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Three valued logic – NULL in Sql Server


In SQL the logical expressions (predicates) can evaluate TRUE, FALSE and UNKNOWN. The third result is unique to “SQL world“ and is caused by NULL value.  This is called Three Valued Logic (3VL).   NULL represents the missing or UNKNOWN value.

The logic could be confusing because Boolean, two valued logic expressions can evaluate only to TRUE or FALSE – a comparison between two KNOWN values gives us TRUE or FALSE.

With NULL as a mark or a placeholder, we have an UNKNOWN value, so the question is how to compare a known with an unknown value or an unknown with an unknown.

The logical expressions with NULLs  can evaluate to NULL, TRUE and FALSE e.g.

nullExpressionResult

The result of NULL negations can be confusing too

  • The opposite of True is False (NOT True = False)
  • The opposite of False is True (NOT False = True)
  • The opposite of UNKNOWN is UNKNOWN (NOT NULL = NULL)

Conjunctive(AND) predicates with Nulls evaluates:

  • True  AND NULL  => NULL
  • False AND NULL => False

and Disjunctive (OR) predicates with Null evaluates:

  • True OR NULL => True
  • False OR NULL => NULL
  • NULL OR NULL => NULL

The logic is confusing and it was source of an ongoing discussion about NULLS for a long time.
The important thing to note is to put the logic in the context of DB design and to understand how Sql Server treats NULLs in different scenarios. Knowing that we can  benefit from using the UNKNOWNS and avoid possible logical errors.

Sql Server is INCONSISTENT when evaluates the logical expressions with NULL values – The same expressions may produce different results TRUE, FALSE and sometimes UNKNOWN.
The following examples shows how SqlServer deals with NULLs in different elements of Tsql language.

 QUERY FILTERS (ON,WHERE,HAVING)

Create a couple of test tables:

..and add some data

Test1: Select all the products over $500.00.
The product with the UNKNOWN(NULL) price will be excluded from the result-set as it was treated as FALSE. Only these rows for which the predicate evaluates to TRUE will be included in the final result-set.

Null1

We’ll get the similar result if we try to select all products with not unknown price like:

The result will be an empty set. Not NULL is still UNKNOWN and is treated as FALSE.

The UNKNOWN result is treated as FALSE when evaluated by ON filter.

As we can see from the result-set, the product “Gyroscope” is excluded from the result as NULL = NULL evaluated to UNKNOWN and was treated as FALSE.

Null2

CHECK Constraint

In case of CHECK constraint, SQL Server treats UNKNOWN result as TRUE.
The product “Gyroscope” has UNKNOWN price. The CHECK constraint on the price column allows only those values which evaluates to TRUE when compared with 0.  In this case Sql Server evaluated;

NULL >0 => True

opposed to the previous example when

.. NULL > 500 => False

To confirm the CHECK constraint behavior we can insert a new product into #Products table without getting a Check constraint error:

This can make sense if we put the NULL logic into the context of the DB design, e.g. All products must have price greater than $0 or NULL if the price hasn’t been decided yet.

Null3UNIQUE Constraint and Set operators

It was mentioned earlier that  NULL = NULL evaluates to UNKNOWN. This means that NULLs are distinct( no two NULLs are equal)

The UNIQUE constraint treats NULL as a distinct value in the context of all other values in the column i.e ProductCategoryId can have only one NULL value and the distinct integers. However, allowing only single NULL value(inserting another NULL value will throw Violation of UNIQUE KEY constraint error), the constraint treats all UNKNOWN values(NULLS) as equal or not distinct.

UniqueConstraintError

The table #ProductCategories has an UNIQUE constraint (enforced by a clustered index) on ProductCategoryId column.

The SET operators UNION, UNION ALL* , EXCEPT,  INTERSECT

Sql Server’s  Set operators treat NULLs as non distinct, in other words  as equal.
e.g. Select the rows from tables @Tab1 and @Tab2 that are common for both tables.

NullsIntersection

As we can see from the result-set, the SET operator treats NULL values as equal e.g  Element (1,NULL ) = Element (1,NULL) and therefore belongs to “the red zone”, whereas e.g Element(5,Bahar) <>(5,Peter) and therefore was left out from the result-set.

*UNION ALL is a multi set operator that combines two sets into one non distinct set (takes all elements from both sets)

 Aggregate Functions

All Aggregate functions except COUNT(*) ignore NULLs.

NullAgg

Note: COUNT(col1) eliminate NULLs as the other aggregate functions. However, if we try something like: SELECT COUNT(NULL+1) the function returns 0 eliminating the result of the expression, which is NULL 🙂
Another interesting thing with this, is that SELECT COUNT(NULL) throws an error – Null does not have any data type (COUNT() always does implicit conversion to INT) and e.g. SELECT COUNT(CONVERT(varchar(max),NULL)) returns 0 – now COUNT() has a datatype to work with.

Cursors (Order by)

When a query requests sorting the result-set becomes cursor – sets are not ordered. Sql Server treats NULL values as distinct and sorts the unknowns first if the sort is Ascending (Descending sort order puts NULL values  last)

NullOrder

*Just a small digression. One way to show NULL values last when  sorting in Ascending direction is to use tsql’s ability to sort record-set by non-selected columns and/or expressions.

The NULL expressions results can explain the “unexpected” results  when used in anti-semi joins, but that will be covered in a separate post.

Conclusion

Three valued logic is unique to DB wold. ANSI SQL Standard provides rules/guidelines about NULL values. However, different RDBMS may handle NULLs  differently in different situations. It is important to understand how Sql Server treats NULL values in different elements of tsql language. Using NULLs  in the context of DB design can make 3VL work for you.

Thanks for reading.

Dean Mincic