Generator

3 circles, and the 8 patterns they realise

A generator in the logic library, called 20 times across 5 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.

Four ellipses, and all sixteen patterns

A smallest model of ∀x (Ax → ∃y (By ∧ ¬Cy)) ∧ ∃x (Ax ∧ Cx) ∧ ∀x (Bx → ¬Ax)

Collapsing a model of ∀x (Ax → ∃y (By ∧ ¬Cy)) ∧ ∃x (Ax ∧ Cx) ∧ ∀x (Bx → ¬Ax) to one thing a region

Every model of ∀x (Ax → ∃y (By ∧ ¬Cy)) ∧ ∃x (Ax ∧ Cx) ∧ ∀x (Bx → ¬Ax), up to what one-place predicates can see

Carroll's babies and crocodiles, on four ellipses

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
after erasing the middle terms, the conclusion's region is empty ×1
and each of the sixteen is a single connected region ×1
cap is read only by the collapse view ×1
cap, the most things kept in one region, is 1 to 3 ×1
every leaf of the formula is a variable ×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
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 collapsed model gives the sentence the same truth value as the large one ×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 number of sets is a whole number between 2 and 5 ×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 two clauses clash on the erased letter ×1
the view is one the family draws ×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

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

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.

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