I was reading chapter-1 The Foundations: Logic and Proofs from this book.
The chapter gives example of translating English sentence : "There is a woman who has taken a flight on every airline in the world." as follows:
- Introducing variables : w for women, f for flight, a for airline
- Let P(w,f) : “w has taken f”
- Let Q(f,a) : “f is a flight on a.”
- Translation : ∃w∀a∃f (P(w,f ) ∧ Q(f,a))
I did understood above. Next it gives example of translating negation of above sentence : "There does not exist a woman who has taken a flight on every airline in the world.", which it solved as follows:
¬∃w∀a∃f (P(w,f ) ∧ Q(f,a)) ≡ ∀w¬∀a∃f (P(w, f ) ∧ Q(f, a)) ≡ ∀w∃a¬∃f (P(w, f ) ∧ Q(f, a)) ≡ ∀w∃a∀f¬(P (w, f ) ∧ Q(f, a)) ≡ ∀w∃a∀f (¬P(w, f )∨¬Q(f, a))
I thought their can be straight approach for this translation instead of going through negation. Anyways though my problem is I am unable to interpret the final translation ∀w∃a∀f (¬P(w, f )∨¬Q(f, a)) in plain English.
"Every woman has either not taken all flights or out of all the flight she has taken are not there on some airline." Is this correct? But if yess, it still does not make me much sense. Anyone?