First order resolution

To understand how resolution works, consider the following example syllogism of term logic:

All Greeks are Europeans. Homer is a Greek. Therefore, Homer is a European.

Or, more generally:

∀X, P(X) implies Q(X). P(A). Therefore, Q(A).

To recast the reasoning using the resolution technique, first the clauses must be converted to Conjunctive normal form. In this form, all quantification becomes implicit: universal quantifiers on variables (X, Y...) are simply omitted as understood, while existentially-quantified variables are replaced by atoms (A, B...) (in the Prolog sense):

¬P(X) ∨ Q(X) P(A) Therefore, Q(A)

So the question is, how does the resolution technique derive the last clause from the first two? The rule is simple:

• Find two clauses containing the same predicate, where it is negated in one clause but not in the other.
• Perform a unification on the two predicates. (If the unification fails, you made a bad choice of predicates. Go back to the previous step and try again.)
• If any unbound variables which were bound in the unified predicates also occur in other predicates in the two clauses, replace them with their bound values (atoms) there as well.
• Discard the unified predicates, and combine the remaining ones from the two clauses into a new clause, also joined by the "∨" operator.

To apply this rule to the above example, we find the predicate P occurs in negated form

¬P(X)

in the first clause, and in non-negated form

P(A)

in the second clause. X is an unbound variable, while A is a bound value (atom). Unifying the two produces the substitution

X => A

Discarding the unified predicates, and applying this substitution to the remaining predicates (just Q(X), in this case), produces the conclusion:

Q(A)

For another example, consider the syllogistic form

All Cretans are islanders. All islanders are liars. Therefore all Cretans are liars.

Or more generally,

∀X P(X) implies Q(X) ∀X Q(X) implies R(X) Therefore, ∀X P(X) implies R(X)

In CNF, the antecedents become:

¬P(X) ∨ Q(X) ¬Q(Y) ∨ R(Y)

(Note I renamed the variable in the second clause to make it clear that variables in different clauses are distinct.)

Now, unifying Q(X) in the first clause with ¬Q(Y) in the second clause means that X and Y become the same variable anyway. Substituting this into the remaining clauses and combining them gives the conclusion:

¬P(X) ∨ R(X)

The resolution rule similarly subsumes all the other syllogistic forms of traditional logic.

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