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Chapter 6. The NOT machine

You need the first five chapters.

A machine with one box

All our machines so far had two boxes. This one has one.

It is called NOT. Its job is to flip: whatever goes in, the light shows the opposite.

box     light
true    false
false   true

The table is tiny, and as always it was agreed on before the game.

If the box is yours

Then there is nobody to play against. You put in "true" — the light says "false". You put in "false" — the light says "true". The trickster has nowhere to put a note: there is one box, and it is yours.

NOT (true)  = false
NOT (false) = true

If the box is the trickster's

Now the only box is his. The box's value is Z. You have nothing to put in.

The trickster looks at the table. He needs "false" on the light. So the box needs "true". That is what he puts in.

NOT (Z) = false

In short: if the box is the trickster's, NOT always shows "false".

"It will" and "it won't" — neither is true

Here is what that means. The class chat is the trickster's box: what will be posted about tomorrow's test is not up to you. Imagine a debate with a friend.

Your friend says: "There will be a test tomorrow." Can ZTL call that true? Remember the rule from Chapter 4: "true" only if the trickster cannot make the sentence false. But here he decides everything. So — not "true".

You answer: "There will NOT be a test tomorrow." That is the NOT machine over the same box: NOT (Z) = false. Not "true" either.

The argument is a draw: ZTL guarantees neither your friend's sentence nor yours.

Careful, a trap! You may want to reason like this: "The sentence 'there will be no test' is false — so there will be a test."

No! As in Chapter 5: "false" here doesn't mean "wrong", it means "the trickster can make this sentence false".

And he can make your friend's sentence false too. So neither of the two sentences is true.

You really can't guarantee anything — and ZTL agrees with you.

NOT — not a secret, but a flip

OR, NOT-AND and IF… THEN had a secret: your note beat the trickster. NOT has no second box, so it has no secret either. If the box is yours, you know everything; if it is his, he decides everything.

Try it with a friend

One of you is the trickster and holds the only box. The other one tells him: "Put in 'false'!" Can the trickster obey and still get "false" on the light? Try every option — there are only two. (Remember: the trickster wins if the light shows "false".)

Puzzles

  1. NOT (false) — what does the light show?
  2. The box is the trickster's. What does he put in, and what does NOT show?
  3. The sentence "it will rain tomorrow", and the forecast is not up to you. Can ZTL call it true? And the sentence "it will NOT rain tomorrow"?

Think it through first, then check.

Answers. 1 — true. 2 — he puts in "true", and the light says "false": NOT (Z) = false. 3 — no and no: the first is the trickster's box, the second is NOT (Z) = false.

On "Try it with a friend". No. If he puts in "false", NOT shows "true", and he loses. So he always puts in "true".

And what if you put into NOT's box not a note but the light of another machine? That is the next chapter: machines in a chain.