The expression ‘grammatical entity’ in this section is a cover term to comprise grammatical categories and their values, grammatical formatives incl. morphemes, constructions etc.
Grammatical entities of a particular language are semiotic in nature, i.e. they are kinds of signs. Consequently, they have a functional (semantic) and a structural (formal) side. Since the association of form and function is a many-to-many relation, polysemy and homonymy, on the one hand, and synonymy and semantic similarity, on the other hand, are the rule rather than the exception. It is therefore difficult even inside a given language to identify a sign only by its significans or only by its significatum. For instance, the Latin relative clause can be identified structurally as the particular subordinate clause that is introduced by a relative pronoun and combines in a particular way with a head nominal. That, however, would exclude headless and head-internal relative clauses, which by semantic criteria should be included. Similarly, we might try to identify the relative clause by its function, e.g. a proposition with an open argument position that enriches the intension of a concept. That, however, would include non-finite constructions and other kinds of attributes that do not count as relative clauses.
At the cross-linguistic level, the situation is essentially the same, except that the gamut of variation gets more densely populated. Signs are generally not strictly synonymous or homomorphous across languages. In other words, the Latin relative clause achieves different functions from the French relative clause, and their structures differ, too. For instance:
Signs exist only inside a language; there are no signs at the interlingual (alias cross-linguistic) level (or at the level of langage). At that level, grammatical entities are abstractions over language-specific entities. Abstractions differ from the individuals they are based on in specificity of detail, but not in nature. That is to say, interlingual (= language-independent) grammatical entities are still semiotic in nature; they have both a structural and a functional side. However, both of these are abstract. Therefore an interlingual grammatical entity cannot be identified with a grammatical entity of a particular language,1 just as the concept of ‘horse’ cannot be identified with a particular horse.
The grammatical entity ‘accusative’ may serve as an example:
If we did not have the semantic part of the definition, we could not tell the accusative apart from other cases. If we did not have the structural part of the definition, we would have to identify an accusative on any direct object nominal or call a transitivizing morpheme on a verb ‘accusative’.2
Interlingual semiotic entities may differ in their degree of abstractness. Sometimes the structural part of the definition is unspecific to such an extent that it only serves to guarantee that a particular concept is grammatical rather than lexical. Similarly, the semantic part of the definition may be very abstract, as for instance in the concept of ‘case’. The logical endpoint of decreasing specificity is, of course, zero intension.
An onomasiological grammar is based on purely functional concepts in this sense; a semasiological grammar is based on purely structural concepts (s. onomasiology vs. semasiology).
1 In terms of underspecification theory one could say that interlingual grammatical entities are underspecified and that the grammar of the individual language supplies the necessary specification to pin them down to a particular grammatical construction.
2 To be sure, such things have been done in the literature, where NPs are assigned cases that are invisible or derivational verbal affixes are called “dative” etc.; but that is a different story.