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Concept

Refutation

A derivation ending in a contradiction, showing that the assumptions it started from cannot all hold at once. It is a certificate of impossibility, checkable line by line without repeating the search that produced it.

Named by 3 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 A resolution refutation of four clauses on three variables. A derivation tree: the given clauses at the top, each later clause obtained by cancelling one variable between two clauses above it, ending in the empty clause.

A 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.

logic · Resolution A implies B, and r ∨ s sits between them. A = ((p ∨ q) ∧ (p → r)) ∧ (q → s) implies B = (r ∨ s) ∨ t. A resolution refutation of A with the negation of B yields the interpolant r ∨ s. It uses only the shared atoms r, s, follows from A and implies B, and lies between the strongest and weakest interpolants.

The sentence between a premise and its consequence

When one formula implies another, something sits between them written only in the words the two have in common: a sentence the first implies and that implies the second. A refutation of the first together with the denial of the second hands such a sentence over, and every possible one lies between a strongest and a weakest that can be computed outright.

logic · Proof systems

Named alongside it

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

Proof systemSatisfiabilityCompletenessLiteralSoundnessBranchingCut eliminationDecision procedureImplicationNormal formResolutionSubformula property

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