# What is the $\tau$ symbol in the Bourbaki text?

I'm reading the first book by Bourbaki (set theory) and he introduces this logical symbol $\tau$ to later define the quantifiers with it. It is such that if $A$ is an assembly possibly contianing $x$ (term, variable?), then $\tau_xA$ does not contain it.

Is there a reference to this symbol $\tau$? Is it not useful/necessary? Did it die out of fashion?

At the end of the chapter on logic, they make use of it by... I don't know treating it like $\tau_X(R)$ would represent the solution of an equation $R$, this also confuses me.

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Just out of sheer interest, what reason do you have for studying Bourbaki's set theory? –  Asaf Karagila Aug 1 '12 at 9:15
@AsafKaragila: I want a concise introduction which states all the things it uses. I'd be thankful for a more modern reference!! Now I'm using Bourbaki and Jech and wikipedia. I try to write the things down while reading it to get a good overview, and it might be a little pedantic but when e.g. Jech starts writing "$a\ne b$" without saying that this symbol "$\ne$" is supposed to express the negation of "$a=b$" I get uneasy. As a remark, I'm really a physicist and I never took a logic/set-theory lecture. –  NikolajK Aug 1 '12 at 9:27
I think $\tau$ is supposed to be some kind of choice operator, but I haven't looked at the definition. (I don't read French.) As you have surmised, its use is unfashionable nowadays. –  Zhen Lin Aug 1 '12 at 9:27
@Nick: You remind me one of my teachers which is extremely pedantic. When I took the advanced logic course we spent two weeks proving that there is a unique way to interpret a sentence, even without parenthesis everywhere. He even had two different symbols for equality, one in the language and one in the meta-language. On the other hand, when I was 12 or so I already knew the symbol $\neq$, so I suppose using it without specifying what its meaning is not the worst thing that can happen in a book. –  Asaf Karagila Aug 1 '12 at 9:33
@AsafKaragila: I know the symbol too of course, it's obvious. But I care about the definition for implementation reasons. A priori "$\ne$", "$\int$" and "$\Gamma$" could all mean the same. Suddently writing "$a\ne b$" is the same as "my_cat_loves_fish(a,b)" for an AI, say. I'd like a book which spares me the work to come up with all these things "$(a\ne b)\ :>\ \text{not}(a=b)$" myself. PS: I think you told me about your teacher in my set builder notation bracket question. –  NikolajK Aug 1 '12 at 9:37

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:

1. form $\tau R$, of length one more than that of $R$;
2. link that first occurrence of $\tau$ to all occurrences of $x$ in $R$
3. 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.

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Thanks! Do you happen to be able a more modern reference, basically for a foundational treatment of first order logic? –  NikolajK Aug 1 '12 at 9:58
I liked Forster's Logic, induction and sets, but you may find it a bit too informal. –  Zhen Lin Aug 1 '12 at 10:11