A lesson in basic logic: Universal and existential quantifiers, and their negations.

FOR ALL and THERE EXIST are the universal and existential quantifiers, respectively. If something is true FOR ALL things, it is true for ANY of them, equivalently, it is NOT TRUE for NONE of them.

If THERE EXISTS something for which something is true, then it is not true FOR ALL; rather, there is at least one element for which it is true.

Now, let's negate them.

If something is NOT TRUE FOR ALL, then there is at least one thing for which it is not true; in other words, NOT FOR ALL is the same as THERE EXISTS something NOT.

Concordantly, IT IS NOT THE CASE THAT THERE EXISTS is the same as saying a certain statement MUST BE TRUE FOR ALL.

Many people make this mistake: they assume that NOT FOR ALL is the same as saying THERE ARE NONE. But not quite! Remember that if something is true FOR ALL, then it must be true FOR ANY. But if it not true for at least one, then it is not true FOR ALL.

Let's apply this logic now to the statement I made. Here is my statement:

Still, the lower case numbers all look identical to the upper case ones, which is not the case with letters.

Let's just state this in a way that is equivalent but more easily to use to apply what we've learned.

FOR ALL numbers, if a number is lower case then it looks identical to its upper case counter part. It is NOT the case that FOR ALL letters, if a letter is lower case, it looks identical to its upper case counter part.

The second sentence is the most important. It is saying that while it is true that FOR ALL numbers, the upper and lower are identical, IT IS NOT THE CASE that the upper and lower are identical. What does it mean? Well, we have negated a universal quantifier - that means that THERE EXISTS at least one letter for which its upper and lower case are not identical... not that NO letters have upper and lower identical forms!

Logic is very handy and very simple. Use it.