2025-02-05 ### **The Relationship Between Soundness and Completeness** Soundness and completeness are **two fundamental properties of logical systems**, particularly in **formal logic, proof theory, and mathematics**. They describe the **connection between syntax (proofs) and semantics (truths)** but operate in **opposite directions**. |**Property**|**Definition**|**Direction of Guarantee**|**Key Question**| |---|---|---|---| |**Soundness**|If a statement is **provable**, it must also be **true**.|**From syntax (proofs) to semantics (truths)**|_Does proving something guarantee that it is true?_| |**Completeness**|If a statement is **true**, it must also be **provable**.|**From semantics (truths) to syntax (proofs)**|_Can we prove everything that is true?_| --- ### **1) Soundness: "No False Statements Can Be Proven"** - Ensures that **the proof system does not allow incorrect conclusions**. - A **sound system** never proves a false theorem. - If AA is provable ( ⊢A\vdash A ), then AA is semantically true ( ⊨A\models A ). - **Example:** In classical logic, if we can prove "All even numbers are divisible by 2," then this must be actually true in the domain of numbers. --- ### **2) Completeness: "All True Statements Can Be Proven"** - Ensures that **the proof system is strong enough to derive every true statement**. - A **complete system** has no missing truths—it proves everything that is true. - If AA is semantically true ( ⊨A\models A ), then AA must be provable ( ⊢A\vdash A ). - **Example:** In a complete arithmetic system, if "There are infinitely many prime numbers" is true, then a proof must exist for it within the system. --- ### **3) The Key Asymmetry: Soundness Holds, Completeness Often Fails** - **Soundness is non-negotiable**—a logical system that is unsound is **useless**, as it could prove falsehoods. - **Completeness is often impossible**—Gödel’s **Incompleteness Theorem** shows that for sufficiently powerful systems (like arithmetic), **there will always be true statements that cannot be proven within the system**. --- ### **4) The Ideal Case: When Soundness and Completeness Both Hold** - If a logical system is **both sound and complete**, then **provability and truth are perfectly aligned**: - If something is provable, it is true (**soundness**). - If something is true, it is provable (**completeness**). - **Example:** Propositional logic is both sound and complete—every tautology (a statement true in all models) can be proven within the system. --- ### **5) Why Completeness Fails in Some Systems (Gödel's Incompleteness Theorem)** - In **first-order logic**, soundness and completeness **both hold** (Gödel’s Completeness Theorem). - In **arithmetic and higher systems**, **completeness fails**—there are true statements that cannot be proven (Gödel’s First Incompleteness Theorem). - The reason: **Truth is infinite, but proof systems are finite**—some truths are too complex to be captured by finite axioms. --- ### **6) Summary: The Relationship in One Sentence** Soundness ensures that **everything provable is true**, while completeness ensures that **everything true is provable**, but powerful mathematical systems (like arithmetic) are **necessarily sound but incomplete** due to Gödel’s theorem. Would you like a deeper dive into Gödel’s proof or an example of an incomplete system?