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All logical operators evaluate to 1 (TRUE), 0 (FALSE) or NULL (unknown, which is in most cases the same as FALSE):

NOT

Logical NOT. Evaluates to 1 if the operand is 0, otherwise evaluates to 0. Exception: NOT NULL evaluates to NULL:

mysql> SELECT NOT 1;
        -> 0
mysql> SELECT NOT NULL;
        -> NULL
mysql> SELECT ! (1+1);
        -> 0
mysql> SELECT ! 1+1;
        -> 1

The last example produces 1 because the expression evaluates the same way as (!1)+1.

Logical OR. Evaluates to 1 if either operand is not 0 and not NULL:

mysql> SELECT 1 || 0;
        -> 1
mysql> SELECT 0 || 0;
        -> 0
mysql> SELECT 1 || NULL;
        -> 1

AND

Logical AND. For non-NULL operands, evaluates to 1 if both operands are non-zero and to 0 otherwise. Produces NULL if either operand is NULL:

mysql> SELECT 1 && 1;
        -> 1
mysql> SELECT 1 && 0;
        -> 0
mysql> SELECT 1 && NULL;
        -> NULL

XOR

Logical XOR. For non-NULL operands, evaluates to 1 if only one of the operators is non-zero. Produces NULL if either operand is NULL:

mysql> SELECT 1 XOR 1;
        -> 0
mysql> SELECT 1 XOR 0;
        -> 1
mysql> SELECT 1 XOR NULL;
        -> NULL

a XOR b is equal to (a AND (NOT b)) OR ((NOT a) and b).