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(1) A < B < C < D
It can be reformulated in the following way: If a language has property D, then it also has properties A, B, and C; if a language has a property C, then it also has properties A and B, etc. In other words, the implicational hierarchy defines the possible combinations of properties A, B, C, and D as listed in matrix (2):
Implicational hierarchies are a useful tool in capturing linguistic generalizations pertaining the different components of the language. They are found in all subfields of grammar.
(3) is an example of an implicational hierarchy concerning the distribution of nasal phonemes across languages, which concerns dental/alveolar, bilabial, and palatal voiced nasals, respectively:
(3) /n/ < /m/ < /?/
This hierarchy defines the following possible combinations of dental/alveolar, bilabial, and palatal voiced nasals in the phoneme inventory of a language:
In other words, the hierarchy implies that there are no languages with /?/ but without /m/ and /n/, or with /?/ and /m/ but without /n/.
Number marking provides an example of implicational hierarchies in morphology.
(5) Number: singular < plural < dual < trial / paucal
On the one hand, the hierarchy implies that no language distinguishes a trial unless having a dual, and no language has dual without a plural. On the other hand, the hierarchy provides implications for the morphological marking: if the plural is coded with a certain number of morphemes, then the dual is coded with at least as many morphemes.
Implicational hierarchies also play a role in syntactic phenomena. For instance, in some languages (e.g. Tangut) the transitive verb agrees not with a subject, or the object, but with the syntactic argument which is higher on the person hierarchy.
(5) Person: first < second < third
See also: animacy.