Week 3 · Chapter

Chapter 3 — Deductive and Inductive Reasoning

Critical Thinking Free sample edition

The module's primary reading · ~35–45 minutes · Objective 3 (with Objective 2 carried forward) · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the Argument Workshop all build on it.

In plain language

This week you start judging arguments, not only laying them out. Some arguments promise that their conclusion cannot miss if their reasons are true; that kind is a deductive argument (reasons meant to guarantee the conclusion). Others promise only that the conclusion is likely; that kind is an inductive argument (reasons meant to make the conclusion probable). You will learn the test that fits each kind: whether a deductive argument is valid (built so true reasons could never lead to a false conclusion), and how strong an inductive argument is (how likely its reasons make the conclusion). You will also learn the counterexample method (imagining one case where the reasons are true and the conclusion is false), which is how you show an argument's pattern does not work.

Why you already care about this

It is 7 a.m. on the day of a 10 a.m. soccer match. A teammate texts the group: "It rained overnight, so the field is wet. Bring your cleats with the long studs." Ten minutes later, a weather app pings 3 of the same teammates: rain is likely again by noon. Both messages give reasons. But they promise different things.

The teammate's message promises more. If it really rained overnight, and rain really wets that field, the field has to be wet. The app promises less. Even a good forecast can miss, and nobody blames the app for being "illogical" when the afternoon stays dry.

Last week you learned to lay an argument out line by line. This week you ask the next question: how much do those lines actually prove? The answer depends on which promise the argument makes, so you first sort the promise, then run the test that fits it.

By the end of this chapter you will be able to:

tell a deductive argument from an inductive one by what its premises are meant to do;
say whether a deductive argument is valid and whether it is sound, and explain why valid is not the same as true;
rate an inductive argument's strength in degrees and say when it is cogent;
show that an argument's pattern is invalid by describing one counterexample;
test each link in the running question's map and name which link is deductive and which is inductive.

Carry this line with you: valid is about the link, not the truth of the lines.

Where we are

Already covered: Week 1 taught you to find an argument; Week 2 taught you to write it in standard form, supply its missing premise, and draw its map.
This week: deductive and inductive arguments, validity and soundness, strength and cogency, and the counterexample method, all run on the running question's map.
How they connect: you can only test a link you can see, so every test this week starts from a standard form and a map like the ones you drew in Week 2.
Where we're going next: Week 4, Conditionals and Deductive Forms, names the common deductive patterns and tests each one with this week's counterexample method.

Week 1 gave you the habit of asking what a writer wants you to accept and what reasons are offered. Week 2 gave you standard form (premises on numbered lines P1, P2, P3, then "So," and the conclusion marked C) and argument maps, where linked premises support a conclusion only together. This week you take each arrow on that map and ask a new question about it: if the boxes at the tail of the arrow were true, how good would the support be?

1. Sort the promise: deductive versus inductive

Think about two friends planning a road trip. One says, "Every gas station on this highway takes cards, and the next stop is a gas station on this highway, so it takes cards." The other says, "The last four stations we passed took cards, so the next one probably will too." Both are arguments. The first is meant to settle the question. The second is meant only to make a good bet.

That difference has names.

A deductive argument is an argument whose premises are meant to guarantee its conclusion, so that if the premises were true, the conclusion could not be false.

An inductive argument is an argument whose premises are meant to make its conclusion probable rather than to guarantee it, so that true premises still leave some chance the conclusion is false.

Notice the words "meant to." You sort an argument by the promise it makes, not by whether it keeps that promise. A deductive argument can fail, and you will see how next section. It is still deductive, because a guarantee is what it was going for.

Worked example — the field and the forecast (illustrative). Sort two short arguments.

The field. "If it rained overnight, the soccer field is wet. It rained overnight. So the soccer field is wet." Ask what the premises are meant to do. They leave no room for the field to be dry. If both premises are true, the conclusion has to be true. This is a deductive argument.
The forecast. "The weather service has called most rainy mornings this month correctly, and it says rain is likely by noon. So it will probably rain by noon." Ask the same question. The word "probably" admits that the conclusion could miss even if every premise is true. This is an inductive argument.
Say it in words. The field argument promises a guarantee, and the forecast argument promises a good chance.

In words: the field argument is deductive because its premises are meant to guarantee the field is wet, and the forecast argument is inductive because its premises are meant only to make rain probable.

Hedge words such as "probably," "likely," and "most" often signal an inductive argument. Words such as "must," "certainly," and "it follows that" often signal a deductive one. Like last week's indicator words, they are clues, not proof.

An honesty note: philosophers disagree about how best to draw the line between these two kinds, and some everyday passages sit right on it. "We've beaten that team every time we've played them, so we'll beat them again" could be read either way. So this course grades the test you apply, not the label you pick for a borderline passage; a borderline case is something to discuss, not a quiz answer.

Misconception check. Students often think a deductive argument is one that works, so an argument that fails "isn't deductive." That is tempting, because "deductive" sounds like a compliment. But the label describes what the premises are meant to do. A deductive argument that fails to guarantee its conclusion is still deductive; it is a deductive argument that is invalid.

Memory hook: ask what the premises promise: a guarantee, or a good chance?

2. Test the guarantee: validity and soundness

Picture a kitchen rule taped to a restaurant's walk-in cooler: "Everything on the top shelf is for tonight's dinner service. This tray of salmon is on the top shelf. So this tray is for tonight's dinner service." Before anyone checks the shelf, you can already see something about this argument. If both premises were true, there would be no way for the conclusion to be false.

That property is the first test for any deductive argument.

An argument is valid when it is impossible for all its premises to be true and its conclusion false; validity depends on the argument's form, not on whether the premises are in fact true.

An argument is invalid when at least one possible case exists in which all its premises are true and its conclusion is false.

Validity is a fact about the link from premises to conclusion. It says nothing yet about whether the premises are true. For that, you need a second standard.

An argument is sound when it is valid and every premise is true; a sound argument's conclusion must be true.

So a sound argument passes two checks: the link holds, and the lines are true.

Worked example — the field on two different mornings (illustrative). Use the field argument from Section 1.

Check the link. Could both premises be true while the field is dry? If it rained overnight, and rain overnight wets the field, the field is wet. There is no case with true premises and a dry field. The argument is valid.
Check the lines on a morning when it rained. Both premises are true. The argument is valid with true premises, so it is sound, and its conclusion must be true.
Check the lines on a morning when it did not rain. The second premise, "It rained overnight," is false. The argument is still valid, because its link has not changed, but it is not sound. Its conclusion might be true or false; the argument no longer settles it.

In words: the same valid argument is sound on a rainy morning and unsound on a dry one, because soundness depends on the premises being true and validity does not.

Now the two surprises this section has been building toward.

Worked example — valid with a false conclusion, invalid with a true one (illustrative). Test each argument's link before you look at its conclusion.

Valid but false. A transit rider argues: "Every bus on Route 9 is electric. The 6:40 bus runs on Route 9. So the 6:40 bus is electric." Suppose two of the Route 9 buses actually run on diesel, and the 6:40 is one of them. The conclusion is false. Is the argument invalid? No. If every bus on Route 9 really were electric, the 6:40 would have to be electric. The link is fine. The first premise is false, and that false premise is what let a valid argument reach a false conclusion.
Invalid but true. A shopper argues: "This grocery store sells fresh bread. So it is open on weekends." Suppose the store really is open on weekends. The conclusion is true. Is the argument valid? No. A store could sell fresh bread and close on weekends; selling bread gives no guarantee about the hours. The conclusion is true by luck, not because the premise supports it.

In words: a valid argument can have a false conclusion when a premise is false, and an invalid argument can have a true conclusion that its premises fail to support, so validity and truth are separate questions.

Misconception check. Students often think "valid" means "true," so a valid argument must have a true conclusion. That is tempting, because in everyday speech "that's a valid point" means "that's right." But in this course, valid describes only the link: true premises would force the conclusion. The Route 9 argument is valid and its conclusion is false. Its trouble is a false premise, not a broken link.

A caveat: "sound" and "valid" are often used loosely as synonyms outside this course. Here they are different. Every sound argument is valid, but a valid argument is sound only when all its premises are true.

Memory hook: valid checks the link; sound checks the link and the lines.

3. Rate the chance: strength and cogency

Inductive arguments need a different test, because they never promised a guarantee. Calling a weather forecast "invalid" misses the point; it was never trying to rule out every dry afternoon. The right question is how much the premises raise the chance of the conclusion.

An inductive argument is strong when its premises, if true, would make its conclusion likely; strength comes in degrees, and an argument that falls short is weak.

A cogent argument is a strong inductive argument whose premises are all true; even a cogent argument can have a false conclusion.

"In degrees" is the key phrase. Validity is all-or-nothing: an argument is valid or it is not. Strength is a dial. New information can turn it up or down.

Worked example — the missing package (illustrative). A resident of an apartment building comes home to find no package, even though the delivery tracker says "delivered." She reasons toward a conclusion: the package was left at the wrong door. Watch the strength change as each premise is added.

One premise. "The tracker says delivered, but the package is not at my door. So it was left at the wrong door." This is weak. The package could have been stolen, held in a mailroom, or marked "delivered" by mistake.
Two premises. Add: "The building's mailroom has no package for me." Two other explanations are now less likely, so the argument is stronger, but theft and a tracking mistake are still open.
Three premises. Add: "My neighbor in the unit with the same number on the floor above got a box with my name on it last month." Now a wrong-door delivery is the most likely explanation, and the argument is fairly strong.
Cogency. If all three premises are true, the argument is cogent. It still does not guarantee the conclusion. The package could, this one time, have been taken from the right door.

In words: the missing-package argument grows from weak to fairly strong as premises are added, and even at its strongest it makes the conclusion likely rather than certain.

Here is a second case from sports radio. A host says, "The home team has won 9 of its last 10 home games, so it will probably win at home this weekend." That is 9 ÷ 10 × 100 = 90 percent of recent home games won. If the record is accurate, the argument is fairly strong and cogent. But a key player's injury, which the host never mentions, could weaken it at once, which is what "in degrees" means in practice.

Misconception check. Students often think strength is all-or-nothing, so an inductive argument is either "proved" or "worthless." That is tempting, because validity works that way. But strength is a matter of degree. The missing-package argument went from weak to fairly strong without ever becoming a guarantee, and that is the normal life of an inductive argument.

Misconception check — cogent is not valid. Students often read "cogent" as a fancy word for "valid." That is tempting, because both words praise an argument. But cogent belongs to inductive arguments: strong, with true premises. A cogent argument can still have a false conclusion, which a sound argument never can.

A limitation worth naming: no formula turns a passage into a strength rating. Judging strength means asking what else could explain the evidence and how likely those other explanations are, which is a skill you will sharpen in later weeks on samples and causes.

Memory hook: deductive arguments pass or fail; inductive arguments come in degrees.

4. Break the pattern: the counterexample method

Suppose someone hands you a deductive argument and you suspect its link is broken. How do you show it? You do not need to prove its conclusion false. You need only one case where the premises are true and the conclusion is false.

The tool works on the argument's pattern.

An argument form is the pattern of an argument's reasoning once its particular subject matter is set aside, so that two arguments about different things can share one pattern.

A counterexample is a case, real or imagined, with the same form as an argument in which every premise is true and the conclusion is false.

The counterexample method is the test that shows an argument form is invalid by describing one case with the same form in which every premise is true and the conclusion is false.

If such a case can exist, the form does not guarantee its conclusion, so every argument with that form is invalid.

Worked example — the field argument run in reverse (illustrative). Here is the field argument with its second premise and its conclusion swapped:

If it rained overnight, the soccer field is wet. The soccer field is wet. So it rained overnight.

Name what the premises say. The first says rain would wet the field. The second says the field is wet.
Look for a case with true premises and a false conclusion. The conclusion is false whenever it did not rain overnight. So ask: can the field be wet when it did not rain?
Describe the counterexample. The sprinklers ran overnight and it did not rain. In that case, "if it rained overnight, the field is wet" is still true, "the field is wet" is true, and "it rained overnight" is false.
Draw the verdict. One case with every premise true and the conclusion false is enough. The reversed field argument is invalid.

In words: both premises can be true while the conclusion is false — the sprinklers ran — so the argument is invalid, even though on many mornings its conclusion might happen to be true.

Because the method targets the form, the same counterexample move works on any argument with that pattern. Written with sentence letters P and Q for whole claims, the pattern is "if P, then Q; Q; so P." A phone user reasons: "If my phone's battery is dead, the screen stays dark. The screen stays dark. So the battery is dead." The counterexample: the screen is cracked and broken while the battery is fully charged. Both premises are true, the conclusion is false, and the argument is invalid.

Compare the original field argument, "if P, then Q; P; so Q." Try as hard as you like; you cannot describe a case where it rained overnight, rain wets the field, and the field is dry. No counterexample exists, because that form is valid. Week 4 gives both of these patterns their names.

Misconception check. Students often think one example with true premises and a true conclusion proves an argument valid. That is tempting, because the example "worked." But a valid argument must hold in every possible case, and one good morning shows only that the conclusion was true that time. Validity is shown by the impossibility of a counterexample; invalidity is shown by finding one.

A caveat about the method: it settles invalidity for deductive arguments only. An inductive argument already admits that true premises could come with a false conclusion, so a counterexample does not refute it. It only reminds you why it was never a guarantee.

Memory hook: one true-premises, false-conclusion case breaks the whole pattern.

5. Test the running question, link by link

Now you run this week's tests on the running question. Here again is its illustrative argument in standard form, as you built it in Week 2:

P1: During finals week many students need a quiet, safe place to study late at night.

P2: The library is the only quiet, safe, well-lit study space on campus large enough to meet that need.

P3 (supplied): The campus should provide study space students need during finals when it can do so at modest cost.

P4: Keeping the library open overnight for the five nights of finals week costs about 1,600 dollars in staff wages.

So,

C: The campus library should stay open around the clock during finals week.

The map from Week 2: P1, P2, and P3 work together to support C, and P4 supports P3's "modest cost" condition. That gives two support links to test.

Worked example — the running question's map, tested link by link (illustrative). Work through it in five steps.

The first link as it stands. Could P1, P2, and P3 all be true while C is false? They could: if keeping the library open overnight did not cost a modest amount, P3 would not require it, and the library could close at its usual hour. So the link from P1, P2, and P3 to C is not valid as it stands.
The cost condition granted. Add the claim "Keeping the library open overnight during finals week costs a modest amount," which none of P1, P2, or P3 says. Call it the cost condition: the claim that keeping the library open overnight during finals week costs a modest amount, which the supplied premise needs and the cost premise is offered to support. With P1, P2, P3, and the cost condition all true, C cannot be false. So the first link is deductive, and valid once the cost is granted to be modest.
The figure behind P4. 2 staff × 8 hours a night = 16 staff hours a night; 16 × 5 nights = 80 staff hours; 80 × 20 dollars an hour = 1,600 dollars; 1,600 ÷ 5 nights = 320 dollars a night.
The second link. Could P4 be true while "the cost is modest" is false? It could: a campus that has already spent its whole study-space budget might not count 1,600 dollars as modest. So no figure guarantees "modest" on its own. P4 makes the cost condition more or less likely, depending on what else the campus spends, which makes the second link inductive, with strength in degrees.
The tally. 2 support links in the map; 1 deductive link (P1, P2, and P3 to C) + 1 inductive link (P4 to the cost condition) = 2 links.

Verdict in words: the step from P1, P2, and P3 to C is deductive and valid once the cost is granted to be modest, while the step from P4 to "modest" is inductive, so the argument can be no better than cogent overall, and the place to look hardest is the inductive step and the truth of P3.

Notice what the test did and did not do. It did not decide whether 320 dollars a night is a lot. It did not decide whether the library should stay open. It showed where the argument's support is a guarantee and where it is a judgment call, so that a supporter and an opponent can argue about the right line.

Misconception check. Students often think that calling a link valid means agreeing with the conclusion, or that calling a link inductive means attacking it. That is tempting, because "valid" sounds like praise and "inductive" sounds like a weakness. But a student who opposes overnight hours should find the same deductive link and the same inductive link as a student who supports them. The course takes no side; the test is the same for everyone. The same goes for the teammates at the soccer field: before arguing about cleats, they can agree on which message promised a guarantee.

Memory hook: one link guarantees once the cost is granted; the other link only makes "modest" likely.

Key terms

deductive argument — An argument whose premises are meant to guarantee its conclusion, so that if the premises were true, the conclusion could not be false. (first used: 1. Sort the promise: deductive versus inductive)
inductive argument — An argument whose premises are meant to make its conclusion probable rather than to guarantee it, so that true premises still leave some chance the conclusion is false. (first used: 1. Sort the promise: deductive versus inductive)
valid — Said of an argument when it is impossible for all its premises to be true and its conclusion false; validity depends on the argument's form, not on whether the premises are in fact true. (first used: 2. Test the guarantee: validity and soundness)
invalid — Said of an argument when at least one possible case exists in which all its premises are true and its conclusion is false. (first used: 2. Test the guarantee: validity and soundness)
sound — Said of an argument that is valid and has every premise true; a sound argument's conclusion must be true. (first used: 2. Test the guarantee: validity and soundness)
strong — Said of an inductive argument whose premises, if true, would make its conclusion likely; strength comes in degrees, and an argument that falls short is weak. (first used: 3. Rate the chance: strength and cogency)
cogent — Said of a strong inductive argument whose premises are all true; even a cogent argument can have a false conclusion. (first used: 3. Rate the chance: strength and cogency)
argument form — The pattern of an argument's reasoning once its particular subject matter is set aside, so that two arguments about different things can share one pattern. (first used: 4. Break the pattern: the counterexample method)
counterexample — A case, real or imagined, with the same form as an argument in which every premise is true and the conclusion is false. (first used: 4. Break the pattern: the counterexample method)
counterexample method — The test that shows an argument form is invalid by describing one case with the same form in which every premise is true and the conclusion is false. (first used: 4. Break the pattern: the counterexample method)
cost condition — In the running question, the claim that keeping the library open overnight during finals week costs a modest amount, which the supplied premise needs and the cost premise is offered to support. (first used: 5. Test the running question, link by link)

Summary

A deductive argument is meant to guarantee its conclusion; an inductive argument is meant to make it probable.
Sort by the promise, not by success; borderline passages are for discussion.
Valid means true premises could never come with a false conclusion.
Sound means valid with every premise true.
A valid argument can have a false conclusion; an invalid one can have a true conclusion.
Strong means true premises would make the conclusion likely, and strength comes in degrees.
Cogent means strong with every premise true, and it can still miss.
One counterexample with true premises and a false conclusion shows a form is invalid.
In the running question, P1, P2, and P3 to C is deductive once the cost is granted modest; P4 to "modest" is inductive.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A gardening club's newsletter (illustrative) says: "Every tomato plant in the north bed gets drip water. This plant is in the north bed. So it gets drip water." A second item says: "The north bed's plants have looked healthy after every drip-water season so far, so they will probably look healthy this season." Sort each argument as deductive or inductive, and say what in the wording leads you there.
A student argues (illustrative): "All music majors play the piano. My lab partner is a music major. So my lab partner plays the piano." Suppose some music majors play only the violin. Is the argument valid? Is it sound? Explain the difference.
A houseplant owner reasons (illustrative): "If a plant gets no water, its leaves droop. My plant's leaves are drooping. So it got no water." Use the counterexample method to test the argument.
A carpool app (illustrative) arrived on time on 46 of the last 50 weekday mornings a nurse used it. She argues: "So it will be on time tomorrow morning." Work out the on-time percentage, then say whether the argument is deductive or inductive, how strong it is, and what would make it cogent.

Answers (worked reasoning — read after attempting)

The first is deductive: if every tomato plant in the north bed gets drip water and this plant is in the north bed, it has to get drip water; nothing in the wording leaves room for a miss. The second is inductive: "so far" and "probably" signal that past seasons only make a healthy season likely. A pest outbreak could prove it wrong even if every premise is true.
Valid: if all music majors really played the piano, and the lab partner is a music major, the lab partner would have to play the piano; no case with both premises true and the conclusion false exists. Not sound: the first premise is false, because some music majors play only the violin. Validity checks the link, and soundness also checks the lines, so the argument is valid and unsound, and its conclusion is left unsettled.
The pattern is "if P, then Q; Q; so P." A counterexample: the plant got plenty of water but sat in a hot, sunny window all afternoon, so its leaves droop. Then "if a plant gets no water, its leaves droop" is true, "the leaves are drooping" is true, and "it got no water" is false. One such case shows the argument is invalid.
The on-time share is 46 ÷ 50 × 100 = 92 percent. The argument is inductive: a strong record makes on-time arrival likely but cannot guarantee it. It is fairly strong, because 46 of 50 is a long record of success. It is cogent if its premise is true, that is, if the app really was on time on 46 of those 50 mornings. Even then it could miss tomorrow, for example in a snowstorm.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the lecture tutorial, and the practice set. Next week, in Conditionals and Deductive Forms, the patterns you tested here get their names, and the counterexample method becomes your test for each one.

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