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Chapter 4. True, false and Z

You need the first three chapters.

A game in one line

A logician can write down a whole round of our game in one short line. By the end of this chapter you'll be able to do it too. For that you need two new words and one letter.

How logicians say it

So far we have written "yes" and "no" on the notes, and the light has shown YES or NO. Logicians say it differently:

yes   →   true
no    →   false

For example, a note says "Pete came". If that is how it is, it is true. If not, it is false. And the light now says not YES and NO but "true" or "false".

From this chapter on, we'll talk this way.

The trickster's box is Z

And what is in the trickster's box? We don't know: he is the one who puts the note in, after you, and he puts in what suits him.

Such a box is marked with the letter Z. Remember this well:

Z is always in the trickster's hands.

Z is not what is written on the note. Z tells you whose box it is: "here the trickster decides". That is how we say it: the value of this box is Z. We called such information dirty: the class chat, the weather forecast, the letter x in a problem. All of it is Z.

Three values in, two out

So in the logic ZTL the inputs — the boxes — can have three values:

true    you know for sure
false   you know for sure
Z       in the trickster's hands

And the output — the light — has only two: true or false. The light never shows a third one. The machine is honest: it always computes by its table. It's just that when a box's value is Z, the trickster picks the note for it — and picks one that makes the light show "false", if he can.

So the rule of ZTL goes like this:

The light shows "true" only if the trickster cannot make it false. Otherwise — "false".

Remember right away: "false" on the light does not mean "wrong". It means "can't be guaranteed". The trickster can make the sentence false — so nobody can guarantee it.

Our three machines in the new language

Let's read one line out loud. The OR machine. Your box holds "true", the other one is Z. Whatever the trickster puts in, OR with one "true" shows "true". So:

OR:  yours — true, the other — Z   →   true

All our discoveries from the first three chapters can be written the same way:

OR. Put in "true" — the light says "true". Put in "false" — the trickster puts in "false", and the light says "false".

The other two machines as a short table: what to put in your box so that the light shows "true".

NOT-AND                                put in "false"
IF… THEN, your box is "if"             put in "false"
IF… THEN, your box is "then"           put in "true"

Put in anything else — and the trickster gets "false".

And one rule for every machine at once: if both boxes have the value Z, the light says "false".

The promised line

And here is how a logician writes down a whole round. First the machine's name, then in brackets what is in the boxes, then after the "=" sign what the light shows:

OR (true, Z)        = true
OR (false, Z)       = false
NOT-AND (false, Z)  = true
OR (Z, Z)           = false

One line — one round. Now you can do it too.

Try it with a friend — in the new words

Play as before, but write "true" and "false" on the slips. On the trickster's box write Z — that means he puts the note in, after you. Take any of the three machines and find what to put in so that the trickster is powerless.

Puzzles

  1. What is the note "yes" called in the logicians' language? And "no"?
  2. What is the value of a box where the trickster decides?
  3. The OR machine: your box holds "false", the other one is Z. What does the light show?
  4. Can the light show Z?
  5. Write in one line: the NOT-AND machine, your box holds "true", the other one is Z.

Think it through first, then check.

Answers. 1 — "true" and "false". 2 — Z. 3 — "false": the trickster puts in "false". 4 — no: the light shows only "true" or "false"; only a box can be Z. 5 — NOT-AND (true, Z) = false.

In the next chapter: machines on which the trickster always wins.