Sound and Complete So I am in a introductory formal logic class, and my professor has asked us questions on our homework about "Sound and Complete rules of inference".
Unfortunately, I am having a hard time finding out what he means by this or where to find an intuitive definition. We are asked to say why the universal instantiation and existential instantiation are not sound and complete rules of inference. 
Any help would be appreciated. Also, I feel that I have a very vague understanding of what a "Rule of Inference" is. 
 A: You need a textbook of mathematical logic ...
See Sara Negri & Jan von Plato, Structural Proof Theory (2001), page 4.
First of all, we have a formal language, with rules for building expression (terms and formulas).
Then we need rules of inference; they are of the form: 

"If it is the case that $A$ and $B$, then it is the case that $C$"

where $A, B$ and $C$ are formulas of the language. Rules allows us to move from given formulas to new ones.
In Hilbert-style systems, (also called axiomatic systems), we have a number of basic forms of assertion (axioms), such as $A \rightarrow (A \lor B)$ and at least one rule of inference (tipically : modus ponens).
In natural deduction systems, there are only rules of inference, plus assumptions
to get derivations started.
With these "ingredients", we may use our proof system : we call theorems of the proof system all the formulas that are deducible, i.e. “produced” starting from the axioms with a finite number of applications of the rules of inference.
The basic properties we are requesting to it are the following : 

1) it  must be sound : assuming a model for our axioms (so that they are true in it), we want that all the theorems must be true in that model (i.e. rules of inference must preserve truth); 
2) it must be complete :  we want that all the logical consequences of our axioms must be deducible from the axioms.

