## Does science have a specific method: Inductivism

### 0.1 Introduction

Is science distinctive in the way it justifies its claims? That is, is there some specific method or approach that scientists take? Intuitively we think that scientific claims are held to a higher standard than non-scientific claims. That is, we think that the method of science proves its claims while other methods lack this kind of strong support.

science is an activity governed by a method that systematically justifies its conclusions (proves them)

In order to determine whether or not science argues in its own distinctive way, it is necessary to introduce several distinctions. The first is the notion of an argument. This course won’t require that we dive deeply into questions like “what is an argument?” or “what are the elements of an argument?” But we do need a compact way to present arguments and a general understanding of how to criticize them. Intuitively, we can think of an argument as a set of reasons (or evidence) given in favor of some thesis.

Definition 1 (argument) An argument is a set of propositions (sentences that can be true or false) where one subset of propositions (called the premises) are given in support of another proposition (called the conclusion).

Consider the example below:

• P1: All pollution is caused by hippies.
• P2: John is a hippie.
• C: Therefore, John causes pollution

The argument above contains two propositions (sentences) called the “premises” that support a third proposition (sentence) called the “conclusion”. We will abbreviate the premises using P and an integer (e.g. P1 or ${P}_{1}$ and C to indicate the conclusion.

We can criticize any argument in one of two ways.

• Option 1: Reject at Least One of the Premises
• Option 2: Argue that the Conclusion Does Not Follow from the Premises.

The first way an argument can be criticized is by citing objections. Objections are some consideration (or reason) against the acceptance of the conclusion, either because the premises are false or because even if the premises are true, they don’t support the conclusion. The argument in ?? can be criticized by arguing that at least one of the premises is false. For example,

O1:
It is false that all pollution is caused by hippies. Why just the other day I was drinking an energy drink and threw it in the street (thereby polluting) and I am not a hippie.

We will use O and an integer ${O}_{1}$ or O1 to abbreviate the term “Objection”.

The second way to criticize an argument is to argue that even if the premises were true, the conclusion does not follow. This mode of criticism is independent of whether the premises of the argument are in true or false. It says, instead, even if we assume that the premises are true, the conclusion does not follow. To get a clearer picture on this second type of criticism, it is necessary to distinguish between two different kinds of arguments: deductively valid arguments and inductively strong arguments.

### 0.2 Deduction and Induction

Tagging an argument as “good” or “bad” involves evaluating the quality of an argument. What makes an argument good? This partly depends upon the purpose of the argument. If we expect arguments to entertain us or be thought-provoking, then whether the premises of the argument are true or not might not be of our concern. Logically, arguments are evaluated in two ways. First, arguments are evaluated according to whether the propositions (premises and conclusion) that compose the argument are true. If the propositions that compose the argument are true, then the argument is considered “good” in that particular sense.

This type of evaluation of arguments is an evaluation independent of the relation between the premises and the conclusion. In short, each proposition of the argument is evaluated independently.

Second, arguments are evaluated according to the type of relation between the premises and the conclusion. In broad strokes, an argument is “good” (in this sense) provided the relation between the premises and the conclusion is one of truth-preservation. This type of relation can be understood in at least two ways. First, an argument can be truth-preserving in that it is impossible for the premises to be true and the conclusion to be false. When an argument is truth-preserving in this sense, the argument is said to be deductively valid.

Definition 2 (Deductive validity) An argument is deductively valid if and only if it is logically impossible for the premises to be true and the conclusion false.

Second, an argument can be truth-preserving in that it is improbable for the premises to be true and the conclusion to be false. When an argument is truth-preserving in this sense, the argument is said to be inductively strong.

Definition 3 (Inductively strong) An argument is inductively strong if and only if it is improbable that its conclusion is false while its premises are true. In other words, were the premises true, it would be unlikely that the conclusion is false.

Deductive validity and inductive strength concern the relation between the premises and the conclusion independent of whether the premises and the conclusion are in fact true. However, when an argument is, deductively valid (inductively strong) and has true premises, then we say that the argument is sound (cogent).

#### 0.2.1 Deductively Valid Arguments

Some arguments are purported to be deductively valid. An argument is deductively valid if and only if it is impossible for the premises to be true and the conclusion false. That is, an argument is deductively valid if and only if it must be the case that on the assumption that the premises are true (not saying they are), the conclusion is also true.

If someone says that a given argument is “deductively valid”, what they express is that there is an impossible state of affairs. Namely, what they are expressing is that there is NO WAY the premises can be true and the conclusion false. So, if the premises are true, we can be confidant that the conclusion is also true.

Table 1: A Deductively Valid Argument is One Where
 Premises True True False False Conclusion True False True False Possible Impossible! Possible Possible

Given that in a deductively valid argument it is impossible for the premises to be true and conclusion false, deductively valid arguments are truth preserving. That is, if an argument is deductively valid, then it is impossible for the premises to be true and the conclusion false, and so, if the premises are (in fact) true, then the conclusion is true too. A deductively valid argument whose premises are, in fact, true is called a sound argument.

As an example, consider the argument in ??. This argument is deductively valid. And, it so happens, its premises are true as well. So, the argument is sound.

• P1: Obama is President or McCain is President of the USA.
• P2: McCain is not President.
• C: Therefore, Obama is President.

One way to criticize an argument other than asserting that its premises are false is to say “the argument is not valid.” That is, the argument is invalid. What this means is that it is possible for the premises to be true and the conclusion false.

Let’s return to the argument above and determine whether the argument is deductively valid or invalid. That argument consists of two premises. Let’s assume that both are true. Does the conclusion still follow? That is, is it impossible for the premises to be true and the conclusion false? No. Even if it were true that “all pollution is caused by hippies” and that “John is a hippie”, it is still possible that “John does textbfnot cause pollution”. Consider the the first premise does not say that “all hippies cause pollution” or “all of the hippies are responsible for all of the pollution”.

Discussion 1 Create a deductively valid argument for why we shouldn’t (or should) hurt puppies. Be sure to label your premises and conclusion clearly.

If an argument is not valid, then it is categorized as “invalid”.

Definition 4 (Deductive invalidity) An argument is deductively invalid if and only if it is not deductively valid.

For example, consider the argument in Table 2. In that argument, notice that it is possible for P1 and P2 to be true yet C to be false. This argument is thus invalid.

 P1 Many astronauts have drinking problems. P2 John is an astronaut. C Therefore, John has a drinking problem.

Table 2: Argument for astronauts having drinking problems.

Note 1 An argument does not need to have true premises in order to be deductively valid. For instance, consider the example in Table 3

 P1 All pigs fly. P2 David Agler is a pig. C Therefore, David Agler flies.

Table 3: Argument for Flying Pigs

Notice that neither P1, P2, nor C are true, but the argument is nevertheless valid. It is deductively valid because were the premises true, it would be impossible for the premises to be false.

Note 2 A deductively valid argument cannot establish the truth of a scientific theory. It can only show that on the assumption that the premises are true, then the conclusion is also true.

#### 0.2.2 Inductive Arguments

Not every argument purports to be deductively valid. Some arguments, instead, purport to be inductively strong. We say that an inductive argument is strong when it is likely that if the premises are true the conclusion is also true. While validity is an all-or-nothing, the strength of an argument is a matter of degree and a comparative notion. Some premises provide more support to their conclusions than others; arguments of this nature are thought to be stronger than those arguments that provide less support.

To illustrate, consider the example below. This particular kind of inductive argument is called an argument by enumeration. These arguments support a universal (general) conclusion by reasoning from singular claims.

• P1: John played football and has bad knees at age 50.
• P2: Frank played football and has bad knees at age 50.
• P3: Liz played football and has bad knees at age 50.
• C: Therefore, everyone who plays football will have bad knees at age 50.

Notice that P1, P2, and P3 are all examples or instances of the general or universal claim found in C. We might then say that the argument above exemplifies a key aspect of the scientific method. Namely, scientists gather observations and then reason to general conclusions.

Discussion 2 Create your own inductive argument for some scientific claim (it need not be a real scientific claim).

Note 3 Inductively strong arguments are not deductively valid. That is, an inductively strong argument is not a strict proof of the conclusion, even if the premises are true. Non-deductive (inductive) modes of inference fall short of strict proof. For example, if our axioms are direct, immediate observations, we cannot prove the universal laws of science.

Example 1 “all observed swans are white” therefore “all swans are white” No, in Australia there are black swans.

Example 2 “whenever we heat water, it boils at 100 degrees” therefore “water always boils at 100 degrees.” No, it has a different boiling point at high and low altitudes.

Example 3 Newton’s laws of motion hold for all matter. No, they need revision for very small (quantum mechanics) and the very fast objects (special relativity)

Experience has shown us that, hitherto, the frequent repetition of some uniform succession or coexistence has been a cause of our expecting the same succession or coexistence on the next occasion. Food that has a certain appearance generally has a certain taste, and it is a severe shock to our expectations when the familiar appearance is found to be associated with an unusual taste. Things which we see become associated, by habit, with certain tactile sensations which we expect if we touch them; one of the horrors of a ghost (in many ghost-stories) is that it fails to give us any sensations of touch. Uneducated people who go abroad for the first time are so surprised as to be incredulous when they find their native language not understood.

And this kind of association is not confined to men; in animals also it is very strong. A horse which has been often driven along a certain road resists the attempt to drive him in a different direction. Domestic animals expect food when they see the person who feeds them. We know that all these rather crude expectations of uniformity are liable to be misleading. The man who has fed the chicken every day throughout its life at last wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken. — Bertrand Russell - “On Induction”

Discussion 3 Scientific claims are often ranked higher than other types of claims because they are thought to be proven. But the above argues that there can be no strict proof for scientific claims. Its foundations and method for justifying claims have irreparable problems. What does this mean for our image of science as an activity that justifies claims? Is there any way to save this image of science

#### 0.2.3 Good and Bad Inductive Arguments

The argument in ?? shows that the strength of an inductive argument is a matter of degree. Some inductive arguments are stronger (better) than others. When an argument is not strong, it is claimed to be weak. To say that an inductive argument is weak is to say that it is not likely that if the premises are true then the conclusion is also true.

Some criteria we might use to evaluate the quality of an inductive argument are the following:

1. the number of observations: whether the generalization is based on a large number of observations or only a few.
2. the diversity of the observations: whether the generalization is based on observations that took place in a variety of contexts and under diverse conditions or whether the observations were confined to a few limited situations.
3. the consistency of the observations: whether there are other observations that conflict with the generalization.
4. the relevance of the observations: whether the observations are relevant to the generalization.

First, note that for we might say that a single case is not enough to make the conclusion convincing but a couple thousand instances would make the conclusion more likely. With respect to arguments by enumeration (see ??) we might say that citing only a few instances of football players who have bad knees is not enough to show that football playing causes bad knees. The bad knees may be the result of some other factor, e.g. some football players may like to dance after their games at clubs and dancing causes bad knees.

Second, as ? points out, one way to increase the sheer number of observations is to repeat the same experiment over and over. For example, one could argue that all metals expand by repeatedly heating the same metal bar over and over in our apartment. This procedure ignores the possibility that different metals might not expand or that certain metals might not expand in certain conditions. As a second example, consider the claim that water at 212 degrees Fahrenheit (100 degrees Celsius). One might boil water over and over again at a certain location and find that it boils at 100 degrees every time. However, the boiling point of water varies depending upon the altitude (atmospheric pressure) at which it is boiled.

Note 4 A Pew Study showed that only 34% of Americans knew that water boils at a lower temperature at higher altitudes.

Another example might be location. Consider that pre-1697, many individuals in the Western world reasoned as follows:

• P1: All swans that I have seen, read about, and those I’ve heard individuals talk about are white
• C: Therefore, all swans are white.

Here there is a large number of observations that point to swams being white. However, despite the truth of the premise, the conclusion is false and was shown to be false to Western people when Dutch explorer Willem de Vlamingh discovered black swans in Australia.

Third, there should not be observations that conflict with the generalization. For example, if the claim is that all metals expanded when heated but there are observations of metals not expanding when heated, then the argument is problematic.

Definition 5 (strong principle of induction) If a large number of As have been observed in a wide variety of conditions, and those As unequivocally have property B, then all As are B (see ?, p.43).

Definition 6 (inductivism) Scientific knowledge is derived by generalizing from observable facts.

Discussion 4 Create an inductively strong argument for why we shouldn’t (or should) hurt puppies. Be sure to label your premises and conclusion clearly. Also, be sure to make your argument as strong as possible.

### 0.3 The Inductivist Picture of the Method of Science

With deductively strong and inductively valid arguments defined, we might now give a picture of how the distinct method of science.

Figure 1: The inductivist picture of science

Essentially, the distinct method of science is that it begins with observations, generalizes these observations into laws, theory, universal propositions, and then uses these generalizations to make predictions about the future behavior of the world.

As an example, suppose I notice that whenever the temperature is below 0 degrees Celsius, water freezes. After repeated observations of these phenomena, I might inductively infer that water always freezes below 0 degrees Celsius. With this generalization, I can now make future predictions about the world. For instance, if weather forecasters tell me that it will be below 0 degrees Celsius tomorrow I can deductively infer that if there is any water on the sidewalk, it will freeze.

Discussion 5 Use the above model to give an illustration of the inductivist picture of science.

### 0.4 Criticisms of Inductivism

Objection 1 (How many observations is enough?) The first criterion is vague. How many observations are required to make an argument strong? 100, 1,000, 10,000? In addition, sometimes a few observations is said to be enough, e.g. we don’t need to test anymore nuclear weapons to know they are destructive.

Objection 2 (What is a sufficient variety of observations?) The second condition is also problematic. With boiling water, since atmospheric pressure effects the boiling point of water and atmospheric pressure varies with altitude, it is relevant to the generalization that water boils at 100 Celsius that the boiling point of water is tested at different altitudes. But, what sorts of conditions are relevant to a generalization? In testing falling bodies, does the color of the object impact the rate at which an object falls? Does the weight? Does the size?

Objection 3 (unobservables) Much of science makes claims about unobservable entities (e.g. electrons, quarks, etc.) and so how can we

Objection 4 (The Problem of Induction) Suppose that the inductivist claims that all scientific knowledge is justified either by an appeal to deductive reasoning (logic) or by experience. Inductive arguments cannot be justified in terms of deductive reasoning (logic) since as inductive arguments are not valid. Can inductive arguments in general be justified by experience? Consider the argument below:

• P1: The principle of induction was successful on occasion ${o}_{1}$ (e.g. determined the position of a planet)
• P2: The principle of induction was successful on occasion ${o}_{2}$ (e.g. determined the effectiveness of a drug)
• P3: The principle of induction was successful on occasion ${o}_{3}$ (e.g. determined the explosiveness of reaction)
• C: The principle of induction will always be successful.

The argument above reasons from successful instances of the principle of induction to the generalization that the principle of induction will always work. The problem with this argument is that it assumes what it is supposed to prove. That is, in order to show the principle the principle of induction is justified, it assumes that the principle of induction is justified.

Response 1 One way that we might respond to this objection is to modify the strong principle of induction. Rather than saying that if a large number of As have been observed in a wide variety of conditions, and those As unequivocally have property B, then probably (it is likely that) all As are B. Let’s call this the “weak principle of induction”

Objection 5 (weak principle of induction is no more justified.) Even an argument for the weak principle of induction will need to be inductive and so it will need to assume what it is supposed to prove