194 Logic as a Tool structures, and also made important contributions to the theory of Lie algebras His most influential legacy, however, was in mathematical logic, where in 1936 and 1941 he proved in full generality the two most fundamental results of model theory: the compactness theorem, stating that if every finite subset of a first-order theory (set of sentences) is satisfiable then the entire theory is satisfiable; and the Löwenheim–Skolem Theorem (see the box on Skolem) 4.5 Resolution for first-order logic We are now almost ready to extend the method of Resolution to first-order logic One more technical topic – unification – is needed for that, described in the following 4.5.1 Propositional Resolution rule in first-order logic The Propositional Resolution rule extended to first-order logic reads: Res0 : C ∨ Q(s1 , , sn ), D ∨ ¬Q(s1 , , sn ) C ∨D This rule, however, is not strong enough For example, the clause set {{P (x)}, {¬P (f (y ))}} is not satisfiable, as it corresponds to the unsatisfiable formula ∀x∀y (P (x) ∧ ¬P (f (y ))) However, the resolution rule above cannot produce an empty clause, because it cannot unify the two clauses in order to resolve them We therefore need a stronger derivation mechanism that can handle cases like this There are two natural solutions4 : Ground resolution: generate sufficiently many ground instances of every clause (over the so-called Herbrand universe of the language) and then apply the standard Resolution rule above In the example, ground resolution would generate the ground clauses {P (f (c))} and {¬P (f (c))} for some constant symbol c This method is sound and complete but inefficient, as it leads to the generation of too many unnecessary clauses It will not be discussed further Resolution with unification introduce a stronger Resolution rule that first tries to match a pair of clauses by applying a suitable substitution that would enable that pair to be resolved We present this method in further detail in the following Both proposed by John Alan Robinson; read more at the end of the section