Soundness of a formal system means that the axioms are true in every intended interpretation and that no rule of inference can derive a false statement from true ones in any of the intended interpretations. From this it follows that every theorem is true in every intended interpretation. Consistency means that a statement and its negation cannot both be theorems. Consistency doesn’t depend on the choice of interpretations or interpretations, except to the extent that we need to identify what negation means in the system. Completeness means that every statement which is true in every intended interpretation is a theorem.
The intended interpretations of our natural deduction system are the ones discussed earlier for zeroeth order logic in general. There is an interpretation for each possible assignment of truth values to variables.
Our natural deduction system is sound. There are no axioms so it’s trivially the case that every axiom is true in every intended interpretation. It’s also impossible to derive a false statement from true ones in any of the intended interpretations.
If you believe that a statement and its negation can’t simultaneously be true then consistency follows from soundness. It’s also possible to prove the consistency of natural deduction directly, but I won’t.
The completeness of the natural deduction system follows from two facts proved earlier, that every tautology has a closed tableau and that any closed tableau can be converted into a formal proof.
For the two axiomatic systems considered earlier it’s fairly easy to prove soundness. For Łukasiewicz it’s also straightforward to prove consistency. For Nicod the main obstacle to proving consistency is the lack of a negation operator in the language. Proving completeness of either system is possible, but is more difficult than for natural deduction.