Question

I am having trouble with Predicate logic and how the ordering of universal quantifiers works

For example:
• V : the domain of the residents of Vilderan
• I: the domain of items
• H(x): x is a hero
• C(x): x is a creature
• S(x): x is skilled
• E(x): x is enchanted
• K(x,y): x defeated y (in battle)
• D(x,y): x dropped y (on the ground)
What's the difference between
∀c ∈V,(C(c) ∧∃i ∈I,D(c,i)) →(∃h ∈V,H(h) ∧K(h,c))
AND
∀c ∈V, ∃i ∈I, ∃h ∈V, (C(c) ∧ D(c,i)) → (H(h) ∧K(h,c))
---------------------------------------------------------------------------
Is there any tips on when to put them at the start and when to incorporate them into the operators?

Answer 1

The main difference between the two statements is in the scope and sequence of the universal and existential quantifiers. The first statement requires the existence of an item for each creature before introducing a hero, whereas the second allows for the simultaneous existence of items and heroes independent of creatures dropping items.

Step-by-step explanation:

The difference between the two predicate logic statements lies in the position and scope of the quantifiers. Quantifiers affect how we interpret the logic and the order of operation within predicate statements.

The first statement translates to: For every creature in the domain of residents of Vilderan, if it dropped an item (existence of some item guaranteed), then there exists some hero in Vilderan that defeated this creature. It implies a certain sequence - first, we assert the existence of an item related to every creature before discussing heroes.

The second statement translates to: For every creature in Vilderan, there exists some item, and there exists some hero, such that if the creature dropped the item, then the hero defeated the creature. This implies items and heroes exist simultaneously but doesn't specify that the hero's existence is predicated on the creature dropping an item. Here, the existential quantifier for heroes doesn't depend on the dropping event.

Using predicate logic and universal quantifiers properly requires understanding the nuances of scope and sequence. When incorporating quantifiers, consider whether an existential quantifier depends on a prior condition or stands independently, as this affects logical interpretations.