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Concept

Syllogism

An argument from two premises about classes to a conclusion about classes, such as all A are B and all B are C, so all A are C. Of the 256 forms Aristotle's scheme allows, only 24 are valid, and a diagram of three circles decides which.

Named by 3 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 Carroll's babies and crocodiles, on four ellipses. Four overlapping ellipses labelled with the four classes of a sorites, the regions emptied by its premises shaded, and the regions its conclusion requires to be empty outlined.

The conclusion is what survives the erasing

Lewis Carroll's puzzles give three premises about four classes — babies, logical people, the despised, crocodile-managers — and ask what follows. Draw all four, shade what the premises rule out, then erase the classes the conclusion is not about: a region survives as empty only if everything above it was. What is left is the conclusion, and erasing a class turns out to be exactly one step of resolution.

logic · Class diagrams Two majorities of one class must overlap. regions (outside, S, M, SM, P, SP, MP, SMP): 3,2,0,4,2,0,4,2.

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.

logic · Class diagrams

Named alongside it

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

Venn diagramCounterexampleQuantifierValidityBoolean functionCategorical statementDecision procedureExhaustive searchExistential importImplicationLinear programmingResolution

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