if-then: the promise rule

a story

A parent says:

“If you clean your room, then you can have screen time.”

That’s a promise. Now — when would your parent have broken that promise?

Take a second and actually think about it. There are four ways the day could go:

1. You clean your room ✅ and you get screen time ✅
2. You clean your room ✅ and you get no screen time ❌
3. You don’t clean your room ❌ but you get screen time anyway ✅
4. You don’t clean your room ❌ and you get no screen time ❌

Only one of those is the parent breaking the promise. Which one?

the idea

An if-then promise — “if A, then B” — is broken in exactly one situation: when you did the “if” part but didn’t get the “then” part.

That’s case 2: you cleaned your room, but got no screen time. Promise broken.

Everything else? Promise kept:

• Case 1 — you cleaned, you got screen time. Obviously kept. ✅
• Case 3 — you didn’t clean, but got screen time anyway. The promise only said what happens if you clean. It said nothing about lazy days. You got lucky, but the promise wasn’t broken. ✅
• Case 4 — you didn’t clean, you got no screen time. Again, the promise never kicked in, so there’s nothing to break. ✅

Here’s the part that trips up almost everyone: if you never do the “if” part, the promise can never be broken. It just sits there, harmlessly true.

the same idea as a checklist

Logicians write “kept” as true and “broken” as false, and line up all four cases in a little table. It’s the exact same story you just figured out:

One “broken,” three “kept.” That’s the whole rule.

spot it in real life

If-then promises are everywhere once you look:

• 🎮 “If you beat the boss, then you unlock the next level.” Broken only if you beat the boss and stay locked out.
• 🔐 “If the password is correct, then you’re let in.” Broken only if it’s correct and you’re kept out.
• 📱 “If it’s a school night, then bedtime is 9pm.” Says nothing about weekends — those are the “didn’t clean your room” case.
• 💻 In code, this is the if (…) { … } you’ve maybe seen: the block only runs if the condition is true.

⚠️ the trap: if-then doesn’t work backwards

This one catches grown-ups too. “If it rains, then the ground gets wet” does not mean “if the ground is wet, then it rained.”

Why? Someone could’ve run the sprinkler 💦. Wet ground, no rain — and no promise broken, because the promise was only about what happens when it rains.

Flipping an if-then around gives you a brand-new claim that might be false. “If A then B” is not the same as “if B then A.”

🎮 you try it

For each promise, decide: was it kept or broken?

1. “If you finish your homework, then you can go to the party.” You finished your homework, but your parents still said no party.

Broken. You did the “if” part (finished homework) but didn’t get the “then” part (the party). That’s the one and only way to break it.

2. Same promise. You did NOT finish your homework, but your parents let you go to the party anyway.

Kept! The promise only said what happens if you finish. It never said you’d be banned otherwise. You got lucky — but nobody broke a promise.

3. “If the light is green, then go.” The light was red, and you stopped.

Kept. The “if” part (green light) never happened, so the promise never kicked in. A promise that never activates can’t be broken.

4. Tricky one: your friend says, “If you study, you’ll pass.” You passed. Does that prove you studied?

No! That’s the backwards trap. Maybe the test was easy, or you got lucky. “If you study, you’ll pass” doesn’t promise the only way to pass is studying. Passing doesn’t prove studying — just like wet ground doesn’t prove rain.

the grown-up name

Logicians call this if-then connector implication, and they write “if P then Q” as $P \rightarrow Q$. The one broken case (did the “if,” didn’t get the “then”) is the only false row in its truth table. Everything you just learned is exactly right — they just use symbols.

⬆️ level up

Ready for the rigorous version with formal truth tables, “vacuous truth,” and notation?

• implication (the full page)
• modus ponens — the thinking move that uses if-then to reach new conclusions
• back to logic, explained simply