Chapter 4: Deriving Theories from the Facts: Induction

Alan F. Chalmers - What is This Thing Called Science?

Today's Agenda: Under the Inductivist Microscope

We have explored:

1. Passive observation and its subjective traps (Chapter 1)
2. Active observation as public, practical intervention (Chapter 2)
3. Contrived, theory-guided experimentation (Chapter 3)

Today's Big Question:
Assuming we have established a secure body of facts through observation and experiment, how do we derive scientific theories and laws from them?

We will critically evaluate Inductivism—the commonsense view that scientific knowledge is logically derived from facts.

Section 1: "Baby Logic"

To understand how scientific laws are derived, we must understand logical deduction.

• Logic is concerned with what follows from what.
• It is truth-preserving: if the premises are true, the conclusion must be true.

Example 1 (Valid Deduction)

1. Premise 1: All books on philosophy are boring.
2. Premise 2: This book is a book on philosophy.
3. Conclusion: This book is boring.

If (1) and (2) are true, it is logically impossible for (3) to be false. To assert the premises and deny the conclusion is to contradict oneself.

Validity vs. Truth

Logic alone cannot establish the truth of factual statements. It only guarantees that valid deductions preserve truth if it was already present.

Example 2 (Invalid Deduction)

1. Premise 1: Many books on philosophy are boring.
2. Premise 2: This book is a book on philosophy.
3. Conclusion: This book is boring.
   This is invalid. Even if P1 and P2 are true, the book might be one of the interesting ones.

Example 3 (Valid Deduction with False Premises)

1. Premise 1: All cats have five legs.
2. Premise 2: Bugs Pussy is my cat.
3. Conclusion: Bugs Pussy has five legs.
   Perfectly valid deduction, but the conclusion is false because Premise 1 is false.

Student Task 1: Active Logic Workout

Form small pairs

Analyze the following arguments. Determine if they are valid or invalid, and if the premises/conclusions are true or false.

1. Argument A:

• P1: All metals expand when heated.
• P2: Copper is a metal.
• Conclusion: Copper expands when heated.

2. Argument B:

• P1: All copper bars expand when heated.
• P2: This bar expanded when heated.
• Conclusion: This bar is made of copper.

3. Argument C:

• P1: All copper bars are made of green cheese.
• P2: This object is a copper bar.
• Conclusion: This object is made of green cheese.

Section 2: Can Scientific Laws Be Derived from Facts?

Scientific knowledge consists of universal statements (laws):

• "All metals expand when heated"
• "Acids turn litmus paper red"
• They refer to all events of a particular kind, at all places, for all time.

Observational evidence consists of singular statements:

• "Copper bar x1 expanded when heated on occasion t1"
• "This specific litmus paper turned red when dipped in hydrochloric acid"
• They are specific claims about a state of affairs at a particular time and place.

The Logical Chasm:
Can we logically deduce a universal statement from a finite list of singular statements?

The Inductive Leap: No Logical Guarantee

Let us schematize the attempt to derive a law from observations:

1. Premise 1: Metal 1 expanded when heated on occasion t1.
2. Premise 2: Metal 2 expanded when heated on occasion t2.
3. ...
4. Premise n: Metal n expanded when heated on occasion tn.
5. Conclusion: All metals expand when heated.

Why this is logically invalid:

No matter how large n is, there is no logical guarantee that some sample of metal might not contract (or melt, or remain unchanged) on some future occasion.

• To assert that all observed metals expanded, and yet assert that the universal law is false, does not involve a logical contradiction.

Russell's Inductivist Turkey

A Cautionary Tale of Pure Observation

Bertrand Russell illustrated this logical gap with a famous, rather gruesome story:

• An inductivist turkey arrived at a new farm. On his first morning, he observed that he was fed at 9:00 AM.
• Being a good inductivist, he did not jump to conclusions. He waited and collected more observations.
• He repeated his observations under a wide variety of conditions: on sunny days, rainy days, warm days, cold days, Wednesdays, and Sundays.
• Each day, he added a singular statement to his list: "Fed at 9:00 AM on day n."
• Finally, his inductivist conscience was satisfied. He made the inductive leap: "I am always fed at 9:00 AM."

The Falsification: On Christmas Eve, instead of being fed, his throat was cut. His inductive generalization led him to a fatally false conclusion.

Section 3: What Constitutes a "Good" Inductive Argument?

If we cannot have deductive proof, how do we distinguish a legitimate scientific induction from a rash, hasty generalization?

Inductivists propose three mandatory conditions:

1. The number of observations forming the basis of a generalization must be large.
2. The observations must be repeated under a wide variety of conditions.
3. No accepted observation statement should conflict with the derived law.

The Principle of Induction:

If a large number of As have been observed under a wide variety of conditions, and if all those As without exception possess the property B, then all As have the property B.

Critiquing Condition 1: What is a "Large" Number?

How many observations are required to make an induction "good"?

• Is it 100? 1,000? 1,000,000? Any specific number chosen would be completely arbitrary.

Furthermore, there are many instances where demanding a large number of observations is absurd:

• The Atomic Bomb: The widespread, reasonable public belief that nuclear weapons cause catastrophic destruction was established on just one dramatic observation at Hiroshima.
• The Flame: A child does not need to put their hand in a fire 100 times to conclude that fire burns.
• Scientific Publishing: If a physicist perfectly replicates a novel experiment and submits it for publication, the journal editor will reject it. Why? Because doing it once successfully is often sufficient!

Critiquing Condition 2: What is a "Wide Variety"?

What counts as a significant variation in circumstances when testing a law?

For the law "metals expand when heated", do we need to vary:

• The type of metal? (Yes—copper, iron, gold)
• The pressure? (Yes—high vs. low atmospheric pressure)
• The shape and length of the bar? (Yes—short vs. long, thick vs. thin)
• The time of day? (No)
• The size of the laboratory? (No)
• The color of the experimenter's socks? (No)

The Inductive Regress: How do we know time of day and sock color are superfluous?

• We draw on our prior theoretical knowledge of the physical world.
• But if we must appeal to prior knowledge to make a "valid" induction, then not all scientific knowledge can be derived from raw facts!

Student Task 2: The Superfluous Variable Game

Group Work

Imagine you are a team of naive inductivists with zero prior knowledge about the world. You are testing whether water boils at 100°C.

1. List 5 variables in the room or environment that you must vary to satisfy the "wide variety of conditions" requirement.
2. Explain how you would logically prove that the experimenter's hairstyle or the country where the test is conducted is irrelevant without appealing to any prior scientific theories.
3. What happens to your experiment if you are forced to test every possible variable combination?

Share your team's struggles with the class.

Critiquing Condition 3: No Exceptions Allowed

Scientific knowledge cannot survive the absolute demand that there be no exceptions.

• If a single accepted observation conflicts with a law, the generalization is invalidated.

The Reality of Science:

• Almost every scientific law is constantly faced with anomalies or apparent exceptions.
• Supercooled Liquids: Inductivists assert that liquids solidify when cooled below their freezing point, yet supercooled liquids can flow uphill and remain liquid under precise conditions.
• If we strictly rejected every law at the first conflict with an observation, we would have no scientific laws left.

Section 4: Further Deep Problems with Inductivism

Even if we patch up the three conditions, inductivism faces three fatal philosophical and practical hurdles:

1. The Problem of the Unobservable:

• Induction generalizes from observable facts to observable properties.
• Modern science is dominated by entities that cannot be observed (protons, electrons, genes, DNA molecules, gravitational fields).
• You cannot inductively generalize from observations of metals to a theory about unobservable subatomic lattices. Induction cannot construct a bridge to the invisible.

2. The Problem of Mathematical Exactness:

• Scientific laws are mathematically exact (e.g., Newton's Law of Gravitation).
• But all experimental measurements are inexact and carry a margin of error ().
• How do you derive an exact, clean mathematical law from fuzzy, inexact empirical evidence?

Further Problems: Hume's Problem of Induction

Can the Principle of Induction itself be justified?
We have only two paths: Logic or Experience.

• We cannot justify it by Logic: Inductive arguments are not deductive. There is no logical contradiction in denying them.
• Can we justify it by Experience? Let's try:
  1. The principle of induction worked successfully on occasion x1 (e.g., optics).
  2. The principle of induction worked successfully on occasion x2 (e.g., gravity).
  3. Conclusion: The principle of induction always works.

The Circularity Trap:
We are using an inductive argument to justify the principle of induction! This is circular reasoning (begging the question) and is totally unacceptable.

Student Task 3: The Unobservable Challenge

Class Debate

The Motion:
"Resolved: Because modern science is heavily dependent on unobservable entities (like quarks and electromagnetic fields), the inductivist account of science must be completely abandoned."

• Team A (Inductivist Defenders): Argue that we can build a safe inductive bridge by correlating macro-behaviors (like cloud chamber tracks) to generalize about the micro-world.
• Team B (Anti-Inductivists): Argue that introducing unobservable entities requires creative, speculative hypotheses that must be proposed before any relevant facts can even be gathered.

Section 5: The Inductivist Loop & Its Appeal

Despite these critiques, inductivism has been immensely popular because of its structured, reassuring picture of scientific progress.

                  LAWS AND THEORIES
                    ^          |
                    |          |
         Induction  |          | Deduction
                    |          v
          OBSERVED FACTS ----> PREDICTIONS & EXPLANATIONS

1. Observe and Record: Gather secure, objective facts (no prior assumptions).
2. Inductive Step: Generalize these facts into laws and theories.
3. Deductive Step: Use these laws to make precise predictions and explanations.

The Inductivist Appeal: Wolfe's Superhuman Mind

The twentieth-century economist A.B. Wolfe captured this idealized view perfectly:

"If we try to imagine how a mind of superhuman power and reach... would use the scientific method, the process would be as follows:

• First, all facts would be observed and recorded, without selection or a priori guess as to their relative importance.
• Secondly, the observed and recorded facts would be analysed, compared and classified, without hypothesis or postulates...
• Thirdly, from this analysis, generalizations would be inductively drawn...
• Fourthly, further research would be deductive as well as inductive, employing inferences from established generalizations."

Why this is a myth: As Chalmers showed in Chapter 1, a mind attempting to record all facts without selection would spend its time measuring the hair length of youths in Sydney instead of measuring atmospheric ozone.

Putting Inductivism to Work: Explaining the Rainbow

Let's look at how the inductivist loop theoretically explains a natural phenomenon:

1. The Fact Base (Observation & Induction)

• We perform hundreds of laboratory experiments reflecting rays of light from mirrors and water surfaces.
• We measure the angles of refraction for light passing from air to water and water to air.
• Under a wide variety of conditions, we inductively derive the laws of reflection and refraction of light.

2. The Initial Conditions

• We describe the current physical state: "It is raining, the sun is shining, and a rain cloud is situated in front of an observer."

Explaining the Rainbow (Deductive Prediction)

3. The Deduction

• We assume raindrops are roughly spherical.
• Using the laws of refraction, we calculate that a ray of white light from the sun incident on a raindrop at a will refract, separating red light along ab and blue light along ab1.
• The law of reflection requires ab to reflect along bc and ab1 along b1c1.
• Refraction at c and c1 separates the colors further.
• Geometrical calculations show a colored arc (angle D) will be visible to the observer.

The Conclusion:
Our general laws, combined with specific initial conditions, deductively yield the explanation of the rainbow.

Summary: The Breakdown of Inductivism

We have subjected Inductivism to critical scrutiny and found that:

1. The logical premise is flawed: Universal laws can never be logically proven by finite singular statements.
2. The conditions are unworkable: "Large numbers" is vague; "wide variety" requires prior theoretical assumptions.
3. The scope is limited: Induction cannot account for unobservable entities or mathematical exactness.
4. The justification is circular: Hume's problem shows induction cannot justify itself without begging the question.

Where do we go from here?

If we cannot prove scientific theories to be true, can we proceed by proving them to be false?

Next Lecture: Chapter 5 and Karl Popper's Falsificationism.