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Chapter 1. The OR machine

You need nothing for this chapter.

The honest machine and the trickster

Picture a machine. It has two boxes and a light. Into each box goes a note — "yes" or "no". The machine reads both notes and lights up its answer: YES or NO.

The machine is honest. It always works by its own rules and deceives nobody.

But next to it stands the trickster. One box is yours, the other is his. You want the machine to show YES. The trickster wants it to show NO, and for that he is ready to play a trick: he puts his note in after you. He sees what you put in and puts in whatever suits him.

You want YES. The trickster wants NO. Who wins?

Can you make sure the trickster gets nowhere?

Sometimes — yes. It all depends on which machine it is.

First — the machine's rules

The machine's rules are known in advance — to you, to the trickster, to everyone. They are its table.

Our first machine is called OR. It shows YES if at least one box holds a "yes":

first note   second note   light
yes          yes           YES
yes          no            YES
no           yes           YES
no           no            NO

NO appears in only one case — when both boxes hold "no".

Let's try

First time. You put in "no". The trickster looks — and puts in a "no" too. The machine simply checks its table: no and no — NO. The trickster got what he wanted.

Second time. You put in "yes". The trickster looks… and thinks. If he puts in "yes", the table says YES. If he puts in "no", the table still says YES. Whatever he does, the honest machine shows YES. The trickster is powerless.

That is the whole secret of the OR machine:

Put in a "yes" — and the trickster can do nothing.

It doesn't matter which box is yours — the left or the right. Your "yes" in either one decides everything.

And if both boxes are the trickster's?

Then he puts "no" in both — and the machine shows NO. Against two of his boxes you can do nothing. You can only stop the trickster if at least one box is yours and you know what to put in it.

Try it with a friend

You don't need a machine — slips of paper will do. One of you is the trickster.

  1. You write "yes" or "no" on a slip and put it down face up.
  2. The trickster looks and writes his slip.
  3. Look at the table: what would the honest machine show?

Try both moves — "yes" and "no" — and see for yourselves.

Where tricksters live

The trickster is everything you don't know and don't decide yourself. Life is full of such boxes.

Your class chat: what will be posted there in an hour is not up to you. A weather app: what it will show is not up to you. Even the letter x in a formula that must hold for every x — for example, x · 0 = 0: anything at all may turn up in place of x, and the formula must work for every one of them.

We will call such information dirty: like water from a puddle — who knows what's in it. It is in someone else's hands, and whoever holds it can slip in what doesn't suit you.

But some things you know for sure — you saw them yourself, counted them yourself. That is your box.

An example. The OR machine: "the test is tomorrow OR today is Monday". Anything may be posted in the chat — that is the trickster's note. But you have seen the calendar yourself — that is your note.

the trickster's box (the chat):  "the test is tomorrow" — what will be posted, nobody knows
your box (the calendar):         "today is Monday" — yes
light:                           YES

Whatever is posted in the chat, the machine shows YES: your box holds a "yes".

A logic you can't trick

There is a logic that always expects the worst: that a trickster sits in every box that isn't yours. It is called ZTL (from zero-trust logic). It answers YES only if the trickster is powerless — whatever he slips in. And if there are no tricksters and everything is known for sure, it is just ordinary logic.

Puzzles

  1. The OR machine. Your box is the left one. What should you put in so that the trickster gets nowhere?
  2. You put in "no". What does the trickster put in, and what does the machine show?
  3. Both boxes are the trickster's. What does the machine show?

Think it through first, then check.

Answers. 1 — "yes": then YES whatever his note. 2 — he puts in "no", and the machine shows NO. 3 — NO: he puts "no" in both.

In the next chapter: a machine on which you stop the trickster the other way round — by putting in a "no".