Bourbaki used a very very strange notation for the epsilon-calculus consisting of $\tau$s and $\blacksquare$. In fact, that box should not be filled in, but for some reason, I can't produce a \Box. 


Anyway, I was wondering if someone could explain to me what the linkages back to the tau mean, what the boxes mean. Whenever I read that book, I replace the $\tau$s with Hilbert's $\varepsilon$s. I mean, they went to an awful lot of trouble to use this notation, so it must mean something nontrivial, right?


You can [see](http://books.google.com/books?id=IL-SI67hjI4C&lpg=PP1&dq=bourbaki&pg=PA15#v=onepage&q=&f=false) it on the first page of the google books link I've posted. I'm not sure if it's supposed to be intentionally vague, but they never introduce any metamathematical rules to deal with linkages except for the criteria of substitution, which pretty much cannot interact with $\tau$ terms. 


Also, of course, since it's a book written to be completely a pain in neck to read, they use a hilbert calculus, and even worse, without primitive equality for determining whether or not two assemblies are equivalent. 



Bourbaki's epsilon-calculus notation
Let me address the part of the question about "what the linkages back to the tau mean, what the boxes mean." The usual notation for using Hilbert's epsilon symbol is that one writes $(\varepsilon x)\phi(x)$ to mean "some (unspecified) $x$ satisfying $\phi$ (if one exists, and an arbitrary object otherwise)." If, like Bourbaki, one wants to avoid quantifiers in the official notation and use $\varepsilon$ instead (specifically, expressing $(\exists x)\phi(x)$ as $\phi((\varepsilon x)\phi(x))$), then any non-trivial formula will contain lots of $\varepsilon$'s, applied to lots of variables, all nested together in a complicated mess. To slightly reduce the complication, let me suppose that bound variables have been renamed so that each occurrence of $\varepsilon$ uses a different variable. Bourbaki's notation (even more complicated, in my opinion) is what you would get if you do the following for each occurrence of $\varepsilon$ in the formula. (1) Replace this $\varepsilon$ with $\tau$. (2) Erase the variable that comes right after the $\varepsilon$. (3) Replace all subsequent occurrences of that variable with a box. (4) Link each of those boxes to the $\tau$ you wrote in (1). So $(\varepsilon x)\phi(x)$ becomes $\tau\phi(\square)$ with a link from the $\tau$ to the boxes (as many boxes as there were $x$'s in $\phi(x)$). 


One might wonder why Bourbaki does all this. As far as I know, the point of the boxes and links is that there are no bound variables in the official notation; they've all been replaced by boxes. So Bourbaki doesn't need to define things like free and bound occurrences of a variable. Where I (and just about everybody else) would say that a variable occurs free in a formula, Bourbaki can simply say the variable occurs in the formula. 


I suspect that Bourbaki chose to use Hilbert's $\varepsilon$ operator as a clever way of getting the axiom of choice and the logical quantifiers all at once. And I have no idea why they changed $\varepsilon$ to $\tau$ (although, while typing this answer, I noticed that I'd much rather type tau than varepsilon).


