Matpic
mathematics, in pictures
Concept

Connective

A word or symbol that builds a compound statement out of simpler ones, taking their truth values to a truth value. There are sixteen of them on two inputs, and a single well-chosen one is enough to build every other.

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 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

Named alongside it

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

Truth functionAffine functionAssignmentClosureDualityFunctional completenessHypercubeMonotone functionNormal formPost classesPropositional formulaSatisfiability

All concepts