I have question about what is difference about two statements below.
$$\forall a,b,c, \ \ a+(b+c)=(a+b)+c $$
or
$$a+(b+c)=(a+b)+c , \ \forall a,b,c$$
I usaually see second in textbooks but my question is why not to write the first.
$\endgroup$ 23 Answers
$\begingroup$Formally, the variant with quantifiers in front is the only correct well formed formula in first order logic. The second one is just the spoken form where "for all $a,b,c$" is said at the end, but written symbolically. I find that form very confusing potentially because it's unclear what the range of the quantifier is.
So only use
$$\forall a: \forall b: \forall c: (a+b) +c = a + (b+c)$$
to use your example. In this case the order of $a,b,c$ does not matter (they yield equivalent statements) so people also write in a shortened form:
$$\forall a, b, c: (a+b) +c = a + (b+c)$$
$\endgroup$ 2 $\begingroup$There is not a difference between the two statement, I prefer the first one with quantifiers declared at first.
Recall that when we have different quantifiers the order of them is very important.
$\endgroup$ $\begingroup$There is no difference in the two version of statements.
I have seen them both and as you noticed the second one is more commonly used than the first one.
Usually the condition is mentioned after the statement, but there is no difference.
$\endgroup$
