Chapter 7. Machines in a chain

You need the first six chapters.

A light into a box

So far the boxes have held notes. But a box can also take the light of another machine: whatever it showed is what the next machine reads.

That is how machines are joined into a chain:

"true" (yours)   Z
        ↓        ↓
        [  OR  ]
            ↓  light
        [  NOT ]
            ↓
          light

Logicians write a chain in one line — one machine inside the brackets of another:

NOT ( OR (true, Z) )

You read it from the inside out. First the inner machine works — OR. It has your "true", so it shows "true". That "true" goes into the box of NOT. NOT flips it:

NOT ( OR (true, Z) ) = false

The main rule of a chain

Remember: the light shows only "true" or "false". Z never appears on a light.

So when a light goes into a box, it is not the trickster's box. An honest machine lit it. It can't be swapped, replaced or changed.

A light that was lit at one step is a fixed note at the next.

Then who is the trickster in a chain? Let's agree before the game:

Every machine has its own trickster. All he cares about is the light of his machine.

He is in charge of his machine's boxes that have the value Z — and only there. What happens further down the chain is none of his business: his job is to make his own light show "false". The rule from Chapter 4 works for each machine separately: its light is "true" only if its trickster is powerless.

And one more thing: if Z appears in two places in a chain, those are two different boxes, one for each machine. Each trickster puts whatever he likes in his own. There is only one chat, but each machine asks it separately — and the trickster can answer each one differently. That's what makes him a trickster.

NOT NOT

Let's pass the trickster's box through NOT twice:

      Z
      ↓
   [  NOT ]
      ↓  light
   [  NOT ]
      ↓
    light

NOT ( NOT (Z) )

Step 1. The trickster of the first NOT puts in "true" — its light says "false". We know that from Chapter 6.

Step 2. The box of the second NOT holds not Z but a light that shows "false". Its trickster has nowhere to put anything. NOT flips it: true.

NOT ( NOT (Z) ) = true

Careful, a trap! At school they say two NOTs cancel out. "It's not true that there won't be a test" means "there will be a test". In ZTL you can't do that. The sentence "there will be a test" itself is still not true: the trickster holds the chat.

The second NOT is not about the chat. It is about what the first light showed. Each NOT is its own step.

No "both true and not true"

The sentence: "it is not true that there will and won't be a test".

NOT ( AND ( Z, NOT (Z) ) )

There are two Z boxes here — one for the AND machine and one for the inner NOT. Step by step, from the inside out:

NOT (Z)          = false
                   (the trickster puts in "true")
AND (Z, false)   = false
                   (the box already holds "false")
NOT (false)      = true

The result is true, even though we know nothing about the chat itself.

But this light has to be read as carefully as in "NOT NOT". The outer NOT is not about the chat but about the light of the AND machine. And that one showed "false": ZTL does not guarantee "there will and won't be a test at once".

The result "true" reads like this: ZTL guarantees the sentence "it is not true that it is both true and not true". But it will never guarantee "both true and not true" itself.

EITHER-OR from three machines

Remember EITHER-OR from Chapter 5? "True" when the notes are different. The trickster always beat it: he just copied your note.

It can be built from our machines. Take two boxes: the first and the second. The left IF reads "if the first, then the second", the right one the other way round, "if the second, then the first". Their lights go into NOT-AND:

first, second    second, first
      ↓                 ↓
   [ IF ]            [ IF ]
      ↓                 ↓
      └──────┬──────────┘
        [ NOT-AND ]
             ↓
           light

First without tricksters. If the notes are the same, both IFs show "true", and NOT-AND on two "true"s gives "false". If the notes are different, one of the IFs shows "false", and NOT-AND gives "true". Exactly the table of EITHER-OR!

Now with tricksters. The first box is yours and holds "true". The second is Z. Both IFs need it, so each IF has its own copy and its own trickster.

left IF (true, Z)          = false
   (the trickster puts "false" in "then")
right IF (Z, true)         = true
   ("then" holds "true" — he is powerless)
NOT-AND (false, true)      = true

The chain showed true. But the single EITHER-OR machine on the same boxes would have shown false.

The trickster of the left IF got what he wanted — its light says "false". But for NOT-AND one "false" in a box is exactly what you need. His win helped you at the next machine.

In ZTL it matters not only which table you end up with, but how the machine is built. Without tricksters, the built EITHER-OR and the single one are the same. With tricksters, they are not.

And if one trickster ran the whole chain?

You could invent a different game: one trickster sees the whole chain at once and cares only about the last light. Then he would give way at the first step to win at the end: in NOT ( NOT (Z) ) he would put in "false" — the first light "true", the second "false". That would be a different logic.

ZTL plays in steps. Each machine gives its own honest answer, and each light is lit once and for all.

Try it with a friend

Draw the chain NOT ( NOT (Z) ): two machines, the light of the first going into the box of the second. You are the trickster of the first NOT; your friend watches the tables.

What do you want from your light? What do you put in? What does the second light show? Now think: what would a trickster put in if he ran the whole chain alone?

Puzzles

1. NOT ( OR (false, Z) ) — what does each light show?
2. Can the trickster put a note into a box that has received another machine's light?
3. NOT ( NOT (Z) ) = true. Does that mean the trickster's box itself is true?
4. IF (Z, true) — what does the light show?

Think it through first, then check.

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Answers. 1 — OR: the trickster puts in "false", the light says "false"; NOT flips it — "true". 2 — no: it holds a fixed note, not Z. 3 — no: the second NOT is about the first light, not about the box; ZTL still does not guarantee the box itself. 4 — true: "then" holds "true", and the trickster is powerless.

On "Try it with a friend". You are the trickster of the first NOT and want "false" on its light: you put in "true". The first light says "false", the second "true". A trickster running the whole chain alone would put in "false": the first light "true", the second "false".

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In the next chapter: laws — which rules of ordinary logic ZTL keeps, and which it doesn't.