# Thread: Can i change the order of quantifiers in this case?

1. ## Can i change the order of quantifiers in this case?

In this sentence:

"No hero is cowardly and some soldiers are cowards."

Assuming h(x) = x is a hero

s(x) = x is a soldier

c(x) = x is a coward.

So the sentence is like this i think:

(∀x (h(x)->¬C(x))∧(∃y (s(y)∧C(y))
In this case, are prenex formulas bellow the same thing?

∀x ∃y (¬h(x)∨¬C(x))∧((s(y)∧C(y))
∃y ∀x (¬h(x)∨¬C(x))∧((s(y)∧C(y))

2. ## Re: Can i change the order of quantifiers in this case?

Your translation is correct. Just personal preference, I prefer to translate "no hero is cowardly" as
$$\neg((\exists x)(h(x)\wedge c(x)))$$

Yes, you can rewrite your symbolic statements in either of the two ways; I'm not sure why you want to do this. I think the original is much clearer. You can formally prove this by going through the business of universal and existential instantiation, some logical equivalences and then existential and universal generalization.

3. ## Re: Can i change the order of quantifiers in this case?

Originally Posted by Lyunth
In this sentence:
"No hero is cowardly and some soldiers are cowards."
Assuming h(x) = x is a hero
s(x) = x is a soldier
c(x) = x is a coward.
So the sentence is like this i think:
(∀x (h(x)->¬C(x))∧(∃y (s(y)∧C(y))
In this case, are prenex formulas bellow the same thing?
∀x ∃y (¬h(x)∨¬C(x))∧((s(y)∧C(y))
∃y ∀x (¬h(x)∨¬C(x))∧((s(y)∧C(y))
The problem of quantifier order usually occurs when a predicate applies to more than one variable such as a relationship (is a class mate of).
Here you have two singular predicates. Isn't true that $\displaystyle \left( {\forall x} \right)\left[ {h(x) \to \neg C(x)} \right] \wedge \left( {\exists y} \right)\left[ {s(y) \wedge C(y)} \right]$ is the same as $\displaystyle \left( {\exists y} \right)\left[ {s(y) \wedge C(y)} \right] \wedge \left( {\forall x} \right)\left[ {h(x) \to \neg C(x)~?} \right]$