Logic — page 5
The plane cut by a list
The continuum hypothesis looks like a statement about sizes: no infinity sits strictly between the whole numbers and the line. Wacław Sierpiński found in 1919 that it is equally a statement about the plane. It is true exactly when the plane splits into two pieces, one meeting every vertical line in only countably many points and the other meeting every horizontal line in only countably many. The split is drawn by listing the line's points in a particular order, and with the rationals in place of the reals it can be drawn exactly — a set that, counted column by column, has no area, and counted row by row, has all of it.
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.