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> In my experience, a language that has only four cases usually has a method for expressing all eight, they just won't be formalized in the same way.

You're getting close to something interesting. Cases are an example of expressing a linguistic distinction through the use of a particular kind of syntax.

It is also common for the same distinctions to be expressed by other means that people generally agree are also syntactic. An English finite passive verb has to be expressed periphrastically, using a dedicated auxiliary verb that exists for that purpose (be or get); the verb will be inflected into a form determined by the governing auxiliary verb, but no passive form exists that can stand as a finite verb.

But this is still essentially a use of syntax to deal with what are generally felt to be "grammatical" categories. What is discussed much less often is that languages may scrupulously observe a standard distinction between grammatical categories without having any syntactic apparatus to accommodate that distinction. The only term I know related to this is "lexical aspect", but the phenomenon exists beyond just aspect. ( https://en.wikipedia.org/wiki/Lexical_aspect )

Two examples between Mandarin Chinese and English:

1. In Mandarin, verbs may be given a "resultative complement" expressing a result of the action. (This has many syntactic consequences.) A complement can be almost anything indicating a state, but there are some standard ones, and the complement that regularly indicates "success of the action" is 到. Thus, by a regular and productive construction of Mandarin grammar, we can observe the following verb pairs:

    看 look   看到 see
    找 search 找到 find
    听 listen 听到 hear

Except of course these aren't pairs at all, if you're a Mandarin speaker; it'd be more natural to call them different verb forms. Seeing is just a particular way of looking, the successful way. By contrast, though English doesn't express this distinction in its grammar... you'd have a hard time claiming that English doesn't observe the distinction. The English verbs in that table are absolutely not interchangeable with their "partners". We have here a grammatical distinction that exists entirely within English's vocabulary.

2. If you study ancient European languages, you'll hear about the locative case, already long dead by the time of the ancient language that you're actually studying. It was used when a noun was conceived of as being a location rather than an object. (And in classical Latin, it survives in fossilized, unproductive form in a few common locational words like "home" and "the ground".)

English does not have a locative case, and also doesn't really observe any distinction between locations and other types of nouns. There are some restrictions, but they are easy to explain as being required by the semantics of locations rather than the grammar of English per se.

Mandarin also does not have a locative case, or any noun cases at all. But it nevertheless maintains a robust distinction between ordinary noun phrases and locational phrases. It is not possible to (grammatically) express in Mandarin that something is "on the table"; you have to say that it is "on the top of the table".

It is surprisingly difficult to talk about this distinction without imagining that locative case is involved. Certainly the background phenomenon is the same. Certainly this is deeply embedded within the syntax of Mandarin. But you would need to posit a "phrasal case"[1] which is entirely unexpressed in order to actually call it a "case".

[1] CGEL attempts to preserve the idea that the English clitic 's is a case marker by stating that it applies genitive case to an entire noun phrase. (This is a necessary analysis, since the clitic attaches to the last word of a noun phrase whether that word is a noun or not, but its meaning applies to the head of the noun phrase, which is not especially likely to also be the last word.) I don't like this; I think a better analysis would be to just give up on the idea of genitive case in English. But if you like the idea of phrasal case, locational phrases in Mandarin are certainly an area where you could look for some support.