At language blog The Stroppy Editor, Tom Freeman takes the Guardian to task for the way it treats part of a submission (for a government consultation process on child poverty) from a team led by Alan Milburn, "ex-Labour minister turned coalition adviser":
(the submission) adds: "It now seems vanishingly unlikely [sic] that this government will hit the targets in the Child Poverty Act."
By adding the expression [sic] after vanishingly unlikely, either the writer or a sub-editor seemed to be suggesting that these words contained an error of some sort, either grammatical or spelling.
Freeman goes into some detail to explain why he thinks the Guardian were wrong, which I won't go into here (there's a link to both articles at the end of this post). What interested me was that I didn't think I'd ever heard the expression vanishingly unlikely before, or vanishingly anything for that matter, and had to think about it for a second or two to work out the meaning.
I checked six British online dictionaries, and it turned out that by no means all of them list the adverb vanishingly. Oxford Online does, but Oxford Advanced Learner's doesn't. Longman and Collins Cobuild do, but Cambridge, Chambers and Macmillan don't. Here is the definition from Oxford Online, with an example sentence:
adverb [as submodifier]
in such a manner or to such a degree as almost to become invisible, non-existent, or negligible:
an event of vanishingly small probability
The adverb vanishingly appears to have been around for some time, in both AmE (blue) and BrE (red):
Collocations with vanishingly
It seems to be almost always used as an intensifier before an adjective (or occasionally a determiner or pronoun). Until relatively recently, however, it was used almost exclusively with 
small. Here are a selection of search results for 
vanishingly in 
Google Books 1938-1996The use of vanishingly unlikely seems to be pretty new, and its use is considerably more common in British books than in American ones. It seems to have really taken off in the UK, for some reason, after 2000:
But this use is still way below that of vanishingly small and to a lesser extent of vanishingly rare:
Here is a list of collocations with 
vanishingly at 
Netspeak that seems to bear this out, 
vanishingly unlikely accounting for only 2.6% of instances of all uses of 
vanishingly (which is itself pretty uncommon - the 56,836th most common word in English, according to 
Wordcount). 
Just the Word only finds one collocation, with 
small.
Collocations with unlikely
When we look at collocations with 
unlikely, the only one with the meaning of change to show up in collocation finders seems to be 
increasingly: see 
Just the Word and 
Netspeak (However, see my comments in the update at the end of this post). 
Vanishingly doesn't even get a look in, which is also the case at Ngram (blue and orange are for American books, red and green for British books):
For somebody (like me) not familiar with the adverb vanishingly, it can look like the opposite of increasingly, and therefore my second or two's confusion. And I wondered if the Guardian writer or sub was also unfamiliar with the expression and had also seen it this way. 
Vanishingly unlikely at the Guardian
The strange thing is that the Guardian is one of the places where we are most likely to see vanishingly unlikely, being beaten only by the Economist, among British newspapers. Here is a league table of instances of vanishingly unlikely as listed in Google site searches on Sunday 23rd March. Interestingly, it doesn't seem to appear in the tabloids at all: the Mail, the Express, the Sun and the Mirror all drew a blank.
So it looks as though someone at the Guardian didn't do their homework very well. And at least I've learnt a new expression, although I don't think I'll be using it very much.
Unlikely - a little grammar (for EFL/ESL students)
Despite its 
-ly suffix, 
unlikely is, like its positive opposite 
likely, an adjective. It sometimes comes before a noun - 
'in the unlikely event', 'an unlikely story', but more often follows a linking verb (e.g. 
be, become, seem) as a subject complement, also known as being in predicate position. Here it is often followed by a 
to-infinitive clause or a 
that clause, as you can see 
here, or is simply used on its own:
- This report is unlikely to have much effect on government policy.
- It is becoming increasingly likely that the company will close down.
- More redundancies seem unlikely at this stage.
Both adjectives are gradable: something can be more or less likely or unlikely. The most common intensifiers are 
highly, very and 
extremely, as you can see 
here and 
here. We wouldn't normally use absolute intensifiers like 
absolutely or 
totally.
Update
I should really read dictionary definitions more carefully! Given that the adverb vanishingly comes from the action verb vanish, which describes a process, I had inferred that vanishingly also described a process, like increasingly: in other words, that vanishingly unlikely meant more and more unlikely. And that is indeed what the Oxford Online definition suggests to me with its use of the word become, as do some of the examples I looked at.
Having read Tom Freeman's comments on his blog in response to this post, however, I now see that the other two dictionaries mention nothing about a changing situation, Longman defining vanishingly as extremely, and Collins as to a very great extent or degree, and Freeman points to examples where no change seems to be in progress.
I still feel privately, that vanishingly ought logically to refer to a changing situation, but in terms of usage it appears that this is not always so. 
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