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In linguistic semantics, a **downward entailing** (**DE**) propositional operator is one that constrains the meaning of an expression to a lower number or degree than would be possible without the expression. For example, "not," "nobody," "few people," "at most two boys." Conversely, an **upward-entailing** operator constrains the meaning of an expression to a higher number or degree, for example "more than one." A context that is neither downward nor upward entailing is *non-monotone*^{[citation needed]}, such as "exactly five."

A downward-entailing operator reverses the relation of *semantic strength* among expressions. An expression like "run fast" is semantically *stronger* than the expression "run" since "John ran fast" entails "John ran," but not conversely. But a downward-entailing context reverses this strength; for example, the proposition "At most two boys ran" entails that "At most two boys ran fast" but not the other way around.

An upward-entailing operator preserves the relation of semantic strength among a set of expressions; for example "more than three ran fast" entails "more than three ran" but not the other way around.

Ladusaw (1980) proposed that downward entailment is the property that licenses polarity items. Indeed, "Nobody saw *anything*" is downward entailing and admits the negative polarity item *anything*, while *"I saw *anything*" is ungrammatical (the upward-entailing context does not license such a polarity item). This approach explains many but not all typical cases of polarity item sensitivity. Subsequent attempts to describe the behavior of polarity items rely on a broader notion of nonveridicality.

Downward entailment does not explain the licensing of *any* in certain contexts such as with *only*:

- Only John ate any vegetables for breakfast.

This is not a downward-entailing context because the above proposition does not entail “Only John ate kale for breakfast” (John may have eaten spinach, for example).

Von Fintel (1999) claims that although *only* does not exhibit the classical DE pattern, it can be shown to be DE in a special way. He defines a notion of Strawson-DE for expressions that come with presuppositions. The reasoning scheme is as follows:

- P → Q
- [[ only John ]] (P) is defined.
- [[ only John ]] (Q) is true.
- Therefore, [[ only John ]] (P) is true.

Here, (2) is the intended presupposition. For example:

- Kale is a vegetable.
- Somebody ate kale for breakfast.
- Only John ate any vegetables for breakfast.
- Therefore, only John ate kale for breakfast.

Hence *only* is a Strawson-DE and therefore licenses *any*.

Giannakidou (2002) argues that Strawson-DE allows not just the presupposition of the evaluated sentence but just any arbitrary proposition to count as relevant. This results in over-generalization that validates the use if *any* in contexts where it is, in fact, ungrammatical, such as clefts, preposed exhaustive focus, and *each*/*both*:

- * It was John who talked to
*anybody*. - * John talked to
*anybody*. - * Each student who saw anything reported to the Dean.
- * Both students who saw anything reported to the Dean.

- Ladusaw, William (1980).
*Polarity Sensitivity as Inherent Scope Relations*. Garland, NY. - Von Fintel, Kai (1999). "NPI-Licensing, Strawson-Entailment, and Context-Dependency".
*Journal of Semantics*(16): 97–148. - Giannakidou, Anastasia (2002). "Licensing and sensitivity in polarity items: from downward entailment to nonveridicality" (PDF). In Maria Andronis; Anne Pycha; Keiko Yoshimura (eds.).
*CLS 38: Papers from the 38th Annual Meeting of the Chicago Linguistic Society, Parasession on Polarity and Negation*. Retrieved 2011-12-15.