The truth table of p → (q → p)
truth-table is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "cube"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- every corner meets one edge per variable, each edge counted once ×1
- every leaf of the formula is a variable ×1
- every opening bracket in the formula is closed ×1
- the cube is drawn for two, three or four variables ×1
- the formula and its disjunctive normal form agree on every row ×1
- the formula has between one and five variables ×1
- the formula is a non-empty string ×1
- the highlighted rows are the true ones, the false ones, or neither ×1
- the table has one row per assignment ×1
- the two ends of an edge differ in exactly one variable ×1
- the whole formula is consumed by the parser ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
LogicA proof with one rule
Two clauses that disagree about exactly one variable can be combined into a third that forgets it; repeat, and if the clauses cannot all be true the empty clause eventually appears — a complete proof system with a single move.
LogicOne 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.
LogicThe 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.
LogicThe middle that is not excluded
Either it is raining or it is not. Drop that as an axiom and what is left is still a logic — one with models made of open sets and of stages of knowledge, in which a set and its negation between them miss the boundary.
LogicThe sentence that says it has no proof
Number every sentence and every proof, and a formal system can talk about itself. Then the diagonal is available one more time, and what it builds is a sentence that is true exactly when it is unprovable.
LogicThe tree that closes
To prove a formula, assume it false and take it apart. Every branch ends in a contradiction, or one of them describes exactly how it could have been false — and either way the tree is the answer, drawn.