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Does it follow?

P: it rains. If P, then Q. Not Q. So: not P? Decide whether a conclusion follows from premises written in letters, checked by the full truth table.

  • Logic
  • Untimed
  • Endless levels

How it plays

Each letter stands for a plain statement. The premises join them — if–then, and, or, not both, if and only if. Decide whether the conclusion must be true whenever every premise is.

A conclusion follows when no way of making the letters true or false makes every premise true and the conclusion false. To show that it does not follow, find that one way; when it does, every way of making the premises true makes the conclusion true too.

Levels add letters, then more connectives, then more premises without end. The answer is checked against every row of the truth table, and a wrong answer is shown the row that decides it.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises judging propositional validity — what must be true given the premises — on arguments like these. It claims nothing wider.

Learn

How to get better at does it follow?: 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

  1. Ignore what the letters stand for; the meanings are paired at random and carry no logic. Work with P, Q, R and the connectives only.
  2. Find the premises that fix letters outright: a lone letter like “Q.” or “Not S.”, and any “and” premise, which fixes both its letters. Write those values down.
  3. Push the known values through the other premises one at a time, using what each connective forces. Keep going until nothing new is fixed.
  4. Now test the conclusion by trying to make it false. If you can choose the letters still free so that every premise holds and the conclusion fails, it does not follow. If every such try breaks a premise, it follows.

Techniques and tricks

  • What each connective forces

    Know one move for each. If P, then Q: P true makes Q true. Or, or both: one false makes the other true. Not both: one true makes the other false. If and only if: same value. Either–or, but not both: opposite value.

    e.g. Not both P and Q. P. So Q is false; then “Either Q or R, but not both” makes R true

  • The silent if-then

    An if-then says nothing when its if-part is false, and nothing about the if-part when its then-part is true. Only P true, or Q false, gets you anywhere.

    e.g. If P, then Q. Not P. Q could be either: the premise is true both ways

  • Hunt the counter-row

    To refute, make the conclusion false first, then try to fill in the rest so every premise holds. That one row is the whole proof, and it is what the working shows you.

    e.g. If P, then Q. Q. So P? Try P false, Q true: both premises hold and P fails — it does not follow

  • Test an if-then by its if

    When the conclusion is “If X, then Y”, assume X is true, add it to the premises, and see whether Y is forced. If Y can still be false, the conclusion does not follow.

    e.g. If P, then Q. If Q, then R. So: if P, then R? Assume P: then Q, then R — it follows

  • Or and not-both shortcuts

    “X or Y, or both” follows as soon as either letter is forced true. “Not both X and Y” follows as soon as either is forced false. You need not know the other.

    e.g. Not R and not S. So “Not both R and Q” follows, whatever Q is

  • Turn it round

    “If X, then Y” always means the same as “If not Y, then not X”. When a premise’s then-part is known false, read it the other way round.

    e.g. If P, then not S. S. So not P

As the levels climb

  • The ladder starts with two letters and only if, and and not. It then adds a third letter, then “or, or both” and “not both”, then an extra premise beyond the ones that link the letters.
  • If Time pressure is on, the next step tightens the clock to 80%. Then a fourth letter, then “if and only if” and “either–or, but not both”, another extra premise, and the clock to 65%.
  • After that the letters grow to the “Most letters” setting, and then only the premises grow, one more each level, without end. More premises leave fewer rows open, so propagate first: the extra lines usually fix more letters for you.
  • With the clock on, each question gets 15 seconds, plus 6 for each premise and 2 for each letter, scaled by the level’s share; “Which conclusion follows?” gets 30% more.

Traps to avoid

  • Reading “or, or both” as “one or the other only”. It is true when both hold. Only “Either X or Y, but not both” rules out both.
  • Reading “Not both X and Y” as “neither”. It allows one of them to be true; it only rules out the two together.
  • Treating “if” as “if and only if”. “If P, then Q” with Q true tells you nothing about P; only “P if and only if Q” runs both ways.
  • Picking a conclusion that is merely possible. In “Which conclusion follows?” the wrong options are the ones the premises still allow — true in some rows but not all. Ask “must it?”, not “could it?”.
  • Losing a “not” inside a premise. “Either not P or T” is a different premise from “Either P or T”; copy each literal with its sign when you propagate.

How to practise

  • Set “Most letters” to 3 while you learn the connective moves, then raise it to 5 or 6 when propagation is quick.
  • Switch “Question” to “Which conclusion follows?” to practise telling must from could; raise “Options shown (which-follows)” to 6 or 8 to make that harder.
  • For each question, say aloud which letters are fixed and which are still free before you read the conclusion. Then the counter-row hunt only has the free letters to try.
  • When you miss one, take the row the working shows and check each premise against it yourself. Seeing every premise hold while the conclusion fails is what fixes the mistake.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. 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.

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