# Soundness and Completeness

Section 4.8 showed us that we can prove two statements are *semantically equivalent* with truth tables and *provably equivalent* with deduction proofs. Does it matter which approach we use? Will there ever be a time when two statements are semantically equivalent but not provably equivalent, or vice versa? Will there ever be a time when a set of premises semantically entails a conclusion, but that the premises do not prove (using our deduction proofs) the conclusion, or vice versa?

These questions lead us to the notions of *soundness* and *completeness*. Formal treatment of both concepts is beyond the scope of this course, but we will introduce both definitions and a rough idea of the proofs of soundness and completeness in propositional logic.

## Soundness

A proof system is *sound* if everything that is provable is actually true. Propositional logic is sound if when we use deduction rules to prove that `P1, P2, ..., Pn ⊢ C`

(that a set of premises proves a conclusion) then we can also use a truth table to show that `P1, P2, ..., Pn ⊨ C`

(that a set of premises semantically entails a conclusion).

**Propositional logic is, in fact, sound.**

To get an idea of the proof, consider the `∧e1`

deduction rule. It allows us to directly prove:

`P ∧ Q ⊢ P`

I.e., if we have `P ∧ Q`

as a premise or as a claim in part of a proof, then we can use `∧e1`

to conclude `P`

. We must also show that:

`P ∧ Q ⊨ P`

I.e., that any time `P ∧ Q`

is true in a truth table, then `P`

is also true. And of course, we can examine the truth table for `P ∧ Q`

, and see that it is only true in the cases that `P`

is also true.

Consider the `∧i`

deduction rule next. It allows us to directly prove:

`P, Q ⊢ P ∧ Q`

I.e., if we have both `P`

and `Q`

as premises or claims in part of a proof, then we can use `∧i`

to conclude `P ∧ Q`

. We must also show that:

`P, Q ⊨ P ∧ Q`

I.e., that any time both `P`

and `Q`

are true in a truth table, then `P ∧ Q`

is also true. And of course, we can examine the truth table for `P ∧ Q`

and see that whenever `P`

and `Q`

are true, then `P ∧ Q`

is also true.

To complete the soundness proof, we would need to examine the rest of our deduction rules in a similar process. We would then use an approach called *mathematical induction* (which we will see for other applications in Chapter 7) to extend the idea to a proof that applies multiple deduction rules in a row.

## Completeness

A proof system is *complete* if everything that is true can be proved. Propositional logic is complete if when we can use a truth table to show that `P1, P2, ..., Pn ⊨ C`

, then we can also use deduction rules to prove that `P1, P2, ..., Pn ⊢ C`

.

**Propositional logic is also complete.**

We assume that `P1, P2, ..., Pn ⊨ C`

, and we consider the truth table for `(P1 ∧ P2 ∧ ... ∧ Pn) → C`

(since that will be a tautology whenever `P1, P2, ..., Pn ⊨ C`

). In order to show propositional logic is complete, we must show that we can use our deduction rules to prove `P1, P2, ..., Pn ⊢ C`

.

The idea is to use LEM for each propositional atom `A`

to obtain `A ∨ ¬A`

(corresponding to the truth assignments in the `(P1 ∧ P2 ∧ ... ∧ Pn) → C`

truth table). We then use OR elimination on each combination of truth assignments, with separate cases for each logical operator being used.