Antonym to "on" (spatial relation)

Question

In the scope of spatial relations, the word "on" in English typically describes the state where one object is positioned directly above another object, where the lower object supports the upper one. For example, "the bottle is on the table".

However, in the same scope of spatial relations, it has another meaning which is not only related to up/down. For example, "the picture is on the wall" describes a similar concept, that does not apply just to the up/down axis.

Is there any antonym for "on" that would capture both cases?

EDIT: To be clear, I'm looking to the antonym in the sense that it's the opposite role. For example, if the "ceiling is above the floor", then the "floor is below the ceiling" and "below" is the antonym of "above".

As the wall is "supporting" the picture (when the picture is on the wall) and the table is "supporting" the bottle (when the bottle is on the table), the question if there's a better word than "support"?

Answer 1

There is no antonym that would apply to both situations. The descriptions of a book being on the table or a picture being on the wall look like descriptions of spatial relations, but they are actually special cases of a more abstract relationship.

For example, the oil stains are on the dress, the news is on the radio, the passenger is on the bus, or the family is on the dole.

There are many questions on this and other sites that deal with when to use in and when to use on (I’m writing this on a phone and it’s awkward to look these up for citation).

The typical reason for using on is that the first thing can move around relation to the second thing, or could have in the past, i.e. a man on a car conjures up a different image than a man in a car.

However, the action that associates the first thing with the second thing is typically more specific, and easier to negate.

The book may be set down on a table, in which case it can be lifted up. A picture may be attached to a wall, after which it can be detached from, etc.

When you look for an antonym to on, however, you’re pretty much stuck with off, which will usually make sense when there’s an imagined action behind it. Without an imagined action, i.e. where on really means associated with, the “opposite sense” of not being associated with can usually cover a lot of possibilities, and therefore doesn’t feel quite right as an antonym.

Answer 2

His office wall holds a picture of his days as a lawyer...(waaytv.com)

Technically, you could say a table holds a bottle, only this is not really used. "Hold" is more commonly used to say that something is in something, like "A bottle holds water."

Support is more general and it fits better to different spatial situations.