A universal claim over a finite universe is a check of every member, an existential claim a search for one. With two variables the order decides whether one partner is shared by all or each member has its own, and the shift from the second to the first is a fallacy that a grid exposes.
Core ideas
The categorical forms had one variable. Statements about relations need two, and the order in which they are quantified is a change of meaning.
With a universe of four members, ∀x P(x) says P holds at each of the four and ∃x P(x) says it holds at one or more. Either can be settled by looking. The lab shows the universe, so no truth is hidden.
In ∀x ∃y P(x, y) the y is chosen after the x is known, so it may differ from one x to the next. In ∃y ∀x P(x, y) the y is chosen first and must serve every x. On a grid, the first is a true cell in every row; the second is a column that is true throughout.
A statement is true or false of a universe, not in itself. The same formula can be true over all the numbers and false over the first four, because a member is missing at the edge.
The material
A relation over four things is a four-by-four grid of true and false. Every quantified statement about it is a question about the grid.
The universal quantifier ∀x says that every member of the universe satisfies what follows; the existential quantifier ∃x says that at least one does. Over a finite universe the first is a conjunction of the cases and the second a disjunction.
With one variable, this is the categorical logic of the last lessons. All S are P is ∀x (S(x) → P(x)), and Some S are P is ∃x (S(x) ∧ P(x)). With two variables the new thing is that the order of quantifiers can change the meaning.
Let P(x, y) mean that y is the successor of x, over the numbers 1 to 4. The grid has rows for x and columns for y, and a true cell where y = x + 1. Read ∀x ∃y P(x, y) as: for every x there is a y. Read ∃y ∀x P(x, y) as: there is a y such that, for every x, P(x, y) holds.
These are different statements, and the lab shows all six forms of two quantifiers at once. Over the infinite numbers, everything has a successor and nothing is everything's successor, so ∀x ∃y P(x, y) is true and ∃y ∀x P(x, y) is false. The finite lab makes one thing visible that the infinite case hides: the grid ends. In the successor preset the number 4 has no partner, so the first form is false at the edge.
To see the separation, close the loop. Click the cell with x equal to 4 and y equal to 1, so that 4 + 1 wraps round to 1, as on a clock. Now every row has a true cell, and no column has more than one. ∀x ∃y P(x, y) is true and ∃y ∀x P(x, y) is false, and the lab's status line says that this is the case that settles the order question.
∃y ∀x P(x, y) implies ∀x ∃y P(x, y): if one y serves every x, it serves each. The reverse fails. One grid with the first true and the second false is a complete refutation of the reverse, whatever the size of the universe.
“Everything has a cause, so something causes everything.” Read P(x, y) as x having y as a cause. The premise is ∀x ∃y P(x, y), a true cell in every row. The conclusion is ∃y ∀x P(x, y), a column all true. The wrapped successor grid is a case with the first and not the second, so the step is invalid. Whether the premise is true of the world is a separate question, and so is whether any serious version of the cosmological argument takes this step: its defenders say it does not, and argue instead that a chain of causes with no first member is itself impossible. That reply grants the verdict on the form and disputes the formalisation, which is the right kind of reply to make.
A finite grid is a legitimate countermodel. The lab does not search every possible universe, and does not need to for this verdict, because one universe with true premise and false conclusion is all invalidity takes. It could not, by looking at a few grids, prove the other direction; that follows from the meaning of the words.
Interactive
A finite universe you can edit
Method
Four steps, and the order of the quantifiers is the first.
The first quantifier is chosen first. If it is ∀, everything is considered; if it is ∃, a single member is picked and held while the rest are tested.
In ∃y ∀x the same y must work for every x. In ∀x ∃y the y may change from one x to the next.
A full row, a full column, a true cell in every row, a true cell in every column or just a true cell somewhere. Each form asks for one of these.
Ask whether the claim would still hold if the universe were larger, and whether it holds only because of its boundary.
Over 1 to 4, let P(x, y) be “y follows x”, with 4 followed by 1. The rows are x and the columns y.
Every row has exactly one true cell, so for each x there is a y: ∀x ∃y P(x, y) is true. No column is true throughout, since each has three false cells, so no one y serves every x: ∃y ∀x P(x, y) is false. Without the wrap the fourth row is empty and the first form fails too, which is the finite universe saying that the claim about the numbers needs the numbers to go on.
Given a relation as a grid, state the values of all six two-quantifier forms and the cell, row or column that decides each, and say which direction between ∀x ∃y and ∃y ∀x holds and why.
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