Week 4 · Chapter

Chapter 4 — Conditionals and Deductive Forms

Critical Thinking Free sample edition

The module's primary reading · ~35–45 minutes · Objective 3 (with Objective 1 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 is about "if" sentences, the kind that run parking rules, refund policies, and course requirements. You will learn to read a conditional (an "if, then" claim) by naming its two parts and the one situation that makes it false. You will meet four patterns of reasoning that never fail, the valid forms (patterns where true reasons always bring a true conclusion), and two look-alike patterns that do fail, the formal fallacies (invalid patterns, whatever the topic). You will also learn to tell a rule's necessary condition (something that must be true) from its sufficient condition (something that is enough on its own). Misreading an "if" sentence is one of the most common reasoning slips there is, and this week gives you the tools to catch it.

Why you already care about this

A campus parking lot has a sign at its entrance: "You may park in Lot C only if you display a permit." A student with a permit pulls in and thinks, "I have a permit, so I can park here." A friend in the passenger seat points at a second sign: the lot is closed after 10 p.m. for sweeping. The permit was needed. It was not enough.

You run into sentences like that sign every day. A store says you get a full refund only if you return the item within 30 days. A course says you can register only if you passed the prerequisite. A weather app says that if the temperature drops below freezing, the roads will be icy. Each one is an "if" sentence, and each one can be read right or read wrong.

Last week you learned the counterexample method: to show a pattern of reasoning is broken, you describe one case where its premises are true and its conclusion is false. This week you aim that method at the most common patterns built on "if" sentences. A few of them always hold. Two of them only look like they do.

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

name the antecedent (the if-part) and the consequent (the then-part) of an "if" sentence, rewrite "only if" as "if, then", and say the one situation that makes the sentence false;
recognize the four valid forms and show why each one has no counterexample;
recognize affirming the consequent and denying the antecedent, and refute each with a counterexample;
tell the converse, the inverse, and the contrapositive apart, and say which one means the same as the original;
state which part of an "if" sentence is necessary and which is sufficient;
read a truth table as a check on a verdict you already reached.

Carry this line with you: a true then-part does not prove the if-part.

Where we are

Already covered: Week 1 taught you to find an argument; Week 2 taught you to write it in standard form; Week 3 taught you validity, soundness, and the counterexample method.
This week: conditionals, the four valid forms, the two formal fallacies, the converse, inverse, and contrapositive, necessary and sufficient conditions, and one truth table as a check.
How they connect: every form this week is tested with Week 3's counterexample method, so a verdict always comes with a case or with the reason no case exists.
Where we're going next: Week 5, Categorical Reasoning, tests arguments built on "all" and "some" with Venn diagrams.

Week 1 gave you the habit of finding a conclusion and the reasons offered for it. Week 2 gave you standard form, with premises on numbered lines and the conclusion marked C. Week 3 gave you the key test: an argument is valid when it is impossible for all its premises to be true and its conclusion false, and one counterexample shows a pattern is invalid. You also met the soccer field and the sprinklers. This week that field comes back, and the patterns you tested on it get their names.

1. Read the rule: the parts of a conditional

Think about a kitchen rule taped above a restaurant's fryer: "If the oil light is red, then the oil is too hot to use." The rule does not say the oil light is red right now. It says what follows whenever the light is red. That is the job of an "if" sentence: it links two claims without asserting either one.

That kind of sentence has a name.

A conditional is a claim of the form if P, then Q, which says the second claim is true whenever the first is; it is false in exactly one situation, when P is true and Q is false.

The letters stand in for whole claims, so you can see the pattern without the subject matter getting in the way.

A sentence letter is a capital letter such as P, Q, or R that stands for a whole claim, so that the pattern of an argument can be written in words without its subject matter.

Each half of a conditional has its own name.

The antecedent is the if-part of a conditional, the claim P in if P, then Q, whose truth the conditional says would bring the then-part with it.

The consequent is the then-part of a conditional, the claim Q in if P, then Q, which the conditional says holds whenever the if-part holds.

In the fryer rule, "the oil light is red" is the antecedent, and "the oil is too hot to use" is the consequent. When would the rule be false? Only if the light is red and the oil is fine to use. A cold fryer with the light off breaks no promise, and neither does a hot fryer with a broken light. The rule only promised something about red-light moments.

Worked example — the parking sign (illustrative). Rewrite and read the sign "You may park in Lot C only if you display a permit."

Find the two claims. One is "you may park in Lot C." The other is "you display a permit."
Rewrite "only if". "P only if Q" is rewritten as "if P, then Q" before anything is tested. Here P is "you may park in Lot C" and Q is "you display a permit." So the sign says: "If you may park in Lot C, then you display a permit."
Name the parts. The antecedent is "you may park in Lot C." The consequent is "you display a permit."
Find the one false situation. The sign is broken only by a car allowed to park in Lot C without a permit. A car with a permit that still cannot park, because the lot is closed for sweeping, does not break the sign.

In words: "only if" puts the claim after it in the then-part, so the sign says every car allowed to park displays a permit, and it says nothing about every car with a permit being allowed.

Arguments built on "if" sentences often carry no indicator word. "The field's wet. It rained overnight, didn't it?" still has a premise and a conclusion. Find the claim the speaker wants you to accept, and the claims offered for it, before you look for a pattern.

Misconception check. Students often think "P only if Q" means the same as "if Q, then P." That is tempting, because the "if" sits right in front of Q. But "only if" marks a requirement, not a trigger. "You may park in Lot C only if you display a permit" means a permit is required for parking; it does not promise that a permit gets you in. Rewrite it as "if you may park in Lot C, then you display a permit," and the trap disappears.

Memory hook: the if-part is the trigger; "only if" names the requirement.

2. Patterns that never fail: the four valid forms

Picture a friend texting before a morning game: "If it rained overnight, the soccer field is wet. It rained overnight." You do not need to see the field to finish the thought. The field is wet. You did not need to know anything about soccer, either. The pattern did the work.

Logicians have named the patterns that work like that. The argument-form set this course uses holds six forms, and four of them are valid.

Modus ponens is the valid form if P, then Q; P; so Q, which moves from a conditional and its if-part to its then-part.

Modus tollens is the valid form if P, then Q; not Q; so not P, which moves from a conditional and the denial of its then-part to the denial of its if-part.

Hypothetical syllogism is the valid form if P, then Q; if Q, then R; so if P, then R, which chains two conditionals that share a middle claim.

Disjunctive syllogism is the valid form P or Q; not P; so Q, which rules out one of two options to conclude the other.

"Not P" is how this course writes a denial: it says that the claim P is false. No symbols, ever; every form is written in words.

To show a form is valid, you try the counterexample method from Week 3 and watch it fail. You search for a case with every premise true and the conclusion false. If no such case can exist, the form is valid.

Worked example — testing the four valid forms (illustrative). Each example comes from the argument-form set.

Modus ponens. "If it rained overnight, the soccer field is wet. It rained overnight. So the soccer field is wet." You look for a case with both premises true and the field dry, and none exists, so the form "if P, then Q; P; so Q" is valid.
Modus tollens. "If it rained overnight, the soccer field is wet. The soccer field is not wet. So it did not rain overnight." Had it rained, the field would be wet, and it is not, so no case gives true premises and a false conclusion: the form "if P, then Q; not Q; so not P" is valid.
Hypothetical syllogism. "If the bus is late, I miss the 8:00 a.m. class. If I miss the 8:00 a.m. class, I miss the quiz. So if the bus is late, I miss the quiz." The two conditionals chain, so no case makes both true while a late bus comes without a missed quiz: the form is valid.
Disjunctive syllogism. "Either the outlet is dead or the charger is broken. The outlet is not dead (a lamp works in it). So the charger is broken." With one option ruled out, the other must hold, so the form "P or Q; not P; so Q" is valid.

In words: each of the four forms holds because no case can give it true premises and a false conclusion, so each is valid whatever its subject matter.

Notice what valid does not mean. Suppose the charger is fine and the phone's own port is broken. Then the premise "either the outlet is dead or the charger is broken" is false. The form is still valid; the argument is unsound, because a premise is false. That is Week 3's lesson, and it holds for every form here.

Misconception check. Students often think a valid form guarantees a true conclusion. That is tempting, because the forms sound so solid. But a valid form only guarantees that true premises bring a true conclusion. Feed modus ponens a false premise, such as "if the bus is late, I miss the quiz" on a day the quiz was moved, and the conclusion can be false. Valid checks the link; true checks the lines.

A caveat: disjunctive syllogism needs the either-or premise to cover the real options. If the phone's port could be broken too, "either the outlet is dead or the charger is broken" is false, and the argument is unsound even though its form is valid.

Memory hook: four forms that never fail, as long as the lines are true.

3. Look-alikes that fail: the two formal fallacies

Here is the soccer field again, with one change. "If it rained overnight, the soccer field is wet. The soccer field is wet. So it rained overnight." Read it quickly and it sounds like modus ponens. It is not. The second premise affirms the then-part instead of the if-part, and that one swap breaks the pattern.

Patterns that break this way have a name.

A formal fallacy is an argument form that is invalid, so that some argument with that pattern has true premises and a false conclusion whatever its subject matter.

The course names two of them.

Affirming the consequent is the invalid form if P, then Q; Q; so P, which treats the then-part being true as proof that the if-part is true.

Denying the antecedent is the invalid form if P, then Q; not P; so not Q, which treats the if-part being false as proof that the then-part is false.

You refute a formal fallacy the same way you refuted the reversed field argument in Week 3: describe one case with every premise true and the conclusion false. One case is enough.

Worked example — one sprinkler morning, two fallacies (illustrative). Test both invalid forms from the argument-form set with a single case.

Affirming the consequent. "If it rained overnight, the soccer field is wet. The soccer field is wet. So it rained overnight." The sprinklers ran overnight and it did not rain: both premises are true and the conclusion is false, so the form "if P, then Q; Q; so P" is invalid.
Denying the antecedent. "If it rained overnight, the soccer field is wet. It did not rain overnight. So the soccer field is not wet." On the same sprinkler morning both premises are true and the field is wet, so the conclusion is false and the form "if P, then Q; not P; so not Q" is invalid.
The tally. 4 valid forms + 2 invalid forms = 6 forms in the set.

In words: both formal fallacies fall to one sprinkler morning, because on that morning their premises are all true and their conclusions are both false, while the four valid forms have no such morning.

The same move works on any topic, because the fault is in the pattern. A gym's ad says (illustrative): "If you train with us, you'll get stronger. Your neighbor got stronger this year. So your neighbor trained with us." The neighbor might have lifted weights at home all year. Same form, same verdict: affirming the consequent.

Misconception check — denying the antecedent is not modus tollens. Students often think denying the antecedent is "the same as" modus tollens, because both contain a "not." That is tempting, but look at what gets denied. Modus tollens denies the then-part ("the field is not wet") and concludes the if-part is false, which is valid. Denying the antecedent denies the if-part ("it did not rain") and concludes the then-part is false, which the sprinklers refute.

Misconception check. Students often think affirming the consequent is valid when its conclusion is true. That is tempting, because on many rainy mornings the conclusion "it rained overnight" really is true. But valid means the pattern can never lead from true premises to a false conclusion. The sprinkler morning shows that it can, so the form is invalid even on mornings when its conclusion happens to be true.

An honesty note: real speakers sometimes say "if" when they mean "if and only if." A manager who says "if you cover the weekend shift, you can leave early on the next workday" may mean that covering the shift is the only way to leave early. Reading a speaker charitably is a judgment about what was meant. The form you test is the one the words actually state.

Memory hook: a true then-part does not prove the if-part.

4. Turn it around: converse, inverse, contrapositive, and conditions

Take a rule from a clinic's front desk (illustrative): "If a visitor has a fever, the visitor waits in the side room." Now flip it: "If a visitor waits in the side room, the visitor has a fever." The flipped version sounds similar, but it says something new. Someone could be in the side room because the main room was full.

There are three ways to turn a conditional around, and each has a name.

The converse is the conditional made by swapping the two parts of if P, then Q to get if Q, then P; it can be false when the original is true.

The inverse is the conditional made by denying both parts of if P, then Q to get if not P, then not Q; it can be false when the original is true.

The contrapositive is the conditional made by swapping and denying both parts of if P, then Q to get if not Q, then not P; it is always true or false together with the original.

Two claims that always share a truth value have a name too.

Two claims are equivalent when they cannot differ in truth value, so that in every possible case both are true or both are false.

Only the contrapositive is equivalent to the original. The converse and the inverse are not.

Worked example — the field turned three ways (illustrative). Start from "If it rained overnight, the soccer field is wet."

The converse. "If the soccer field is wet, it rained overnight." On the sprinkler morning the field is wet and it did not rain, so the converse is false while the original stays true. Not equivalent.
The inverse. "If it did not rain overnight, the soccer field is not wet." On the same sprinkler morning, it did not rain and the field is wet, so the inverse is false too. Not equivalent.
The contrapositive. "If the soccer field is not wet, it did not rain overnight." If the field is dry, rain cannot have fallen, because rain would have wet it. This says the same thing as the original. Equivalent.

In words: the converse and the inverse can both fail on the sprinkler morning while the original holds, but the contrapositive cannot come apart from the original, so it alone is equivalent.

Turning a rule around also tells you what it requires and what it guarantees.

A sufficient condition is a condition whose truth is enough to bring another with it; in if P, then Q, the if-part P is sufficient for Q.

A necessary condition is a condition that must hold for another to hold, though it may not be enough on its own; in if P, then Q, the then-part Q is necessary for P.

Worked example — the prerequisite (illustrative). A course listing says: "You can register for the course only if you passed the prerequisite."

Rewrite "only if". "If you can register for the course, you passed the prerequisite."
Read the necessary condition. The then-part, passing the prerequisite, is necessary for registering: no one registers without it.
Read the sufficient condition. The if-part, being able to register, is sufficient for knowing you passed: anyone who can register passed.
Check what is not promised. Passing is not sufficient for registering. You also need an open seat.

In words: passing the prerequisite is necessary for registering, but it is not sufficient: you also need an open seat.

The parking sign from Section 1 works the same way. Displaying a permit is necessary for parking in Lot C. The sweeping hours show it is not sufficient.

Misconception check. Students often think the converse of a rule means the same as the rule. That is tempting, because "if it rained, the field is wet" and "if the field is wet, it rained" use the same words. But the converse is a new claim, and the sprinklers make it false. Only the contrapositive, "if the field is not wet, it did not rain," always matches the original.

Misconception check — necessary is not sufficient. Students often read a necessary condition as a sufficient one: "I passed the prerequisite, so I can register." That is tempting, because a requirement feels like a ticket. But a requirement only rules people out. The open-seat rule shows that meeting one requirement does not get you in.

A limitation worth naming: everyday rules often hide extra conditions, such as the sweeping hours or the open seat. The logic tells you exactly what the stated rule requires and guarantees. It cannot tell you what the rule's writer left unsaid.

Memory hook: flip and deny both parts, or you have changed the claim.

5. Check the verdict: the truth table, once

You have already settled every verdict in this chapter with counterexamples. A truth table is a second way to look at one of them, the way you might check a long sum by adding the column again from the bottom. It is shown once, as a check.

A truth table is a small table that lists every combination of true and false for the sentence letters and shows whether a claim built from them is true or false in each row.

The table below lists each combination of truth values for P and Q, and the last column gives the conditional's truth value in that row.

Truth values of if P, then Q for every combination of P and Q
Row	P	Q	if P, then Q
1	true	true	true
2	true	false	false
3	false	true	true
4	false	false	true

Worked example — the truth-table check (illustrative). Use the table to check the verdict on affirming the consequent.

Count the rows. 2 sentence letters, each true or false, give 2 × 2 = 4 rows.
Find the false row. "If P, then Q" is false in 1 row (P true, Q false), so 4 − 1 = 3 rows make it true.
Find the counterexample row. The third row (P false, Q true) is the sprinkler morning, where "it rained overnight" is false and "the field is wet" is true.
Read the verdict. In row 3 the premises of affirming the consequent, "if P, then Q" and "Q", are both true, and its conclusion, "P", is false. That is a counterexample row, so the form is invalid.

In words: the conditional is false in exactly one row, P true and Q false, and the third row is the sprinkler morning, so the table confirms the counterexample verdict without adding a new one.

Notice how rows 3 and 4 match the fryer rule from Section 1. A cold fryer with the light off, or a hot fryer with a broken light, does not break the rule. The conditional only promised something about the rows where P is true.

Misconception check. Students often think a conditional with a false if-part must be false, so rows 3 and 4 "should" say false. That is tempting, because nothing in those rows seems to confirm the rule. But a conditional only promises something when its if-part is true. A rule about red-light moments cannot be broken on a morning with no red light.

A caveat: this course never asks you to build a truth table for a grade. Your test is the counterexample method; the table is a way to see why it works.

Memory hook: one false row: the if-part true and the then-part false.

Key terms

conditional — A claim of the form if P, then Q, which says the second claim is true whenever the first is; it is false in exactly one situation, when P is true and Q is false. (first used: 1. Read the rule: the parts of a conditional)
sentence letter — A capital letter such as P, Q, or R that stands for a whole claim, so that the pattern of an argument can be written in words without its subject matter. (first used: 1. Read the rule: the parts of a conditional)
antecedent — The if-part of a conditional, the claim P in if P, then Q, whose truth the conditional says would bring the then-part with it. (first used: 1. Read the rule: the parts of a conditional)
consequent — The then-part of a conditional, the claim Q in if P, then Q, which the conditional says holds whenever the if-part holds. (first used: 1. Read the rule: the parts of a conditional)
modus ponens — The valid form if P, then Q; P; so Q, which moves from a conditional and its if-part to its then-part. (first used: 2. Patterns that never fail: the four valid forms)
modus tollens — The valid form if P, then Q; not Q; so not P, which moves from a conditional and the denial of its then-part to the denial of its if-part. (first used: 2. Patterns that never fail: the four valid forms)
hypothetical syllogism — The valid form if P, then Q; if Q, then R; so if P, then R, which chains two conditionals that share a middle claim. (first used: 2. Patterns that never fail: the four valid forms)
disjunctive syllogism — The valid form P or Q; not P; so Q, which rules out one of two options to conclude the other. (first used: 2. Patterns that never fail: the four valid forms)
formal fallacy — An argument form that is invalid, so that some argument with that pattern has true premises and a false conclusion whatever its subject matter. (first used: 3. Look-alikes that fail: the two formal fallacies)
affirming the consequent — The invalid form if P, then Q; Q; so P, which treats the then-part being true as proof that the if-part is true. (first used: 3. Look-alikes that fail: the two formal fallacies)
denying the antecedent — The invalid form if P, then Q; not P; so not Q, which treats the if-part being false as proof that the then-part is false. (first used: 3. Look-alikes that fail: the two formal fallacies)
converse — The conditional made by swapping the two parts of if P, then Q to get if Q, then P; it can be false when the original is true. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
inverse — The conditional made by denying both parts of if P, then Q to get if not P, then not Q; it can be false when the original is true. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
contrapositive — The conditional made by swapping and denying both parts of if P, then Q to get if not Q, then not P; it is always true or false together with the original. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
equivalent — Said of two claims that cannot differ in truth value, so that in every possible case both are true or both are false. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
sufficient condition — A condition whose truth is enough to bring another with it; in if P, then Q, the if-part P is sufficient for Q. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
necessary condition — A condition that must hold for another to hold, though it may not be enough on its own; in if P, then Q, the then-part Q is necessary for P. (first used: 4. Turn it around: converse, inverse, contrapositive, and conditions)
truth table — A small table that lists every combination of true and false for the sentence letters and shows whether a claim built from them is true or false in each row. (first used: 5. Check the verdict: the truth table, once)

Summary

A conditional "if P, then Q" is false only when P is true and Q is false.
P is the antecedent (the if-part); Q is the consequent (the then-part).
Rewrite "P only if Q" as "if P, then Q" before you test it.
The four valid forms are modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism.
Affirming the consequent and denying the antecedent are invalid; the sprinkler morning refutes both.
Only the contrapositive is equivalent to the original; the converse and the inverse are not.
In "if P, then Q", P is sufficient for Q and Q is necessary for P.
A truth table checks a counterexample verdict; it is not a graded skill.

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 bakery's training sheet says (illustrative): "If the bread is fully baked, it sounds hollow when tapped." A new baker taps a loaf, hears a hollow sound, and concludes the loaf is fully baked. Name the form and say whether it is valid.
A smoke alarm's manual says (illustrative): "If there is smoke in the room, the alarm sounds." Write the converse, the inverse, and the contrapositive, and say which one is equivalent to the manual's sentence.
A climbing gym posts (illustrative): "You may use the top-rope wall only if you have passed the belay check." Rewrite the rule as "if, then", then say which condition is necessary, and whether passing the belay check is sufficient for using the wall.
A coworker reasons (illustrative): "If the printer is out of paper, the tray light blinks. The tray light is not blinking. So the printer is not out of paper." A second coworker reasons: "If the printer is out of paper, the tray light blinks. The printer is not out of paper. So the tray light is not blinking." Name each form, say which is valid, and give a counterexample for the invalid one.

Answers (worked reasoning — read after attempting)

The pattern is "if P, then Q; Q; so P," with P "the bread is fully baked" and Q "it sounds hollow when tapped." That is affirming the consequent, and it is invalid. A counterexample: a loaf with a hard crust and a doughy middle sounds hollow but is not fully baked. Both premises are true and the conclusion is false.
The converse: "If the alarm sounds, there is smoke in the room." The inverse: "If there is no smoke in the room, the alarm does not sound." The contrapositive: "If the alarm does not sound, there is no smoke in the room." Only the contrapositive is equivalent. Steam from a shower could set off the alarm with no smoke, which makes the converse and the inverse false while the manual's sentence stays true.
Rewritten: "If you may use the top-rope wall, then you have passed the belay check." Passing the belay check is the necessary condition, because no one uses the wall without it. It is not sufficient: the rule never promises that passing gets you on the wall, and the gym might also require a waiver or an open lane.
The first coworker uses modus tollens, "if P, then Q; not Q; so not P," which is valid: if the tray were empty the light would blink, and it is not blinking. The second uses denying the antecedent, "if P, then Q; not P; so not Q," which is invalid. A counterexample: the printer has paper but a jammed tray sets the light blinking. Both premises are true and the conclusion is false.

Next up in this module: the lecture, the slide deck, the Readings & Resources page, the lecture tutorial, and the practice set. Next week, in Categorical Reasoning, the Venn diagram becomes your test for arguments built on "all" and "some".

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