Inference in first-order logic involves making logical deductions based on a set of logical sentences represented in first-order logic. One common approach to perform inference is by converting logical sentences into a specific form called clause form, which simplifies the reasoning process. Here’s an overview of inference in first-order logic and the clause form conversion process:

Inference in First-Order Logic:

1. Logical Sentences:
   • In first-order logic, logical sentences are represented using quantifiers, predicates, variables, and logical connectives.
   • For example, a sentence like “All humans are mortal” can be represented as ∀xHuman(x)→Mortal(x).
2. Inference Rules:
   • Inference rules are used to derive new logical sentences from existing ones.
   • Common inference rules in first-order logic include Modus Ponens, Universal Instantiation, Existential Instantiation, and Universal Generalization.
3. Proof Methods:
   • Proof methods involve systematically applying inference rules to a set of logical sentences to derive new conclusions.
   • Methods such as resolution, backward chaining, and forward chaining are commonly used for theorem proving and reasoning in first-order logic.

Clause Form Conversion:

Clause form is a specific representation of logical sentences that simplifies the inference process, particularly when using resolution-based methods. The conversion process involves transforming logical sentences into a disjunction of literals, where each literal is either a predicate or the negation of a predicate. Here’s how it’s done:

1. Eliminate Implications:
   • Use the logical equivalence P→Q≡¬P∨Q to eliminate implications.
   • Replace each implication with its equivalent disjunction of the negation of the antecedent and the consequent.
2. Move Quantifiers to the Front:
   • Bring quantifiers to the front of the sentence, distributing them over the rest of the formula.
   • This step is crucial for standardizing the quantified sentences.
3. Skolemization (if needed):
   • Skolemization is used to eliminate existential quantifiers by introducing Skolem functions or constants.
   • It ensures that the resulting clause form does not contain any existential quantifiers.
4. Distribute Quantifiers (if needed):
   • Distribute existential quantifiers over conjunctions and universal quantifiers over disjunctions.
   • This step simplifies the resulting clause form and prepares it for resolution-based inference.
5. Negation Normal Form (Optional):
   • Convert the sentence into negation normal form if needed. This step is often performed before or during the clause form conversion process.

Once the logical sentences are converted into clause form, inference can be performed using resolution-based methods, such as resolution refutation, to derive new conclusions or prove the validity of logical arguments.

Example:

Consider the sentence: “All humans are mortal.”

• Original: ∀xHuman(x)→Mortal(x)
• Clause Form: ¬Human(x)∨Mortal(x)

In this example, the implication is eliminated, and the quantifier is brought to the front, resulting in a clause form representation.

By converting logical sentences into clause form, inference becomes more straightforward, enabling the use of efficient resolution-based methods for theorem proving and logical reasoning in first-order logic.