[carodgers]

"We’ve established that the truth or falsity of a statement depends on its context — that is, on the assumptions we take as true or false in order to justify it."

I am inexplicably and disproportionately irritated by people like this. They've an a priori commitment to epistemic relativism. Unnaturally allergic to any claim that certainty is possible, and dedicated to undoing the careful work of those who are building up what may be known.

No, not every statement requires assumptions in order to hold. "The assumptions A implies B and B implies C, taken together, yield A implies C." This statement contains assumptions and makes observations about them, but it is true regardless of whether the assumptions it describes are true. The statement as a whole is "true" in the exact sense that the no counterexample to it can ever be given in any universe, under any set of assumptions.

I cannot wait for the day that Lean and other proof systems become accessible enough, conversant enough, and interdisciplinary enough to put these eternal confusion peddlers out of business.

[matt_kantor]

> No, not every statement requires assumptions in order to hold. "The assumptions A implies B and B implies C, taken together, yield A implies C." This statement contains assumptions and makes observations about them, but it is true regardless of whether the assumptions it describes are true. The statement as a whole is "true" in the exact sense that the no counterexample to it can ever be given in any universe, under any set of assumptions.

"The assumptions A implies B and B implies C, taken together, yield A implies C" is a statement using propositional logic. Like any formal system, propositional logic has axioms (for example, as defined by Frege[0]). Those axioms plus rules like modus ponens are things you need to "assume" to decide the truth value of that statement.

[0]: https://en.wikipedia.org/wiki/Hilbert_system#Frege's_Begriff...

[carodgers]

Modus ponens holds regardless of whether it is assumed correct or not. By that, I mean that excepting contradictions, not only are no counterexamples known, but it is known that none will ever be found.

So I decline to adopt your convention of classifying it as an assumption. It might be more accurate to call it an observation.

[matt_kantor]

The domain of your claim was "in any universe, under any set of assumptions". I now take it that you assume (see what I did there?) that all universes are sound according to classical logic and can admit no contradictions. You might be interested in <https://plato.stanford.edu/entries/impossible-worlds>, an overview of philosophical thinking on that subject.

Separately, if you scrub "plus rules like modus ponens" from my previous comment, do you agree with the rest of it?

[carodgers]

No, again, I have made no assumptions. I have made the observation (as others may, just as easily) that no one will ever succeed in describing a scenario or a universe where modus ponens is violated.

[matt_kantor]

It seems like you don't actually want to engage with me in discussion. If I'm mistaken about that, please correct me (I enjoy talking about stuff like this and would love to understand your perspective). If not, that's fine too, but I don't want to waste my time.

[carodgers]

On other topics I imagine the conversation would flow better and, I hope, be just as lively and engaging. But to be direct, no, I'm not eager to continue when universes that admit contradictions are on the table. A conversation that permits contradictions is not likely to be productive. So I'll be content to just restate my point. Cheers!