# Lecture 39: soundness and completeness

We have completely separate definitions of "truth" (⊨) and "provability" (⊢). 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. 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*.

## Sketch of proof of soundness

To prove that the set of natural deduction rules introduced in the previous lecture is sound with respect to the truth-table semantics given two lectures ago, we can use induction on the structure of proof trees.

Let *X* be the set of well-formed proofs. Then *X* is an inductively defined set; the set of rules of the proof system are the rules for constructing new elements of *X* from old.

Let *P*(*x*) be the statement ``if *x* is a valid proof tree ending with *φ*_{1}, …, *φ*_{n}⊢*ψ* then *φ*_{1}, …, *φ*_{n}⊨*ψ*''. We can prove ∀*x*∈*X*, *P*(*x*) by structural induction; we simply have to consider each inference rule; for the rules with subgoals above the line we can inductively assume entailment.

## Sketch of proof of completeness

The idea behind proving completeness is that we can use the law of excluded middle and ∨ introduction (as in the example proof from the previous lecture) to separate all of the rows of the truth table into separate subproofs; for the interpretations (rows) that satisfy the assumptions (and thus the conclusion) we can do a direct proof; for those that do not we can do a proof using reductio ad absurdum.

In other words, we can build a proof tree corresponding to each row of the truth table and snap them together using the law of excluded middle and ∨ elimination.

## First order logic

We also introduced the syntax and started discussing the semantics of first-order logic, see the slides for the next lecture for details.