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I'm trying to understand Bourbaki's definition of universal quantification. The definition is on Page 36 in Theory of Sets as follows:

$(\exists x R) \equiv (\tau_{x} \mid x) R$

$(\forall x R) \equiv \lnot ((\exists x) \lnot R)$

For example:

$R = \in x y$

The resulting formula is: $\lnot (\lnot \in (\tau \space \lnot \in \square y) y)$

The resulting tree is: Bourbaki Tree Universal Quantification

It's my understanding the $\square$ is a distinguished object that satisfies the evaluated truth of $\tau$. Therefore, if there is a distinguished object $\square$ that satisfies $\tau$, i.e. a object that does not satisfy $\in x y$, the universal quantification is false.

Does the quantification function $\tau$ try every object in the domain of discourse? If so, and all the objects in the domain of discourse do not satisfy $\tau$ (i.e. universal quantification should be true), what does $\tau$ evaluate to?

It seems that $\tau$ must evaluate to an object that satisfies $\in x y$ for the formula to be true, however I'm unsure how this occurs as every object in the domain of discourse must be tested first for universal quantification to be true.

Any guidance here appreciated. Thanks

Asaf Karagila
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    Why are you reading this in Bourbaki? – Asaf Karagila May 14 '19 at 16:07
  • @Asaf Karagila Of course there is an alternate definition given by Church: $\forall x$ ___ is True if the value of ____ is True for all values of x. $\forall x$ ___ is False if the value of ____ is False for any value of x. Also defined $\lnot ((\exists x) \lnot R)$. –  May 14 '19 at 16:24
  • You haven't answered my question. Also, I believe that is due to Tarski, not Church. – Asaf Karagila May 14 '19 at 16:26
  • @Asaf Karagila I'm interested in the Bourbaki view. The definition is in Church's Introduction to Mathematical Logic, however could be from someone else. –  May 14 '19 at 16:27
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    @Nick Except for historical curiosity, there's almost no reason to read Bourbaki's Theory of Sets. A term of length 4,523,659,424,929 gives a pretty clear critique that the formal system presented is poorly made resulting in large amounts of needless complexity. The discussions referenced and had here point to other issues. Theory of Sets is not particularly necessary for the later volumes if those are your real goal. – Derek Elkins left SE May 14 '19 at 19:19
  • @Derek Elkins I'm aware of the post. Historic curiosity. It has also been noted elsewhere that Mathias's paper on the length of the term may be more because other formalisms do not reduce quantifiers further into symbols, at least in Bourbaki symbols of meta-mathematics. Thanks –  May 14 '19 at 19:29
  • @Nick Okay. I just wanted to be sure that you aren't going through this for the wrong reasons. It's very easy to imagine someone deciding to read through Bourbaki for reasons that it's understandable to believe but happen to be false. – Derek Elkins left SE May 14 '19 at 20:07

2 Answers2

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The concept here is that by some means, $\tau_x(R)$ picks - once for all time - an otherwise completely arbitrary value of $x$ such that $R$ is true. If there is no value of $x$ for which $R$ is true, then $\tau_x(R)$ simply picks an arbitrary object.

Think of it as some munificient deity has compiled a list of all possible relations involving $x$, and for each assigned at random a value that makes it true, provided that such a value exists. Otherwise, it assigned a value completely at random. Once this list and assignments are established, $\tau_x(R)$ will always be the value assigned to $R$ in the list. Since the values assigned are random, when $R(\tau_x(R))$ is true, the only things that are knowable about $\tau_x(R)$ are the theorems that can be proved from $R(\tau_x(R))$. And when $R(\tau_x(R))$ is false, the only things knowable about $\tau_x(R)$ are things than can be proven for all values.

Of course, these are just the intuitions behind the operator. From the formalist view of Bourbaki in this book, it is really just strings of symbols being manipulated in accordance with certain rules.

But say what you will about their shortcomings (and I agree), I will still hold a fond spot for formalism in general and Bourbaki's in particular, for first demonstrating to me that the question of "what is the true mathematics" is meaningless. As long as the axioms you've chosen are consistent, your theory is just as "true" as any other. That is mathematical freedom.

Paul Sinclair
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  • I get the first part about the random value that makes it true. I'm confused on the statement that false "it assigned a value completely at random". The statement in the book says if an object doesn't satisfy $\tau_x (R)$ then it "represents an object about which nothing can be said". So something other than empty set? Also, I'm still confused on how the tree above evaluates to true, which would show $\forall x$ is a true relation. Thanks –  May 15 '19 at 00:52
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    @Nick $\tau_x(\bot)$ is basically like adding a new constant to the set theory, call it $c$. This new constant has no axioms associated with it. You don't know anything about it. This includes negative information. You don't know if $c=\varnothing$, but you also don't know if $c\neq\varnothing$. For every predicate, $P$, that isn't either true for every set or false for every set, $P(c)$ is undecidable. That's what it means by "an object about which nothing can be said". – Derek Elkins left SE May 15 '19 at 01:12
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    Exactly as Derek says. When $R$ is always false, $\tau_x(R)$ is just a random object. It does not need to be some new thing. "An object about which nothing can be said" just means we are unable to prove anything about this object from $\lnot R(\tau_x(R))$. It can have properties, but we have no way of knowing what those properties are. – Paul Sinclair May 15 '19 at 01:25
  • As for your tree, it is the tree of the expression $(\forall x)(x \in y)$. Why would you expect this to evaluate as true? You've not even defined $y$ yet, so how would you expect to prove anything about it. Later on, Bourbaki will prove $(\forall y)(\exists x)(\lnot(x \in y))$. – Paul Sinclair May 15 '19 at 01:38
  • @Paul Sinclair For the tree, $y$ can be defined to be a set that all $x$ is a member of. –  May 15 '19 at 20:35
  • I'm looking at this as an expression tree which is evaluated truth functional. It's my understanding the evaluation of the tree is $(\forall x) R$ asserts or denies truth. Where $((\forall x) R) = F$ it is my understanding that the $\tau_x (R)$ in the figure is evaluated to a substantific value which does not satisfy the $\in$ predicate and $(\forall x) R$ is evaluated as $F$. –  May 15 '19 at 21:31
  • I'm still confused about $(\forall x R) = T$. When mentioning random, do you mean that $\tau_x (R)$ is evaluated to an indeterminate object? I.e. it represents the expression partially evaluated? In this, “nothing can be said” of the object. Or, do you mean that $\tau_x(R)$ is evaluated to be a random substantific object that is disjoint from the all values selected that did not satisfy $\tau_x (R)$? In the latter case, I can see how the total expression evaluates $T$. If I'm looking at this wrong, appreciate any guidance you may have! –  May 15 '19 at 21:48
  • @Nick - "For the tree, $y$ can be defined to be a set that all $x$ is a member of" - No, it most certainly cannot be defined as such. If fact, as I already stated, Bourbaki later proves that no such universal set exists. Your understanding of your "expression tree" is not something I can comment on, as you didn't get it from Bourbaki, and I am not familiar with it. But your understanding of $(\forall x)R = F$ is correct. – Paul Sinclair May 15 '19 at 23:35
  • For $(\forall x)R = T$, this evaluates to $\lnot\lnot R[\tau_x(\lnot R[x])]$, so it is $t = \tau_x(\lnot R)$ we are discussing here. In this case, there is no object $x$ making $\lnot R[x]$ true (since $(\forall x)R = T$). $t$ is simply an arbitrarily chosen object. It does not "represent" anything. $t$ is not some special object. It could be anything in the theory. It could be $\emptyset$, it could be $\aleph_7$. The reason "nothing can be said" is not because of some special property of $t$, but simply because we have no information about which object $t$ is. – Paul Sinclair May 15 '19 at 23:51
  • @Paul Sinclair "For the tree, y can be defined to be a set that all x is a member of": is this because of a paradox? If so, I would have to think of a formula which would assert truth for $(\forall x) R$. Do you have a suggestion which might make this easier to understand? Thanks again –  May 16 '19 at 01:48
  • I've said twice already that Bourbaki proves that for any $y$, there is some $x$ that is not an element of $y$. So if you assume that such a $y$ exists, that would be a contradiction, not a paradox. And as Bourbaki also demonstrates, all statements can be proved from a contradiction. So everything would be both true and false (both the statement and its negation are provable). If you want a relation $R$ involving $x$ such that $(\forall x)R$ is true, the most common choice is $(\forall x)(x = x)$. – Paul Sinclair May 16 '19 at 02:21
  • @PaulSinclair Theory of Sets, S2 Criteria of Substitution, CS4: "Let $A$ and $B$ be assemblies and let $x$ and $y$ be distinct letters. If $x$ does not appear in $B$, then $(B|y) \tau_x A$ is identical with $\tau_x (A')$, where $A'$ is is the assembly $(B|y) A$." You state above that $\tau_x (R)$ picks - once for all time", then the above statement would be false, because $y$ is substituted in after the value for $x$ is selected. Yet, CS4, says both formulas are identical. The end resulting formulas are the same, but intermediate different. Again, perhaps I'm missing something... Thanks –  May 16 '19 at 22:43
  • When our munificent deity lists all the relations in $x$, and makes the picks, $x$ is the only letter allowed in the assemblies. One thing I didn't mention (because I was only communicating an intuitive concept, not a concrete theory) is that these assemblies can contain unlinked $\square$, in which case, the $\tau$ deity picks a value of $x$ for each possible value of each square (the values of linked squares are controlled by the $\tau$s they are linked to, so the deity does not worry about their values. This does not violate Substitution, because substition has to be finished beforehand. – Paul Sinclair May 17 '19 at 00:40
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See page 20 :

Let us consider the assertion $B$ as expressing a property of the object $X$; then, if there exists an object which has the property in question, $\tau_X(B)$ represents a distinguished object which has this property; if not, $\tau_X(B)$ represents an object about which nothing can be said.

The source is the so-called Hilbert's Epsilon Calculus :

The intended interpretation is that $ε x A$ denotes some $x$ satisfying $A$, if there is one.

Quantifiers can be defined as follows:

$$∃x A(x) \equiv A(ε x A)$$

<p><span class="math-container">$$∀x A(x) \equiv A(ε x (¬ A)).$$</span></p>

Compare with page 36 :

$(\tau_x(R) \mid x)R$ is denoted by "there exists $x$ such that $R$".

The intuition is (quite) simple : if there are some objects $X$ such that $A$ holds of them, the "choice operator" $\tau_X$ will pick up one of them and obviously $A$ will hold of it.

Consider now the example with the universal quantifier : $\forall x R(x)$ [where $R(x)$ is $\in (x,y)$, i.e. $(x \in y)$].

If $\forall x R(x)$ holds, this means that $\lnot R(x)$ holds of no object.

According to the above specification, $\tau_x (\lnot R)$ will pick up an object whatever, and $\lnot R(x)$ does not hold of that object, i.e. $\lnot [((\tau_x (\lnot R)) \mid x) (\lnot R)]$.

In plain language : if $\lnot R$ does not hold of an object whatever, this means that $R$ holds of every object.

Mauro ALLEGRANZA
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  • Given this, in the tree above where an object does not satisfy $\tau$, $\tau$ evaluates to "an object about which nothing can be said". I'm confused however, because I would suspect that $\forall x$ evaluates to true if every $x$ satisfies this quantification. –  May 14 '19 at 17:51