##  [Resolution Refutation](/resolution-refutation-0) 

 Definition

A proof method that demonstrates the unsatisfiability of a set of clauses by repeatedly applying the resolution rule to produce new clauses until the empty clause is derived, indicating a contradiction.

 

 

 

 

 

 





## Principle

Principle

Use the binary resolution rule to combine clauses containing complementary literals, optionally with unification in the first-order case, and control the search (selection, ordering, subsumption) so that derivation of the empty clause is reachable if the clause set is unsatisfiable.

 

 

 

 

 





## Demonstration

Demonstration

Starting from clauses {p ∨ q, ¬q ∨ r, ¬r, ¬p}, resolve p ∨ q with ¬p to derive q, resolve q with ¬q ∨ r to derive r, resolve r with ¬r to derive the empty clause, thereby refuting the original set.

 

 

 

 

## Misapplication

Misapplication

Applying propositional resolution procedures verbatim to unconstrained first-order clause sets without unification or without strategies for termination, or assuming resolution always produces short or human-readable proofs; naive expansion can explode combinatorially.

 

 

 

 

 





## Consequence

Consequence

Resolution refutation is sound and, for propositional logic and appropriately configured first-order settings, refutation-complete: if the clause set is unsatisfiable, a refutation exists; it underpins many automated theorem provers and SAT solvers (after CNF transformation and sometimes with additional mechanisms).

 

 

 

 

## Reversal

Reversal

The inversion is constructive proof search (e.g., tableau or natural deduction producing a direct derivation of the formula) rather than refutation by contradiction; in the reversed approach one attempts to build a model or direct proof instead of deriving a contradiction.

 

 

 

 

 





## Boundary

Boundary

Applicable to clause form (CNF) representations; pure resolution does not by itself construct models for satisfiable sets and, in first-order logic, may not terminate without suitable strategies or restrictions (e.g., ordering, selection, or loop check).

 

 

 

 

 





## Semantic Tension

Semantic Tension

There is tension between resolution refutation and sequent or natural deduction styles: resolution is oriented towards refutation and algorithmic search over clauses, while other styles emphasize constructive witness extraction and different proof shapes and normal forms.

 

 

 

 

 





## Synthesis

Synthesis

Resolution refutation combines the local inference rule of resolution with unification (in first-order) and global search control so that contradictions are derived from clause sets; it transforms the task of proving validity into a systematic clause-elimination search that either yields the empty clause or fails to refute.