About the position of "for all" quantifier

Question

I'm not expert in logic, but as far as I know, quantifiers comes before the predicates they refer to. Still, if written in english, there are statements which sounds better when you don't put all the quantifiers before. For example, the definition of a sequence of functions $f_n:\mathbb{R}\to\mathbb{R}$ converging uniformly to some function $f:\mathbb{R}\to\mathbb{R}$:

For all $\varepsilon > 0$, exists $n_0\in\mathbb{N}\ $ such that $\ n>n_0 \implies |f_n(x) - f(x)| < \varepsilon$ for all $x\in\mathbb{R}$

I have seen teachers writing this as

$$\forall\varepsilon > 0,\ \ \exists n_0\in\mathbb{R}\ \ \text{ such that } \ \ n>n_0 \implies |f_n(x) - f(x)| < \varepsilon,\ \ \forall x\in\mathbb{R}$$

Maybe an even more simple example, from probability. It's not hard to find something like

$$P[X_n=Y_n, \forall n\in\mathbb{N}]$$

where $(X_n)_{n\in\mathbb{N}}$ and $(Y_n)_{n\in\mathbb{N}}$ are collection of random variables.

I understand the motivation: if you talk about this probability, you probably will say the probability of have $X_n=Y_n$ for all $n$, and not the probability of, for all $n$, have $X_n=Y_n$. The same reasoning applies for the statement about uniform convergence. That last for all quantifier will fit better at the final of the sentence if you have to say it in words.

So my question is: are this "informal" formulas just wrong (in the formal logic point of view)? Or the formal language of logic can handle this kind of writing?

Thanks.

Answer 1

The informal, natural language examples are neither right nor wrong form the formal viewpoint, because the informal examples are not written in a formal way.

The general rule is that the "last quantifier" in natural language becomes the innermost quantifier in formal language. Here are some common examples of informal phrasings and their formal counterparts:

• A sequence $(x_n)$ converges to $L$ if for all $\epsilon > 0$ there is an $N$ such that $|x_n - L| < \epsilon$ for $n > N$. Formally: $$(\forall \epsilon > 0)(\exists N)(\forall n)[ n > N \to |x_n - L| < \epsilon]$$

• A function $f$ is continuous at a point $x$ if for every $\epsilon > 0$ there is a $\delta > 0$ such that $|f(x) - f(y)| < \epsilon$ for all $y$ with $|y-x| < \delta$. Formally: $$ (\forall \epsilon> 0)(\exists \delta > 0)(\forall y)[|x-y| < \delta \to |f(x) - f(y) | < \epsilon].$$

This is something that is rarely mentioned explicitly in textbooks, but which you have to learn as a student in order to read informal mathematics correctly. But that does not make the informal mathematics "wrong" and the formal mathematics "right". They are simply different ways of approaching the same topic.

Answer 2

I think you're correct that in precise logical statements the quantifier should come first, and the more colloquial mathspeak is an abuse. This bothered me for a while when I went from my first logic class to my first analysis class.

Answer 3

Carl Mummert's answer is very good, but I'd like to refer specifically to the examples you have given.

Examples such as

$$\forall\varepsilon > 0,\ \ \exists n_0\in\mathbb{R}\ \ \text{ such that } \ \ n>n_0 \implies |f_n(x) - f(x)| < \varepsilon,\ \ \forall x\in\mathbb{R}$$

are perfectly fine on a blackboard: writing on it takes time, and it is a dynamic process (you see it being written), probably assisted by the lecturer's comments. Actually spelling everything out in words would take up too much time and blackboard space, and the "dynamic" plus the commentary make the text completely understandable (hopefully).

Things are very different with a stand-alone mathematical text. In there, you should (as a rule of thumb) either spell out quantifiers (and logical connectives!) in words, as in

For all $\varepsilon > 0$, there exists $n_0\in\mathbb{N}\ $ such that whenever $\ n>n_0$, we have that $\lvert f_n(x) - f(x)\rvert < \varepsilon$ for all $x\in\mathbb{R}$.

(and this is preferable), or spell everything out purely formally, like:

$$\forall\varepsilon > 0\ \ \exists n_0\in\mathbb{N}\ \ \forall n\in {\mathbb N} \ \left( n>n_0 \implies \forall x\in\mathbb{R}\ \ |f_n(x) - f(x)| < \varepsilon,\right).$$

(In the latter case, you should also probably avoid writing the formal part inline (unless it is very short).)

You can, of course, use various notational shorthands for formal writing, like writing $\forall n>n_0$ or similar, and likewise, when writing in natural language, there is also some wiggle room: for instance, you could write "for all real $x$" instead of "for all $x\in {\mathbb R}$", so there is some wiggle room in how much to spell out and how formal to be, but mixing formal language and natural language too much is bad form IMHO, makes the text harder to read and easier to write sloppily.

Answer 4

Strictly in terms of formal logic, quantifiers are at the beginning of any formula. However, no one gives a proof that is written in the formal language. Even simple proofs would be very long and unreadable. The point is that statements like the probability example you gave can be written in the formal language.

Answer 5

The proper usage of a formal notation or of a more informal one depends particularly on the context of presentation. It is essential to whom we communicate an idea and this should guide us to use a suitable level of formal notation.

Here is an excerpt from P.R. Halmos' instructive paper How to write Mathematics regarding the aspect:

What about the usage of logical symbols?

P. R. Halmos: Here is a sample :

"Prove that any complex number is the product of a non-negative number and a number of modulus $1$."

...

One way to recast the sample sentence of the preceding paragraph is to establish the convention that all "individual variables" range over the set of complex numbers and then write something like $$\forall z\exists p\exists u [(p=|p|) \wedge (|u|=1) \wedge (z=pu)]. $$

I recommend against it. The symbolism of formal logic is indispensable in the discussion of the logic of mathematics, but used as a means of transmitting ideas from one mortal to another it becomes a cumbersome code. The author had to code his thoughts in it (I deny that anybody thinks in terms of $\exists, \forall, \wedge$, and the like), and the reader has to decode what the author wrote ; both steps are a waste of time and an obstruction to understanding. Symbolic presentation, in the sense of either the modern logician or the classical epsilontist, is something that machines can write and few but machines can read.

Hint: You might also have a look at the related question Why there is no sign of logic symbols in mathematical texts?

Answer 6

I think that, at this point, it's useful to point out what "formal logic" actually is. Formal logic is reasoning based on the form of the statement, i.e., its syntax. For example, the statement $(X\land Y)\rightarrow X$ is true not because you know something about the individual meaning of $X$ and $Y$ but because all statements of that form are true.

In formal logic, we use a fixed syntax, to make this reasoning based on form simpler. In particular, this syntax places quantifiers ahead of the thing being quantified over: we write "$\forall x\,\varphi$" not "$\varphi,\, \forall x$".

When writing informally (and, here, informal really means "casual", rather than "not based only on the form of the sentence"), this order is often reversed because, everyday English is more flexible than the syntax of first-order logic.