Since TeX is also a Turing-Complete programming language as well, …
Note that while indeed TeX macros turn out to be Turing complete, TeX was not initially designed as a programming language; this is more an instance of “accidentally Turing-complete” — it became so despite Knuth's original intention not to make it a programming language. I've written in other answers about this, so won't elaborate here.
…language specification (syntax, semantics, primitives), etc. that described TeX.
The “official” specification of TeX's syntax, semantics, primitives, etc., is the book called The TeXbook (the “definitive user manual and reference manual for TeX”), which is also available in hardcover as Volume A of Computers and Typesetting. You may also be able to find it in a library (including online at the Internet Archive's library, where usually there's a waiting list—only one person in the world can borrow the book at a time—but this is temporarily lifted).
But if your question is primarily historical:
Before there was The TeXbook, which documents the current TeX (written in WEB, also known as TeX82 to distinguish it from TeX78 which was written in the SAIL language), there was a manual for TeX78. It is very similar in contents to The TeXbook, and you can find it in, for instance, the book TeX and METAFONT: New Directions in Typesetting, along with the METAFONT manual and the article Mathematical Typography.
If you want to go back even further in history and just want the truly first TeX “proposal” or specification, then you're in luck! In May 1977, before a single line of code had been written, Knuth typed up a document called “Preliminary preliminary description of TEX”, and saved it as a file named
TEXDR.AFT. In July he revised it to “Preliminary description of TEX” as the file
TEX.ONE. Both of these have been published in the book Digital Typography, and are also available online at the SAILDART historical archive. Note that the first of these doesn't even have backlashes for control sequences (for example), and even the second of these is really preliminary: Knuth's intention was that a couple of his graduate students would code up TeX from this specification while he was away for the summer, but it turns out that the specification was really incomplete (but they got a prototype capable of printing a simple page anyway), so when he came back he spent the next several months writing the program, which became later known as TeX78 and is the one documented in the manual mentioned above.