In linguistic

Linguistics

Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....

 semantics

Semantics

Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....

, a downward entailing (in short: "DE") expression is one that denotes

Denotation

This word has distinct meanings in other fields: see denotation . For the opposite of Denotation see Connotation.*In logic, linguistics and semiotics, the denotation of a word or phrase is a part of its meaning; however, the part referred to varies by context:** In grammar and literary theory, the...

 a monotone decreasing

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

 function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. A downward entailing expression reverses the relation of semantic strength among expressions. An expression like "run fast" is semantically stronger than the expression "run" since "run fast" is true of fewer things than the latter. Thus the sentence:

John ran fast.

entails

Entailment (pragmatics)

In pragmatics , entailment is the relationship between two sentences where the truth of one requires the truth of the other ....

 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.

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