Introduction to Logic

Exercise 12.1 - Clausal Form

Consider a language with two object constants a and b and one function constant f. For each of the following sentences in this language, say which of the alternatives shown is its clausal form.

1.y.∀x.p(x,y) 
  {p(a,y)} 
  {p(c,y)} 
  {p(x,b)} 
  {p(x,c)} 
  {p(x,f(x)} 
  {p(x,g(x)} 
 
2.x.∃y.p(x,y) 
  {p(a,y)} 
  {p(c,y)} 
  {p(x,b)} 
  {p(x,c)} 
  {p(x,f(x)} 
  {p(x,g(x)} 
 
3.x.∃y.(p(x,y) ∧ q(x,y)) 
  {p(a,b), q(a,b)} 
  {p(a,b)} and {q(a,b)} 
  {p(c,d), q(c,d)} 
  {p(c,d)} and {q(c,d)} 
  {p(x,y), q(x,y)} 
  {p(x,y)} and {q(x,y)} 
 
4.x.∀y.(p(x,y) ⇒ q(x)) 
  {¬p(a,b), q(a)} 
  {¬p(c,d), q(c)} 
  {¬p(x,b), q(x)} 
  {¬p(x,d), q(x)} 
  {¬p(x,y), q(x)} 
  {¬p(x,g(x)), q(x)} 
 
5.x.(∃y.p(x,y) ⇒ q(x)) 
  {¬p(a,b), q(a)} 
  {¬p(c,d), q(c)} 
  {¬p(x,b), q(x)} 
  {¬p(x,d), q(x)} 
  {¬p(x,y), q(x)} 
  {¬p(x,g(x)), q(x)}