Standard logic notation in mathematics

Question

My profesor is always complaining that my proofs are very long and difficult to read because I never use notation, meaning I say everything in words. Tired of that I decided to study logic by myself and develop my proofs by using the methods of logic. The problem to me now is that I don't know exactly the point of being pedantic in writing everything in symbols of logic and writing in words. For example I don't know if it's ok or how to simplify my logic statement

$\forall X\forall Y(X\in Cam(R)\wedge Y\in Cam(R)\wedge (X,Y)\in R\longleftrightarrow (F(X),F(Y))\in S)$.

or for example if it's the case of having a lot of quantifiers like

$\forall X \forall Y \forall Z(Z\in \mathbb{N} \wedge Y\in \mathbb{N} \wedge Z\in \mathbb{N}\wedge \phi(X,Y)\longrightarrow \psi(X,Z))\wedge Z>Y)$

Also I always see that, when doing deductions, people write things like

$\phi(X)\longrightarrow \psi(X)$

$\longrightarrow \psi_{1}(X)$

$\longrightarrow \psi_{2}(X)$

.

.

. $\longrightarrow \psi_{n}(X)$

$\therefore \psi_{n}(X)$

This means we suppose $\phi$. Then we get $\psi$. This implies $\psi_{1}$ . Then this implies $\psi_{2}$, and then it implies $\psi_{3}$... etc.

So, basically what I need is "tips" (the more,the better) to develop my proofs in a more or less standard way, just like you would do in your tests or the way you would write your proof on the blackboard, or to hand in your homework, etc. (If you know a book, a paper, or something that might help me with this I will be very gratefull).

Answer 1

It’s possible that your proofs are too wordy and don’t use enough notation, but even if that’s true, what you’re proposing here is not the solution: arguments expressed almost entirely in formal logical notation are at least as hard to read as arguments that drown the reader in verbiage. Of course you want to be sure that what you write is correct, but after that the most important thing is saying it clearly. Generally that requires a well-chosen mixture of symbols and words; the ideal mixture is different for different people, but for most people both extremes are hard to read. Here’s an example, modified from an earlier question, of three ways of defining a certain function:

1. Define a function $f$ by $$f:\wp(\Bbb N)\setminus\{\varnothing\}\to\Bbb N \\ x\mapsto y \owns y\in x\land\forall z:z\in x\to z\geq y\;.$$

2. Define a function $f$ by $$f:\wp(\Bbb N)\setminus\{\varnothing\}\to\Bbb N:x\mapsto\min x\;.$$

3. Given a nonempty set $A$ of natural numbers, denote its least element by $f(A)$.

The first is horrible to read. The second is much clearer and would be appropriate wherever a concise but readable definition is wanted. And the third, though by far the most wordy, is instantly clear. I would never use the first; the choice between the second and third would depend on the context and the intended audience. When writing for your professor, for instance, you might want to lean towards the second version.

Besides tending to make mathematics hard to read, excessive use of formal notation can get in the way of thinking about a problem. For a possible example of this phenomenon, see this question and my answer to it.

Please note that I do not mean to deny the possibility that you really are muddying things by using words when symbols would be preferable: a version of the quadratic formula and its derivation that used no mathematical symbols would be extremely hard to follow.

Because judgement of readability is highly subjective, it’s very difficult to give concrete, objective advice. The best that I can suggest is to pay attention to the way proofs are written in textbooks; you’ll find considerable variation, but in general it will be within the range of acceptable practice, especially in your more advanced courses.

Answer 2

Typically, we only use a handful of logical symbols in proofs. Instead of saying 'for all' we (sometimes) write $\forall$. Instead of 'there exists' we (sometimes) use $\exists$, etc. But this is not really the point. The art of writing good proofs involved good notation, but that is more of a side issue (unless you 'choose' some obscure, non-standard, and anti-intuitive notation). A good proof should be easy to read, follow a clear path, and explain the logical steps followed, with clear references to theorems. There are some general principles that can be followed to both discover a proof and for writing it in a pleasant way after you found it. I'll try to show-case some.

First principle: Give it a name! Consider the following: prove that a natural number is even if, and only if, the square of the number is even. Here is a proof: Suppose that a given natural number is even. Then it is twice some other natural number. But then the square of the first natural number is 4 times the square of the second natural number, and thus the square of the first number is even. Now assume that the square a given natural number is even, yet the number itself is odd. Then, the number can be written as twice some natural number plus one. But the square of twice some natural number plus one is one plus an even number, and thus is odd. This contradicts the assumption that the square of the first number is even. Thus, we conclude that the number was even in the first place.

This proof is correct but is horrible. Instead, consider the following: Let $n$ be a natural number. We wish to show that $n$ is even if, and only if, $n^2$ is even. Let us first assume that $n$ is even. Then we may write $n=2\cdot k$ for some $k\in \mathbb N$. And then, $n^2=4k^2$, an even number. Thus if $n$ is even it follows that $n^2$ is even. For the implication in the other direction, assume that $n^2$ is even but $n$ is odd. Then, we may write $n=2k+1$ for some $k\in \mathbb N$, and then $n^2=4k^2+4k+1$, an odd number. This contradicts the assumption that $n^2$ was even, thus $n$ can't be odd. This completed the proof that $n$ is even if, and only if, $n^2$ is even.

The two proofs are roughly the same length, but it is much easier to follow the second one. The reason is that as soon as I'm talking about an entity, I give it a name. After all, it's a lot easier to say "Hi Daniela" instead of "Hi to the person with the hat standing next to the supermarket, wearing a purple T-shirt running away from the t-rex while singing 'twinkle twinkle' in a strong Irish accent".

Notice also how I used short phrases to parse the logical flow of the proof. I announce which direction in the implication I'm about to prove. I also announce clearly when an implication was established. At the end of the proof I clearly state the proof ended, and why.

So, give things a name. But, don't give each and every thing a name, or your scrap of paper will soon be full of useless entities. Trial and error is your friend. It's also worth mentioning that in the proof above I could have used more logical symbols, such as "$n$ is even $\iff$ $n^2$ is even". This would make little difference, and is a quite of minor changes in style.

Next principle: unpack! Definition are there to encapsulate information for us. The human brain loves encapsulating information. It loves it so much that sometimes it refuses to un-encapsulate. But quite often, you have to. For instance, let us prove that $\lim _{n\to \infty }\frac{1}{n}=0$. First, we'll use the hand-waving technique. We'll wave our hands sufficiently well to convince one of the argument, and produce some levitation as well. So, we need to show that the quantity $\frac{1}{n}$ can be made arbitrarily small, when $n$ is large enough. Clearly ('clearly' is a great word to use in hand-wavy arguments), when $n$ is very big, the fraction $\frac{1}{n}$ is very small, and get closer and closer to $0$. Thus, the desired limit is $0$.

This is of course not a proof. To turn this intuitively correct argument into a proof, we need to unpack the meaning of $\lim _{n\to \infty }\frac{1}{n}=0$. Here goes then. Proof of the above: Well, it means that for all $\epsilon >0$ there exists $K\in \mathbb N$ such that $|\frac{1}{n}|<\epsilon$ for all $n>K$. So, let (i.e., give it a name!) $\epsilon>0$. By the archamedean property of the natural numbers, there is a natural number $K$ such that $K>\frac{1}{\epsilon }$. But then, $\frac{1}{K}<\epsilon$ and so for all $n>K$ it holds that $\frac{1}{n}\le \frac{1}{K}<\epsilon$. The proof is complete.

Again, I could have used more logical symbols above, like $\exists$ and $\forall$, but that is minor stylistic issues.

I hope this answer is useful.

Answer 3

Just because your professor is complaining that you never use notation, that does not mean you should switch to writing everything in notation. You should find out what your professor wants you to do.

I say this because what you're proposing in your question -- using symbols for the logical quantifiers and connectives -- is, in my opinion, not a good idea. I find it much easier to read if the logical symbols are written out in words, and symbols are reserved for the objects and equations that form the content of whatever you're proving.

Take a look at some of the answers on this site from experienced members. You'll notice that even though there may be a lot of symbols, there are also a lot of words. Even if you don't understand an answer, you should notice that there usually aren't any logical symbols in the answer -- they write out the words instead.

But again, if this is for a particular class, I would recommend asking your professor for more details.

Answer 4

If you have a habit of describing everything in words, try to use words once to introduce a symbol, and then use that symbol everywhere else. Many proofs begin with something along the lines of "Let $S$ be a set of..." followed by many uses of $S$.

Also, stick to symbolic notation for as long as you possibly can until you need to introduce a new symbol or explain a particularly tricky step. If a casual reader can follow a simple derivation without having to pause at every line to figure out what you're doing, then you don't need to justify yourself.

You might be interested to read some old mathematical proofs from Greek mathematicians. You should be able to find a copy of the Elements if you google it. That will be filled with lots of very wordy proofs of very basic results. Try to work your way through a few of those, and then compare them to the proofs in your textbooks. The differences should be obvious.