"If ___ then ____" means: "if you provide me with evidence that ____, then I can give you evidence that _____", so I see it as a kind of challenge. I swear that if you prove to me that the sky is yellow, I will give you a pot with red grass.

See the related discussion here. Roughly speaking, the point made there is logical connectives such as "if--then" need to formalized by having rigorous, consistent truth tables, in order to be useful in mathematics.

@RobArthan what an unfair characterization of OP's question. There is nothing wrong with questioning the notion of vacuous truths or the LEM in good faith. This area of mathematics (conventions & notations) is a social discipline, and OP is right to note that, in their view, there is a disparity between the mathematical and linguistic notions of implication. Indeed, the mathematical OR is inclusive and the English OR is not!

@JackGallagher: this has nothing to do with the law of the excluded middle. The OP is using heated words like "gibberish", not me. Your observations about the word "or" in English are wrong.

@RobArthan Given the subject of the thread, ironic that you ignore the protasis of that section of OP's remark... he states that "few people [who aren't mathematicians]" wouldn't think of such a statement as gibberish, which is obviously a true sociological observation.

This is not a sociology forum.

We are humans communicating, so unless you're currently passing the Turing Test, yes, yes it is.

Does this answer your question? In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?

And also math.stackexchange.com/q/70736

"If the milk you sold me this morning was fresh, then I'm a monkey's uncle." Colloquially, this is a statement that the milk was not fresh.

If... then is ubiquitous in math, really. This interpretation is second nature. Consider the following Theorem from calculus: A function that is continuous in the interval $[a,b]$ is integrable over $[a,b]$. The theorem is simply TRUE. We don't want the theorem to be sometimes true + sometimes gibberish. But, the theorem really claims that If $f$ is continuous, then $f$ is integrable. The if...then structure was simply semi-hidden in the first phrasing. The same holds for all the theorems in mathematics. This forces us to adopt the usual convention. The theorem must be true. Always.

(cont'd) But the theorem only claims something when $f$ actually is continuous. When $f$ is not continuous it may or may not be integrable. But the need to have a TRUE theorem dictates the need to declare that "if $p$ then $q$" is always true should $p$ be false. Irrespective of veracity of $q$.

I do confess that I'm being fuzzy with the difference of propositions and predicates. My point kind of is that mathematical truths are written, used and interpreted as implications between predicates. And this forces the truth table at the level of propositions to be the way it is.

@JyrkiLahtonen You seem to be making the assumption that "if ___, then" needs to be a function of truth values.

@JyrkiLahtonen Also, the question you sent me doesn't address the first part of my question---at least not directly---which is why we should think of "if ___, then" as a function of truth values at all

@PiKindOfGuy The point was that a mathematical theorem needs a truth value (after universal/existential quantifiers are in place). "If... then" needs one as a consequence of that.

@JyrkiLahtonen I'm not sure I understood you correctly, but I'd like to point out that (function of truth values) $\neq$ (truth-valued function)

Just check out the soda machine comments. That is a type of meaning/use we need to do math. Also this answer, making the point that it may be better to see this as a description of how "if...then" is used by mathematicians to communicate. Anyway, why do you think that things would improve if instead of true/false we had 3-valued functions true/false/gibberish?

@JyrkiLahtonen I never claimed that things would improve if we were to use a 3-valued function.

Then I don't know how do you want to "improve" things? Anyway, I think that a colloquial "if...then" often also implies a causal connection, making "if sky is blue then..." also suspect. Unless the color of the grass is somehow dependent on the color of the sky (a live possibility!) Too bad SpikedMath webcomic is down. There was a fitting strip trying to explain the math usage. A more useful variant of this question would be how to make this easier for students to understand this.