Inference Theory of Predicate Logic - The Logic Tutorial

Inference Theory of Predicate Logic

Introduction

Predicate logic, also known as first-order logic, is a formal system used in mathematics, philosophy, linguistics, and computer science to express statements about objects, their properties, and relationships between them. Inference theory in predicate logic deals with the rules and procedures for deriving valid conclusions from given premises.

Basics of Predicate Logic

Predicate logic extends propositional logic by introducing quantifiers and predicates. A predicate is a function that maps objects to truth values, and quantifiers specify the extent of the objects being considered.

For example, consider the following predicate logic statement:

∀x (P(x) → Q(x))

This statement asserts that for all objects x, if P(x) is true, then Q(x) is true.

Inference Rules

Various inference rules are used in predicate logic to make valid deductions. Some of the common rules include:

• Universal Instantiation: From a universally quantified statement, derive an instance by substituting a specific object.
• Existential Introduction: From a statement about a specific object, derive an existentially quantified statement.
• Modus Ponens: If P implies Q, and P is true, then Q is true.
• Modus Tollens: If P implies Q, and Q is false, then P is false.
• Generalization: Derive a universally quantified statement from a specific instance.
• Existential Instantiation: Derive an instance from an existentially quantified statement by introducing a new object.

Example

Let's consider the following premises:

• 1. ∀x (Human(x) → Mortal(x))
• 2. Human(Socrates)

We aim to infer the conclusion: "Mortal(Socrates)".

Using the inference rules, we can proceed as follows:

1. Apply Universal Instantiation to premise 1 with x = Socrates, we get: "Human(Socrates) → Mortal(Socrates)".
2. Apply Modus Ponens to the derived statement and premise 2, we conclude: "Mortal(Socrates)".

Conclusion

Inference theory of predicate logic provides a systematic way to draw valid conclusions from given premises. By applying inference rules correctly, we can ensure the soundness and validity of logical deductions.