Completeness Soundness Example

Question

I am confused between Completeness and Soundness.

I know that in Soundness every derivable formula is true and in completeness every true formula is derivable.

Provide Simple examples(in layman terms) of system which is:

1. Complete and Sound
2. Complete but Not Sound
3. Not Complete but Sound
4. Not Complete and Not Sound

Answer 1

For formal proof systems that try to prove purely logical theorems:

1. Pretty much any formal derivation system in any text as these are usually vetted to be both sound (whatever you formally derive is in fact a logical theorem) and complete (the rules are 'strong' or 'powerful' enough to prove every logical theorem).

2. A proof system that contains the Hokus Ponens inference rule: from nothing derive any statement $P$. This system can formally derive everything, so it can prove all logical theorems (hence it is complete) but also statements that are not logical theorems (hence it is not sound). It is too powerful!

3. A proof system that contains no inference rules at all. Since it cannot prove anything at all, it follows that it can't prove anything that is not true, so it is sound. On the other hand, it can't prove any of the true things either (and there is at least one logical truth), so it is not complete. It is too weak.

4. A proof system that contains exactly one inference rule Modus Bogus: from $P \rightarrow Q$ and $Q$ infer $P$. This system can derive falsehoods, while at the same time is unable to prove various truths. So it is neither sound nor complete. ... It is pretty useless, really.