Quantifier
Named by 19 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
One block of quantifiers, read off an algebra
Which properties of words can be stated with existential quantifiers alone — 'there are positions, in this order, carrying these letters' — and Boolean combinations of such statements? Imre Simon answered in 1975: exactly those whose syntactic monoid is J-trivial. Of the 57,068 languages that automata of up to four states recognise, 848 pass the test, and every one of them is decided by which words of length three or less it contains as scattered subwords.
The sizes at which a sentence can come true
Write a sentence about a relation and ask for which numbers of points some relation makes it true. "Every two points have exactly one common neighbour" comes true at every odd size and no even one, and every graph that satisfies it is a windmill of triangles. "Every two players are both beaten by some third" fails for every tournament of two to six players and holds from seven on. A search can settle one size at a time; only a proof settles all of them, and the two sentences need different kinds.
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.
Four quantifiers, one of them blind
In 'for every x there is a y, and for every u there is a v', the last choice may depend on everything before it. Leon Henkin asked in 1961 what happens if v may depend on u alone. Using nothing but the equals sign, four such quantifiers then say that a set has an even number of points, or that it is infinite — things no ordinary sentence can say — and the game behind the sentence can end with neither player able to win.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchDecision procedureExpressive powerElementary equivalenceModelQuantifier orderCompletenessCounterexampleDecidabilityEhrenfeucht fraisse gameSatisfiabilityStrategy