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. 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.

    rung 1 · logic
  2. 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.

    rung 2 · logic
  3. 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.

    rung 3 · logic

All ladders