First-Order Logic

First-order logic (FOL), also known as predicate logic or first-order predicate calculus, represents a fundamental advancement in formal logical systems, providing a rich framework for expressing mathematical statements and reasoning about objects and their properties.

Core Components

Syntax

1. Logical Connectives

   • Inherited from Propositional Logic
   • Includes ∧ (and), ∨ (or), ¬ (not), → (implies), ↔ (if and only if)

2. Quantifiers

   • Universal quantifier (∀): "for all"
   • Existential quantifier (∃): "there exists"
   • Forms the basis for Quantification Theory

3. Basic Elements

   • Constants and variables
   • Predicate Symbols
   • Function Symbols
   • Well-Formed Formulas

Semantic Framework

Model Theory

• Mathematical structures that interpret formulas
• Model Theoretic Semantics
• Truth Conditions
• Semantic Interpretation

Interpretation

1. Domain of discourse
2. Variable Assignment
3. Satisfaction Relation
4. Formal Semantics

Proof Systems

Formal Deduction

• Natural Deduction
• Sequent Calculus
• Hilbert Systems
• Resolution

Key Properties

1. Soundness
2. Completeness Theorem
3. Compactness Theorem
4. Löwenheim-Skolem Theorem

Applications

Mathematics

• Foundation for Set Theory
• Peano Arithmetic
• Group Theory
• Real Analysis

Computer Science

• Automated Theorem Proving
• Logic Programming
• Database Theory
• Formal Verification

Artificial Intelligence

• Knowledge Representation
• Expert Systems
• Automated Reasoning
• Semantic Networks

Limitations

Expressiveness Boundaries

• Cannot quantify over predicates
• Limited to first-order properties
• Leads to Higher-Order Logic
• Type Theory

Computational Aspects

• Undecidability
• Computational Complexity
• Decision Problems
• Algorithm Analysis

Historical Development

Key Contributors

1. Gottlob Frege

   • Begriffsschrift
   • Foundation of modern logic

2. Alfred Tarski

   • Model Theory
   • Semantic theory

3. Kurt Gödel

   • Completeness Theorem
   • Formal Systems

Modern Extensions

Contemporary Variants

1. Many-Sorted Logic
2. Modal First-Order Logic
3. Fuzzy Logic
4. Temporal Logic

Advanced Applications

• Program verification
• Formal Methods
• Specification Languages
• Software Engineering

Relationship to Other Logics

Hierarchy

1. Extends Propositional Logic
2. Subset of Higher-Order Logic
3. Related to Modal Logic
4. Distinct from Non-Classical Logic

See Also

• Mathematical Logic
• Proof Theory
• Model Theory
• Formal Languages
• Philosophy of Logic

This entry builds upon the foundations established in Mathematical Logic while focusing specifically on the structure, capabilities, and significance of First-Order Logic within the broader landscape of formal systems and their applications.