Propositional calculus forms the bedrock of logical reasoning. It introduces atomic propositions, logical connectives, and well-formed formulas, providing a framework for constructing complex logical statements. These building blocks allow us to represent and analyze arguments systematically.
The syntax and semantics of propositional logic work hand in hand. While syntax defines the structure of logical statements, semantics assigns meaning through truth values and truth tables. This interplay enables us to evaluate the validity of arguments and identify tautologies, contradictions, and contingencies.
Foundations of Propositional Calculus
Syntax of propositional calculus
- Atomic propositions form simplest units represented by lowercase letters (p, q, r)
- Logical connectives include negation (¬), conjunction (∧), disjunction (∨), implication (→), biconditional (↔)
- Well-formed formulas (WFFs) defined recursively, atomic propositions serve as base case, combinations of WFFs using connectives form inductive step
- Parentheses disambiguate complex formulas (p ∧ (q ∨ r))
- Precedence rules establish order of operations for logical connectives (¬, ∧, ∨, →, ↔)
Semantics of propositional formulas
- Truth values operate in binary system: True (T) and False (F)
- Truth tables evaluate compound propositions (p ∧ q, p → q)
- Connectives interpreted: negation reverses truth value, conjunction true when both operands true, disjunction false when both operands false
- Implication false only when antecedent true and consequent false, biconditional true when operands have same truth value
- Tautologies always true regardless of input (p ∨ ¬p)
- Contradictions always false regardless of input (p ∧ ¬p)
- Contingencies depend on truth values of components (p → q)
Advanced Concepts and Applications
Soundness and completeness proofs
- Soundness: provable formulas are valid, proved by induction on proof length
- Completeness: valid formulas are provable, proved by contrapositive method
- Consistency ensures no contradiction derived from axioms
- Independence of axioms prevents derivation of one axiom from others
- Compactness theorem: satisfiability of every finite subset implies satisfiability of entire set
Natural deduction in propositional logic
- Introduction rules: conjunction (p, q ⊢ p ∧ q), disjunction (p ⊢ p ∨ q), implication (conditional proof)
- Elimination rules: conjunction (p ∧ q ⊢ p), disjunction (proof by cases), modus ponens (p, p → q ⊢ q)
- Derived rules: modus tollens (p → q, ¬q ⊢ ¬p), hypothetical syllogism ((p → q) ∧ (q → r) ⊢ p → r)
- Proof strategies include direct proof, proof by contradiction, proof by cases
- Fitch-style notation uses vertical and horizontal lines to indicate scope
- Subproofs form nested arguments within larger proof structure