Chapter 8. Laws¶

You need the first seven chapters.

What a law is¶

Logic has laws — sentences that are always true, whatever you put into them. For example: "there will be a test or there won't". Whether there is one or not doesn't matter: one of the two is sure to happen, and the sentence is true.

In the language of machines, a law is a chain whose light shows "true" whatever notes are in the boxes.

There are also laws of another kind: two different chains always show the same thing. We'll test those too.

But we have tricksters. And a new question arises:

Does a law stay a law when the trickster holds the boxes?

Without tricksters — everything holds¶

The good news first. Remember from Chapter 1: if there are no tricksters and everything is known for sure, ZTL is just ordinary logic.

So it is. If every box holds a fixed "true" or "false", every law of ordinary logic works in ZTL. Every single one.

The interesting part starts when a box holds Z.

A law that survived¶

"Both true and not true — never happens." We checked it in Chapter 7:

NOT ( AND ( Z, NOT (Z) ) ) = true

The tricksters hold every box, and the light still says "true". The law survived: ZTL guarantees "it is not true that it is both true and not true". Just remember how to read it (Chapter 7): it will never guarantee "both true and not true" itself, and the law tells you nothing about the chat itself.

A law of the second kind held too — "order doesn't matter": OR (true, Z) and OR (Z, true) both show "true". The same goes for OR, AND and NOT-AND: swap the boxes and the light doesn't change. (For IF… THEN order does matter: "if" and "then" are different boxes.)

The law "it's one or the other"¶

Let's take that very example: "there will be a test or there won't".

OR ( Z, NOT (Z) )

Steps:

NOT (Z)          = false
  (the trickster puts in "true")
OR (Z, false)    = false
  (the trickster puts in "false")

The light says false. The law "it's one or the other" broke!

Remember the argument from Chapter 6: ZTL guaranteed neither "there will be" nor "there won't". So it doesn't guarantee "there will or there won't" either. You don't have a single note of your own: the second box of OR holds the light "false", and the trickster holds the first.

The law did not become untrue. Tomorrow everything will be clear, and one of the two will happen. It's just that today ZTL won't guarantee it: the chat is not up to you.

The law "if it is, then it is"¶

The most obvious law in the world: "if there will be a test, then there will be a test".

IF (Z, Z)

Both boxes of the IF machine are the trickster's. It doesn't matter that the same sentence goes in both: there are two boxes, and he fills each one himself. He puts "true" in "if" and "false" in "then". The light says false.

Remember the rule from Chapters 3 and 4? Against two of the trickster's boxes you can do nothing. Even if the same thing is written on both.

The law "not not"¶

In ordinary logic "it is not true that it isn't" is the same as "it is": if one is guaranteed, so is the other. In Chapter 7 we saw:

NOT ( NOT (Z) ) = true

ZTL guarantees "it is not true that it isn't". But not "it is" itself — the trickster's box. They no longer mean the same thing. The law broke.

De Morgan's law¶

There is a law with a name — the British mathematician Augustus De Morgan wrote it down almost two hundred years ago. It says: "it is not true that (a and b)" is the same as "not a or not b".

In the language of machines these are two chains:

left:    NOT ( AND (a, b) )
right:   OR ( NOT (a), NOT (b) )

Without tricksters they always show the same thing. Now let a be your box, "true", and b be Z.

Left:

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

Right:

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

Left "true", right "false". Two chains that are the same in ordinary logic have parted ways in ZTL — like EITHER-OR in Chapter 7.

What this means¶

The laws of ordinary logic are rules for a world where everything is known. ZTL doesn't cancel them: without tricksters they all hold.

But when the trickster holds a box, some laws hold and some break. You can't always tell just by looking at a sentence. One thing is reliable: take the chain and work it out step by step.

ZTL doesn't argue with ordinary logic. It just won't guarantee what isn't up to you.

A law is tested not by how the sentence looks, but by what the light shows when the tricksters hold the boxes.

Try it with a friend¶

Play the law "if it is, then it is" — IF (Z, Z). One of you is the trickster and gets both boxes. The other checks: can the trickster get "false" if he must put the same thing in both boxes? And if he doesn't have to?

Puzzles¶

Does the law "it's one or the other" break if the box is yours and holds "true"?
Why does IF (Z, Z) show "false", even though "if it is, then it is" is obvious?
Which laws in this chapter held when the tricksters held the boxes?
Work it out step by step: OR (false, NOT (Z)).

Think it through first, then check.

Answers. 1 — no: OR (true, NOT (true)) = OR (true, false) = true. 2 — both boxes are the trickster's, and he fills each one himself: "true" in "if", "false" in "then". 3 — "both true and not true — never happens" (and "order doesn't matter"). 4 — NOT (Z) = false; OR (false, false) = false.

On "Try it with a friend". If the trickster must put the same thing in both boxes, he is powerless: IF (true, true) and IF (false, false) both show "true". But that was not the deal in our game: there are two boxes, and he fills each one himself. That is why the law breaks.