# Does the order of quantifiers matter in propositional calculus?

1. May 5, 2012

### Arian.D

I think it matters, for example when I think of examples that I encounter in life it seems that the order of quantifiers matters and if we change the order the meaning could be interpreted differently, but does the order of quantifiers matter in propositional calculus? If yes, how could we show that it matters?

2. May 5, 2012

### DonAntonio

Exactly because of what you said it matters: it is enough one single example that shows that we can have different

meaning in order to deduce the order of quantifiers matters, and a lot, in fact.

DonAntonio

3. May 9, 2012

### 920118

There aren't any quantifiers in propositional calculus...? If you mean predicate calculus, then yes, the order matters. e.g.,

$\exists$x$\forall$yR(x, y)
$\forall$y$\exists$xR(x, y)

The first case says that there exists something such that it stands in the relation R to everything. The second case says that everything stands in the relation R to something.
e.g., given the domain U = {1, 2, 3, ...} and R = {(x, y) | x is larger than y}, case one is false since there is no number that it is larger than every number. But case two is true, since every number has some successor. It's a matter of convention rather than something you show.

4. May 10, 2012

### mbs

The order can be reversed in the special case of two existential or two universal quantifiers, one entirely within the scope of the other, as long as no free variable occurrences become bound and no bound variable occurrences become free in the process of reversing the order.

Quantifiers of different types can never be reversed.