Deduction, Induction and the Best Explanation — Critical Thinking 3 (C1)

CRITICAL THINKING · LESSON 3

Deduction, Induction and the Best Explanation

C1 (B2–C2) · 90 minutes

How sure are you?

Task Give each conclusion a score out of 100 for how sure you are. Then decide: if everything before "So" is true, could the conclusion still be false?

1. Every medicine in the blue cabinet is past its expiry date. This box of tablets came from the blue cabinet. So this box is past its expiry date.

2. Lucía has eaten at the same small restaurant in Lisbon thirty times and has never been ill. So she won't be ill if she eats there tonight.

3. You come home. There's a broken glass on the kitchen floor beside a puddle of water, and the cat is sitting on the shelf above, licking its paw. So the cat knocked the glass over.

READING

Guaranteed, if: deduction

An argument is deductive when it claims that its conclusion follows necessarily from its premises: if the premises are true, the conclusion cannot be false.

An argument is valid when it is impossible for all its premises to be true and its conclusion false. Validity depends on form, the shape of the reasoning, not on its content. "All A are B; all B are C; so all A are C" is valid whatever we put in place of the letters.

Validity surprises people in two ways. A valid argument can have a false premise and a false conclusion: "All planets are made of rock; Jupiter is a planet; so Jupiter is made of rock" is perfectly valid, although Jupiter is a giant ball of gas. And an invalid argument can have true premises and a true conclusion: "Some

metals are liquid at room temperature; mercury is a metal; so mercury is liquid at room temperature." Everything in it is true, but only by luck.

An argument is sound when it is valid and all its premises are true. Only a sound argument proves its conclusion. So there are two separate ways to attack a deductive argument: show that the conclusion doesn't follow, or show that a premise is false. Logic can settle the first. The second usually needs evidence from the world.

Deduction has a price. A valid conclusion only makes explicit what the premises already contained. To learn something new about the world, we need riskier kinds of inference.

Valid? Sound?

Task Copy the table. Write each letter in one column. First ask: could the premises be true and the conclusion false? Then ask: are the premises actually true? Circle any letter you can't agree on.

A. All cats are mammals. All mammals breathe air. So all cats breathe air.

B. Every capital city has more than ten million people. Canberra is a capital city. So Canberra has more than ten million people.

C. Some birds cannot fly. Penguins are birds. So penguins cannot fly.

D. All fish are mammals. All whales are fish. So all whales are mammals.

E. All birds can swim. Ducks can swim. So ducks are birds.

F. Botanists call any part of a plant that grows from a flower's ovary and holds its seeds a fruit. A tomato grows from a flower's ovary and holds its seeds. So botanists call a tomato a fruit.

G. All squares have four sides. All rectangles have four sides. So all rectangles are squares.

H. On every morning for which we have records, the sun has risen over Ulaanbaatar. So it will rise over Ulaanbaatar tomorrow.

Valid? Sound?

Valid, true premises (sound)

Valid, a false premise

Invalid, true premises

Invalid, a false premise

Which argument would you most like to use in real life? Is it a valid one?

READING

More than the evidence: induction

Inductive arguments move from observed cases to unobserved ones: from the past to the future, or from a sample to a whole population. Even the best can have true premises and a false conclusion. Instead we judge their strength: how probable the premises, if true, make the conclusion. A strong argument with true premises is called cogent.

Strength depends on the evidence. A survey of 2,000 randomly chosen adults supports a claim about a country far better than a survey of twenty friends. Size matters, but so does

representativeness: a huge sample drawn from one city, one age group or one website can mislead as badly as a tiny one. And new evidence can overturn even a strong induction. Europeans had long treated "all swans are white" as safe; in 1697 a Dutch expedition in Western Australia named the Swan River after the black swans it found there.

The Scottish philosopher David Hume asked a deeper question in A Treatise of Human Nature (1739). Every inductive inference assumes that the unobserved will resemble the observed.

READING · CONTINUED

More than the evidence: induction

How do we know that? Not by deduction: there is no contradiction in supposing that nature might change. Not from experience either: "nature has been uniform so far, so it will stay uniform" is itself inductive, and assumes what it sets out to prove. Hume concluded that our expectations rest on

custom, or habit, not on reason. Bertrand Russell later made the point with a chicken that is fed every morning of its life, until the morning the man who feeds it wrings its neck instead. Philosophers still disagree about whether this problem of induction can be solved, or only lived with.

1. Which matters more in a sample: its size or how representative it is?

2. If Hume is right, is it irrational to expect the sun to rise over Ulaanbaatar tomorrow?

READING

The surprising fact: abduction

The American philosopher Charles Sanders Peirce (1839–1914) argued that deduction and induction leave out the reasoning that produces new ideas. In an 1878 paper he used a bag of beans. Deduction: all the beans from this bag are white; these beans are from this bag; so these beans are white. Induction: these beans are from this bag; these beans are white; so all the beans from this bag are white. The third pattern, which he then called hypothesis and later abduction, runs: all the beans from

this bag are white; these beans are white; so these beans are from this bag.

As deduction this is invalid, since white beans could come from anywhere, yet it is how we reason from an effect back to a cause. Peirce later set it out like this:

"The surprising fact, C, is observed; But if A were true, C would be a matter of course, Hence, there is reason to suspect that A is true." (Peirce, Collected Papers 5.189)

READING · CONTINUED

The surprising fact: abduction

Note the modesty of "reason to suspect". For Peirce, abduction proposes a hypothesis; deduction then works out what should follow if it were true, and induction tests those predictions.

In 1965 Gilbert Harman named a related idea inference to the best explanation: when several hypotheses could explain the evidence, infer the one that explains it best. Peirce himself never spoke of a "best" explanation. What makes an explanation best?

Common answers include simplicity (fewer extra assumptions), scope (it explains more of the facts, ideally some it was not built to explain), fit with what we already know, and testability (it predicts something we could check).

Critics ask why the simplest explanation should be the true one. Bas van Fraassen's "bad lot" objection adds that the best explanation we have may only be the best of a poor set; the truth may be one nobody has thought of.

The dog that did nothing

Sherlock Holmes calls his method "deduction". Many logicians disagree. In Arthur Conan Doyle's "Silver Blaze" (1892), a famous racehorse disappears from its stable on the moor at night, and its trainer is found dead nearby. The stable lad on guard had been drugged. The police arrest a stranger seen near the stables that evening. Then Holmes points to something nobody else thought worth mentioning.

1. Inspector Gregory: Is there any other point to which you would wish to draw my attention?

2. Holmes: To the curious incident of the dog in the night-time.

3. Inspector Gregory: The dog did nothing in the night-time.

4. Holmes: That was the curious incident.

The dog that did nothing

The stable dog had not barked, so whoever took the horse out was probably someone the dog knew well. This is not deduction: a dog might stay silent for other reasons. It is reasoning from a surprising fact to the hypothesis that would make it "a matter of course". In The Sign of Three (1983), edited by Umberto Eco and Thomas Sebeok, several contributors read Holmes's method as Peirce's abduction. Not all agree: Jaakko and Merrill Hintikka, in the same book, model it instead as skilled questioning combined with deduction.

1. What other explanations could there be for the silent dog?

2. Why should something that did not happen count as evidence?

3. Does it matter whether we call Holmes's method deduction or abduction?

The Thawed Archive

Task Read the clues. Agree on your best explanation of what happened, give it a likelihood out of 100, and name one clue that would change your minds. Copy the table and fill in one row per round. Don't read ahead.

Kraków, Monday, 08:05. The Institute of Glacier Science (a fictional institute) keeps its most valuable samples in Lab 4: ice cores drilled in the Pamir mountains, each a record of centuries of climate.

The Thawed Archive: Round 1

1. Dr Amara Osei opens Lab 4. Freezer B, which should hold minus 30 degrees, reads plus 4 degrees. The ice cores inside have melted.

2. Freezer A stands next to it, is plugged into the same wall socket, and reads minus 30 degrees as normal.

3. Both freezers are linked to an alarm that texts the technician on call if either one rises above minus 20 degrees. No text was sent this weekend.

4. Freezer B's door looked closed, but it swung open at a touch, without the usual pull of the rubber seal.

5. The corridor's door log shows two visits at the weekend. Saturday, 14:10 to 14:40: Pavel Nowak, the technician on call, reason "monthly alarm test". Sunday, 22:15: the swipe card of Dr Kenji Mori, a researcher from Lab 3 next door.

6. The cores belong to Professor Inés Duarte. She and Kenji have both applied for the same large research grant. The decision is due next month, and colleagues say the two have argued about it.

Round

Our best explanation

Likelihood (out of 100)

What would change our minds

1

The Thawed Archive: Round 2

Task Read the new clues with the old ones. Keep, change or replace your explanation. Write a new row in your table, with a new likelihood.

7. The alarm panel's history shows: Saturday, 14:15, "TEST MODE ON: texts paused". There is no "TEST MODE OFF" entry.

8. The instructions printed on the panel say: after a test, press OFF. Otherwise, test mode switches itself off after 48 hours.

9. Pavel says he only tested the panel on the wall. "I never touched the freezers."

10. The corridor camera shows Kenji going into Lab 3 at 22:16 on Sunday and leaving at 22:41 with a cool box. A connecting door between Lab 3 and Lab 4 is never locked, and the camera can't see it.

11. Kenji says he came in to collect his own samples for a flight on Monday morning. "Lab 4 was humming very loudly. I thought it was strange, but it isn't my lab."

Round

Our best explanation

Likelihood (out of 100)

What would change our minds

2

The Thawed Archive: Round 3

Task Read the last clues and write your final row. Then agree one sentence: what can you still not be sure of, and why?

12. Freezer B has its own battery-powered temperature logger, recording every 10 minutes. It shows minus 30 degrees until Saturday 14:30, then a steady rise: minus 20 at Saturday 21:00, zero at Sunday 09:00, plus 4 from Sunday 20:00 onwards.

13. Freezer B's door sensor recorded "door open" at Saturday 14:28. It never recorded "door closed".

14. Step 3 of the monthly alarm-test checklist says: "Open each freezer and check that the alarm probe is in its holder." On Pavel's signed checklist for Saturday, step 3 is ticked for both freezers.

15. The service engineer reports that Freezer B's motor had been running non-stop since Saturday afternoon, and its cooling coils were blocked with ice.

Round

Our best explanation

Likelihood (out of 100)

What would change our minds

3

Explanation contest: the early swallows

Task List at least three rival explanations of the observation. Score each from 0 to 2 on every criterion, add up the totals, and choose a winner. Then design one check that could prove your winner wrong.

A fictional case. In the river town of Lindavár, in Hungary, the first swallows of spring have been recorded at the town's two bridges, on average, 9 days earlier each year since 2019 than they were from 1995 to 2018. At farms 20 kilometres away, the date of the first swallow has not changed.

Explanation contest: the early swallows

1. The bridge records come from a local volunteer group. The farm records come from a university team that has used the same method since 1995.

2. In 2018, the town lit both bridges with bright lamps that stay on all night.

3. In 2019, after a newspaper story and a free bird-recording app, the volunteer group grew from 3 members to about 40.

4. Spring temperatures have risen by about the same amount in the town and at the farms.

5. The number of swallow nests under the bridges has stayed about the same.

Explanation

Makes the facts "a matter of course"

Simplicity

Scope

Fit with what we know

Testability

Total

Talking it over

Task Choose three or four questions. Spend most of your time on the one your group disagrees about.

1. Is deduction the only reasoning that gives real knowledge, or is that standard too high?

2. If induction can't be justified, why do we trust it? Should we?

3. Is a court verdict "beyond reasonable doubt" an inference to the best explanation? Is that enough?

4. Is the simplest explanation more likely to be true, or just easier to think with?

5. Should we trust a medical AI that picks the "best explanation" of our symptoms?

6. Some explanations explain absolutely everything. Is that a strength or a warning sign?

Private reflection

Task Choose one prompt and write privately. Nobody will read what you write. Sharing afterwards is optional.

Prompt 1

Think of a time you were sure of an explanation and it turned out to be wrong. What did you miss, and why?

Prompt 2 (lighter)

Which kind of reasoning do you use most in your work or studies: deduction, induction or abduction? Note one example from this week.

Before Lesson 4

Look back at your three scores from the start of the lesson. Would you change any of them now?

1

2

This week, catch one person, advert or article making an inference to the best explanation.

Write down the surprising fact, the explanation offered, and one rival explanation they didn't consider.

3

Next lesson: Fallacies and Foul Play. Arguments that look good, feel good, and fail.

TEACHER'S REFERENCE · SKIP OR DELETE BEFORE CLASS

Answer key (1)

HOW SURE ARE YOU?

1 Deduction. Valid: if the premises are true, the conclusion must be true.

2 Induction. Strong, perhaps, but the premises could be true and the conclusion false.

3 Abduction. A plausible explanation of a surprising scene; a rival (a person, an earthquake, a glass left on the edge by the sink) could still be the right one.

VALID? SOUND?

• Valid, true premises (sound): A, F. F surprises people who think a tomato "is a vegetable": in cooking it is treated as one, but the argument is about botanists' definition, and both premises are true.

• Valid, a false premise: B (Canberra has fewer than half a million people; the premise is false, and so is the conclusion), D (both premises are false, yet the conclusion is true: a valid argument with false premises can reach a true conclusion by luck).

• Invalid, true premises: C (premises true, conclusion true, but "some birds" says nothing about penguins in particular), G (premises true, conclusion false: sharing a property doesn't make two things the same), H (the premise is true, but no number of past sunrises guarantees the next one; as a deductive argument it is invalid, but as an inductive argument it is very strong).

• Invalid, a false premise: E (the first premise is false, the conclusion is true, and the argument is invalid anyway: many animals that swim are not birds). E has the same shape as Peirce's third bean pattern.

THE THAWED ARCHIVE the solution: everything is explained, but one point stays open

• What happened: an accident during the alarm test, two mistakes in a row. On Saturday, Pavel followed step 3 of the checklist and opened both freezers (clue 14). Freezer B's door did not latch when he closed it (clues 4 and 13), so warm air got in and the motor ran non-stop until the coils iced up (clues 11 and 15). He also left the alarm in test mode, so no text was sent (clues 3, 7 and 8).

TEACHER'S REFERENCE · SKIP OR DELETE BEFORE CLASS

Answer key (2)

• Why not Kenji: the temperature began rising at 14:30 on Saturday and passed zero at 09:00 on Sunday (clue 12), more than 13 hours before his card was used (clue 5). By the time he arrived, the cores had already melted. His motive (clue 6) is a red herring, and his "loud humming" (clue 11) fits the solution: it was the overworked motor.

• Why not a power cut: Freezer A, on the same socket, was fine (clue 2).

• What the evidence can't settle: whether Pavel is lying (clue 9) or simply forgot opening the freezers during a routine he does every month. Honest forgetting explains the facts at least as well and needs no hidden motive, but the clues cannot prove it.

• Fairness check: the solution was available in outline from Round 1: an unsealed door (clue 4) and a visit for an alarm test (clue 5).

EXPLANATION CONTEST: THE EARLY SWALLOWS no single answer; strong candidates

• More watchers, earlier sightings: with about 40 volunteers and an app instead of 3 people, someone is far more likely to see the very first swallow. This explains the timing (from 2019), why only the bridges changed, and why the number of nests did not change. Simple and testable: compare the average arrival or the date most nests are occupied, not the first sighting, or ask the university team to record the bridges for a year with its own method.

• Night lighting: lamps from 2018 might attract insects and so draw birds earlier. Weaker on scope: it explains why the bridges differ, but not why the change starts in 2019 rather than 2018, and it adds an untested assumption about insects and arrival.

• Climate warming: fits what we know in general, but fails on scope: temperatures rose equally at the farms, where nothing changed.

• A different population of swallows: possible, but needs an extra assumption with no evidence, and does not explain why the change began in a particular year.