My book on Russian claims that in phrases like здесь нет телефо́на the нет is a contraction of не and есть .
Is this etymologically true?
(I realize that it is translated as such, but this is not my question.)
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Currently no Russian speaker feels that word нет is compressed form of не есть. The нет and есть are thought as antonyms. However looks like long ago (over 600 years ago) the connection between them existed.
According to Vasmer's dictionary, the etymology of нет:
народн. не́ту, др.-русск. нѣту (РП, Владим. Моном., Лаврентьевск. летоп.) и нѣтъ (Лаврентьевск. летоп., Договорная грам. Димитр. Донск. 1372 г., Полоцк. грам. 1399 г.; см. Соболевский, Лекции 94 и сл.). Из более старого *не ѥ ту; см. Соболевский, там же, 249; Преобр. I, 218, 596; Цв. Тодоров, "Slavia", 14, 519 (где о -ту, с литер.); ср. также тут (см.).
So, before 14th century it had a form *не ѥ ту (asterisk means that the written form is reconstructed).
On the other hand the word есть etymologically has a prototype in Old Slavonic:
ѥстъ, редко ѥ
Etymology of тут by Vasmer is connected with Old Russian word ту:
др.-русск. ту - "там, туда, тут"
So looks like the word нет before 14th century was expressed as:
*не ѥ ту = не есть тут, не есть там, не есть туда
I always used that simple rule teaching English to Russian children.
And yes, "нет" was sometimes the contraction of "не"+"есть". The old form was "несть", and it can be met in classic Russian poetry of XIX Century. But nobody feels that connection anymore, it is too hidden, in contrast to Slovenian, for example, where it is obvious even now. But Slovenian and Czech languages are the most conservative Slavic languages.