What an Implication Promises

Last timeAnd, Or, and Not

An if-then claim rules out exactly one combination and says nothing about the rest. Reading it as though it ran backwards is the most expensive mistake in the subject.

A rule is posted. If an order is over a hundred pounds, shipping is free.

An order arrives for forty pounds with free shipping. Has the rule been broken?

It has not, and most people hesitate before answering. The rule promised

something about orders over a hundred pounds and said nothing whatever about

orders under it. Anything at all may happen to those without the promise being

touched.

FIG 1If A then B, every possible case

	A	B	if A then B	if B then A
order over 100, shipping	1	1	1	1
order over 100, shipping	1	0	0	1
order under 100, shippin	0	1	1	0
order under 100, shippin	0	0	1	1

One means true. The third column is the posted rule and the fourth is its reverse. The rule is false in exactly one row, the second, where the condition holds and the promise fails. The third row is the one worth sitting with: an order under a hundred pounds given free shipping keeps the rule, because the rule said nothing about such orders. And the two marked cells in that row show the point of the lesson: the rule is true there while its reverse is false, so the two are genuinely different claims and neither follows from the other.

The one thing it forbids

The last two rows deserve a name, because they are the cases where the

condition never arises. A promise about a situation that does not occur cannot

be broken, so the claim holds. This is why a statement about every item of an

empty collection is true, which seems like a trick until you notice that it is

simply the third and fourth rows again.

FIG 2Three claims that look related, two of which are the same

: if A then B, false only when A holds and B fails
: the reverse, which swaps the two parts and is an independent claim
: swap the parts and negate both, which is true in exactly the same cases as the original
: not, reversing the truth of what follows

The middle claim shares no truth with the first: it can hold when the first fails and fail when the first holds, as the table above shows in rows two and three. The third claim is the first one wearing different clothes, true in every case where the original is true and false in every case where it is false, which can be checked row by row against the table. The rewrite is free and always available, and it is worth reaching for whenever the original direction is awkward to argue, which happens often enough that proving the rewrite instead is a standard move rather than a trick.

The reverse is a different claim

Read the rewrite in the shipping example. If the shipping was charged, the

order was not over a hundred pounds. That is a legitimate backwards inference

and it is the only one the rule supports. From free shipping, nothing follows.

The lesson stops here

3 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

What an Implication Promises

The one thing it forbids

The reverse is a different claim

3 more paragraphs to go

This is the reading half

Read alongside