http://en.wikipedia.org/wiki/Null_(SQL)#Three-valued_logic_.283VL.29
Basically, three value logic is true/false/null, and the boolean/comparison operators will function in certain ways when comparing true == null, null == null, etc.
Boolean logic by definition uses only two values. To me, this question doesn't make sense. If he would asked how can you define or extend Boolean logic so that it could use three-value system, it would be clearer.
Boolean values by definition consist of two options: True and False, or some representation of them like 1/0, ON/OFF, YES/NO, etc.
Many database systems also allow you to store a NULL value in fields designated as Boolean to allow them to represent three values: Yes, No, and Unknown (NULL).
The Boolean Operators are AND, OR, and NOT.
Comparison Operators are some form of EQUALS or NOT EQUALS.
Operations with TRUE/FALSE values on both ends are obvious:
TRUE or FALSE -> TRUE
NOT TRUE -> FALSE
TRUE=TRUE -> TRUE
What he is getting at are the consequences of adding the NULL (Unknown) value:
TRUE or UNKNOWN(NULL) -> ???
NOT UNKNOWN(NULL) -> ???
TRUE=UNKNOWN(NULL) -> ???