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Concept

Truth function

A rule computing the truth value of a compound statement from the truth values of its parts and nothing else. There are sixteen of them on two inputs, and a single one can generate all the rest.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

(p ∨ q) ∧ ¬r, drawn on the cube of 8 assignments. The assignments as corners of a cube, joined when they differ in one variable, with the satisfying corners filled.

A formula is a corner of a cube

A formula about three letters is a set of eight rows. Written as a table that is a list; drawn on a cube it is a shape — and the shape is what almost every later question in this field turns out to be about.

logic · Truth functions Post's five classes, and which connectives escape them. A table of connectives against the five closed classes, with the completeness verdict for each.

One connective is enough

Of the sixteen ways to combine two truth values, exactly two can build all the others by themselves. Which two is not obvious, and the reason turns out to be five properties that a connective either has or escapes.

logic · Truth functions ((p ∧ q) ∨ (r ∧ s)) ∨ (¬p ∧ ¬r), covered by 3 rectangles. A grid of the assignments arranged so that neighbouring squares differ in one variable.

The map that puts neighbours side by side

Reorder the rows of a truth table so that neighbouring squares differ in one letter, and finding a short formula stops being algebra and becomes the problem of covering a shape with rectangles.

logic · Truth functions

Named alongside it

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

ConnectiveHypercubeNormal formAffine functionAssignmentClosureCoveringDualityFunctional completenessGray codeKarnaugh mapMinimality

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