What is the difference between these two propositions?

Question

My text says:

Let Evens be the set of even integers greater than 2, and let Primes be the set of primes. Then we can write Goldbach’s Conjecture in logic notation as follows:

$ \forall n \in Evens. \exists p,q \in Primes. n = p + q $

Then later it reports:

Swapping quantifiers in Goldbach’s Conjecture creates a patently false state- ment; namely that every even number 2 is the sum of the same two primes:

$ \exists p,q \in Primes. \forall n \in Evens. n = p+q $

What exactly is the difference between these two notations? They look the same to me, with the same quantifiers -- just written in a different order

Answer 1

The outer quantifier[s] set the context for the inner quantifiers. In the first case, given different even integers, we are allowed to take different prime numbers. So for $n=10$ we may take $p=3,q=7$ and for $n=24$ we may take $p=7,q=17$.

In the second case we claim that there are two concrete prime numbers $p,q$, that every even integer is their sum. So we have that $p+q=2=4=6=8=10=\ldots$

So while the first claim is plausible, the second claim is clearly false.

Answer 2

I'll try another example. Consider the two following sentences:

For any natural number, there is another that is bigger. $$\forall n\in\mathbb{N}.\ \exists m \in \mathbb{N}.\ m > n \tag{$\spadesuit$}$$

There is a natural number, that is bigger than any other. $$\exists m \in \mathbb{N}.\ \forall n\in\mathbb{N}.\ m > n \tag{$\clubsuit$}$$

The first is true, because you can set $m = n+1$. On the other hand, the second is false, because it fails for $n = m$. In fact, negation of $(\clubsuit)$

$$\forall m\in\mathbb{N}.\ \exists n \in \mathbb{N}.\ m \leq n$$

can be made true by $n = m$.

I hope this helps $\ddot\smile$

Answer 3

What exactly is the difference between these two notations? They look the same to me, with the same quantifiers -- just written in a different order

That really is a rather astonishing thing to say and I'm afraid it betrays a real failure of understanding at the most basic level: the whole point of Frege's quantifier-variable notation is that the order of quantifiers tells us unequivocally which quantifier is in the scope of which, thus avoiding the kind of ambiguities that we can get in ordinary language. Any good introductory logic text will explain this cleary and carefully: try Paul Teller's A Modern Formal Logic Primer (freely downloadable) or my Introduction to Formal Logic. Or look at Daniel Velleman's How to Prove It.

Other answers here give clear specific examples to illustrate the difference that the order of quantifiers can make: but do also get clearer about the general principles involved.