This doesn't seem quite right to me:

  In the modern academic practice, the question of where a particular idea came from, or whether an axiom is ontologically correct, is considered vacuous and out of scope. For the most part, you’re just handed a rulebook to play someone else’s game.

I very much had the opposite problem with Munkres's Topology or Dummit and Foote's Abstract Algebra: those authors hand you the ontological / scientific justifications for "everyday" ZFC without actually telling you the precise rules. I had to read a formal book on mathematical logic before I really understood point-set topology (at which point my misconceptions were clearly trivial confusion).

To be clear I think the standard intuitive semi-naive set theory is the correct approach for most math students. But it didn't work for me. I needed to see the axioms and formal language.

Oh man, that resonates with me. One of the constant frustrations for me was that once you get foundations in a topic, the rest follows, but the foundations are often full of things that are axioms under one metatheory and theorems under another metatheory. When they were axioms, I remember always being comfortable, like "sure I can assume things," but as theorems there's always that bit of "wait hold up you can't just do that without saying more."

The one that I remember most strongly that way was the unique mapping from the empty set/object/whatever as a theorem.

> Formal logic usually isn’t taught in high school

Have things changed? Last century, this was a key part of (in the US) high school geometry courses. I won't argue that it was as in depth as you'd get in a college course (like you'd be exposed to in a math or philosophy degree program), but it was formal logic and it was taught.

Not in geometry for me, but it was required in my linear algebra, discrete math and computer architecture classes, all of which were required for my CS degree.

I've found that any time you try to generalize about what is or isn't "usually" required in school, there are a lot of exceptions, to say the least. Curriculum is all over the place.

I would say high school geometry is still mostly "syllogistic" and not the formal philosophical / mathematical logic worked out between Kant and Gödel which forms the backbone of modern mathematics. It is good solid logical thinking, and mathematically correct! - but not really what the author means here.

> "If you don't finish house chores, you can't play Minecraft"

is equivalent to "Do finish the house chores, or you can't play Minecraft".

By the traditional translation of if/then sentences to classical logic, it is. If you want to go further down the rabbit hole, several things are debatable here: When people use if/then sentences, do they really mean material implication, where (A -> B) is equivalent to (~A OR B)? Also, people often use the word "or" in a sense that's closer to exclusive OR, as opposed to the inclusive OR that the logical operator indicates. Overall, can the meaning of a sentence with imperative intent be fully captured by a proposition that is just meant to indicate a state of affairs?

"Do finish the house chores" is more naturally taken as a command, so the question is whether it should be assigned a truth value at all. It is linked to some normative claims that would plausibly be truth-valued: "you are permitted to play Minecraft only if you finish the chores", or perhaps "you ought to finish the chores, or you are not permitted to play Minecraft". The former is equivalent to the original sentence only if we take "can't" as expressing lack of permission (which it may not): "if you don't finish house chores, you can't/are not permitted to play Minecraft".

And “If you don’t play Minecraft, you don’t have to finish house chores.”

"xor you can't play Minecraft" is correct :)

you sure?

I think "if you don't you can't" does not preclude other don'ts leading to you can'ts, but "Do or you can't" means that if you Do you can, although in normal vernacular usage you are right that they are interchangeable.

You sure? I am pretty sure you need something like, "Child! If you do not finish your chores, I will not allow you to use the PC to do anything other than read the HowTo.com webpages on how to do housechores. Ergo, you will be unable to play Minecraft."

> "Do or you can't" means that if you Do you can

You have this backwards IMO. In logical terms (not vernacular usage), "Do or you can't" being true means that if you CAN (play), then you do/did the chores. It definitely does not preclude other don'ts leading to you can'ts.

It only makes "do" a necessary (not sufficient) condition for "can".

Another way to see it : "Do or you can't" is a form of "A or B", so "you can" means B is false, so A must be true.