Breaking a Claim With One Case

Last timeCollections and What Is In Them

One case is enough to destroy a claim about everything, which makes disproof cheap. The skill is constructing awkward cases on purpose rather than waiting to be surprised by one.

A claim about everything is expensive to establish and cheap to destroy. This

lesson spends the cheap half properly, because knowing how to break a claim is

most of knowing which claims are worth proving.

The licence comes straight from the previous lesson. The negation of every case

works is at least one case fails, so producing one failing case proves the

negation. Not suggests, not casts doubt on: proves.

One case is the whole argument

Somebody claims that reversing a list twice gives back the original list. That

is probably true. Somebody claims that sorting a list of names alphabetically

puts them in the order a reader expects. That one falls to a single name

beginning with an accented letter, and the single name is the entire

refutation.

The temptation after finding one is to find more, which is wasted effort for

the purpose of refuting, though it can be worth doing to learn the shape of the

failure.

FIG 1Standard awkward inputs against four plausible claims

	the empty case	exactly one item	at the boundary	a repeated item	a negative or zero
the average of the list	1	0	0	0	0
sorting then taking the	1	0	0	0	0
removing duplicates leav	0	0	0	1	0
the midpoint of two numb	0	0	0	0	0

One means that category of input breaks the claim. The first two claims are both destroyed by the empty list, where there is no average and no first element, which is the single most productive case to try and the one almost nobody tries. The third claim fails on a list with no duplicates in it, where the result is the same length rather than shorter, and the fix is the word strictly being removed. The fourth survives all five categories and is a reasonable candidate for proof, though it fails on a sixth category not shown, which is arithmetic that overflows, since adding two large numbers before halving them can exceed the largest representable value.

Where to look

The list of categories is the thing to carry: nothing at all, exactly one, the

boundary and just past it, repeats, negatives and zero, the largest

representable value, and an item of a kind the claim never considered.

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

Breaking a Claim With One Case

One case is the whole argument

Where to look

1 more paragraph to go

This is the reading half

Read alongside