The ways in which mathematical logic has been used in database theory and database systems are likely to require extension for phenomenal data mining..
Database theory and database system commonly use mathematical logic to represent facts. However, subsets of logic are adequate for most present database systems. For example, the databases can often be considered as collections of ground literals, i.e. predicate symbols applied to constants. More general sentences are used as rules and given a special status.
One example that immediately arises in the supermarket problem is the fact that the customers who bought particular baskets are unknown, and it is not known a priori whether two given baskets were bought by the same customer. In Prolog and similar systems, the unique names hypothesis, i.e. that distinct constant expressions represent distinct objects, is usually built into the system.
Consider
which asserts that baskets b1 and b2 were purchased by the same
customer. Unlike the common situation in database systems, the truth
of this assertion is not in the database. Neither is a unique
identifier for 
available.
The set of customers is unknown, although it is known to exist. Facts about it may be known or conjectured.