Downward entailing

In linguistic semantics, a downward entailing (in short: "DE") expression is one that denotes a monotone decreasing function. A downard entailing expression reverses the relation of semantic strength among expressions. An expression like "run fast" is semantically stronger than the expression "run" since the former is true of fewer things than the latter. Thus the sentence:
 * John ran fast.

entails the sentence:
 * John ran.

Examples of DE expressions include "not", "nobody", "few people", "at most two boys". They reverse the entailment relation of sentences formed with the predicates "run fast" and "run, for example. So the following sentence:
 * Nobody ran.

entails that:
 * Nobody ran fast.

And the sentence:
 * At most two boys ran.

entails that:
 * At most two boys ran fast.

Negative Polarity Items are said to occur in DE environments:
 * Only one person saw anything.

Is a DE expression which entails that:
 * Only one person saw any wrongdoing.

While the following is ungrammatical:
 * Everyone saw anything / any wrongdoing.

Because any is occurring in a non-DE environment.

An upwards entailing expression is one that preserves the relation of semantic strength among expressions.