Quantifiers — the series

j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

part 1 · logic

Six sentences from two quantifiers

One relation, two variables, 'for every' and 'there is': there are eight ways to arrange them and six different sentences come out. Which of them imply which is a small, complete diagram, found by checking all 512 relations on three points — and the diagram crosses over in the middle, which is where every confusion about the order of quantifiers lives.

part 2 · logic

"There is an x" is a shadow: x² + ax + 1 = 0 has a solution exactly when |a| ≥ 2. A grid with a horizontal and x vertical, marking the cells the curve x squared plus a x plus one equals zero passes through; beneath it, a strip marking the columns that contain a mark, which are exactly those with a at least two in size.

A quantifier is a shadow

'There is an x such that …' asks whether a column of a grid contains a mark — which is the same as asking whether a shape casts a shadow on the axis below it. Over the real numbers every such shadow can be described without the quantifier, by polynomial inequalities: 'x² + ax + 1 = 0 has a solution' is just a² ≥ 4. Over the whole numbers the same kind of shadow can carve out the primes, and any set a computer can list.

part 3 · logic

Whole-number points in a strip, and the shadow they cast. The lattice points satisfying 2x ≤ 5y ≤ 2x + 1 for x from 0 to 30, and their projection onto the x-axis, which repeats every 5.

Arithmetic with addition alone

Over the real numbers, a quantifier's shadow is described by inequalities. Over the whole numbers with addition and multiplication, a shadow can be any set a computer can list. In between lies arithmetic with addition and no multiplication, and there the shadows are always the same kind of thing: a finite exception, then a pattern that repeats. The whole numbers made from coins worth 6, 9 and 20 are every number from 44 on; the squares, which need multiplication, never repeat at all.

part 4 · logic

Addition as a machine with two states. Two-state carry automaton over columns (x, y, z): from carry 0 staying on 000 011 101, to carry 1 on 110; from carry 1 staying on 010 100 111, back on 001.

A machine that carries one bit

Write three numbers in base two, one above another, and read the columns from the right. Whether the bottom number is the sum of the top two can be checked with one bit of memory — the carry. That two-state machine is the whole of addition, and machines can be combined, negated and made to guess a missing number. So every sentence about whole numbers built from addition can be decided by building a machine for it, and the same machines can also handle 'x is a power of two', which addition alone cannot say.

part 5 · logic

Three-colourability as a sentence: there exist three sets such that …. Petersen graph coloured with three colours; witness sets R = {0,2,6}, G = {1,3,5,9}, B = {4,7,8}; all 25 first-order checks pass.

There is a relation such that

A graph can be coloured with three colours exactly when there are three sets of its points such that every point lies in one and no edge joins two points of the same set. The sets are a guess; once guessed, the rest is a list of simple checks. Ronald Fagin proved in 1974 that this shape of sentence — 'there exist relations such that' followed by first-order conditions — describes exactly the problems whose solutions can be checked quickly, the class called NP. The central question of computer science is therefore also a question about what sentences can say.

part 6 · logic