So I just wanted to ask the forum, as I'm a bit conflicted and likely uninformed on this issue.

Is the phrase, "A tomato is a vegetable" as wrong as the phrase, "A tomato is a lump of coal"? Are there degrees to which something can be incorrect, or is 'wrongness' a binary value?

Degrees. If I said this was the 5th warmest year on record I'd be wrong. If someone responded with "climate change is a hoax", they'd be more wrong. A tomato is a lot closer to being a vegetable than a lump of coal.

It's confusingly simple.

In botanical terms, it's a berry with pericarp walls that develop directly from the ovary, and this contains seeds.

In culinary terms, it's used AS a vegetable. From a taste perspective, treating it like a fruit would be disgusting.

So I don't think it's much of a problem to say that it's both, with the botanical sense employing somewhat more objective criteria than umami-tasting ground-apes.

And what about cucumbers? Won't somebody think of the cucumbers?

From the fields of ontology and epistemics the assertion that a tomato is a lump of coal is what is known as a 'category mistake' (or category error).
Here is a quick outline:

https://psychology.wikia.org/wiki/Category_error

As far as I understand some of the ontology, the difference between your two examples is not one of degree, but one of kind.

Interestingly enough, there may be a category error in your original question, in that you speculate that 'wrongness' might have the quality of being gradable whereas, in pure terms, gradability is not a quality that can be ascribed to the concept 'wrong'.

Of course in conversational terms, we can apply gradability to utterances that have multiple components, and entailements and implicatures.

If a class of 100 students take a test and 100 students pass the test then the two utterances

a) All of the students passed the test

and

b) Some of the students passed the test

can both be said to be 'true', but (a) is "truer" than (b).

The speaker of (b) is not wrong per se but he/she failed to apply the Gricean maxim of quantity which enjoins participants to 'make your contribution as informative as is required.'

Language is not logic.

I once encountered an interesting example of the difference between the scientific and the normal use of language, when my eldest daughter was doing a maths exercise and was asked whether a particular statement "was true" ("stimmt"). The statement was numerically close to the correct statement, so she wrote "es stimmt fast" (it's almost true). Since it was a maths exercise, this was, naturally, marked wrong.

Evolving wrote:I once encountered an interesting example of the difference between the scientific and the normal use of language, when my eldest daughter was doing a maths exercise and was asked whether a particular statement "was true" ("stimmt"). The statement was numerically close to the correct statement, so she wrote "es stimmt fast" (it's almost true). Since it was a maths exercise, this was, naturally, marked wrong.

But "almost true" implies false. She was robbed.

“Knowledge is knowing that a tomato is a fruit. Wisdom is knowing not to put it in fruit salad. Philosophy is wondering whether that means ketchup is a smoothie”

—Tired internet meme

Evolving wrote:I once encountered an interesting example of the difference between the scientific and the normal use of language, when my eldest daughter was doing a maths exercise and was asked whether a particular statement "was true" ("stimmt"). The statement was numerically close to the correct statement, so she wrote "es stimmt fast" (it's almost true). Since it was a maths exercise, this was, naturally, marked wrong.

This brings to mind a chapter in a book I read some while back. (See reference below).

It is well known that in mundane spoken interactions speakers often prefer to use vague expressions for numbers, times, amounts, etc.

'Oh, I dunno, like ten, fifteen, something like that.'

'Yeah, I should be there about sevenish.'

'It's probably gonna cost around 3,000 yen.'

It is assumed that in mathematics, precision is the key and vagueness is to be avoided. However, in the research carried out by Rowland, it seems to be the case that in mathematics classrooms, vagueness is a key interactional resource. Rowland quotes a pamphlet (pp. 17-18) by the Association of Teachers of mathematics. (p.79)

"Because it is a tolerant medium, everyday language is necessarily ambiguous... Now, mathematising is also a form of action in the world. And its expressions, however carefully defined, have to retain a fundamental tolerance...Because it is a tolerant medium, mathematics is also necessarily and ambiguous one."

Rowland goes on to note,

"As a 'product' (polished, final), mathematics may be presented, particularly in writing but also in speech, as though it lacked ambiguity, representing truths about the world - or at the very least, about itself - in a sure, exact and unequivocal kind of way. By contrast the 'process' of mathematics production (mathematising) is characterized by a number of forms of vagueness." (p.80)


The rest of Rowland's chapter gave detailed accounts of how language is used in 'Mathematising'. (The title of the chapter says it all, really.)
Reference:

Rowland, T. (2007). Well maybe not exactly, but it's around fifty basically? Vague language in mathematics classrooms. In J. Cutting. (Ed.), Vague language explored. Plagrave. (pp. 79-96.)

Once again, the mismatch between language use in science and daily life is highlighted.