2.4 Soundness and completeness

Soundness and completeness are crucial concepts in logic. Soundness ensures a system only proves valid formulas, preventing false conclusions. Completeness guarantees all valid formulas can be proven, making the system comprehensive.

These properties are vital for logical systems' reliability. Proving soundness involves showing axioms and inference rules preserve validity. Proving completeness often uses countermodels or proof-theoretic methods to demonstrate all valid formulas are provable.

Soundness and Completeness in Propositional and First-Order Logic

Soundness and completeness in logic

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• Soundness means a logical system proves only valid formulas
  • If a formula is provable using the system's inference rules, it must be true under all interpretations satisfying the axioms
  • Soundness prevents the logical system from proving any false statements (inconsistencies)
• Completeness means a logical system can prove all valid formulas
  • If a formula is true under all interpretations satisfying the axioms, it can be derived using the system's inference rules
  • Completeness ensures the logical system is capable of proving all true statements (tautologies)

Proving soundness of logical systems

• To prove soundness, demonstrate that for every formula $A$, if $A$ is provable in the system, then $A$ is valid
• Proceed by induction on the structure of the derivation of $A$
  • Base case: Show all axioms of the system are valid
  • Inductive step: Show if the premises of an inference rule are valid, the conclusion is also valid
• By showing each derivation step preserves validity, conclude the entire derivation produces a valid formula
• Examples of sound logical systems:
  • Classical propositional logic
  • First-order logic with standard inference rules

Proving completeness of logical systems

• To prove completeness, demonstrate that for every formula $A$, if $A$ is valid, then $A$ is provable in the system
• Use the contrapositive approach:
  1. Assume $A$ is not provable
  2. Construct a countermodel satisfying the axioms but not $A$
  3. The countermodel serves as a witness to the non-validity of $A$
• Alternatively, use proof-theoretic methods:
  • Construct a canonical model
  • Use a complete proof calculus (sequent calculus, natural deduction)
• Examples of complete logical systems:
  • Classical propositional logic
  • First-order logic with standard axiomatization

Significance of soundness vs completeness

• Soundness theorem guarantees the logical system is reliable and trustworthy
  • Ensures any proven formula within the system is true
  • Prevents deriving false or contradictory statements
  • Gives confidence in the results obtained through logical deduction
• Completeness theorem demonstrates the logical system is sufficiently expressive
  • Ensures all valid formulas can be proven within the system
  • Establishes the system captures all logical consequences of its axioms
  • Allows relying on the system as a tool for establishing truth and validity
• Together, soundness and completeness provide a strong foundation for using logical systems in reasoning and argumentation
  • They establish the correspondence between semantic truth and syntactic provability
  • They justify the use of logical systems for drawing reliable conclusions and verifying arguments