The way of presenting this (family of) puzzle(s) is due to American philosopher and logician, W.V.O. Quine. It concerns possibilia. I will ignore the vagueness of “fat” and take “doorway” as understood. So, let’s take a look. People say things like:

A: Is X coming to the meeting?
B: I don’t know. It is possible, I guess.

and

A: A ball dropped near my floor will hit the floor and not fly off somewhere and turn feathered.
B: I don’t know. Maybe it is really possible, just unlikely.

and

A: God possibly exists.
B: Oh really?

and

A: Nothing goes faster than light.
B: Why not?
A: It is impossible!

and finally

A: It is impossible that there is a river of Coca-Cola elsewhere (not on Earth) in the universe.
B: Why is that?
A: Because Coke is necessarily made under license by a corporation based in the U.S. here on Earth, and as yet it has not licensed any manufacturing elsewhere.

So what do we see in these small dialogues? Many uses of “possible” and its negation, “impossible.” We also see “necessarily,” a related word. What’s the relationship between necessary and possible?

The conventional identity is that something is necessary if it is not possible that it is not. For example, let p be a proposition, then p is necessary if it is not possible that not p.

The next thing to notice is our puzzle: What do all the uses of “possible” (including the one “embedded” in the use of “necessarily”) have in common? You can start by asking yourself what is the domain of the operator in question. Are they all the same? What does linguistic analysis tell you? What else might bear on this question? Once you have done that, you have done more work than found in a lot of discussions of possibility.

Then ask: How do you evaluate the truth value (if there is one!) of the propositions using “possible” (etc.), and what in the world would “truth make” that truth value? (What would the world have to be like for there even to be a truth value?) As one naïve example (not a theory of truth), “Electrons have charge (thus and so)” is true because (for example) electrons have charge (thus and so). Your answers to all these questions should answer Quine’s question, or “dissolve” it.

Recommended reading: Bunge’s Treatise on Basic Philosophy (Volume 3) and Armstrong’s A World of States of Affairs. Schaum’s Outline of Logic, is a very easy way into the formal systems often used here, and yet also briefly addresses some of the philosophical topics as well. Pay attention to its discussion of the ontological argument (a fragment of which is in the third dialogue).