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I understand why naïve is spelled with two dots, and that those dots are called a diaeresis.

What I do not understand is whether the use of a diaeresis is legal in English; is it?

Other than naïve, I am hard pressed to come up with another example of a word that uses a diaeresis.

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Yes, of course it is “legal”. The Oxford English Dictionary includes all these words with either diaereses or umlauts in them:

achroö-, aïoli, Ångström (unit), arsinoïtherium, Aufklärer, Aufklärung, bastnäsite, beënt, Boötes, Böttger, bratwürste,bröggerite, Brontëan, Brontëana, Brontëism, Brückner, cacoëpistic, cacoëpy, caïque, caïquejee, Chün, cinnameïn, Danaïdean, denkmäler, dzongpön, einfühlung, Eötvös, epiploön, faröelite, faux-naïf, flügelhorn, föhn, fräulein, Fröbel, Fröhlich, Führer,gefüllte fish, gemütlich, Gemütlichkeit, Gestaltqualität, glögg, glühwein, Gödel, Gödelian, Gödel(’s) theorem, Götterdämmerung, gräben, Grübelsucht, grünauite, grünerite, grünlingite, gümbelite, Günz, hälleflinta, hälleflintoid, haüyne, haüynite, haüynophyr, heröa, heröon, heveëne, hexaëmeron, hübnerite, hügelite, hühnerkobelite, hydantoïn, hyperoödon, Identitätsphilosophie, jäger,kaïd, kämmererite, kësterite, Klebs-Löffler, knödel, kochubeïte, kulturträger, kümmel, Kümmell’s disease, Künstlerroman, Kuo-yü, lagerstätte, lagerstätten, Länder, ländler, langue d’oïl, Laocoön, lekë, Lieberkühn, Lieberkühnian, löllingite, löweite, löwigite,lünebergite, Lung-ch’üan, märchen, Marie-Strümpell, Marinesco-Sjögren syndrome, mässig, Meistersänger, Meistersängers, melopoeïa, microaërophilous, mitteleuropäisch, Möbius, Moët, monoïdeism, Mössbauer, muisvoël, Müller, müller, Müllerian, Müller-Lyer, Müller’s larva, Müller-Thurgau, naïd, naïdes, naïf, naïfly, naïs, naïve, naïvely, naïveness, naïveté, naïvety, nordenskiöldine,onomatopoësis, ouabaïn, panzoöty, penniä, Perrier-Jouët, ploïmate, Plücker, Plückerian, poë-bird, Prägnanz, prosenneëdrous, purpureïn, Qualitätswein, Querflöte, Ragnarök, Reichsführer, rohrflöte, Rörstrand, rötheln, rumänite, salonfähig, Salpausselkä, salpingo-oöphorectomy, salpingo-oöphoritis, Sängerfest, Schabzieger Käse, Schätzi, Schimpfwörter, Schiötz, schmierkäse, Schönlein,schröckingerite, Schröder, Schrödinger (wave, ψ-) function, Schrödinger, Schrödinger(’s) (wave) equation, Schüller-Christian, Schutzbündler, schwärm, schwärmer, schwärmerei, schwärmerin, schwärmerisch, Sjögren, sjögrenite, Sörensen, Spätlese, Spätzle, Spielräume, Spitzflöte, Spörer, Spörer minimum, Spörer’s law, Sprachgefühl, Standartenführer, staphisaïne, Steinhäger, strüverite, Sturmbannführer, tetraëterid, tetrastoön, thebaïcine, Thebaïd, thebaïne, thebaïsm, thymoïl, thymoïlamide, Töpler, triacontaëterid,trögerite, Türkmen, über alles, Überfremdung, überhaupt, Übermensch, übermenschlich, Übermenschlichkeit, ulvöspinel, väyrynenite, Versöhnung, Volapük, Volapüker, Völkerwanderung, Völkerwanderungen, völkisch, Vöslauer, wabaïn, Waldenström, Wandervögel,Widmanstätten, Wöhler, wölsendorfite, Wörter und Sachen, Würm, Würmian, Würtemberg siphon, Württemberger, wüstite, xanthoxyloïn, yüeh, Yüeh, yüeh ch’in, Zöllner.

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    @Sabuncu You should definitely see this answer here. You can still click the outlined checkmark on this current question to accept it even though it is marked as a duplicate. They may merge it later, though. – tchrist Mar 25 '13 at 14:18
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    Adding umlauts to an answer about diaereses is wantonly confusing; the former indicate a foreign origin in writing (and may or may not affect pronunciation in English), while the latter affect pronunciation but are effectively optional in writing. – Tim Lymington Mar 25 '13 at 15:35
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    @TimLymington I’m well aware of the difference, but I could think of no simple way to grep out which is which. I think the questioner was really just asking whether using those two little dots above is legal in English. You will note that Unicode does even have a separate code point for an umlaut that is distinct from a diaeresis, and the OED does not encode those differently either. – tchrist Mar 25 '13 at 15:50
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    @Sabuncu Yes, of course it is “legal” — whatever that means. Those are all from the OED, which documents what people actually use. – tchrist Mar 25 '13 at 17:56
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    Has Noël been cancelled? – Edwin Ashworth Mar 25 '13 at 18:20