"We want our beliefs to be true, but if we don’t even know what they mean, we’re in trouble. A single sentence might be true in some sense, false in some other sense, and meaningless in a third. If you believe 'St. Trinian’s is a pretty little girls’ school,' what do you believe?"

When someone claims they believe that statement, they have a meaning in mind. Amphiboly is a feature of syntax and spoken language, not beliefs. Confusion, on the other hand, is.

"In 'the eggplant is a fruit,' probably what is meant is that all eggplants are fruits. In 'the dog is a Samoyed,' probably what is meant is that some dog is a Samoyed."

What he's describing is equivocation. However, something similar occurs when words or phrases are analogical (for example "healthy body" vs. "healthy sandwich"). Which brings me to...

"This problem is pervasive. Linguists catalog many distinct ways a sentence can be ambiguous. On analysis, almost any sentence can be read with multiple meanings. [...] natural languages—English, Chinese, Tamil—are hopelessly broken. They are incapable of adequately expressing true beliefs."

Why broken? By what measure? And why? That we must grapple with ambiguities doesn't mean natural language isn't serving its proper purpose. And incapable? I do hope that's hyperbole because, as the author has no doubt noticed, the entire article, including the quoted claim, is written in English.

"Modern rationalism’s first major improvement on traditional logic replaced natural language sentences with mathematical formulae."

Modern rationalism? Also formal languages can help, but they aren't magic bullets. I am still vulnerable to equivocation.

And why the dismissal of Aristotelian logic for which the author offers no tangible justification, like:

"Frege’s 1879 invention of modern formal logic fixed several outstanding defects in traditional, Aristotelian logic"

"Outstanding defects" is, to riff on the article's fixation, ambiguous. Three features of Aristotelian term logic that Fred Sommers identifies as "missing" are the inability to deal with particulars in a systematic way, the absence of relations and the absence of compound propositions. If those are defects, then fine, but even without those features, the logic is immensely useful in a broad range of applications. Also, recall that it dominated logic for two millennia (I highly recommend (Joyce 1916)). I think it deserves a bit more credit. Furthermore, Sommers updated term logic with his own "algebraized" term functor logic which accounts for these three "deficits". Worth also noting is that Sommers saw Fregean logic as a regression to Platonic views of subject and predicate and rejects putting them in separate universes (Sommers 1982).

"He solved several long-standing technical problems, in which Aristotelian logic gave outright wrong answers"

Wrong answers? Like?

"In Fregean epistemology, we can eliminate the ambiguity of 'is' [examples follows]"

How does that depart from the copula in term logic in any relevant way? Chapman's examples transcribed into term logic are "every Eggplant is a Fruit" and "some Dog is a Samoyed". Each is governed by laws of immediate inference. So what ambiguity is he talking about here that his FOL examples don't also suffer from?

"But what if you see a dog, and it’s obviously a Samoyed, but you don’t know its registry number? What do you believe then?"

You believe exactly what you've described.