Outline

Predicate Logic
Resolution

Part 1: What is Predicate Logic

• To solve the limitations in the prepositional calculus, you need to analyze propositions into predicates and arguments, and deal explicitly with quantification.

• Predicate Logic provides formalism for performing this analysis of prepositions and additional methods for reasoning with quantified expressions.

Examples

Remember

• When translating Universal Quantifier $\mathrm{\forall }$ , usually with implication $⟹$
• and for Existential Quantifier $\mathrm{\exists }$ , with Conjunction $\wedge$
• When negating, apply Demorgan ( $\mathrm{\neg }\mathrm{\forall }$ turns to $\mathrm{\exists }$ and vice versa)

Part 2: Resolution

Resolution is a technique for proving theorems in the predicate calculus using the resolution by refutation algorithm. The resolution refutation proof procedure answers a query or deduces a new result by reducing the set of clauses to a contradiction.

The Resolution by Refutation Algorithm includes the following steps:-

1. Convert the statements to Predicate Logic. (Translation)

2. Convert the statements from Predicate Logic to Clause Forms. (Skolemization)

3. Add the negation of what is to be proved to the clause forms. (Negate the conclusion)

4. Resolve the clauses to producing new clauses and producing a contradiction by generating the empty clause. (Apply resolution till contradiction)

Part 3: Predicate logic to clausal form:

In step 3,9: Standardation means to replace any repeated variable names for each quantifier, as for each quantifier, same name $\ne$ same variable

See Examples in Slides