Lecture 39: soundness and completeness **A proof system is sound if everything that is provable is in fact true**. In other words, if φ_{1}, …, φ_{n}⊢ψ then φ_{1}, …, φ_{n}⊨ψ. A proof system is complete if everything that is true has a proof. In other words, if φ_{1}, …, φ_{n}⊨ψ then φ_{1}, …, φ_{n}⊢ψ.

## What is the soundness and completeness of the rule?

**Soundness means that you cannot prove anything that’s wrong.** **Completeness means that you can prove anything that’s right**. In both cases, we are talking about a some fixed system of rules for proof (the one used to define the relation ⊢ ).

## What is soundness and completeness in logic?

Soundness is among the most fundamental properties of mathematical logic. **The soundness property provides the initial reason for counting a logical system as desirable.** **The completeness property means that every validity (truth) is provable**. Together they imply that all and only validities are provable.

## How do you prove a proof system is sound?

To show that our proof system is sound, we prove something stronger: **if φ1,φ2,⋯⊢ψ then φ1,φ2,⋯⊨ψ**. Assume φ1,φ2,⋯⊢ψ, so that there exists a proof tree T terminating with this line. Note that proof trees are inductively defined structures, so we can actually do a meta-inductive proof on the structure of the object proof.

## What is the difference between sound and complete?

**Soundness is the property of only being able to prove “true” things.** **Completeness is the property of being able to prove all true things**. So a given logical system is sound if and only if the inference rules of the system admit only valid formulas.

## What is soundness of A proof?

We would like them to be the same; that is, we should only be able to prove things that are true, and if they are true, we should be able to prove them. These two properties are called soundness and completeness. **A proof system is sound if everything that is provable is in fact true**.

## How do you prove completeness?

Any proof of the Completeness Theorem consists always of two parts. First we have show that all formulas that have a proof are tautologies. This implication is also called a Soundness Theorem, or soundness part of the Completeness Theorem. The second implication says: **if a formula is a tautology then it has a proof**.

## Is propositional logic complete?

Truth-functional propositional logic and first-order predicate logic are **semantically complete, but not syntactically complete** (for example, the propositional logic statement consisting of a single propositional variable A is not a theorem, and neither is its negation).