What is the $\tau$ symbol in the Bourbaki text?
Adrian Mathias offers the following explanation here:
Bourbaki use the Hilbert operator but write it as $\tau$ rather than $\varepsilon$, which latter is visually too close to the sign $\in$ for the membership relation. Bourbaki use the word assemblage, or in their English translation, assembly, to mean a finite sequence of signs or leters, the signs being $\tau$, $\square$, $\lor$, $\lnot$, $=$, $\in$ and $\bullet$.
The substitution of the assembly $A$ for each occurrence of the letter $x$ in the assembly $B$ is denoted by $(A|x) B$.
Bourbaki use the word relation to mean what in English-speaking countries is usually called a well-formed formula.
The rules of formation for $\tau$-terms are these:
Let $R$ be an assembly and $x$ a letter; then the assembly $\tau_x(R)$ is obtained in three steps:
- form $\tau R$, of length one more than that of $R$;
- link that first occurrence of $\tau$ to all occurrences of $x$ in $R$
- replace all those occurrences of $x$ by an occurrence of $\square$.
In the result $x$ does not occur. The point of that is that there are no bound variables; as variables become bound (by an occurrence of $\tau$), they are replaced by $\square$, and those occurrences of $\square$ are linked to the occurrence of $\tau$ that binds them.
The intended meaning is that $\tau_x(R)$ is some $x$ of which $R$ is true.
The Hilbert operator was introduced by Hilbert in 1923. Hilbert was conviced of the axiom of choice as an indispensable principle but refrained to introduce it in its logical system. Instead he postulated another axiom akin in content to the axiom of choice, introducing new kind of terms: the $\tau$-terms (in fact Hilbert used the letter $\epsilon$ instead of $\tau$ but Bourbaki preferred the latter to avoid any confusion with the set-theoretical $\in$ symbol).
Hilbert's operator offers a systematic mechanism of attaching to a first-order predicate $R[x]$ a $\tau$-term by the procedure described above, i.e. $\tau_x(R)$. Formally this is a string where all the occurences of $x$, if any, are muted and tightly bound to the initial $\tau$ symbol by an upper link (the largest link indicates the scope of the bounding).
The intented intuitive meaning is according to Bourbaki the following: If there exists some object for which the predicate $R[x]$ is true then $\tau_x(R)$ represents some kind of an "ideal" (Hilbert's terminology) "prefered" (Bourbaki's terminology) object for which the predicate $R[x]$ is certainly true. Otherwise, strictly nothing can be said about it".
Accordingly to this naïve interpretation $(\exists x)R$ can be considered as an alias for the relation $R[\tau_x(R)]$ where the $x$ variable is in fact muted and bound in all of its occurrences in $R$. Similarly $(\forall x)R$ can be considered as an alias for the string $\neg(\exists x)(\neg R[x])$, which becomes now expandable into an explicit formula based on Hilbert's $\tau$-term.
Two logically equivalent 1st-order predicates in variable $x$ should lead to the same ideal $\tau$-term. The axiom schema about $\tau$-terms and equality ensures that: $(R\leftrightarrow S)\rightarrow \tau_x(R)=\tau_x(S)$
The net result of that schema is that when the predicate is one-to-one in $x$ (i.e. $R[x] \wedge R[x'] \rightarrow x=x'$ then $R[x]\rightarrow$ $x=\tau_x(R)$. : the $\tau$-term attached to predicate $R$ is the obliged logical solution of $R[x]$, if such a solution does exist. This was the axiom postulated by Hilbert. If furthermore this solution exists, i.e. $(\exists x) R[x]$ that is to say $R[\tau_x(R)]$ is true, then $R[x]$ logical equation is equivalent to $x=\tau_x(R)$, i.e. the converse of Hilbert's axiom is also true.
$\tau$-term $\tau_x(R)$ can thus be regarded as the result of some particular choice (an ideal one) among all possible logical solutions of predicate $R[x]$ and can thus be construed from a choice function which usage is set-theoretically justified by the axiom of choice.