Matpic
mathematics, in pictures
Series

Class diagrams — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 4 circles, and the 14 patterns they realise. Closed curves overlapping in the plane, with each region of the arrangement identified by which curves contain it.

    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.

    part 1 · logic
  2. The 256 syllogistic forms, and the 24 that work. A grid with one cell per syllogistic form, marked according to whether it is valid and what it needs to be valid.

    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.

    part 2 · logic
  3. A smallest model of ∀x (Ax → ∃y (By ∧ ¬Cy)) ∧ ∃x (Ax ∧ Cx) ∧ ∀x (Bx → ¬Ax). Three overlapping circles with some regions shaded as empty and a single dot in each occupied region, forming a model of a sentence of monadic first-order 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.

    part 3 · logic
  4. Carroll's babies and crocodiles, on four ellipses. Four overlapping ellipses labelled with the four classes of a sorites, the regions emptied by its premises shaded, and the regions its conclusion requires to be empty outlined.

    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.

    part 4 · logic

All series