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Whenever I come across the word paradigm, I have to make a small conscious effort not to pronounce the letter ‘g’.

In Italian, it is spelled paradigma and each letter is individually pronounced i.e. /pa·ra·dìg·ma/. But in English, paradigm is pronounced paradime, which is written phonetically as /ˈparədʌɪm/

(You can hear its pronunciation in this Oxford Dictionary video)

Its etymological roots are Greek; παράδειγμα (parádigma/parádeigma) meaning pattern, model, precedent, example, instance, and from παραδείκνυμι (paradeiknynai) meaning show, present, compare/confront.

However, I could find no explanation as to why the letter ‘g’ is silent nor why the letter ‘a’ was dropped.

I know in English there are many, many words that have silent letters. For example, the silent ‘e’ in minute, mouse, goose, have, etc. is a vestige of Old and Early Middle English when the suffix denoted whether a word was a noun or an adjective, singular or plural, or if it was the subject of a sentence or an object. The ‘h’ was pronounced in the middle of a word in Old English and its spelling changed to ‘gh’ in Middle English when it was preceded by a vowel. Eventually, the ‘h’ sound was dropped but its peculiar spelling still persists in words such as night, sight and thought.

If we look at the etymology of the term phlegm (which also ends with ‘gm’) we find

“Middle English fleem, fleume, from Old French fleume, from late Latin phlegma ‘clammy moisture (of the body)’, from Greek phlegma ‘inflammation’, from phlegein ‘to burn’. The spelling change in the 16th century was due to association with the Latin and Greek.”

Was there a similar change in spelling for paradigm? I didn't find anything. And why the /ʌɪ/ sound and not /ɪ/?

Other words of Greek origin such as gnomon, gnostic, and gnosis the ‘g’ is silent but in prognosis, the ‘g’ is instead pronounced.

Similarly, the letter ‘g’ in paradigmatic is pronounced
/parədɪɡˈmatɪk/ or /ˌpær.ə.dɪɡˈmæt.ɪk/ (BrEng) and /ˌper.ə.dɪɡˈmæt̬-/ (AmEng). Phonetic transcriptions courtesy of Cambridge Dictionary
_{(I don’t understand what happened to the missing k symbol in the American transcription but you can definitely hear it in the audio.)}

• Why was the last letter ‘a’ in the Greek word parádeigma omitted in English?


• Why is the letter ‘g’ in paradigm (paradime) silent but not in paradigmatic?

Sources
Why is "night" spelled with "gh"?
Silent "e" at the end of words
https://en.wikibooks.org/wiki/Old_English/Nouns

With paradigm and paradigmatic, just as with phlegm and phlegmatic, English only allows that g to be sounded when you can split a syllable. (The unassimilated version with a final ‑a technically does still exist, but the OED calls it “rare”.)

This is all because the phonotactics of English (the rules for how one can arrange its phonemes) do not permit a /g/ followed by a nasal at the end the word. You see the same thing occurring with diaphragm, or for that matter with the simpler phlegm, about which the OED writes that:

The g has probably never been pronounced except in disyllabic forms in ‑a.

Indeed, this same thing happens with all the words with a g and a nasal at their end, including
align,
apophthegm/apothegm,
arraign,
assign,
benign,
campaign,
consign,
deign,
design,
diaphragm,
ensign,
feign,
foreign,
hypodigm,
malign,
phlegm,
reign,
resign,
sign,
sovereign,
syntagm.

None of those is pronounced with a /g/, and probably never has been in English. At most they once had a /ɲ/ sound there way back when they ended with an unstressed vowel that we no longer write, all thanks to French.

When these words came into English, standard spelling did not yet exist. Many passed through French, where they were written with a final ‑e, or were modelled after words that were. French for example has paradigme, diaphragme. And words with a final unstressed ‑e like those came to be written without it once it stopped being said, at which point there was no chance to put the /g/ in one syllable and the nasal is the next.

You might as well ask why the g is “silent” in the Italian city of Bologna, pronounced of course with a geminated or “long” /ɲ/ in Italian or as /nj/ in English. In Italian, just as in French, the wheels of time have ground it down so that the ‹gn› spelling is now a digraph (a two-letter combo) representing /ɲ/, not as two separate letters each with their own sound.

Notice how that same thing happens with the French region of Bourgogne (Burgundy in English). Even when it’s spelled Borgogna in Italian, nobody “says” that g in French or Italian. It’s got a /ɲ/ phoneme there, which is why the Spanish sensibly spell it Borgoña to avoid confusing people.

We tend to keep the written ‹g› in English words like this, even though we “can’t” say it there at the end of the word right before that final nasal. This helps us understand the shared relationship with longer words like paradigmatic that have a vowel after the nasal, which allows the /g/ to “reappear”. But we probably no more ever said it in paradigm(e) than we ever said it in phlegm. Our phonotactic rules forbid it.

This likely also explains why apophthegm from Greek ἀπόϕθεγμα is more often spelled apothegm these days. We can’t say a lot of those letters, so we’ve given up writing them.

In comments, someone asked why aligning doesn’t cause the /g/ to “reappear” in split syllables as it seems to with paradigmatic (cf. French paradigmatique). The answer is that ‑ing is an English verbal inflection not a Classical or Romance one, but align the verb came to us from Middle French alinher, now¹ spelled aligner in Modern French. That means ther was never a /g/ phoneme for anyone there historically the way there was in the Greek παραδειγματικός. Part of the real, spoken language, our ‑ing verbal inflection in English is 100% regular with zero exceptions, so it cannot produce something out of nothing.

In comments, the asker further inquired as to whether the word went through a bunch of different spellings historically the way phlegm did, and why the last syllable has a diphthong:

“Middle English fleem, fleume, from Old French fleume, from late Latin phlegma ‘clammy moisture (of the body)’, from Greek phlegma ‘inflammation’, from phlegein ‘to burn’. The spelling change in the 16th century was due to association with the Latin and Greek.” was there a similar change in spelling for paradigm? I didn't find anything. And why the /ʌɪ/ sound and not /ɪ/?

The answer to these questions is that paradigm didn’t show up in English until hundreds of years after the word we now spell as phlegm did. For paradigm, the first actual English citation (rather than Latin) is from 1493 in Caxton, written paradygmes and translating French.

And the reason it’s today pronounced [ˈpæɹədʌɪm] or [ˈpeɹəˌdɑɪm] with a diphthong in the last syllable is because the Great Vowel Shift² notoriously changed the “long” i sound /iː/ into a diphthong. English spelling was more or less frozen before the GVS, which explains a great deal of confusion compared with the standard Latin values for vowels that everyone else but us uses.

Footnotes

1. Because the ‹nh› and ‹gn› digraphs represent the same /ɲ/ phoneme under different orthographic traditions. Old Occitan / Provençal, being langues d’oc, used ‹nh› for it, as does Catalan which is also in that group. The Galician–Portuguese language deliberately broke from Castilian orthographic habits and instead adopted ‹nh› during the early 1300s thanks to the famous poet Dinis I, king of Portugal and the Algarve, because of the prestige position which the troubadours’ tongue held in that age. In contrast, the “langues d’oïl” branches of French, of which Modern French is a descendant, used the ‹gn› digraph just like Modern Italian does for this /ɲ/ sound. Castilian and Asturian use ‹ñ› (earlier ‹nn›) for the same /ɲ/ sound in those languages as the ‹nh› and ‹gn› digraphs do in theirs. Uncountably many research papers and scholarly tomes have been written about the origin of the palatalized nasal [ɲ] and lingual [ʎ] sounds during the transition from Latin to Modern Romance, no matter the spelling.




2. The GVS was a chain shift that reassigned new sounds to all the long vowels in English in a bizarre way that all other users of the Latin alphabet forevermore hate us for. :)

Why it isn't spelled with "a"

As tchrist mentioned, the lack of final "a" is probably based, at least in part, on the change of Latin word-final "a" to French "e" /ə/, and the loss of word-final /ə/ in the latter part of Middle English.

Why "paradigm" isn't pronounced with /gm/

The word "paradigmatic" is pronounced with /g.m/ in separate syllables; this isn't possible in the word "paradigm", where the "m" is word-final. I would say that the phonotactic constraint that tchrist mentioned can be described as a ban on /gm/ as a coda (syllable-final) consonant cluster.

I'm not sure exactly how to explain why English speakers used /m/ rather than something like /gm̩/ as a way of getting around this ban (compare and contrast the treatment of -sm and -thm words), but it seems plausible that the use of g as a "silent" letter in older words was a factor. Wikipedia says that the simplification of word-initial /gn/ to /n/ occurred somewhere around the 17th century, so words like gnostic and gnosis may not have been an influence until fairly late, but I believe words like sign that come from French and are spelled with the digraph gn that is used in French to represent the palatal nasal /ɲ/ had g-less pronunciations from the beginning in English.

The pronunciation of the vowel in the final syllable

There is actually an alternative pronunciation of "paradigm" with the "short i" sound in the last syllable. It seems to be uncommon nowadays, but I have the impression that it was more common in the past. I'm not sure that we can be confident about tracing the pronunciation with the "long i" sound back to Middle English.

The American Heritage Dictionary lists both pronunciations as follows: "păr′ə-dīm′, -dĭm′".

Walker's Critical Pronouncing Dictionary (1791) gives the pronunciation as "pa⁴r′a⁴-di²m", where "a⁴" is described as the vowel in "fat" and "i²" is described as the vowel in "pin". The lack of vowel reduction in the second syllable may be an artificial artefact of Walker's transcription habits, but it's nevertheless clear that this indicates a pronunciation with "short i" in the final syllable. Walker compares the pronunciation to that of other words ending in "-gm" in some other places (e.g. rule #389 in the initial section laying out principles and patterns of English pronunciation, and in the entry for impugn).

Charles Harrington Elster's Big Book of Beastly Mispronunciations (2nd edition, 2005) summarizes the sources he looked at by saying

PAR-uh-dim is the traditional pronunciation in both England and the United States; it was preferred by Walker (1791), Worcester (1860), Funk & Wagnalls Standard (1867), and the Century (1914). Sometime before 1900 the variant PAR-uh-dym gained currency and was sanctioned as an alternative by Webster's Collegiate (1917), Phyfe (1926), OED 1 (1928), and Vizetelly (1929), and preferred by Fowler (1926). Webster 2 (1934) preferred -dim and labels -dym "especially British," but subsequent sources listed -dym without comment, and current dictionaries — with the exception of OED 2 (1989) — list PAR-uh-dym first. Everyday Reader's (1985) favors -dim, but other recent authorities — including Lass & Lass (1976), the NBC Handbook (1984), Barnhart (1988), Jones (1991), and Burchfield (1996) — prefer -dym, and WNW Guide (1984) says that "fewer and fewer people are saying PAR uh dim." For what it's worth, I am one of them.

Second edition update: I'm changing my tune. PAR-uh-dim is dead. Two of the six major current American dictionaries don't even bother to list it, and ODP (2001) gives PAR-uh-dym as both the British and American pronunciation. The -dym has come; I'm going with the flow.

(p. 366)