More Points About Predicate Calculus

• Different variables do not necessarily designate different objects.
  • $\forall x \forall y Loves(x,y)$ asserts that for any x and any (not necessarilydistinct) y, x loves y. It not only means every thing loves every other thing, but it also means everything loves itself.

• Choice of variable makes no difference in meaning.
  • $\exists x \exists x Loves(x,x)$ is symbolized just as correctly using $\exists y \exists y L(y,y)$.

• Mixing universal and existential quantifiers matters.
  • If one qunatified variable is within the scope of another quantified variable the meaning is different.
  • $\exists y \forall x Blah(x,y)$ is not the same as $\forall x \exists y Blah(x,y)$
  • When both quantifiers are the same it does not matter.
  • However, $\forall x \exists x Loves(x,x)$ is illformed and ungrammatical.

• Same variable used with 2 different quantifiers doesn't necessarily designate the same object.
  • $\exists x Loves(Bob,x) \wedge \exists x Loves(Cathy,x)$ may or may not mean the same thing is loved by both Bob and Cathy