  The logic of nulls in databases has been subject of investigation since their
introduction in Codd's Relational Model, which is the foundation of the SQL
standard. We show a logical characterisation of a first-order fragment of SQL
with null values, by first focussing on a simple extension with null values of
standard relational algebra, which captures exactly the SQL fragment, and then
proposing two different domain relational calculi, in which the null value is a
term of the language but it does not appear as an element of the semantic
interpretation domain of the logics. In one calculus, a relation can be seen as
a set of partial tuples, while in the other (equivalent) calculus, a relation
is horizontally decomposed as a set of relations each one holding regular total
tuples. We extend Codd's theorem by proving the equivalence of the relational
algebra with both domain relational calculi in presence of SQL null values.
