Predicate Calculus Examples Cont.

• All S are P
  $\forall x(S(x) \rightarrow P(x))$
  • Wrong: $\forall x(S(x) \wedge P(x))$
    this says everything is both S and P
    
  $\neg \forall x(S(x) \rightarrow P(x))$
  • it is not true that all S are P
    rephrased: Some S are not P
  $\exists x \neg(S(x) \rightarrow P(x))$
  $\exists x \neg(\neg S(x) \vee P(x))$
  

  $\exists x S(x) \wedge \neg P(x)$

• Some S are not P

• Some S are P
  $\exists x(S(x) \wedge P(x))$
  
  • Wrong: $\exists x (S(x) \rightarrow P(x))$
    this says for at least one x if x is S then x is P.
    True: If a person has 7 heads then that person is peculiar
    False: Some seven headed people are peculiar
    (Since there are no seven headed people)
    
• $\neg \exists x(S(x) \wedge P(x))$ . . . It is not the case that some S are P
  $\forall x \neg(S(x) \wedge P(x))$ . . . rephrased: No S are P
  $\forall x(\neg S(x) \vee \neg P(x))$
  

  $\forall x S(x) \rightarrow \neg P(x)$
  

• No S is P