R_S's example with "put" is more about the fact that obliques may be arguments instead of adjuncts, and arguments may be obliques instead of core. If you have to mention something when using a particular verb, that something is one of the verb's arguments; if you don't even have to have one implicitly in mind, it's an adjunct.
The major difference between core arguments and oblique arguments, AIUI, is that:
* Core arguments are marked for the grammatical and syntactic relations and functions rather than, or at least moreso than, their semantic roles. A core argument's semantic role is likelier to be shown by voice-marking or agreement-marking on the verb.
* Obliques (both arguments and adjuncts) are marked for their semantic roles, not for their grammatical or syntactic functions or relations.
Everything in the core is (theoretically?) an argument, and every adjunct is (theoretically?) oblique; but some arguments can also be oblique, like the "where would you put it?" prepositional phrase in "put" clauses.
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Whether or not the core vs oblique distinction among arguments is actually considered universal, may depend on the school of the linguist you ask.
If it is universal, it means that in every language at least some clauses have at least one core argument, which means that in every language at least some clauses have a subject.
Clearly some languages don't have any core arguments besides subjects.
Also, there's a lot of agreement between, on the one hand, those linguists who think the distinction is universal but is important in some languages and unimportant in others, and, on the other hand, those linguists who think the distinction occurs in many languages but doesn't occur in many others.
At the extremes, there are linguists who think it's universal, and there are linguists who think it occurs only in some languages and is virtually useless even in those.
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As for applicatives:
Note that it's about promoting oblique arguments to "direct or primary object", not about promoting adjuncts.
In some schools, among them "MAPping theory", there are considered to be in many languages such things as "Morphosyntactically-Assigned Argument Positions" or "MAP"s. And one of the main typological parameters for languages is thought to be "How many MAPs does this language have?"
I know you're aware of Klaiman's supertype of "Derived Voice Systems", in which the main function of grammatical voice marking of a verb is to promote various arguments into and demote various arguments out of various grammatical or syntactic relations or functions.
According to Mapping Theory (and some other schools), such valency-rearranging operations (whether or not they raise or lower valency) tend to target either the most-syntactically-privileged MAP or the least-syntactically-privileged MAP. That is, they either move something into or out of the top core position, or into or out of the bottom core position.
Applicativization moves oblique arguments into the second MAP, and may move whatever was already in the second MAP, if it was already occupied, into an oblique position, or else make it implicit.
Whether or not what Mapping Theory says in the paragraph before the last is unconditionally universally true, it's a very strong statistical correlation; Applicativization tends to be restricted to 2-MAP languages (languages with just two grammatical or syntactic relations or functions).
(BTW: Actually, the same is true of dechticaetiativity.)
Languages with three MAPs tend to have valency-rearranging operations like "dative applicativization" (promoting some oblique argument into the 3rd MAP), and/or "dative movement" (promoting what's in the 3rd MAP into the 2nd MAP).
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If a language such as Tagalog, or what David Peterson says the average conlanger thinks is meant by "trigger language", has only one MAP (i.e. the Subject), then the only way to promote an oblique argument into the core is to promote it directly to Subject. So such languages may be "voice-prominent", since the Subject may fill one of many semantic roles and the verb's voice is what tells the addressee which role the Subject fills.
2-MAP languages with applicativization often require two steps to get an oblique argument into the Subject position; applicativization moves it into the Object position, and then passivization moves it from there into the Subject position.
3-MAP languages, OTOH, often need two steps to get an oblique argument into the primary (or direct) object position; "dative applicativization" moves it into the secondary (or indirect) object position, then "dative movement" moves it into the primary (or direct) object position. Whether or not it's then possible to passivize the result and thus move what was originally an oblique argument into the subject position, I don't know; if so, I imagine it depends on the language.
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Obviously all the mapping-theory stuff I spoke about applies only to languages which have what Klaiman called "a Derived Voice system". And, if there's more to mapping-theory than what I already said, I haven't found out about it yet.
But, even leaving aside the voice system, it's still possible that a language has one or more MAPs, though it might be that it's hard to conclude how many it has. The "subject properties list" and the "correlates of the absolutive" in your pbwiki and in the Kneequickie are good guides to at least two of them; but note they don't say every language has one.
As you can see from reading them, though, the syntactic subject, if there is one, is highly likely to be a core term (probably the same goes for the absolutive); and, if there are any core terms, it's highly likely that one of them is either a syntactic subject or an absolutive.
But I don't know that a language has to have either a syntactic subject or an absolutive, nor do I know if one has to have a core argument. If a language does have core arguments, I don't know that it must also have oblique arguments.
I highly suspect, though, that it's psychologically easier to speak a language that has either syntactic subjects or absolutives or both, than one that has no such thing. There may be common diachronic forces that frequently cause such things to evolve into core terms or MAPs; I suspect there are, but I doubt they always operate.
In every language, very many clauses cross-linguistically semantically seem to have more than one argument. For a language with only one MAP, such a clause would be easier to express if the remaining arguments could be oblique arguments.
In most languages, a numerous minority of clauses cross-linguistically semantically seem to have more than two arguments. For a language with only two MAPs, such a clause would be easier to express if the remaining arguments could be oblique arguments.
In several languages, a few clauses cross-linguistically semantically seem to have more than three arguments. For a language with only three MAPs, such a clause would be easier to express if the remaining arguments could be oblique arguments.
I've never read any report or rumor of any language anyone has said has more than four MAPs. I also do not know any clause which, no matter what language it's in, would semantically require five or more arguments, though I vaguely remember reading that someone claimed they had one. If that's so, then to express any clause that needs five or more arguments, it's going to be nearly impossible to do it without some of the arguments being obliques.
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HTH?
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