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Truth table

A table listing every assignment of true and false to a formula's variables and the value the formula takes on each. With n variables it has 2^n rows, which are the corners of an n-dimensional cube, and it decides any question about a formula by exhaustion.

Named by 2 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 Weighted votes: two functions with a cut, and parity without one. For three functions of three letters, the eight assignments placed on a line by a weighted count, with true assignments filled and the threshold marked where one exists.

A plane through the cube

Some truth functions are weighted votes: give each letter a weight, add the weights of the true letters, and say yes when the total passes a threshold. On the cube of assignments, such a function is a plane cutting the true corners from the false. Majority is one. Exclusive-or is not, and never can be — and of the 65,536 functions of four letters, only 1,882 are. The ones that are are exactly what a single artificial neuron can compute.

logic · Truth functions

Named alongside it

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

HypercubeAssignmentBoolean functionConnectiveExclusive-orExhaustive searchMajority ruleNormal formPropositional formulaSatisfiabilityTautologyThreshold function

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