Every, and At Least One

Last timeWhat an Implication Promises

Claims about all cases and claims about some case are opposites of each other, and negating one turns it into the other. Getting the order of two of them wrong changes the claim entirely.

Two shapes of claim cover most of what anybody wants to say about a collection.

Every order in the batch was processed. At least one order in the batch failed.

They are opposites, and the way they are opposites is the most useful thing in

this lesson.

FIG 1What each shape costs to establish and to refute

	examples needed to estab	examples needed to refut	true of an empty collect
every case works	0	1	1
at least one case works	1	0	0
every case fails	0	1	1
at least one case fails	1	0	0

A nought in the first two columns means no finite number of examples will do it, so an argument covering all cases at once is required; a one means a single example settles it. The two marked cells are the asymmetry the whole course turns on: a claim about every case cannot be established by examples and can be destroyed by one of them. Rows one and four are negations of each other, as are rows two and three, and the costs swap accordingly, which is not a coincidence but the same fact read twice. The last column is the vacuous case: with nothing in the collection there is nothing to break an every claim and nothing to support a some claim.

The two shapes, and their costs

The asymmetry is practical rather than philosophical. If you want to refute a

claim about everything, go looking for one bad case and stop as soon as you

find it. If you want to establish one, no amount of looking will finish the

job, and that is the subject of the two lessons after this.

FIG 2The negation moves inwards and swaps the shape

: for every x in the collection under discussion
: there is at least one x in that collection
: whatever is being claimed about x, such as this order was processed
: not, reversing the truth of what follows

Read the left one as: it is not the case that every order was processed, which is exactly the claim that at least one order was not processed. Read the right one as: no order failed, which is exactly the claim that every order did not fail. This is the same pattern as the rule for and and or in the second lesson, and for a reason: a claim about every case is a long chain of ands, a claim about some case is a long chain of ors, and the negation flips the chain. The rule to carry is that the negation moves inwards, swaps the shape, and lands on the part inside.

Negating them

Applying it mechanically beats thinking about it. Take a claim with some

structure in it: every request is answered within a second. Negating gives at

least one request is not answered within a second, which is a request taking a

second or longer. Note where the boundary went, which is the same care the

second lesson asked for.

The lesson stops here

1 more paragraph 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

Every, and At Least One

The two shapes, and their costs

Negating them

1 more paragraph to go

This is the reading half

Read alongside