Generator

3 circles, and the 8 patterns they realise

A generator in the logic library, called 43 times across 9 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

venn is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

3 circles, and the 8 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

One try: letters set in a random order, forced or guessed

Every letter's critical clauses, and how often they force it

How many letters one try has to guess

The chance one random-order try succeeds, against its guarantee

Two ways to beat trying everything, by clause length

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

3 circles realise every one of the 8 patterns ×2
4 circles cannot realise all 16 patterns ×2
a successful try is found ×1
a try that walks several steps and succeeds ×1
adding clauses satisfied by the plant eventually leaves it the only solution ×1
after erasing the middle terms, the conclusion's region is empty ×1
and each of the sixteen is a single connected region ×1
and it finds the one solution ×1
cap is read only by the collapse view ×1
cap, the most things kept in one region, is 1 to 3 ×1
each letter comes last in a critical clause at least a third of the time ×1
each step flips exactly one letter, so crosses exactly one curve ×1
every invalid classical form fails with at most one thing in each region ×1
every invalid form has a counterexample with at most four in each region ×1
every leaf of the formula is a variable ×1
every letter has a critical clause, because flipping it must break some clause ×1
every letter in the set expression names a drawn curve ×1
every region the drawing uses is present in it ×1
exactly two premises mention the first erased class ×1
faces are crossings plus two, by Euler's formula ×1
fifteen classical forms are valid ×1
for every clause length the walk's base is below this method's, and both are below 2 ×1
guessing right every time reaches the solution ×1
most M are P and most M are S together force some S to be P ×1
most S are M and most M are P do not make most S P ×1
ninety-nine per cent of M are P and no S is P ×1
ninety-nine per cent of S are M ×1
no form with 'most' in both premises and conclusion is valid ×1
no occupancy the premises allow refutes the conclusion ×1
no pattern occupies more pieces than the arrangement has ×1
puzzle is read only by the sorites, erase, clauses views ×1
resolving on the erased classes derives the conclusion's clause ×1
sentence is read only by the model, collapse, search views ×1
sets is read only by the circles view ×1
the 32 combinations fall into 8 orbits under rotation: 2 fixed and 6 of five each ×1
the average number of guesses is at most two thirds of the letters ×1
the bound is (3/4)ⁿ up to a factor of the walk's length ×1
the chance from distance j is at least 2^(−j) up to a factor the length of the walk ×1
the clauses leave exactly one satisfying assignment ×1
the collapsed model gives the sentence the same truth value as the large one ×1
the drawn counts satisfy both premises and deny the conclusion ×1
the drawn counts satisfy both premises and the conclusion ×1
the forced overlap is a + b − 1 ×1
the formula has exactly one satisfying assignment ×1
the formula is a non-empty string ×1
the four ellipses realise all sixteen patterns ×1
the last resolvent is the conclusion's clause ×1
the linear programme's solution scales to a whole-number counterexample ×1
the measured rate is at least the pessimistic bound ×1
the measured success rate is at least the bound 2^(−2n/3) ×1
the number of sets is a whole number between 2 and 5 ×1
the option is one this view reads ×1
the pieces the circles cut the plane into match Euler's count ×1
the premises empty every region the conclusion needs empty ×1
the puzzle is one this family knows ×1
the search is over one-per-region models, so the sentence has no ≠ ×1
the sentence has a model to draw ×1
the sentence is one this family knows ×1
the success rate matches the average of 2 to the minus guesses ×1
the two clauses clash on the erased letter ×1
the view is one the family draws ×1
the walk's base at three letters is 4/3 ×1
the whole formula is consumed by the parser ×1
with ≠ in the sentence, one thing per region is too few ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Logic

A letter the clauses already decide

Set the letters of a formula one at a time, in a random order, and guess each one with a coin — unless some clause has already had its other two letters set to false, in which case the clause decides. A try succeeds when every guess is right, so what matters is how many letters get guessed. On a formula with one solution, every letter has a clause that forces it in at least one order out of three, so at most two thirds of the letters are guessed on average, and about 1.587ⁿ tries suffice. It is slower than the random walk. Refined, it is the fastest method known.

Logic

A walk that beats trying everything

To decide whether clauses of three letters can all be satisfied, the obvious method tries all 2ⁿ assignments. Uwe Schöning's method, from 1999, starts at a random assignment and wanders: pick a clause that is false, flip one of its letters at random, and repeat three times as many times as there are letters. A single try usually fails, but it succeeds with chance at least about (3/4)ⁿ, so about (4/3)ⁿ tries are enough — and the reason is a walk on a line that goes the wrong way two times in three.

Logic

Four circles cannot do it

Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.

Logic

One thing in each region is enough

Give first-order logic its full apparatus of nested quantifiers but only one-place predicates, and every question about truth is still settled by the regions of a diagram. A predicate cannot tell apart two things in the same region, so no model ever needs more than one thing per region — and with three predicates there are only 255 models to try.

Logic

Syllogisms that say most

The syllogism knew four words — all, no, some, some not — and two thousand years of logic decided which of its 256 forms were valid. Add 'most', a word everyone uses in arguments, and the 256 become 864. A diagram with marks for 'empty' and 'occupied' can no longer decide them, because 'most' compares sizes; but writing the number of things in each region turns every form into a linear programme, and the programme decides all of them at once. Thirty-one are valid, including one Augustus De Morgan found in 1847: if most M are S and most M are P, then some S are P. And 'most' cannot be passed along a chain: ninety-nine per cent of S can be M, ninety-nine per cent of M can be P, and no S be P at all.

Logic

The conclusion is what survives the erasing

Lewis Carroll's puzzles give three premises about four classes — babies, logical people, the despised, crocodile-managers — and ask what follows. Draw all four, shade what the premises rule out, then erase the classes the conclusion is not about: a region survives as empty only if everything above it was. What is left is the conclusion, and erasing a class turns out to be exactly one step of resolution.

Logic

Twenty-four out of two hundred and fifty-six

Aristotle's syllogisms are four sentence forms in four arrangements, which makes 256 patterns of argument. Fifteen of them are valid. Nine more become valid if you assume the things being talked about exist, and the gap between those numbers is a two-thousand-year-old disagreement.

Logic

Venn diagrams that turn into themselves

A Venn diagram shows every combination of in and out exactly once. Ask for one that looks the same after a turn of 1/n, each curve rotating onto the next, and a counting argument forbids it unless n is prime. Five congruent ellipses manage it, certified by Euler's formula; four rotated copies cannot, and it took until 2004 to show that every prime can.

Algebra

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

The whole library · What the figures prove