Existential Quantification - SQL
Existential quantification-stating that something is true of at least one object under consideration-can be expressed by OR(r,c), meaning that at least one object that satisfies a predicate for r also satisfies c, and IS_NOT_EMPTY(r). The names for the aggregate operators AND and OR reflect the facts that when we confine our attention to finite sets, universal and existential quantification are equivalent to repeated invocations of dyadic AND and dyadic OR, respectively. Note that AND(r,c) is equivalent to COUNT(r) = COUNT(r WHERE c), and OR(r,c) is equivalent to COUNT(r WHERE c) > 0 and also to IS_NOT_EMPTY(r WHERE c).
Quantification also appears in various guises in SQL, but its meaning is muddied by those same two violations of relational theory that we have already seen muddying the waters: duplicate rows and NULL. For example, SQL's (SELECT COUNT(*) FROM r), a so-called scalar subquery (because it is an expression denoting a table with one row and one column, enclosed in parentheses), denotes the number of rows in the table r, but can we really say that this represents the number of objects that satisfy a predicate for r, if the same row can be counted more than once, or if NULL appears in place of a column value in some row of r? In fact, what might it mean to say that a row does or does not satisfy a predicate? In 2VL we say that object a satisfies predicate P(x) exactly when P(a) is true. Does this still hold in 3VL, or might SQL deem a to satisfy P(x) also when P(a) is unknown? Well, it turns out that SQL uses both interpretations, depending on the context, as we shall discover.