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I saw in the Farlex Grammar Book an explanation of gradable adjectives and graded adverbs. It lists the following words as examples of each category:

Gradable adjectives

small

cold

hot

difficult

sad

Non-gradable adjectives

tiny

freezing

boiling

impossible

devastated

Grading adverb

absolutely

utterly

fully

virtually

completely

Non-grading adverbs

a bit

dreadfully

unusually

extremely

slightly

The rule it states is that non-grading adverbs "can generally only" modify non-gradable adjectives, and grading adverbs "are generally only paired with" gradable adjectives. So:

• absolutely small
  
  (wrong) non-grading adverb and gradable adjective
  




• a bit small
  
  (correct) grading adverb and gradable adjective
  




• completely sad
  (wrong) non-grading adverb and gradable adjective
  




• slightly sad
  (correct) grading adverb and gradable
  adjective
  

It contains this note:

Note that in informal speech or writing, many grammar rules are often ignored, misused, or misunderstood, so you may come across non-grading adverbs used with gradable adjectives (e.g., “utterly surprised,” “absolutely interested”) or grading adverbs used with non-gradable adjectives (e.g., “extremely certain,” “very tiny”). However, other than the exceptions listed above, this usage should be avoided, especially in formal or professional writing.

The exceptions it's referring to are the adverbs really, fairly, pretty and quite, which can modify both gradable AND ungradable adjectives. And so considers the following acceptable:

• pretty freezing
  
  


• fairly impossible
  
  

I consider it strange that it implies "utterly surprised" or "very tiny" to be ungrammatical (or at least bad style), and yet considers "pretty freezing" and "fairly impossible" to be fine.

I started experimenting with mixing mismatched adverbs and adjectives from the above source, along with a couple of others, including (englishclub.com [grammar]) that is, gradables with non-gradeds and vice versa. Some examples I found to be fine (by my standards, at least) were:

perfectly happy

(non-graded adv. + gradable adj.)

very awful / extremely awful

(graded adv. + ungradable adj.)

nearly dead / almost dead

(non-graded adv. + ungradable adj.)

virtually unknown

(non-graded adv. + ungradable adj.)

more terrible / more terrifying

(graded adv. + ungradable adj.)

The Farlex source seems to imply my above examples are ungrammatical or at the very least should be avoided in formal writing.

The same thing seems to be implied at the other source (englishclub.com).

I'm wondering if either is true, whether they're ungrammatical or to be avoided in formal writing?

Also, a quick second question, if I may:

At learnenglish.britishcouncil.org it says:

Adjectives like ‘terrifying’, ‘freezing’ ‘amazing’ are also non-gradable adjectives. They already contain the idea of ‘very’ in their definitions – ‘freezing’ means ‘very cold’.

And at englishclub.com

Non-gradable adjectives do not normally have comparative and superlative forms.

And gives the example that you can't have "more dead" or "more freezing". I agree with that, but that definitely doesn't seem to be the case for amazing, terrifying, and terrible, which are all considered ungradables. Also I see nothing wrong with saying that Village A was more devastated by the war/drought than Village B (devastated being an ungradable adjective).

Am I wrong in thinking that there's nothing wrong with "more amazing" or "more terrifying"? Example: "It's more amazing/terrifying than you can imagine." It seems the sources are telling me these words shouldn't be comparable. Are they right?

This is a perfect example of a prescriptivist approach to grammar teaching as opposed to a descriptivist approach. As long as the rules put forth actually match how language is used and how it is actually perceived by educated speakers, such rules are fine, but in this case Farlex's rule neither matches common, accepted usage, nor is it accurate enough to serve as guidance on use in different registers or text genres. The rule they propose is imperfect, as your own analysis has shown.

The concept of gradable and ungradable (absolute) adjectives and adverbs is correct, but Farlex's definition is too inflexible. There are adjectives which conceptually do not have degrees or gradations and there are those that do, but there are many adjectives which can be understood either in absolute terms, or with some amount of gradation.

Examples:
Dead. An organism is either alive or dead, there is no in between. However, it can be conceived of as being an ongoing process or state. A battery, for example, can be too low on power to run a device, but still have enough power to become "more dead".
"That battery is nearly dead," vs. "That battery is completely dead."

Perfect. Either something is perfect or it is not. However, this too can be subject to a different conception. Achieving perfection can be seen as an ongoing process which can have degrees of completion. Something with no flaws or mistakes can gain greater detail.
"That was a perfect answer" "Adding usage makes the answer even more perfect."

While certain conceptions of particular adjectives are more informal or artistic in nature, meaning it would be best to steer clear of them in formal texts or registers, others are more dependent on having a clear understanding of the concept involved. This added nuance does not lend itself to a simple list as the grammar book attempted.

So, you are right in your analysis--the book's grammar rule here is flawed.

In my academic experience, College Professors stressed with me they are to be avoided in formal writing. Here's the answer they gave me as why:

Formal or professional writing: Doctoral disertation; Graduate thesis; college essay: Writing a Medical/Scientific Journal for publication. Preparing a Scholarly Textbook, etc.] The more formal it goes, the better it gets. Clarity;concise; direct--essential to stating or arguing complex ideas. Stating argument/one's position when a reader, or submitted for peer review by a group of scholars have preconceived belief. This grueling work where a published work survives an entire community of professionals critical eye: tiny is tiny. Freezing..freezing. The terms are not gradable. Redundant with grading adverb and weaken paper position. Formal writing is not creative writing, and the creativity is in accurately using English grammar. This tells the reader, the writer's stated opinion have worth to listen to.

In my creative English Undergrad/Graduate classes, Professors stressed with me Creative writers stick to the rules of grammar, just like in formal writing; but, they often immediately introduce a bend of these rules since most is fiction, or non-fiction biography true people stories. The audience is not academic. Fiction can introduce made up words. A story that is fiction must gab the reader, and hold their attention. This instant of making something of fantasy believable. A S. Coleridge "suspension of disbelief" must occur by the writing style of the author for the reader to believe it, or who reads the work? Essays in these classes allowed me to use gradable adverbs with non gradable adjectives.

Just as the texts you refer to state: human speech in informal settings often completely ignore the rules. It has been extremely challenging for me at times to shift when in one community it is absolutely to the point of forbidden to do, then shift to my kids or talking to the neighbor where not doing it makes you sound like an oddity.

So, best might be just answer your own question by asking yourself, "Who is my audience?"