Downward entailing

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In linguistic semantics, a downward entailing (DE) propositional operator is one that denotes a monotone decreasing function. 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.

Examples of DE contexts 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. The proposition “Nobody ran” entails that “Nobody ran fast”. The proposition “At most two boys ran” entails that “At most two boys ran fast”.

Conversely, an upward entailing operator is one that preserves the relation of semantic strength among a set of expressions (for example, “more”). A context that is neither downward nor upward entailing is non-monotone, such as “exactly”.

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:

  1. P → Q
  2. [[ only John ]] (P) is defined.
  3. [[ only John ]] (Q) is true.
  4. Therefore, [[ only John ]] (P) is true.

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

  1. Kale is a vegetable.
  2. Somebody ate kale for breakfast.
  3. Only John ate any vegetables for breakfast.
  4. 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.

See also[edit]


  • 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. CLS 38: Papers from the 38th Annual Meeting of the Chicago Linguistic Society, Parasession on Polarity and Negation. Retrieved 2011-12-15.