When we use disposed as an adjective, we mean "inclined or willing" in some contexts. E.g. James didn't seem disposed to take the hint.
I think the primary meaning of dispose is get rid of. Why does it have a meaning of incline? Is it related to its primary meaning?
I can understand why the connection between these two senses of "dispose" seems so mysterious. The key to understanding it is: "to get rid of" is actually a secondary sense, even though today it is very common. The primary sense is "to put things into their proper places". The primary sense is now rare, but it still colors and explains the secondary senses.
When you dispose of garbage, you put it into its proper place: a garbage can, the city dump, or someplace like that. When you dispose of radioactive waste, you put it someplace very far away from people. However, throwing away garbage is only one of many ways that things can be "disposed of".
I marveled at the ability of a South China crime boss to dispose of these resources and organizational refinements with such precision. [Source]
Here, "dispose of" doesn't mean "get rid of" those resources, it means to put them to work in a coordinated manner—that is, to bring them to the proper places for performing some large, coordinated action.
In the Order Disposal Problem:
A manufacturer … may receive and dispose many production orders during a time period [but] he can dispose at most one order at any time due to resource constraints.
the word "dispose" means "to complete a job"—another way in which something can be put into its settled and proper place.
Here is an example of saying that someone is disposed to act in a certain way:
Virtually all soldiers at all times have been strongly disposed to obey orders to go to war irrespective of whether the war has been just or unjust. [Source]
This calls upon another secondary sense of dispose: to put into a state of readiness to do some action—the action that one is disposed to or toward. The elements of one's mind, one's habits, one's abilities, one's resources, one's knowledge, one's surroundings, etc. are in a configuration that makes one especially ready to perform the action. This sense also occurs in sentences like this:
His generation and his origins dispose him to be a democrat. [Source]
The word "dispose" gets used for many, widely varying meanings—more than I can list here, and indeed more than you'll find in dictionaries. Once you understand the primary meaning, it becomes a lot easier to understand how people are extending that meaning to fit different situations.
I think it helps to know the etymology, both for understanding the many senses of "dispose" and for many other words that contain the same two roots: dis- and pose. "Dis-" (in "dispose") is a Latin root meaning to break apart and spread out, as in disperse, disseminate, dissolve. "Pose" is a Latin root meaning to put something in a certain place or state, as in position, opposite, repose.
Knowing that, you might be able to figure out that a disposition can mean both an inclination toward doing something and the result of doing something—"where" things ultimately got put. The latter sense is rare, but an example is that in computers, a task is sometimes said to have a "disposition" of succeeded/failed/aborted. Another one is:
I can put five employees at your disposal.
This means that I can bring five employees to the appropriate place, where they will be ready to do as you command.
And finally, here's a stock phrase which is worth knowing, because it sheds a lot of light on the other senses: propose and dispose. Here's an example:
Men propose; women dispose. [Source]
This means that men offer possibilities to consider (propose), and women decide which proposals to carry out and which to reject (dispose—that is, sort out the proposals and bring some but not all to action).
I was about to tell you that the two meanings were unrelated, but then I looked it up and learned something.
According to the OED, "dispose" means "arrange, order, control, regulate" and derives partially from the same root as "position." (http://www.etymonline.com/index.php?term=dispose)
"Disposition" means "ordering, management;" but also "tendency of mind." The second meaning - and the one we're addressing here - is from astrological use of the word for "position of a planet as a determining influence." (http://www.etymonline.com/index.php?term=disposition)
So, to dispose of something is to arrange it or put it into order. In one sense of this, you can put your trash into order by putting it into the trash can; and in the sense above, John can arrange his mind against taking the hint.
Although "get rid of" is the sense of "dispose (of)" that I would first think of, it comes fifth or sixth in most dictionaries.
The OED gives the sense of "disposal (of waste material)" as coming from around 1960 (http://www.etymonline.com/index.php?term=disposal). That seems much too recent to me, especially since the patent for the first in-sink "Garbage disposal device" was published in 1933.
Another example: “Disposal of Unapproved” Minutes does not mean “finish off” unapproved minutes or “get rid of” unapproved minutes! Here the word “disposal” means "to put things into their proper places." The phrase "Disposal of Unapproved" Minutes means approve the minutes and put them in their proper place in the permanent records.
