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Concept

Quantifier

A word settling how many things a statement is about — every one of them, or at least one. Which one is used, and in which order, is where most of the content of a mathematical statement lives.

Named by 15 essays across one field — each of them below, with the objects they name alongside it.

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.

logic · Class diagrams 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.

logic · Quantifiers 3 rounds on chains of 4 and 5. Two chains of dots with pebbles placed in turn, and the transcript of a play: Spoiler picks an element of one chain, Duplicator answers in the other, and the pebbles must keep the same order.

A game that decides what can be said

Two players take turns pointing at elements of two structures; if the second can survive k rounds, then no sentence with k quantifiers tells the structures apart — a statement about infinitely many formulas, settled by a finite search.

logic · Ehrenfeucht–Fraïssé games Three properties that leave the middle, and one that cannot. Four measured curves of the share of random graphs having a property, plotted against the number of points: three first-order properties running to zero or one, and the parity of the edge count sitting on a half throughout.

Nearly always, or nearly never

Toss a coin for every pair of points and ask whether the graph that results has some property. For a property a first-order sentence can state, the answer in the limit is never a genuine probability — it is zero or it is one, and the game is what proves it.

logic · Ehrenfeucht–Fraïssé games What a sentence of depth 2 can reach. Two rings of points, of 14 and 19 points, each with a run of 9 consecutive points marked as the neighbourhood a sentence of depth 2 can inspect.

The distance a sentence can see

A first-order sentence with three quantifiers cannot notice anything about a graph beyond a fixed distance from the points it names. That single limitation is why it cannot say connected, and why the failure survives every attempt to add more quantifiers.

logic · Ehrenfeucht–Fraïssé games Closed after 3 uses of the universal. The Herbrand expansion of a first-order question at 5 stages, with the number of remaining models at each. It reaches nought after 3 instantiations.

The instance that has to be guessed

Every rule of a propositional tableau replaces a formula by shorter ones, which is why it stops. The rule for a universal claim does not replace it — it keeps it and adds an instance — and one word changing turns a decision procedure into a search that may run forever.

logic · Proof systems Who survives with two pebbles and who with three, over 4 rounds. A table of three pairs of graphs with, for each, whether the duplicating player survives a two-pebble game and a three-pebble game played to a fixed depth.

The boundary at three variables

Restrict a sentence to two variable names and it can still be arbitrarily long, because the names are reused. What it cannot be is deep: every satisfiable two-variable sentence has a small model, so asking whether one is satisfiable is a bounded search. Allow a third name and the question becomes undecidable.

logic · Ehrenfeucht–Fraïssé games A countable structure grown by adding witnesses. Stages of a structure built from 0 and 1 by adding sums, products, negatives and roots of quadratics: 2, 4, 12, 158 elements between −3 and 3.

A countable field that passes for the line

The real numbers are uncountable, and every first-order sentence about their addition, multiplication and order is also true of a countable field inside them — the real algebraic numbers. Löwenheim and Skolem showed this is no quirk of the reals: every theory with an infinite model has a countable one, including set theory, which then contains sets it calls uncountable.

logic · Models Six sentences from two quantifiers, and which imply which. A diagram of the six sentences that can be built from two quantifiers and a relation, arranged from strongest to weakest with arrows for implication, each labelled with how many of the 512 relations on three points satisfy it.

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.

logic · Quantifiers "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.

logic · Quantifiers 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.

logic · Class diagrams 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.

logic · Quantifiers Unifying f(x, g(x)) with f(h(y), g(z)). Three term trees: f(x, g(x)), f(h(y), g(z)), and their common instance f(h(y), g(h(y))) under the most general unifier x ↦ h(y), z ↦ h(y).

Two terms made equal, and no more

Resolution with variables needs two literals to clash, and they clash only after something has been substituted for their variables. There are infinitely many substitutions that would do. One of them is the most general — every other is it followed by something more — and an algorithm of four rewriting rules finds it or proves there is none. That single computation turns the search for instances from guessing into arithmetic.

logic · Resolution 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.

logic · Quantifiers 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.

logic · Quantifiers

Named alongside it

The objects these essays reach for when they reach for this one.

Decision procedureExhaustive searchElementary equivalenceExpressive powerModelCompletenessQuantifier orderSatisfiabilityStrategyCounterexampleDecidabilityEhrenfeucht fraisse game

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