Matpic
mathematics, in pictures
Ladder

Truth functions — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. (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.

    rung 1 · logic
  2. 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.

    rung 2 · logic
  3. ((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.

    rung 3 · logic

All ladders