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Chapter 9. Reasoning

You need the first eight chapters.

Can we reason at all?

In the last chapter even the law "if it is, then it is" broke. That's worrying: maybe you can't reason with tricksters around at all? Maybe nothing can be deduced?

It can. This chapter has two rules the trickster cannot break.

What "deducing" means

Reasoning is when you get something new from what you already know.

In the language of machines: something is already definitely "true" — your note or the light of another machine. You look at it and find out what else has to be "true".

We'll use the letters a, b, c for sentences. A sentence can be your note, the light of another machine or the trickster's box. Writing "a = true" means: a is your note "true", or a light that shows "true". You can't say that about the trickster's box: its value is Z.

The bridge rule

Say you know two things:

a              = true
IF (a, b)      = true

What can you say about b?

Remember the table of IF… THEN from Chapter 3 (back then we wrote "yes" and "no", now "true" and "false"). Take only the rows where "if" holds "true". There are two. The light says "true" in only one of them: when "then" holds "true" too.

But what if b is the trickster's box? Then the trickster of this IF would put "false" in "then", and the light would say "false". But it says "true". So b is not the trickster's box but a fixed "true".

The bridge rule: if a = true and IF (a, b) = true, then b = true as well.

IF (a, b) is like a bridge from a to b. You stand on a, the bridge holds — you cross over to b:

a  ══bridge══►  b

A bridge isn't built from nothing

Notice what the bridge rule does not do. It doesn't make true what the trickster holds.

If b is the trickster's box, then IF (true, b) will never show "true": the trickster will put in "false". There is no bridge — and nothing to cross.

Reasoning passes "true" along, but it doesn't make it out of thin air.

The chain of bridges rule

Now you know two other things:

IF (a, b)      = true
IF (b, c)      = true

A bridge from a to b and a bridge from b to c:

a  ══►  b  ══►  c

So you can go from a straight to c:

IF (a, c)      = true

Let's check all three cases for a.

a = false. Then IF (a, c) = true by the table, whatever c holds.

a = true. By the bridge rule b = true. By the bridge rule again c = true. And IF (true, true) = true.

a is the trickster's box. Every machine has its own trickster and its own copy of a. The trickster of the machine IF (a, b) would put "true" in "if" and win — unless b were "true". Since its light says "true", b = true. By the bridge rule c = true. And the machine IF (a, c) has "true" in "then" — its trickster is powerless.

The chain of bridges rule: from IF (a, b) and IF (b, c) follows IF (a, c).

What about "if it is, then it is"?

Strange: the law "if it is, then it is" broke, but the bridge rule stands. Why?

IF (Z, Z) is a machine where you have not a single note of your own. Both boxes are the trickster's, and he gets "false".

The bridge rule starts from what you already have: a = true and bridge = true. That is what b is deduced from. In IF (Z, Z) nothing at the input is yours — so the trickster puts "true" in "if" and "false" in "then".

For the curious: the rule as a law

The bridge rule can be written as a chain, like the laws in Chapter 8. Notice: a appears in it twice — those are two boxes.

IF ( AND (a, IF (a, b)), b )

Let the tricksters hold every box. Step by step:

IF (Z, Z)          = false
AND (Z, false)     = false
IF (false, Z)      = true

If you go through all the cases — with your notes and with tricksters — the light always says "true". The same goes for the chain of bridges.

But to be honest: when the tricksters hold the boxes, the law held because the inner IF showed "false". The law tells us nothing about the boxes themselves. It confirms something else: this way of reasoning is reliable.

Try it with a friend

This time without a trickster. The IF machine; your friend puts notes in both boxes however they like, and you write down what the light showed.

Find all the ways of filling the boxes where "if" holds "true" and the light showed "true". What is in "then" every time? That is the bridge rule.

Puzzles

  1. a = true and IF (a, b) = true. What can you say about b?
  2. IF (a, b) = true and IF (b, c) = true. What can you say about IF (a, c)?
  3. b is the trickster's box, a = true. What does IF (a, b) show? Can you deduce b by the bridge rule?
  4. Why did "if it is, then it is" break while the bridge rule didn't?

Think it through first, then check.

Answers. 1 — b = true. 2 — IF (a, c) = true. 3 — "false": the trickster puts "false" in "then". No, you can't: there is no bridge. 4 — in IF (Z, Z) you don't have a single note of your own, and the trickster wins; the bridge rule starts from what is already "true".

On "Try it with a friend". There is only one such way: IF (true, true). "Then" holds "true". And IF (true, false) = false.

In the next chapter: long sentences — many boxes at once.