Every person likes all food" and ∀x ∀y (P(x) ∧ F( y ) ∧ L(x,y))
When we have a universal quantification, it imposes that all objects of the domain has a certain property. Is there a way to rewrite this to make the predicate logic formula true?
For example: Every person likes all animals
P(x) - x is a person
A(x) - x is an animal
L(x,y) - x likes y
∀x ∀y (P(x) ∧ A( y ) ∧ L(x,y)).
This answer is not correct, because the use of the universal quantifiers in the formula imposes that all objects are person and animal. Is there a way to make the formula correct?
The formula you gave litteraly says "everyone is a person and everyone is an animal and everyone loves everyone".
The formula expressing "Every person likes all animals" would be : $$\forall x \, \big( P(x) \longrightarrow (\forall y \, A(y) \longrightarrow L(x,y) \big) $$ This formula is equivalent to $\forall x \, \forall y \, \big(P(x) \longrightarrow (A(y) \longrightarrow L(x,y) \big) $, or even to the more readable: $$\forall x \, \forall y \, \big( (P(x) \wedge A(y)) \longrightarrow L(x,y) \big) $$