Term for if-then statements where the "then" part does not logically follow

First, I'd like to apologize for the awkwardly-phrased title.

Instead of trying to explain the structure I'm thinking of, I'd like to give some context and then an example or two. I've used and seen this construction used conversationally, and if my memory and intuition serve, conversationally only. I'm curious to know if there is a formal term for it.

Ex 1

"If you don't like what I'm offering, the door is that way."

A purely logical interpretation would make little sense with this example. In this context, the inference is that the speaker is indicating to whomever he is speaking that they should leave, which suggests that there might be an elision, e.g.

"If you don't like what I'm offering, ... [then you may leave. T]...he door is that way."

Ex 2

"If you're hungry, I have food."

This is a similar context. The implication is that the speaker will offer food to whom she is speaking if they are hungry. A (strictly) logical interpretation would, again, make little sense (if you are hungry, then I have food doesn't convey any intent to provide it, nor does it make sense; the speaker always has food, she just will offer it if her guest is hungry).

I recognize that this might be too specific to have a term associated with it. I'm just curious to know what it is if it does exist.

This type of if is known by a few names: relevance conditionals¹, pragmatic conditionals², and, if you're hungry, biscuit conditionals³.

It includes sentences where if comes after:

I have food, if you're hungry.

Also, mandatory XKCD: