Matpic
mathematics, in pictures
Concept

Tableau

A proof by decomposition: assume the opposite of what is wanted and take it apart. Every branch either contradicts itself or describes a way for the assumption to hold, so every branch closing is a proof, and a branch staying open hands over a counterexample rather than merely failing.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

A tableau for ((p → q) ∧ (q → r)) → (p → r). A branching tree of formulas, each branch ending in a contradiction or in a description of a counterexample.

The 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.

logic · Proof systems 7 steps and 3 withdrawn assumptions. A natural-deduction derivation of (p → q) → (¬q → ¬p). Each horizontal bar is one inference, named on its right; the bracketed formulas are assumptions, and each is withdrawn at the step that names it.

The assumption a proof pays back

A tableau assumes the opposite once and takes it apart. Natural deduction assumes things freely, uses them, and then withdraws them — and the withdrawal is what turns a derivation of a consequence into a proof of an implication.

logic · Proof systems A lemma, and the proof that never mentions one. Two derivations of ((p → q) ∧ (q → r)) → (p → r) compared: the cut-free one uses 8 nodes and only subformulas of the goal, and the one through a lemma uses 21 and mentions a formula the goal does not contain.

A lemma, and the proof that never mentions one

Proving something by first proving a lemma is what makes mathematics readable, and it is exactly what makes a proof system impossible to search — because the lemma can be any formula at all. Gentzen proved the step can always be removed, and the removal is not free.

logic · Proof systems Closed after 3 uses of the universal. The Herbrand expansion of a first-order question at 5 stages, with the number of remaining models at each. It reaches nought after 3 instantiations.

The instance that has to be guessed

Every rule of a propositional tableau replaces a formula by shorter ones, which is why it stops. The rule for a universal claim does not replace it — it keeps it and adds an instance — and one word changing turns a decision procedure into a search that may run forever.

logic · Proof systems

Named alongside it

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

CompletenessProof systemDecision procedureSoundnessBranchingNatural deductionSatisfiabilityCut eliminationDischargeImplicationLiteralQuantifier

All concepts