With Latin dat/abl, I've always related them to the prepositional phrases in English (to the car, from the church, toward the rear etc), but my Latin is scary at the best of times.
There are prepositions in latin (many of tham, actually) that take the accusative. Famously:
Hannibal ante portas!
Cases work quite differently in different languages, and (as Medievalist indicates) they can work
very differently in different language families.
As a native speaker of English, you might encounter case discussions with controversial topics. For example:
This is me/I.
People may argue for any of the pronoun forms, and depending on their preference they may use different case saystems to do so.
Latin and other languages related to English apply the nominative case to subject complements, thus people who think that English should do this, too, might argue that "This is I," is correct and "me" is incorrect.
Many languages that rely on word-order, though, have a two-case system - nominative/oblique. So you could argue that English does this, too, and treats subject complements with the oblique case, rather than the nominative. If so, you would expect "me" to be correct, not "I". Since we're used to analysing English in Latin terms, this can lead to strange-seeming situations. For example, "nominative" would no longer be the "default". (See: "Who wants a cookie?" - "Me!!!")
The problem is that English seems to use both systems, though maybe not to the same extent, and that we have - apart from pronouns - nothing much to check it against. No verb endings. Some constructions may make more sense in one system, others in another - but we might be missing the real system. Or it's just messy, and we're imposing regularity for analysis sake (in which case we shouldn't hate the language for not working the way we'd like it to).
There is no easy way to transport cases from one language to another, because there is nothing outside any concrete language to give us any help. We can only measure languages against each other. It's not surprising that attempts to provide a universal frame for that tend to come from transformative grammars (since they tend to believe in language universals). There's case grammat, which posits a deep structure that underlies all languages, where languages then select according the concrete languages requriement:
For example, you have a semantic event structure, such as opening a door with a key. Say, the participants are John (agent), key (instrument), door (object/patient). You assign cases to the situation according to the "universals", and then you describe how language arises through "transformation rules".
So you can have:
John opens the door with the key.
or
The key opens the door.
etc.
Note that the latter, "The key opens the door," involves "agent deletion", and the agent has no place in this sentence. It would be odd to say: "The key opens the door with John." You cannot insert the agent into the instrumental slot.
I'm not that fond of transfromative grammars, but I also don't see much else we can (from a certain point onwards) to compare the case systems in different languages. (This is where I am a relativist; I think there's nothing wrong with assuming a common thought-background that helps you get at the core of the relation between two languages, as long as you remain conscious that your doing it from your own position.)
So what
is the evidence for case systems?
1. morphological (different word endings or word forms - in English: pronouns and possessive -'s)
2. Syntax (this one's more complicated, and I won't attempt to explain - try the wikipedia article on
absolutive-ergative languages - written from a nominative-accusative point-of-view (which makes sense since it's written in English)).
Apart from that you'll need to make assumptions about a language's semantic base. I'm not aware of any clear-cut consensus on such assumptions, but then comparative linguistics is not really my strongest area.