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Concept

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 5 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 Which chains of truth values validate Gödel's disjunctions. A grid of Gödel's pigeonhole formulas in m letters against chains of n truth values, marking which chains validate which formula, forming a staircase where m exceeds n.

No table of truth values is enough

Two truth values decide classical logic. Gödel asked in 1932 whether some longer list of values could decide the constructive system, and answered with a pigeonhole: with n values, some two of n + 1 statements must share one, so a formula saying exactly that holds in every n-valued table and is not a theorem. The chains of truth values then descend forever, and what they share is a logic of its own — the logic of the real interval.

logic · Non classical logic Six formulas on the interval of truth values. Graphs on [0, 1] of six one-variable formulas of Łukasiewicz logic, each a zigzag of whole-number slopes with values 0 or 1 at the ends.

The zigzags a formula can draw

Let truth be any number from 0 to 1, and Łukasiewicz's connectives turn every formula in one variable into a graph. Every graph that appears is a zigzag of straight pieces with whole-number slopes, ending at 0 or 1 — and McNaughton proved in 1951 that every such zigzag appears. A logic of degrees of truth turns out to be a theory of piecewise-linear functions with integer coefficients.

logic · Non classical logic Three truth values, two of which count as holding. Truth tables of negation, conjunction and disjunction in the logic of paradox, with the values that count as holding — both and true — shaded.

A contradiction that stays where it is

In classical logic one contradiction proves everything: from p and not-p, any q at all. Add a third truth value — both true and false — and count it as holding, and the explosion stops. The classical laws all survive as laws; what goes is exactly the reasoning that carries a contradiction somewhere it was not. Run the other way, the same three tables make a logic of gaps instead of gluts.

logic · Non classical logic

Named alongside it

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

Exhaustive searchMany valued logicTruth functionHypercubeNormal formAlgebraAssignmentBoolean functionChainCharacterisationCompletenessConnective

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