The One Backwards Step That Is Always Safe

You were handed a warning and it was a good one. If A then B does not give you if B then A. A dragon with four legs need not be blue. A wet patch of ground need not mean rain, because a burst pipe would do it too.

So reading a rule backwards is a trap. That is where it was left, and it leaves a question sitting there: is every backwards reading a trap, or only that one?

Only that one. There is exactly one way to read a rule backwards, and it is not a guess or a good bet. It is as certain as the rule you started with.

Start with a dragon that has the wrong number of legs

Here is the rule again.

If a dragon is blue, then it has four legs.

Now you meet a dragon with two legs. Is it blue?

Try assuming it is. If it were blue, the rule would take over and hand it four legs — that is what the rule promises, and rules do not make exceptions. But it has two. Both cannot be true.

So the assumption was wrong. The dragon is not blue. Not probably. Not usually. It cannot be, because being blue would force a number of legs it does not have.

You have just read the rule backwards and got a certain answer out of it, which is the thing you were told could not be done. The difference is what you carried backwards.

Four sentences, and only two of them are safe

One rule can be twisted into four different sentences. Two are safe and two are not, and the pattern is worth learning once.

Look at the two safe ones together. They are not two rules. They are the same rule facing two directions, and if you believe one you have already believed the other.

Look at the two unsafe ones and you can see what went wrong in both. Each of them takes the then half and treats it as though it were the if half — as though four legs were a thing that only blue dragons were allowed to have. The rule never said that. It said what blue dragons must have, and stayed silent about everybody else.

The safe backwards step keeps the halves in their places. It only turns each one into its opposite: no four legs, so no blue.

Sort these

Two rules are in play. If a dragon is blue, it has four legs. And if it is raining, the ground is wet.

Each line below claims something. Some follow with certainty and some do not.

The two dry-and-wet lines are the ones people get wrong most often, and they are worth staring at. Wet ground could be a burst pipe, so wet proves nothing. Dry ground proves a great deal: rain would have wet it, and it is not wet, so there was no rain. The weaker-sounding fact is the one carrying all the weight.

Why this is the shape of nearly every clue

Go and look at a logic grid puzzle and you will notice something odd about the clues. Hardly any of them tell you what is true. They queue up to tell you what is not: Alice's dragon is not red, Bob's is not green, David's is neither blue nor yellow.

That looks like a mean way to write a puzzle. It is not. Every one of those crossings-out is a backwards step being made safely, and they stack up until only one arrangement survives.

You can see the same move in a story. A detective who says "the dog did not bark, so the visitor was somebody the dog knew" has not guessed. There was a rule underneath — a strange visitor makes the dog bark — and no bark means no stranger. The clue is a thing that did not happen, and it is the strongest clue in the case.

Try to break it

Take one of your own rules — the ones you wrote for your own creatures work well — and hand somebody the backwards version.

"If a creature has wings, then it can fly. This one cannot fly."

Watch them hesitate, then get there: no wings. Then try the trap on them the other way round, with something that flies, and see whether they claim it must have wings.

The gap between those two answers is the whole idea, and once you can feel it you will start catching it everywhere — in arguments, in instructions, and in your own reasoning about half an hour after you finish reading this.