Week 2 · Chapter

Chapter 2 — Arguments and Thought Experiments

Introduction to Philosophy Free sample edition

The module's primary reading · ~40–50 minutes · Objective 2 (identify arguments, put them in standard form, judge whether they are valid, sound, or inductively strong, and test claims with counterexamples and thought experiments), with Objective 1 (explain what philosophy is, name its main branches, and describe the Socratic practice of examining beliefs through questioning and argument) carried forward · Read it first this week; the lecture, the slide deck, the tutorial, the quiz, and the Philosophy Workshop all build on it.

In plain language

People give reasons all the time, and some reasons hold up better than others. This week you learn to lay out an argument (a set of reasons offered for a claim) so you can check it one step at a time. You learn to ask whether the reasons, if true, would guarantee the claim, which is what makes an argument valid (built so the claim cannot be false if the reasons are true). You also learn two tools philosophers use to test ideas: the single case that breaks a general claim, and the imagined scenario called a thought experiment (a made-up case used to test an idea). These skills carry every later week of the course.

Why you already care about this

Picture a jury room after a long day of testimony. One juror leans forward and says: "If the defendant was in the building, the camera would show someone in a red coat. The camera shows someone in a red coat. So the defendant was in the building." A few heads nod. It sounds tidy. Yet something about it bothers you, and you cannot quite say what.

You have met this feeling before. A friend says a movie must be good because it made a lot of money. A coworker says the new schedule will fail because the last change failed. Each one gives you a reason and a claim. Each time, you sense a gap between them. This chapter gives you the words for that gap and a way to show it to someone else, calmly and fairly.

Here is the short version of the juror's problem. Someone else in a red coat would also show up on the camera. The reasons could all be true while the claim is false. By the end of this chapter you can say that in one plain sentence, and you can name the pattern it follows.

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

put an argument in standard form, with numbered premises and a conclusion marked C;
tell a valid argument from a sound one, and explain why validity depends on form alone;
name five valid forms and three invalid ones, and find each in an argument;
build a counterexample that shows a form is invalid;
tell a deductive argument from an inductive one and judge an inductive argument's strength;
refute a general claim with one counterexample;
work the barber puzzle as a reductio;
restate an argument in its strongest form before you criticize it.

Carry this line with you: valid is about the shape, not about the facts.

Where we are

Already covered: Week 1, What Philosophy Is: Socrates and the Examined Life, showed you philosophy as careful questioning, named its main branches, and gave you the Socratic habit of testing a definition with a counterexample.
This week: what an argument is, how to write it in standard form, how to judge whether it is valid, sound, or inductively strong, and how counterexamples and thought experiments test claims.
How they connect: Socrates tested people's definitions one case at a time. This week turns that habit into tools you can use on any argument.
Where we're going next: Week 3, Plato: Reality and the Forms, opens the unit on reality and knowledge, and you will put Plato's reasoning in standard form with the tools you learn here.

Week 1 matters here in one specific way. Socrates did not win debates by announcing the truth. He asked a person for a definition, then found a case the definition got wrong. That move, the counterexample, returns this week in two forms. It can break a general claim, and it can show that an argument's shape is broken. Every later week leans on both. The pack's ten-argument set, Dataset B, will come back in Weeks 4, 6, 7, 9, 10, 11, 12, and 13, so the work you do on it now pays off all term.

1. Standard form: lay out the reasons, then the claim

Start with the juror again. The juror gave two reasons and one claim, all in one breath. Spoken arguments often run that way. The claim may come first, last, or in the middle. Reasons hide inside questions or asides. To check an argument, you first have to see its parts.

An argument is a set of statements, the premises, offered as reasons for another statement, the conclusion. A premise is a statement offered as a reason. The conclusion is the statement the premises are offered to support.

Standard form writes the argument so each part sits on its own line. You number the premises P1, P2, and so on. Each premise is one declarative sentence, a sentence that can be true or false. The conclusion comes last, marked C and introduced by "So". If you add a premise the arguer left unsaid, you mark it "(added)" so no reader mistakes your words for theirs.

Worked example — the juror in standard form. Find the claim the juror wants the room to accept: the defendant was in the building. That is the conclusion. The two reasons are the premises. Drop the extra words and number the lines.

P1: If the defendant was in the building, the camera would show someone in a red coat.

P2: The camera shows someone in a red coat.

C: So the defendant was in the building.

In words: two premises are offered as reasons, and the conclusion comes last. Now you can look at each line by itself, and at how the lines fit together.

Sometimes the arguer skips a step. A hospital volunteer says, "The waiting room is full, so we should open the second check-in desk." The volunteer is relying on a reason they never said: when the waiting room is full, the second desk should open. In standard form you write it out and mark it.

P1: The waiting room is full.

P2 (added): If the waiting room is full, we should open the second check-in desk.

C: So we should open the second check-in desk.

Misconception check — premise or conclusion? Students often think the first sentence a person says must be a premise. It is tempting because we expect reasons to come first. The cure is to ask, "What is this person trying to get me to accept?" Whatever answers that question is the conclusion, wherever it sits in the speech. In standard form it moves to the last line, marked C.

2. Validity and soundness: does the shape lock in the claim?

Think of a vending machine that works perfectly. Put in the right coins, press the button, and the snack drops every time. The machine does not care whether you wanted that snack. It only guarantees the output if the input is right. A valid argument works like that machine.

A deductive argument aims to guarantee its conclusion. Two words judge it.

Valid — an argument whose conclusion cannot be false if its premises are true. Validity depends on the argument's form, not on whether its premises are in fact true.
Sound — an argument that is valid and whose premises are all in fact true.

So you ask two separate questions. First, if the premises were true, would the conclusion have to be true? That is validity. Second, are the premises true? Only when both answers are yes is the argument sound.

Worked example — valid but not sound. Look at this argument about fish.

P1: Every fish can fly.

P2: Every salmon is a fish.

C: So every salmon can fly.

Ask the first question. If every fish could fly, and every salmon were a fish, then every salmon would have to fly. There is no way out. So the argument is valid. Now ask the second question. P1 is false: most fish cannot fly. So the argument is not sound. In words: the shape is perfect, but one premise is false, so the argument is valid and unsound.

Worked example — valid and sound. A tile setter picks up a tile and reasons:

P1: If this tile is a square, it has four sides.

P2: This tile is a square.

C: So this tile has four sides.

If both premises are true, the conclusion must be true, so the argument is valid. Both premises are true of the tile in hand, so it is also sound. In words: a good shape plus true premises gives a sound argument.

Misconception check — valid does not mean true. Students often think a valid argument must have true premises. It is tempting because "valid" sounds like praise, like "good" or "correct". The cure is the salmon argument. It is valid, and its first premise is false. Validity is a promise about form: if the premises are true, the conclusion is too. It promises nothing about whether they are true.

3. The valid forms: five patterns you can trust

You do not have to test every argument from scratch. Some shapes are valid every time, whatever the sentences are about. Learn them once and you can spot them anywhere.

The letters help. P, Q, and R stand for whole statements. In "if P then Q", P is the antecedent, the part after "if", and Q is the consequent, the part after "then". "Not P" denies P. "P or Q" says at least one of them is true. In the categorical forms, the letters A, B, and C name groups of things instead.

Here are the five valid forms, each with an example from Dataset B, the argument set (illustrative). The set's premises are written for teaching; they are not claims this course endorses.

Modus ponens — if P then Q; P; so Q. Argument 1: "If knowledge requires certainty, then I know very little. Knowledge requires certainty. So I know very little." Argument 7 has the same form.
Denying the consequent — if P then Q; not Q; so not P. Argument 2: "If the mind is the brain, then mental states have locations. Mental states have no locations. So the mind is not the brain."
Hypothetical syllogism — if P then Q; if Q then R; so if P then R. Argument 3: "If an act maximizes happiness, it is right. If it is right, we ought to do it. So if an act maximizes happiness, we ought to do it."
Disjunctive syllogism — P or Q; not P; so Q. Argument 4: "Either free will is an illusion or determinism is false. Free will is not an illusion. So determinism is false."
Categorical syllogism — every A is B; every C is A; so every C is B. Argument 9: "Every person deserves respect. Every prisoner is a person. So every prisoner deserves respect."

Worked example — naming the form of argument 2. Match the parts. P is "the mind is the brain". Q is "mental states have locations". The first premise is "if P then Q". The second premise says mental states have no locations, which is "not Q". The conclusion says the mind is not the brain, which is "not P". So the pattern is if P then Q; not Q; so not P. In words: argument 2 denies the consequent, and that form is valid.

Misconception check — a valid form settles the big question. Students often think that because argument 2 is valid, it proves the mind is not the brain. It is tempting because the conclusion follows so neatly. The cure is the second question from Section 2. Validity tells you the conclusion follows if the premises are true. Whether mental states really have no locations is an open question you will meet in Week 6. The form is settled; the premise is not.

4. Invalid forms and counterexamples: show the shape is broken

Some shapes look almost like the valid ones, but they leak. The premises can all be true while the conclusion is false. Three of them come up again and again.

Denying the antecedent — if P then Q; not P; so not Q. Argument 6: "If the universe was designed, it shows order. The universe was not designed. So it shows no order."
Reasoning from the consequent back to the antecedent — if P then Q; Q; so P. Argument 8: "If a machine understands Chinese, it can answer questions in Chinese. This machine answers questions in Chinese. So it understands Chinese." It runs the conditional backward.
The invalid categorical pattern — every A is B; this C is B; so this C is A. Argument 5 has this shape: "All just laws protect the vulnerable. This law protects the vulnerable. So this law is just." The conclusion could be false with true premises, because an unjust law could also protect the vulnerable.

How do you show a form is invalid, rather than only saying so? You build a counterexample. A counterexample to a form is a case with premises of the same form that are true and a conclusion that is false. One such case proves the form can fail.

Worked example — a counterexample to the juror's form. First, find the form. The juror said: if the defendant was in the building, the camera shows a red coat; the camera shows a red coat; so the defendant was in the building. That is if P then Q; Q; so P, the backward form. Second, keep the form and change the content to something everyone can check:

P1: If an animal is a dog, it has four legs.

P2: A cat has four legs.

C: So a cat is a dog.

Both premises are true, and the conclusion is false. So this form can carry true premises to a false conclusion, and the juror's argument, which has the same form, is invalid. In words: someone else in a red coat would also show up on the camera, so the camera alone cannot place the defendant in the building.

Misconception check — invalid does not mean false. Students often think an invalid argument must have a false conclusion. It is tempting because "invalid" sounds like "wrong". The cure is an example. "Every dog is a mammal. This beagle is a mammal. So this beagle is a dog." The conclusion is true, yet the form is the invalid categorical pattern: the same shape gives "this cat is a dog" when you swap in a cat. An invalid argument fails to guarantee its conclusion. The conclusion may still happen to be true.

5. Inductive strength: support that comes in degrees

Some arguments never try to guarantee their conclusion. A commuter says, "The bus on this route has run late on every trip I have taken. So the next one will run late too." That is a reasonable bet. It is not a promise. The next bus could be on time.

An inductive argument is one whose premises are offered to make its conclusion probable rather than to guarantee it. An inductive argument is strong when its premises make its conclusion probable without guaranteeing it. Strength comes in degrees. Validity does not: an argument is valid or it is not.

Worked example — judging the commuter's argument. First, ask the deductive question. Could the premise be true and the conclusion false? Yes: the bus could run late every time before and be on time tomorrow. So the argument is not valid, and it was never meant to be. Second, ask the inductive question. Do many late buses make another late bus likely? Yes, if the trips were many and typical. So the argument is fairly strong. Third, ask what would make it stronger or weaker. More trips at more times of day would strengthen it. Learning that the route was fixed yesterday would weaken it. In words: the commuter's argument is inductive and fairly strong, and its strength depends on how much the past trips tell us about the next one.

Misconception check — deductive or inductive? Students often call a strong inductive argument "valid". It is tempting because both words sound like approval. The cure is to keep the two questions apart. Valid belongs to deductive arguments and allows no exceptions. Strong belongs to inductive arguments and comes in degrees. A strong inductive argument can have true premises and a false conclusion, and it is still strong.

6. Counterexamples to general claims: one case is enough

Week 1 showed you Socrates testing a definition with a case it got wrong. Here is the same move in an ordinary setting. A hiking club's notice board says: "Every member has paid this year's dues." Then a new member joins this morning, before paying. That one person makes the notice false.

A counterexample to a definition or a general claim is a case the claim gets wrong. For a claim about every member of a group, one such case is enough to refute it.

Worked example — testing the club's claim. First, find the form of the claim. "Every member has paid this year's dues" says something about every member, with no exceptions. Second, describe a possible case: a member who joined this morning and has not paid yet. Third, check the case against the claim. The new member is a member, and has not paid, so the claim gets this case wrong. In words: one unpaid member is enough to show that "every member has paid" is false, and the club should reword the notice, perhaps to "every member who joined before this month has paid".

The same move works on definitions. Suppose someone defines a game as "an activity with a winner". Two friends tossing a ball back and forth are playing a game with no winner. If you accept that case, the definition is too narrow and needs repair.

Misconception check — is one case too few? Students often think one counterexample is too few to refute a universal claim, as if you need many. It is tempting because, for an inductive claim about what usually happens, one odd case proves little. The cure is to read the claim. A claim about every case makes a promise with no exceptions, so one genuine exception breaks the promise. Ten more cases cannot repair it.

7. Thought experiments: the barber puzzle as a reductio

Not every test case has to happen. Sometimes you imagine one. A thought experiment is an imagined scenario used to test a claim, a definition, or an intuition. The case does not need to be real. It needs to be clear enough that you can follow where it leads.

This course has a roster of 15 thought experiments, one at the center of each instructional week. This week's is the barber puzzle, a standard logic puzzle usually associated with Bertrand Russell. It asks: can there be a barber who shaves all and only those who do not shave themselves?

You work it as a reductio: you assume a claim, show that the assumption leads to a contradiction, and conclude that the claim is false.

Worked example — the barber puzzle as a reductio. First, make the assumption: there is a barber who shaves all and only those who do not shave themselves. Second, ask one question about him: does he shave himself? There are only two cases.

Case 1: he shaves himself. Then he is someone who shaves himself, so by the rule he does not shave himself. That contradicts the case.
Case 2: he does not shave himself. Then he is someone who does not shave himself, so by the rule he shaves himself. That contradicts the case too.

Third, draw the conclusion. Both cases end in a contradiction, so the assumption is false: there can be no such barber. In words: the description sounds like a person you could meet, but no one could ever fit it.

The puzzle is useful because it shows that a description can sound fine and still be impossible. Philosophers use the same test on definitions. If a definition leads to a contradiction, it cannot be right as stated.

Misconception check — dodging the puzzle. Students often answer, "The barber grows a beard," or "Someone from out of town shaves him." It is tempting because it feels like a clever escape. The cure is to label those answers for what they are: variations. A variation changes the case, and the course always labels variations as variations. The puzzle as stated asks about a barber who is one of the people his rule covers, and for that barber the contradiction stands.

Pause and think. Choose one of two prompts. (a) Describe a rule from your own life, at work, at school, or at home, that sounded sensible but turned out to contradict itself in one case. (b) Invent your own variation of the barber puzzle, label it as a variation, and explain whether it escapes the contradiction or meets it again.

8. Reading charitably: the strongest version first

Imagine a group project for a business course. One teammate says, "We should split the work evenly." Another teammate snaps back, "So you think the person who already did the research should do more work? That's unfair." That reply attacks a weak version of the first teammate's point, a version they never meant.

Reading an argument charitably means stating it in its strongest form, the way its best defenders would, before criticizing it. You choose the most reasonable reading the words allow. If a premise is missing, you add the one that makes the argument work best, and you mark it "(added)".

Worked example — the teammate's argument, read charitably. First, find the conclusion: we should split the work evenly. Second, find what the teammate most likely meant by "evenly": each person does an equal share of the remaining work, with work already done counted. Third, add the premise that makes the reasoning work.

P1: Each of us has an equal stake in the project's grade.

P2 (added): When people have an equal stake in a shared result, they should share the remaining work equally, counting work already done.

C: So we should split the remaining work evenly.

Now the objection has a fair target. A teammate who still disagrees can question P2, perhaps by pointing out that people have different amounts of free time this month. In words: the charitable reading turns a quick remark into a clear argument, and the real disagreement is about P2.

Misconception check — charity is not agreement. Students often think a charitable reading means going easy on an argument, or agreeing with it. It is tempting because "charity" sounds like kindness. The cure is to see what charity is for. You build the strongest version so that your criticism hits the real argument. If your objection beats the strongest version, it has done real work. If it only beats a weak version, it has done none.

9. Keying the argument set: valid or invalid, never sound

Now put the whole week together. Dataset B, the argument set (illustrative), holds ten short arguments. Each one is keyed for validity and given its form. The course keys validity only, never soundness, because the premises are open questions.

This table lists the ten arguments with their keys and forms.

Dataset B, the argument set (illustrative): the 10 arguments keyed for validity, with each one's form
#	Argument	Key	Form
1	If knowledge requires certainty, then I know very little. Knowledge requires certainty. So I know very little.	valid	modus ponens
2	If the mind is the brain, then mental states have locations. Mental states have no locations. So the mind is not the brain.	valid	denying the consequent
3	If an act maximizes happiness, it is right. If it is right, we ought to do it. So if an act maximizes happiness, we ought to do it.	valid	hypothetical syllogism
4	Either free will is an illusion or determinism is false. Free will is not an illusion. So determinism is false.	valid	disjunctive syllogism
5	All just laws protect the vulnerable. This law protects the vulnerable. So this law is just.	invalid	the conclusion could be false with true premises: an unjust law could also protect the vulnerable
6	If the universe was designed, it shows order. The universe was not designed. So it shows no order.	invalid	denying the antecedent
7	If Socrates is wise, he admits his ignorance. Socrates is wise. So he admits his ignorance.	valid	modus ponens
8	If a machine understands Chinese, it can answer questions in Chinese. This machine answers questions in Chinese. So it understands Chinese.	invalid	runs the conditional backward, from the consequent to the antecedent
9	Every person deserves respect. Every prisoner is a person. So every prisoner deserves respect.	valid	categorical syllogism
10	Most people in this town favor the law. So the law is just.	invalid	moves from what people believe to what is right

Worked example — the tally. Count the valid arguments first: arguments 1, 2, 3, 4, 7, and 9 make 6. Count the invalid ones next: arguments 5, 6, 8, and 10 make 4. Check the total: 6 + 4 = 10 arguments. Now count the valid forms. Modus ponens appears twice (arguments 1 and 7), and denying the consequent, hypothetical syllogism, disjunctive syllogism, and the categorical syllogism appear once each, so 2 + 1 + 1 + 1 + 1 = 6 valid arguments. Both counts agree. In words: 6 of the 10 arguments are valid and 4 are invalid, and since validity depends on form alone, none of the 10 is shown to be sound or unsound by its form.

Argument 10 deserves one more look. It has only one premise, about what most people believe, and a conclusion about what is right. Even if the premise is true, the law could still be unjust, so the argument is invalid. Week 10 returns to this move when it asks whether what a community believes settles what is right.

An honesty note: the key never says which conclusions are true. Whether knowledge requires certainty, whether the mind is the brain, and whether free will is an illusion are questions later weeks take up, and the course gives no verdict on them. The key tells you only whether each conclusion follows from its premises.

Misconception check — the key is not a verdict. Students often read "valid" on argument 4 as the course saying determinism is false. It is tempting because the key sits right next to the conclusion. The cure is to remember what a key covers. It covers form. A caveat worth carrying: someone who thinks free will is an illusion can agree that argument 4 is valid and still reject its second premise.

Key terms

Argument — A set of statements, the premises, offered as reasons for another statement, the conclusion. (first used: 1. Standard form: lay out the reasons, then the claim)
Premise — A statement offered in an argument as a reason to accept its conclusion. (first used: 1. Standard form: lay out the reasons, then the claim)
Conclusion — The statement an argument's premises are offered to support; in standard form it comes last, marked C and introduced by So. (first used: 1. Standard form: lay out the reasons, then the claim)
Standard form — A way of writing an argument with its premises numbered P1, P2, and so on, each one declarative sentence, and its conclusion given last, marked C and introduced by So; a premise the reconstruction adds is marked as added. (first used: 1. Standard form: lay out the reasons, then the claim)
Deductive argument — An argument that aims to guarantee its conclusion, so that the conclusion cannot be false if the premises are true. (first used: 2. Validity and soundness: does the shape lock in the claim?)
Valid — Said of a deductive argument whose conclusion cannot be false if its premises are true; validity depends on the argument's form, not on whether its premises are in fact true. (first used: 2. Validity and soundness: does the shape lock in the claim?)
Sound — Said of a deductive argument that is valid and whose premises are all in fact true. (first used: 2. Validity and soundness: does the shape lock in the claim?)
Antecedent — The part of a conditional that follows if; in if P then Q, the antecedent is P. (first used: 3. The valid forms: five patterns you can trust)
Consequent — The part of a conditional that follows then; in if P then Q, the consequent is Q. (first used: 3. The valid forms: five patterns you can trust)
Modus ponens — The valid form: if P then Q; P; so Q. (first used: 3. The valid forms: five patterns you can trust)
Denying the consequent — The valid form: if P then Q; not Q; so not P. (first used: 3. The valid forms: five patterns you can trust)
Hypothetical syllogism — The valid form that chains two conditionals: if P then Q; if Q then R; so if P then R. (first used: 3. The valid forms: five patterns you can trust)
Disjunctive syllogism — The valid form: P or Q; not P; so Q. (first used: 3. The valid forms: five patterns you can trust)
Categorical syllogism — The valid form that reasons about groups: every A is B; every C is A; so every C is B. (first used: 3. The valid forms: five patterns you can trust)
Denying the antecedent — The invalid form: if P then Q; not P; so not Q. (first used: 4. Invalid forms and counterexamples: show the shape is broken)
Counterexample to a form — A case in which premises of the same form are true and the conclusion is false; one such case shows that the form is invalid. (first used: 4. Invalid forms and counterexamples: show the shape is broken)
Inductive argument — An argument whose premises are offered to make its conclusion probable rather than to guarantee it. (first used: 5. Inductive strength: support that comes in degrees)
Strength — The degree to which an inductive argument's premises make its conclusion probable without guaranteeing it; strength comes in degrees, while validity does not. (first used: 5. Inductive strength: support that comes in degrees)
Counterexample to a definition — A case that a definition or general claim gets wrong; one such case is enough to refute a claim about every member of a group. (first used: 6. Counterexamples to general claims: one case is enough)
Thought experiment — An imagined scenario used to test a claim, a definition, or an intuition. (first used: 7. Thought experiments: the barber puzzle as a reductio)
Reductio — Reasoning that assumes a claim, shows that the assumption leads to a contradiction, and concludes that the claim is false. (first used: 7. Thought experiments: the barber puzzle as a reductio)
Reading an argument charitably — Stating an argument in its strongest form, the way its best defenders would, before criticizing it. (first used: 8. Reading charitably: the strongest version first)

Summary

Standard form: numbered premises P1, P2, then the conclusion last, marked C and introduced by So; added premises are marked.
Valid means the conclusion cannot be false if the premises are true; sound means valid with all premises true.
Validity depends on form, not on whether the premises are true.
Valid forms: modus ponens, denying the consequent, hypothetical syllogism, disjunctive syllogism, and the categorical syllogism.
Invalid forms: denying the antecedent, reasoning from the consequent back to the antecedent, and the invalid categorical pattern.
A counterexample to a form has true premises and a false conclusion in the same shape.
An invalid argument can still have a true conclusion.
Inductive arguments are strong or weak, in degrees; deductive arguments are valid or invalid.
One counterexample is enough to refute a claim about every case.
A thought experiment tests a claim with an imagined case; the barber puzzle is a reductio.
Read charitably: strongest version first, then criticize.
Dataset B, the argument set (illustrative): 6 valid, 4 invalid, keyed for validity and never for soundness.

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 worker reasons: "If the oven is on, the kitchen is warm. The kitchen is not warm. So the oven is not on." Put the argument in standard form, name its form, and say whether it is valid.
A gardener reasons: "If a plant is a cactus, it needs little water. This aloe needs little water. So this aloe is a cactus." Build a counterexample to the form.
A library sign says, "Every book in this room is a novel." Describe one case that would refute the sign, and explain why one case is enough.
A lab technician says, "Every sample from this batch has tested clean so far. So the next sample will test clean." Is this argument deductive or inductive? Say what would make it stronger.

Answers (worked reasoning — read after attempting)

P1: If the oven is on, the kitchen is warm. P2: The kitchen is not warm. C: So the oven is not on. P is "the oven is on" and Q is "the kitchen is warm". The pattern is if P then Q; not Q; so not P, which is denying the consequent. That form is valid, so if both premises are true, the oven cannot be on.
First find the form: if P then Q; Q; so P, the backward form. Then keep the form and use content everyone can check: "If an animal is a fish, it lives in water. A dolphin lives in water. So a dolphin is a fish." Both premises are true and the conclusion is false, because a dolphin is a mammal. So the gardener's form is invalid, even though the aloe's conclusion could be checked separately.
One book in the room that is not a novel, such as a cookbook or an atlas, refutes the sign. The sign makes a claim about every book in the room, with no exceptions, so one genuine exception is enough to make it false.
It is inductive: the technician offers past results to make the next result likely, not certain, since the next sample could still fail. It would be stronger if the samples so far were many and drawn from every part of the batch, and weaker if they all came from one corner of it.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, the quiz, the discussion, and the assignment. Next week, the method you learned here, standard form and the test for validity, meets Plato's theory of Forms and his allegory of the cave.

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