An Example of Converting a FOL Sentence to Clause Form

Convert the sentence (Ax)(P(x) => ((Ay)(P(y) => P(f(x,y))) ^ ~(Ay)(Q(x,y) => P(y))))

1. Eliminate <=>
   Nothing to do here.
2. Eliminate =>
   (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ ~(Ay)(~Q(x,y) v P(y))))
3. Reduce scope of negation
   (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ey)(Q(x,y) ^ ~P(y))))
4. Standardize variables
   (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Ez)(Q(x,z) ^ ~P(z))))
5. Eliminate existential quantification
   (Ax)(~P(x) v ((Ay)(~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
6. Drop universal quantification symbols
   (~P(x) v ((~P(y) v P(f(x,y))) ^ (Q(x,g(x)) ^ ~P(g(x)))))
7. Convert to conjunction of disjunctions
   (~P(x) v ~P(y) v P(f(x,y))) ^ (~P(x) v Q(x,g(x))) ^ (~P(x) v ~P(g(x)))
8. Create separate clauses
   
   • ~P(x) v ~P(y) v P(f(x,y))
   • ~P(x) v Q(x,g(x))
   • ~P(x) v ~P(g(x))
9. Standardize variables
   
   • ~P(x) v ~P(y) v P(f(x,y))
   • ~P(z) v Q(z,g(z))
   • ~P(w) v ~P(g(w))