While I am not interested in hearing the common distinction made in introductory text-books, I am interested in hearing what meaningful distinction there can be between morphology and syntax. Is there any structural, or rather formal, difference between the two? Also if that distinction requires the concept of word, how does one formally describe that?
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The Oxford Concise Dictionary of Linguistics defines morphology as 'The study of the grammatical structures of words and the categories realised by them', and syntax as 'The study of grammatical relations between words and other units within the sentence'.
As you point out in your question these (pretty standard) definitions depend on the notion 'word' to delimit the domains of each subject. The notion of 'word' is problematic as there's no good definition that applies across all languages. It seems that it's best (or easiest) to define 'word' on a language by language basis, although even then there are problems as different kinds of 'word' can often be distinguished depending on the sets of criteria used: often there is a 'prosodic word' (defined on phonological bases) which differs from the 'grammatical word' (defined in terms of morphosyntax).
If 'word' can be precisely defined in a given language then the domains of morphology and syntax can be fairly clearly distinguished, but if not then there is not a clear boundary.
So cross-linguistically the notion of 'word' is fuzzy, as are the terms 'morphology' and 'syntax'. But for most languages, and in most cases, it's still pretty clear what's meant by these terms; and of course there's the term 'morphosyntax' for boundary cases. So while these two terms are not precise they are still useful as a general way of referring to certain domains within a language.
The very first thing on the Syntax Topics handout for my linguistics classes was:
Syntax and morphology are the two parts of grammar.
By which is meant that things that take place within the word are morphological, and things that take place between words are syntactic. Without a sense of "word", there would be no easy way to distinguish. But we'd need to distinguish anyway, because there are big differences between grammar as expressed morphologically and grammar as expressed syntactically. And plenty of examples to deal with.
In particular, there are many analytic languages where grammar is almost all syntax, like English or Mandarin; and also many synthetic languages where grammar is almost all morphology, like Russian or Inuit. In an analytic language word order is terribly important, and there are many little auxiliaries and articles and particles to augment the word orders. In a synthetic language, word order is mostly a matter of style or taste, because the structure of the sentence is all tied up and color-coded with the agreement phenomena that morphology enables.
There are other differences as well. Bound morphemes are very fussy about their order, and normally don't tolerate any variance. Syntax, however, spins off variation at the drop of a syllable, resulting in many competing patterns which frequently die off, but just as frequently fission into separate constructions. These are very different kinds of things, even if they do somehow "cover the same ground" semantically. It's not necessary to restrict one's observations to only the generalizations one wants to capture. Generalizations are nice, but facts are more useful.
You may encounter people who insist that "syntax" means "grammar". That's just the theory talking; normally they're just parroting somebody else's dictum, and normally this marks them as generative theory adherents. That's harmless unless you think they mean what they say; really they just want you to use their terminology as a mark of its superiority. Sort of like saying that "Mercedes-Benz" means "automobile". I.e, it's not true, or even really helpful, but it lays out your allegiance clearly and that's the important thing.
Let me fail to answer your question by asking another question -- what do we GAIN by drawing the distinction? By "ironing out" the distinctions, we can avoid having to postulate quite different grammatical architectures for polysynthetic vs. analytic languages, and we can capture the fact that there are generalizations across these languages. If we have distinct notions of morphology and syntax, the fact that subjects and objects agree with the verb in certain ways regardless of whether there's a word boundary between them or not is completely accidental. Put differently, why should we have phonological atoms be the same as syntactic/semantic atoms at the cost of generality across typologically distinct languages? Remember, although it gets debated a lot in various circles in Linguistics, the standard analyses of verbal morphology in English has pretty much always been syntactic at its core in the Chomskyan generative enterprise since LSLT!
I think another way of getting at the same kind answer is to ask what can be done in morphology that can't be done in syntax. There are approaches to morphology where much of what is traditionally considered with morphology is handled with the same kind of formalism as is used in syntax, and morphology may be referred to as "word syntax" on occasion. However, I do not know of approaches to syntax that strive to account for most common syntactic phenomena using the language of morphology.
When word derivations have hierarchical structures, and affixes are given the status of functional heads, it may seem like it's syntax all the way down. There are some facts about inflectional morphology, however, which make syntax-like accounts difficult to pursue to their logical end. (See Ch.1 of Stump 2001 "Inflectional morphology" for discussion) One of these is non-concatenative morphology. In affixal morphology, it is not a real problem to treat the stem and affix as constituents of a tree: [[dog]N.root [-z]PL]N -> dogs. When the morphology is non-affixal, there is a dilemma. But how is "men" derived? In Distributed Morphology, stem-changes are handled in the following way: a zero affix triggers a "readjustment rule" which changes the stem vowel: men = [[man]N.root [zero]PL] -> /man-zero/ -> men. Stump considers this a resort to "extraordinary means", since it seems like quite a large coincidence that so many languages should have an identical (zero) affix triggering stem changes. Non-concatenative morphology is a challenge for syntax-like approaches to morphology because the basic units of syntax are considered to have a strict precedence relationship: for two constituents X and Y, X must either precede Y or follow it; tertium non datur.
Another issue is morphologically-conditioned phonological rules. Particular morphemes often trigger specific types of phonological alternations in a language (e.g. changes induced by suffixation of -ity on English nouns), but comparable syntactic phenomena, where, say, a particular type of category triggered an idiosyncratic phonological process in a sister constituent, seem very rare.
A third issue is variable exponence. A given property, when realized morphologically, may have a wide range of unrelated exponents. In a single language, plural on nouns may be realized by either a prefix, a change in the stem vowel, or a change in the tone, depending on the word. Parallel phenomena are hard to find in syntax.
If the issue is whether there should be a distinction at all, I suspect that it is in theory possible to remodel our theory of language structure so that syntax and morphology are modeled in the same way, but at the moment there are quite a few details of morphology which do not lend themselves to a syntactic explanation.
Actually, the traditional distinction between morphology and syntax is pretty clear. Morphology studies how words are inflected and new forms created, while syntax studies how they are combined into phrases and sentences. So, for example, do > does would be an issue in morphology while dog barks : does dog bark? would be an issue in syntax.
The whole confusion arose in the Anglo-Saxon world because English expresses with syntax most of what non-analytic languages express with morphology. As long as the main objects of study for linguistis were Latin and Greek, I don't think anyone mixed the two terms. But after WW2 English became the dominant language and with it, the confusion spread.
Linguists who are under weaker influence of American linguistics, however, i.e. those who did not study in an Anglo-Saxon country, or who do not specialize in theoretical linguistics, usually make a very clear distinction between syntax and morphology, often combined with looking down on those who don't.
You ask if the distinction is meaningful. Absolutely. It is only in a purely analytic language that syntax can tackle the issues of morphology. But not even English is purely analytic, so if you stick to the traditional understanding, even in English there are problems (e.g. the formation of plural) that syntax can't deal with. It is only because there is so little morphology in English that the term syntax came to mean for Anglo-Saxons both syntax and morphology simultaneously.
I come from computer science background and only dabble in linguistics when it comes to programming languages. But it seems to me that (in common usage) "syntax" is the set of rules that lead to "morphology." In natural language, don't we infer syntax from morphology? But can we consider one cause (syntax) and the other effect (morphology)? Maybe I'm making a distinction that only applies to languages intentionally designed by humans. In programming, the creator or the language decides on the syntactic rules. And the morphology is whatever results.