If–then
If P then Q. Q. So P? Tell the valid if–then arguments from the fallacies that look just like them, from one rule to long chains.
How it plays
You see a rule — “If Asha is at home, then Bilal drinks tea” — a fact, and a conclusion. Decide whether the conclusion must be true whenever the lines above it are.
The sentences are paired at random, so nothing sounds likely or unlikely; only the form decides. From the rule and its first half, the second half follows. From the rule and the second half, nothing follows — Bilal may drink tea for his own reasons.
Levels add reasoning backwards (not Q, so not P), the contrapositive against the converse and the inverse, negated clauses, and then chains that grow a line at a time, sometimes shuffled, sometimes with one link broken. The working shows the path of rules, or a case where every line holds and the conclusion fails.
Keys
Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.
What it trains
This practises telling valid conditional arguments from invalid ones of the same shape. It trains this judgement on these sentences; it is not evidence of better reasoning in general.
Learn
How to get better at if–then: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.
Your approach to each question
- Turn each rule into an arrow in your head: “If Asha is at home, then Bilal drinks tea” is home → tea. Ignore whether it sounds likely; the sentences are paired at random on purpose.
- Look at the fact and ask which end of an arrow it matches. A fact that matches the start of an arrow lets you move along it. A fact that matches the end of an arrow tells you nothing on its own.
- Remember every arrow also runs backwards with both ends negated: home → tea also means no tea → not home. That is the only legal way to go backwards.
- Before you say “it follows”, try to break it: picture a case where every line is true and the conclusion is false. If you can build one, it does not follow.
Techniques and tricks
The four forms
With the rule “if P, then Q”, two moves are valid and two are not. P, so Q — valid. Not Q, so not P — valid. Q, so P — invalid. Not P, so not Q — invalid.
e.g. Bilal drinks tea, so Asha is at home? No: Bilal may drink tea for his own reasons
Flip and negate
The contrapositive swaps the two halves and negates both. It says exactly what the rule says. Use it to follow a rule backwards or to rewrite a line so it joins a chain.
e.g. “If Asha is not at home, then Bilal drinks tea” becomes “If Bilal does not drink tea, then Asha is at home”
Converse and inverse are twins
The converse swaps the halves; the inverse negates both. Each is the other’s contrapositive, so they stand or fall together — and neither follows from the rule.
e.g. Rule: if P, then Q. Converse: if Q, then P. Inverse: if not P, then not Q. Both fail when Q is true and P is false
Build the counter-case
To show a conclusion does not follow, make every line true and the conclusion false. If the rule’s if-part is false, the rule is true whatever happens after it — that is where counter-cases live.
e.g. If Kiran wears a hat, then Sana wins. Sana wins. So Kiran wears a hat? Counter-case: no hat, Sana wins anyway
Link the chain
In a chain, start from the fact and look for a line whose if-part is exactly it — or, flipped, a line whose then-part is its opposite. Step along, and repeat until you reach the conclusion or get stuck.
e.g. If Asha is at home, Bilal is cooking. If Chen is not awake, Bilal is not cooking. Asha is at home. Flip the second: if Bilal is cooking, Chen is awake. So Chen is awake
Cancel double negatives
With negated clauses, the opposite of “is not late” is “is late”. Negating a negated clause removes the “not”; never stack two.
e.g. Rule: if Ravi is not late, then Uday is not awake. Uday is awake. So Ravi is late — that is not Q, so not P
As the levels climb
- The ladder opens with forwards only — P so Q, against Q so P. Next come the backward forms, not Q so not P against not P so not Q, and the clock closes to 80%.
- Then the swapped rules arrive — the contrapositive against the converse and the inverse — and the clock goes to 65%. After that, clauses can be negated, so a rule may read “if not P, then Q”; the clock then reaches 55%.
- From there, chains join the mix, five lines long at first, each level adding a line, without end. Some lines in a chain are written as their contrapositive, and half the time the lines are shuffled.
- Each question’s clock is 7 seconds plus 4 for each rule plus 4 more — 15 seconds for a single rule — scaled by that level’s share, so a long chain gets more time. Use it to find the fact’s line first rather than reading top to bottom.
Traps to avoid
- Affirming the consequent: the fact matches the then-part, and you run the arrow backwards. The rule only speaks when its if-part is true.
- Denying the antecedent: the if-part is false, and you conclude the then-part is false. Nothing is said about that case — the then-part can still be true.
- Treating the converse or the inverse as the same as the rule. Only the contrapositive is; with “What follows?” on swapped rules, the right option is the one that both swaps and negates.
- Missing the break in a chain. One link may run the wrong way, start from the opposite of what the last one ended with, or the fact may sit at the wrong end. Check each step joins exactly, word for word, including every “not”.
How to practise
- If one form keeps catching you, untick the others in “Kinds of question” — say, leave only “Not Q, so not P? — and not P, so not Q?” — until that form is automatic, then tick them all again.
- Switch “Question” to “What follows?” once “Does it follow?” is easy. Choosing between X, not X and “Neither can be concluded” makes you derive the answer instead of checking one.
- Untick “Chains of rules” to stay on single rules while you learn the four forms; without chains the ladder is eight levels long. Tick it again to train linking.
- When you get one wrong, read the counter-case in the working and say it out loud. Seeing the case where every line holds and the conclusion fails is what fixes the fallacy.
Levels
Each question has a clock that comes from the work it asks for. There are 325 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.
All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.
More logic
All games →Which cards to turn
A rule, some cards, and only one side of each showing. Pick exactly the cards that could break the rule — the test most people fail.Overlapping sets
Seventy like tea, fifty like coffee, thirty like both — how many like neither? Count overlapping groups without counting anyone twice.Logic grid
Three to five people, each with a different pet, drink or home town. From a handful of clues, work out who has what.Seating arrangement
Who sits where? From clues like “Asha sits at one end” and “Bilal is immediately left of Chen”, place everyone in a row or round a table.Blood relations
Ravi is Meena’s mother’s brother — what is Ravi to Meena? Follow chains of family words, from two steps to as many as you can hold.Truth tables
P AND NOT (Q OR R), with P = 1, Q = 0, R = 1 — is it 1 or 0? Evaluate logic expressions and gate circuits, find the inputs that light them, count the 1s.