So far every question has been about the stamp a statement gets in the world we live in. This section adds a second question: could the stamp have been different? Keeping those two questions apart is one of the most useful habits in the course, and the second question turns out to be a tool for testing claims.
Two questions about any statement
Up to now the fact-checker has asked one question: which stamp, TRUE or FALSE, does this sentence get in the real world? Now add a second question, asked with the stamps set aside: could this sentence have been true?
Take "The bus was on time today," said on a day the bus was 20 minutes late. First question: the stamp is FALSE. Second question: could it have been true? Sure. Picture the same morning with lighter traffic. Nothing about an on-time bus is self-contradictory. So the sentence is false, but possible.
A way the world could be, told all the way through without contradicting itself, is called a possible situation. The real world is one of them. A world where the bus came on time is another. A world where your dog speaks French is another. It's odd, but you can describe it without contradicting yourself.
Some sentences fail the second question too. "Mia is taller than Jo, and Jo is taller than Mia." Try to picture it. You can't, because the sentence takes back what it says. No possible situation makes it true. That kind of sentence is impossible.
So every statement lands in one of three places: true, false but possible, or impossible. A true sentence is always possible, because the real world is one place where it's true. Two mix-ups to avoid: "it could be true, so it is true," and "it's false, so it couldn't have been true." Both treat the two questions as one.
True and false are the two truth values a statement can have, and which one it has is fixed by how things actually are. A separate question is whether things could have been such that the statement is true.
A possible situation is any complete way the world could be that does not contradict itself. It may match the actual world, or differ from it in any number of details. The actual world is one possible situation among many.
Relative to possible situations, a statement is:
- possible if it is true in at least one possible situation;
- impossible if it is true in none, which happens when the statement contradicts itself: "Ana is taller than Ben, and Ben is taller than Ana."
Every true statement is possible, since the actual world is one possible situation where it is true. A false statement may still be possible: "The bus was on time today," said on a day it was late, is false, yet nothing about it contradicts itself. So there are three categories, not two: true, false but possible, and impossible. There is no fourth category of "true but impossible."
"Possible" here is broader than "likely" and broader than "allowed by today's technology." A dog that speaks fluent French is possible in this sense: the description is strange, but nothing in it takes itself back. Everyday speech also uses "possibly" to mean "for all we know," as in a headline's "It is possible the battery lasts two days." This course's "possible" asks only whether a description contradicts itself. Two mistakes follow from blurring the questions. One treats a statement as true because it could be true. The other treats a statement as impossible because it is false.
True, false but possible, or impossible?
Sort each statement by what it is in the actual world and whether it could have been true.
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Worked example
False, but it could have been true
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1basic
On a day the school bus arrived 20 minutes late, someone says, "The bus was on time today." Answer the two questions separately.
basic
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First question: is it true? The bus was late, so the stamp is FALSE.
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Second question: could it have been true? Picture the same morning with no traffic jam. The bus arrives at 7:40 as planned. Nothing in that picture contradicts itself.
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So the statement is false but possible. Its being false tells you about this morning. It says nothing about whether an on-time bus was possible.
Worked example
“It could be, so it is”
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1common trap
Noor says, "The store might have my size in these shoes. So it has my size." What went wrong?
common trap
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Split the two questions. "The store has my size" is possible: nothing contradicts itself in a store stocking a size 8.
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Being possible is not the same as being true. Plenty of possible things never happen. To learn the stamp, Noor has to check the shelf or call.
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4
The move from "might" to "does" answers the first question with the second question's answer. That leap is the mistake to avoid: "could be" never settles "is."
Worked example
A sentence that takes itself back
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1impossible statement
In a race with five runners, a sports report says, "Priya finished first, and Priya finished last." Is that possible?
impossible statement
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Try to describe a situation where both parts hold. For Priya to finish first, the other four runners must finish behind her.
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For Priya to finish last, the other four must finish ahead of her. Both can't be true of the same race with five runners.
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No situation makes the whole sentence true, so it is impossible. You don't need to know anything about the actual race to see this. The words alone rule it out.
Check your understanding (counts toward your progress)
Which statement is impossible?
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That could easily be so. Nothing about it contradicts itself.
Four inches in a year is a lot, but a growth spurt like that contradicts nothing.
Two people can share a height. Nothing here takes itself back.
Why
“Ana is taller than Ben, and Ben is taller than Ana” contradicts itself, so no situation makes it true. The others are all possible, whether or not they are actually true.
Which sentence takes back what it says?
A headline says, “It is possible that the new phone’s battery lasts two days.” What does that headline promise?
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That swaps “could be” for “is.” The headline reports no test, and nothing in it settles the actual battery life.
“Possible” is weaker than “probable.” A long shot is still possible.
“Possible” doesn’t hint the opposite either. The battery might well last two days.
Why
“Possible” answers only “could it be true?” It says this phone’s battery could last two days, and nothing about how long it does last. You need a test or a review to learn the stamp.
Which of the two questions does “possible” answer?
What two separate questions should you ask about a statement?
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Belief plays no part in either question. What someone believes doesn’t change the stamp or the possibility.
Likely and popular are different questions again. A statement can be unlikely and still true.
Those decide whether a sentence is a statement at all. This section is about statements you already have.
Why
The first question asks for the stamp in the real world. The second asks whether any possible situation makes the statement true. Answer them one at a time.
One question is about the real world. The other is about ways the world could have been.
Noor says, “The store might have my size, so it has my size.” What is wrong with this?
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Lots of possible things never happen. Checking the shelf is the only way to get the stamp.
Stocking a size contradicts nothing, so it is possible. The problem is the jump, not the claim.
It says something is the case, so it has a truth value even before she checks. The mistake is in how Noor decides that value.
Why
“Might” answers the second question: it could be true. Noor used that answer for the first question. “Could be” never settles “is.”
Which of the two questions does “might” answer?
Practice: Is it true? Could it be true? (5 problems · doesn’t affect your progress)
It is sunny all day. “It rained this afternoon” is…
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An afternoon shower contradicts nothing. It just didn’t happen today.
It was sunny all day, so the stamp is FALSE.
Why
The sentence is false today, but a rainy afternoon is perfectly possible.
Answer the two questions one at a time.
false but possible.
One of these reports about a shop can’t be true in any situation. Which one?
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An odd day, but nothing contradicts itself. Five minutes of business is possible.
Slow days happen. Selling nothing is unusual, not impossible.
Old shops exist. Two hundred years contradicts nothing.
Why
Opening at 9 and not opening at all can’t both hold on the same day, so that sentence is impossible.
Find the sentence that takes back what it says.
The shop opened at 9 a.m. and did not open at all that day.
Sort each statement.
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Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
Ice is frozen water. Not every bird is blue and people bake bread every day, so those are false but possible. A circle with corners contradicts what “circle” means.
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Why
A circle has no corners by what the word means.
Theo says, “It’s possible our team wins the league, so we’ll win the league.” Which question did Theo answer, and which did he skip?
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“It’s possible” is the answer to the second question, not the first.
Winning is possible, but that doesn’t settle whether the team will win.
Why
Theo showed only that a league win is possible. Whether it will actually happen is a separate question the season will settle.
What does “it’s possible” tell you?
He answered “could it be true?” and skipped “is it true?”
Which of these can never happen?
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This happens all the time, like “the bus was on time” on a day it was late.
Every true statement is also possible, so this is the most common case of all.
Every impossible statement is false, like “Ana is taller than Ben and Ben is taller than Ana.”
Why
Anything true holds in the actual world, which is a possible situation. So nothing can be true and impossible at once.
Is the real world one of the possible situations?
A statement that is true but impossible
Imagining a scenario to test a claim
Imagining a possible situation, a way the world could be, is not only for daydreaming. It's a test you can run on a claim.
Say a friend claims, "Every café on Main Street sells tea." How could that claim fail? You need just one café on Main Street with no tea. That one café would be enough. So picture it: a tiny espresso bar called Grind that sells only coffee. Now the claim is FALSE in that picture. You've shown the claim could be false. Whether it is false means walking down Main Street and checking.
Notice the lopsided math. One café without tea can sink the claim. But a hundred cafés with tea can't prove it, unless those are all the cafés there are. An unchecked café might be the one without.
The same trick shows when one claim doesn't lock in another. Take these two:
- Every student who studied passed.
- Every student who passed studied.
They seem alike. Does the first guarantee the second? Build a small class. Ana studied and passed. Ben didn't study and failed. Dev didn't study, but passed anyway. Check claim 1: everyone who studied, just Ana, passed. TRUE. Check claim 2: Dev passed without studying. FALSE. One imagined class, and the two claims come apart.
Make your scenarios specific. Give people names. Decide every fact the claims mention. "Maybe somebody got lucky" is a hunch. "Dev didn't study and passed" is a test anyone can check.
The second question becomes a tool once you point it at a claim. To show that a claim could be false, describe one possible situation in which it is false. To show that a claim could be true, describe one in which it is true. The situation need not be actual. It needs only to be consistent and detailed enough to settle the claim.
This is the most efficient way to test a general claim. "Every café on Main Street sells tea" is false in any situation containing one café on Main Street without tea. A single such café is enough. No number of cafés with tea can show that the claim couldn't be false.
The same tool shows when one claim does not guarantee another. Consider:
- Every student who studied passed.
- Every student who passed studied.
To show that (1) can be true while (2) is false, describe a class in which Ana studied and passed, Ben neither studied nor passed, and Dev did not study but passed. Every student who studied passed, so (1) holds. Dev passed without studying, so (2) fails. The scenario proves that (1) alone does not guarantee (2).
A good scenario is specific: named people, definite facts, and every detail the claims mention settled one way or the other. A vague scenario ("maybe someone was lucky") leaves the reader to check whether it really works.
Which scenario breaks the claim?
The claim is “Every student who studied passed the exam.” Sort each scenario: would it make the claim false, or not?
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Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
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Why
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Worked example
Sinking a general claim with one case
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1typical
A travel blog says, "Every hotel in this town has free parking." Describe a scenario that shows the claim could be false.
typical
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Ask what would make it false: one hotel in town without free parking.
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Build it with details: the Harbor Inn on Front Street charges $15 a night to park. Every other hotel parks free.
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In this scenario the claim is false, because of the Harbor Inn alone. We have shown the claim could be false. Whether it is false in the real world still needs a look at the real hotels.
Worked example
Two claims that look alike
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1recognition
Does "Every student who studied passed" guarantee "Every student who passed studied"? Use a scenario to decide.
recognition
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2
To show the first doesn't guarantee the second, we need a scenario where the first is true and the second is false.
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The second fails when someone passed without studying. Add Dev: did not study, passed. Add Ana: studied, passed. Everyone who studied, Ana, passed.
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Claim 1 is true in this class. Claim 2 is false, because of Dev. So claim 1 does not guarantee claim 2. One consistent scenario settles it.
Check your understanding (counts toward your progress)
The claim is “Only employees may enter the storeroom.” Sort each scenario.
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Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
The claim fails whenever someone who isn’t an employee enters, like the driver or the customer. An employee who is kept out doesn’t break it, since “only” never promised every employee gets in.
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Why
“Only employees” means everyone who enters is an employee.
Does a scenario that tests a claim have to describe what really happened?
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An imagined scenario shows what could be. That is exactly what you need to show a claim could be false.
It works the other way around: it may be imagined, but it must settle every detail the claim mentions.
Why
To show a claim could be false, any consistent scenario that makes it false will do. Whether it actually is false is a separate question about the real world.
Is the question “is it false?” or “could it be false?”
The claim is “Every student who studied passed the exam.” Sort each scenario.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
Only Tom, who studied and failed, breaks the claim. Ravi is tempting, but the claim says nothing about students who didn’t study.
Correct answer
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Why
The claim only talks about students who studied.
A friend says, “Maybe somebody got lucky” to show that “Every person who bought a ticket won a prize” could be false. What is weak about this?
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A hunch leaves the reader to figure out whether it really works. Getting lucky doesn’t even point in the right direction here.
A scenario needs only to be possible, not actual. That is what makes it a test you can run anywhere.
Nothing about the claim contradicts a ticket buyer going home empty-handed. It can easily be false.
Why
The claim fails if some ticket buyer won nothing. A good scenario says so specifically: “Ines bought a ticket and won no prize.” Vague hunches can’t be checked.
What exactly would have to happen for the claim to fail?
Practice: Build a scenario (5 problems · doesn’t affect your progress)
To show “Every restaurant on Pine Street is open on Mondays” could be false, what do you need?
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That works, but far less is needed. One closed restaurant is enough.
Oak Street isn’t covered by the claim.
An open restaurant fits the claim. It can’t break it.
Why
One Pine Street restaurant closed on Mondays breaks the claim. That single case is all the scenario needs.
How many exceptions does an “every” claim survive?
One restaurant on Pine Street that is closed on Mondays
The claim is “Every dog at the shelter has had its shots.” Sort each scenario.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
Only Bella, a shelter dog without shots, breaks the claim. The cat and the pet-store dog fall outside what the claim talks about.
Correct answer
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Why
The claim covers only dogs at the shelter.
Which scenario shows that “Everyone who lives in Riverton works in Riverton” doesn’t guarantee “Everyone who works in Riverton lives in Riverton”?
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Here both claims are true. That shows they can agree, not that one forces the other.
The resident who works in Lakeside makes the first claim false, so this scenario can’t test what the first claim guarantees.
Why
With every resident working in town and Dana commuting in, the first claim holds and the second fails. So the first doesn’t guarantee the second.
Make the first claim true and the second false in the same scenario.
Every Riverton resident works there, and Dana commutes in from Lakeside.
Describe a specific scenario in which “Every phone in the store comes with a charger” is false. Name at least one phone and say what is missing.
AI grading isn't configured on this deployment.
Explanation
Put the steps in order for showing that “Every intern got a laptop” doesn’t guarantee “Everyone who got a laptop is an intern.”
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
First fix the goal: first claim true, second false. Then make every intern get a laptop, add a non-intern with a laptop, and check that the two claims really come apart.
Correct answer
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Why
Decide what the scenario has to do before building it.
Where this shows up later in the course
Where this goes next
This course
Chapter 3 turns the scenario test into the course's main tool. Its sections 3.2 and 3.6 ask one question of a set of reasons and the claim they back. Can you describe a possible situation where every reason is true and the claim is false? If you can, the reasons don't guarantee the claim, just as Dev's scenario showed above. Section 3.3 is built on this section's split between "is it true?" and "could it be true?"
Chapter 12 makes the idea mechanical. Each row of the tables you'll build there is one possible situation: one way of stamping every basic statement TRUE or FALSE. Checking every row is checking every scenario at once. Everyday reading uses the same move. When an ad says "every customer saves," one customer who didn't is all it takes.