.
4. This marble probably came from jar #1.

1. This marble came from jar #2.
2. Of 100 marbles from jar #2, 100 are red.
3. This marble is red.
4. All the marbles in jar #2 are red.

hint

The conclusion to an inductive argument should show that the argument is inductive. For example, “I always find a parking place. So, we will find a parking place," has an inductive premise but the wording of the conclusion suggests an absolute truth. The conclusion should be, "So, we will probably find a parking place."

try it

For each of the following arguments, determine whether it is inductive or deductive.

1. The average winter temperature in Saskatchewan is well below freezing, so it’s safe to schedule an outdoor hockey match in Saskatoon in January.

This argument is inductive, suggesting what is likely to happen.

2. It was well below freezing last night, so the lake will be frozen over.

This argument is deductive, coming to a firm conclusion about what’s true based on the premises.

3. Mice live in colonies, so if you see one mouse in your house, there are a lot more.

This argument is deductive, coming to a firm conclusion about what’s true based on the premises.

4. Mice actually don’t like cheese, but they do love peanut butter. Bait your traps with peanut butter to have a better chance of catching them.

This argument is inductive, suggesting what is likely to happen.

terms to know

Deductive Reasoning

Constructing or analyzing arguments where the conclusion must follow from the premises.

Inductive Reasoning

Constructing or analyzing arguments where we look for evidence that the conclusion is probably or likely but may not be 100% certain.

Strong

An indication that an inductive argument has a conclusion that is well supported by its premises. In inductive reasoning, these are not absolute statements like valid and invalid.

Weak

An indication that an inductive argument has a conclusion that is not well supported by its premises. In inductive reasoning, these are not absolute statements like valid and invalid.

2. Assessing an Argument's Strength

As with validity, the determination of a strong argument is based entirely on how well the premises logically support the conclusion. However, we don’t have the absolute certainty that we have with sound arguments. In judging inductive arguments, then, we don’t ask if we can find a single counterexample. We instead ask if the premises provide probable support for the conclusion. Like the arguments themselves, these judgments may not be straightforwardly true or false. For example, we may use statistics or probability.

But the premises can be accurate or inaccurate. For example, the statement, “It’ll probably rain tomorrow,” could be based on strong meteorological indicators (accurate) or completely on a hunch (inaccurate). Either way, it is not an absolute certainty.

If the premises are known to be accurate and provide logical support for the conclusion, we call an inductive argument cogent, or convincing. This is similar to the concept of soundness for deductive reasoning, but there is a key difference. In a sound argument, the conclusion is objectively true, the premises are true, and the conclusion follows. The concept of cogency suggests only that the argument is convincing to a reasonable person, but because the conclusion is only likely, or probable, we cannot consider it to be objectively true.

Moreover, the conclusion of a cogent argument may end up being false, but this doesn’t make the argument itself “wrong” or “weak.” For example, if you caution your brother that playing the lottery has an extremely low probability of paying off, and he does strike it rich, the argument itself is not proven wrong. The logic and reasoning were good, but an extremely unlikely event occurred. Hope he still shares his winnings with you anyway and there are no hard feelings!

However, if correctly phrased, the conclusion itself is not stated as an absolute fact. In this example, you might say, “It is extremely unlikely you will win the lottery.” Even if the person does, the conclusion (correctly phrased) is correct. It was extremely unlikely that he won!

think about it

Can you have a strong inductive argument with a wrong conclusion?

You may in fact have a wrong conclusion to a strong inductive argument if the premises supported your argument logically but were not reliable.

Can you have a cogent inductive argument with a wrong conclusion?

You may have a conclusion to a cogent inductive argument that is contradicted by actual outcomes. Because they are usually not phrased as absolutes, they would not be “wrong,” but may have been inaccurate predictions even if they were based on reliable and logical premises.

The point of all this is that inductive arguments—even cogent inductive arguments—are not (deductively) valid. They are not watertight. Unlikely though it might be, it is possible for their conclusions to turn out to be wrong, even when all of their premises are true and the logical support is strong.

EXAMPLE

Election analysts are sometimes said to “get it wrong” if there is an upset in a political race. However, this judgment is not necessarily fair. If an analyst says, for example, that candidate A has a 80% chance of winning, that acknowledges there is a 20% chance that candidate B will pull it off. Their conclusion is not really disproven by the outcome, as the argument may have been based on reliable data and good reasoning.

think about it

Think back to Unit 1.

What kind of cognitive bias would be exhibited in the previous example if someone publicly rebukes an analyst for "getting it wrong"?

Somebody second-guessing an election analyst’s predictions the day after the election is exhibiting hindsight bias.

term to know

Cogent

In an inductive argument, indicates that the argument is strong and the premises are true, and so is convincing to a reasonable person. However, unlike soundness, the conclusion in an inductive argument is not a 100% certainty.