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For example, is "Donald" a morpheme of the English language?

I can see reasons for and against.

Reasons for:

• It allows us to say stuff like "a language is a function from sequences of morphemes of it to meanings". If "Donald" is not a morpheme of English, then how does English assign a meaning to the phrase "Donald Trump is president of the USA"?

Reasons against:

• It is extremely arbitrary. If someone decides to name their kid some name that had never been given before, is it suddenly a new morpheme of English? That seems way too arbitrary.

"Donald" isn't a morpheme. Morphemes have grammatical meaning, not lexical meaning.

You seem to want to ask whether "Donald" is a lexeme. There is a category of words that includes names: proper nouns (or proper names). Proper names have meaning, and they behave like other English nouns in that you can make them possessive (Donald Trump's presidency) or even plural (as in Dr. Seuss' story Too Many Daves). Grammatically, it behaves similarly to other lexemes. However, proper names are usually not included in lexicons.

It is extremely arbitrary. If someone decides to name their kid some name that had never been given before, is it suddenly a new morpheme of English? That seems way too arbitrary.

All words are arbitrary, in that there is no relation to the sound of any particular word to what it represents. New words, and also new proper names, are invented all the time. If you decided to invent a new name and put it in a sentence in place of an existing proper name, you would still be understood. Inventing a word or name would indeed add a lexeme to English; but since language is a medium of communication, it only works if it becomes part of the vocabulary of the community of speakers. No individual can change others' language until the others adopt the change. See Saussure on the immutability and mutability of the sign.

It depends on your definition of morpheme.
S. Anderson cites an 1880 characterization by Baudoin de Courtenay (Stankiewcicz translation) that a morpheme is "that part of a word which is endowed with psychological autonomy and is for the very same reason not further divisible. It consequently subsumes such concepts as the
root (radix), all possible affixes, (suffixes, prefixes), endings
which are exponents of syntactic relationships, and the like". This is different from Bloomfield's definition as "'a linguistic form which bears no partial phonetic-semantic resemblance to any other form' i.e. a form that contains no sub-part that is both phonetically and semantically identical with a part of some other form". However, in both cases, a root is a morpheme (and a proper name is a root).

The question then is whether there is some way to empirically determine whether a root is in fact a morpheme. This is analogous to trying to determine whether a particular sound is "marked" – how do you determine that one definition is correct and the other is wrong? Indeed, it is not obvious that "morpheme" is a necessary technical concept.

There is no requirement in linguistic theory that the set of roots be closed, though they might be in some language. In English, they class "noun" is certainly open.