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What is compositionality?

What is compositionality? Here are some definitions.

Noun
  1. The property of being compositional.
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Examples
Montague suggested a perspicuous way to capture the principle of compositionality formally.
That compositionality is the key notion for the analysis of idioms is emphasized in most accounts of idioms.
This trait of phrasal verbs is also what makes them interesting for linguists, since they appear to defy the principle of compositionality.
The main burden of this chapter is to argue against strict compositionality and for partial compositionality.
It's hard for me to believe that any creature could develop anything much like human language without at least some limited form of recursive compositionality.
Other exceptions to compositionality are idioms, figures of speech, and expressions which are subject to pragmatic interpretations.

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