Propositional Logic
A fundamental system of formal logic that analyzes the relationships between propositions using truth-functional connectives and rules of inference.
Propositional Logic
Propositional logic, also known as sentential logic or statement logic, forms one of the foundational systems of formal logic. It deals with propositions (statements that are either true or false) and the logical relationships between them.
Core Components
Atomic Propositions
- Basic statements that cannot be broken down further
- Typically represented by letters (p, q, r)
- Must have definite truth values
- Examples: "It is raining," "The sun is shining"
Logical Connectives
- Conjunction (AND, ∧)
- Disjunction (OR, ∨)
- Negation (NOT, ¬)
- Implication (IF-THEN, →)
- Equivalence (IF AND ONLY IF, ↔)
Truth Functions
The truth value of complex propositions is determined through truth tables, which systematically show how the truth values of component propositions combine under different logical operations.
Fundamental Laws
Propositional logic upholds several key principles:
Inference Rules
Basic Rules
- Modus Ponens
- Modus Tollens
- Hypothetical syllogism
- Disjunctive syllogism
Derived Rules
- Simplification
- Conjunction
- Resolution principle
Applications
Computer Science
- Boolean algebra implementations
- Circuit design
- Programming languages
Mathematics
- Proof theory
- Mathematical logic
- Foundation for set theory
Artificial Intelligence
Limitations
While powerful, propositional logic has certain constraints:
- Cannot analyze internal structure of atomic propositions
- Requires extension to predicate logic for quantification
- Cannot handle modal or temporal aspects (requires modal logic)
Historical Development
The modern form of propositional logic emerged from:
- Ancient Greek contributions (Stoic logic)
- Medieval developments (scholasticism)
- 19th-century formalization (George Boole)
Contemporary Significance
Propositional logic remains crucial for:
- Foundation for more complex logical systems
- Computer architecture
- Formal verification
- Critical thinking
Extensions and Variations
Modern developments include: