Concept

Truth function

A rule computing the truth value of a compound statement from the truth values of its parts and nothing else.

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

000001010011100101110111(p ∨ q) ∧ ¬r on the 3-cube of assignments — 3 of 8 cornerscorners next to each other differ in one variable, which every edge here was checkedagainst

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
keeps 0keeps 1monotoneself-dualaffineenough alone?∧ and··no∨ or··no¬p not p···no↑ nand·····yes⊕ exclusive or···no→ implication····no6 connectives against Post's five classes — a tick means the connective stays insidenand escapes all five, and is therefore enough alone

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
rspq00011110000111101110111011110010((p ∧ q) ∨ (r ∧ s)) ∨ (¬p ∧ ¬r) covered by 3 of its 6 primeimplicantsr∧s ∨ ¬p∧¬r ∨ p∧q — checked against the formula on all 16assignments

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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