Think like a careful detective. Some promises hold every single time. Some only sometimes. And flipping one around can quietly break it.
Follow the clueLogic is the art of testing promises. You take a statement shaped like "if P, then Q" and ask one sharp question: does the promise hold every time, only some times, or never?
That little word "if" is doing a lot of work. It says: whenever the first part is true, the second part must follow. A detective doesn't take that on trust — she hunts for a single case where the promise breaks.
A conditional is a statement built from two pieces joined by "if" and "then" — a promise that says whenever the first part is true, the second part follows. We can call the first part P (the "if" part) and the second part Q (the "then" part):
If it is raining, then the ground gets wet.
The "if" part is the trigger. Until it's true, the promise just sits there, quietly waiting. The statement above isn't claiming it's raining — it's only telling you what must happen on the days it does. A conditional is less a fact and more a rule that kicks in under certain conditions, like "if you finish your homework, then you can go outside."
Here's the key move. To judge a conditional, you only ever check the cases where the "if" part is true — the rainy days. On every rainy day, does the ground really get wet? If yes, every time, the promise holds. If you can find even one rainy day with dry ground (say, you're standing under a roof), the promise wobbles. You don't have to worry about the dry days at all; the promise made no claim about those.
So every "if… then…" lands in one of three buckets:
Always true — the promise holds in every single case. No exceptions, anywhere.
Sometimes true — it holds in some cases but breaks in others. True for a few, false for the rest.
Never true — the promise fails every time. The "then" part can't go with that "if" part at all.
Your whole job in logic is to figure out which bucket a statement belongs in — and to back it up with a reason, not a guess.
Here's a statement that trips a lot of people up:
If a shape is a square, then it is a rectangle.
Surprised? Most people picture a rectangle as "long and thin" and a square as "the same on all sides," so they sound like opposites. But look at the actual rules. A rectangle is any shape with four straight sides and four right-angle corners. A square has four straight sides, four right-angle corners… and all sides equal.
A square already does everything a rectangle has to do — it just does a little extra. So every square fits the rectangle rule perfectly. Test it: pick any square at all, and it will always pass the rectangle check. There is no square anywhere that fails. That makes this promise always true.
That's the part worth slowing down on: to call this promise "always true," you can't just check one square. You have to be sure no square anywhere could ever fail the rectangle test. And because the rule for a square already includes everything the rectangle rule asks for, you can be sure — not by checking a million squares one by one, but by checking the definitions. That's how you earn an "always."
Think of it like this: a square is a special, fancy rectangle — the one that happened to get all its sides the same length.
Here's where careful thinkers earn their magnifying glass. Let's flip the square statement — swap the "if" part and the "then" part. The flipped version has a name: the converse (the same statement turned around, with its two halves traded):
If a shape is a rectangle, then it is a square.
Is this always true? Picture the door of your classroom, or this screen you're reading — both rectangles, neither anywhere close to a square. So we've found a rectangle that is not a square. The flipped promise breaks.
But it doesn't break every time. A square is a rectangle too, so some rectangles really are squares. The converse is true for those and false for the rest — which makes it sometimes true.
Why does flipping change things so much? Because the converse is asking a genuinely different question. "Square → rectangle" asks, "do all squares pass the rectangle test?" — and they do. "Rectangle → square" asks, "do all rectangles pass the square test?" — and most don't, because they're missing the equal-sides part. Same two shapes, opposite direction, completely different answer.
A true "if… then…" does not promise anything about its flip. "Square → rectangle" being always true tells you nothing about whether "rectangle → square" is true. You have to test the converse on its own.
Imagine someone claims: if a number is bigger than 10, then it is bigger than 100. They point at 200 and say, "See? 200 is bigger than 10, and bigger than 100 — told you!" But one lucky example doesn't make an "always" rule true. You can always find one case that fits by chance.
To break an "always true" claim, you only need a single counterexample — one case where the "if" part is true but the "then" part fails. And here it's easy: 50 is bigger than 10, but it is not bigger than 100. One counterexample, and the rule is sunk. So that claim is only sometimes true.
It works the same with animals. Say someone insists, "if an animal lives in the ocean, then it is a fish." A dolphin lives in the ocean and is not a fish — it's a mammal. That single counterexample is all you need; you don't have to survey every creature in the sea. One clear exception does the whole job, which is why counterexamples are a detective's favourite tool: they're quick, and they're final.
This is the detective's rulebook:
To prove a promise is always true, you need a reason that covers every case — a single happy example is never enough.
To disprove it, one counterexample is plenty. Find one crack and the "always" falls apart.
This isn't just shapes. Look at this everyday promise:
If it is raining, then the ground gets wet.
That's a solid promise — rain falling really does wet the ground. Now flip it into its converse: if the ground is wet, then it is raining.
Step outside and you'll spot the trap fast. The ground can be soaking wet because a sprinkler ran, someone tipped out a bucket, or snow melted in the sun. Wet ground is a clue that it might have rained — but it is not proof. The flipped promise is only sometimes true.
You'll find the same shape everywhere once you start looking. "If an animal is a penguin, then it is a bird" is always true — every penguin really is a bird. But flip it: "if an animal is a bird, then it is a penguin"? A robin, an owl and a duck all say no. The forward promise is rock-solid; the flip falls apart the moment you picture a different bird.
Detectives live in this gap. "The window is broken" doesn't prove a burglar — a stray football explains it just as well. A good clue points you somewhere; it rarely closes the case on its own.
Read the "if… then…" card, then tap Always, Sometimes, or Never. You'll get instant feedback and a one-line reason for every card. Watch some true statements and their flipped versions land in different bins.
Tip: when in doubt, try to imagine a counterexample. If you can picture even one case where the promise fails, it isn't "always."
Pick a statement that's true. Hit Flip it to make its converse, then guess: is the flipped version still always true, or only sometimes? Most flips don't survive — but watch the last one.
The square/rectangle pair is the classic: the original is rock-solid, but its flip needs only one long, thin rectangle to fall apart.
This is the single most common slip in logic, and now you can name it. People hear a true "if P then Q" and quietly assume "if Q then P" comes free. It doesn't. The forward promise and its converse are two separate claims, and each has to earn its own verdict.
Sometimes the flip happens to be true as well — "if today is Monday, then tomorrow is Tuesday" flips just fine, because the days of the week run in a fixed order. But that's a bonus you have to check, never something you can assume. Squares and rectangles, dogs and mammals, rain and wet ground — far more often, the flip quietly breaks.
"P leads to Q" does not mean "Q leads to P." Wet ground doesn't prove rain. A mammal isn't always a dog. Test the flip on its own evidence — every time.
Pick the best answer for each. You'll know right away if your reasoning holds up.
Spot the "if" part and the "then" part. The claim is: whenever "if" is true, "then" follows.
Always, sometimes, or never? One counterexample knocks a claim out of "always."
The converse is a brand-new case. Wet ground isn't proof of rain. Test it on its own.