Is criterions a valid plural for criterion?
If you use a word like criterion anywhere you are on the safe side by using the Greek plural, criteria. People who understand the former will get the latter, and vice versa.
Moreover, criterions is really not common according to google n-grams.
Let me emphatically echo FumbleFingers' comment: NO!
(I’ll add this somewhat meandering discussion as an answer to bring together in one place my various comments posted elsewhere.)
Let me add that the regularized -ion > -ions inflection in English for Greek 2nd-declension nouns instead of the Greek *-ion > -ia is comparatively rare. The OED attests only four instances of this that I can discover:
Then again, any such noun is pretty rare in English. I think the only even moderately commonplace Greek -ion word apart from criterion is the medical term ganglion, and it too was historically on occasion given a regularized English plural, even though ganglia is now standard.
There are just not that many -ion nouns in English that come from Greek nouns. An OED search looking for English headwords ending in -ion and derived from Greek uncovers only 96 terms, and some of these do not count. For example, although ion is indeed < Greek ἰόν, that was the neuter present participle of ἰέναι to go.
If you relax the language of origin restriction, searching for -ion nouns in English headwords returns around 6900 terms, showing how uncommon the Greek ones actually are.
Most Greek -ion words occur in technical English, where retention of classical inflections is always higher than in the general populace, a well-documented phenomenon of ‘precisionist technicalese’, as it were. Only criterion alone is moderately commonplace outside technical circles, although a scant few other -on words do occur, like phenomenon in the previous sentence. The rest are all technical terms such as anthemion, aphelion, hapax legomenon, idolon, lepton, mitochondrion, noumenon, polyhedron, and taxon.
In computational linguistics, all the popular stemmers and lemmatizers (with one still-unpublished exception) do a distressingly poor job on technical writing, because they are ignorant of the classical inflections specialists continue to use with technical terms. The biomedical literature is particularly rich in Greek terms, and computer programs attempting natural language processing on these corpora must be specially trained to handle things like stoma, stomata if you expect to unify down to common lemmas.
The route by which these Greek words were introduced to English is not uniform. It may be directly from Greek; it may have first gone through Latin to get here; and it may have spent a time on holiday in France or even in Germany before finally making its way to us. For some words, such as parhelion for example, more that one route of origin applies. This multi-sourcing has led to a plethora of alternate forms. Here’s one example from the OED of how confusing the historical record of forms can be:
As I mentioned in a comment to Barrie, the OED also attests a Latinized form of the singular criterium to go with the Greek criterion, with three citations one from each of the 17–19th centuries and starting with John Donne. Both forms conveniently become criteria in the plural.
There is however another, much more recent sense of criterium in contemporary English. The crowd-sourced Wiktionary entry gives this sense as
Apparently, the plural for this particular sense is always criteriums, as grating as that may be to some ears. The crowd-sourced Wikipedia entry also gives the abbreviation crit for the circuit race, but I can find no history of the origin of this criterium for this sense.
Long-term stability of classical inflections in English plurals is dubious. Most eventually get regular forms in English, but this takes much longer if it is only a technical term used by specialists. In words of common usage, there is often a reanalysis of the classical form to better match the native form. On the other hand, technical literature is now more accessible than ever before, and this may interfere with regularization moreso than in previous centuries.
For example, data is often reanalysed to be a singular instead of a plural. This becomes a problem if it is to be used as a count noun instead of mass noun, since that would lead to the double-plural *datas, which has caused the more regularly behaving dataset and datapoint to spring up to work around that particular problem. Since we now have workable alternatives to address the mass- vs count-noun problem, I begin to doubt datum/data will be around that much longer in even technical English. No harm done, of course.
Similarly, casual speakers and writers, especially in newspapers and nontechnical magazines, are often found using bacteria and phenomena in the singular. You need only check the prevalence of ‘a bacteria’ or ‘this bacteria’ in uncurated, crowd-sourced Wikipedia to see how commonplace such errors are.
Finally, I should note that occasionally both forms persist, potentially with different nuance, such as a multidimensional array having several indices but a book having multiple indexes. It wouldn’t surprise me if casual speakers wrote ‘a criteria’ just as they so often write ‘a phenomena’ or ‘a bacteria’, unaware of or unconcerned with the chalkboard-scraping effect this has on some of us.
However, it seems to me that anyone conscious enough of the root lemma in -ion will choose the classical inflection rather than selecting the courageously regular ?criterions, let alone the Latinized ?criteriums (apart from the race).
According to the Merriam-Webster dictionary, criteria and criterions can both serve as plural to the singular criterion. Here's the link.. http://www.merriam-webster.com/dictionary/criterions
Is criterions a valid plural for criterion? In a hundred years it might be, with the tendency towards anglicizing everything, but not now. If your native language is not English, then just take it from native speakers that criterions sounds to their ear as unpleasant as the proverbial fingernails on the blackboard.
"Criterions" does, however, sound like it might be a great name for a ball team...:)