# How can I translate this quantified logical expression into english

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?

• I dont know if this is somewhat not very precise question and I may be at risk of getting banned if this question is closed for not being precise cause cs.stackexchange is already warning me. But logic always proves fuzzy to me and making a confusing logic more clear is itself an effort. So please help. I wish there could have been some place on stackexchange where I can ask my initial questions and promote them to actual questions if they are valid. Chat rooms are there but cs chatroom is not active. – anir123 Oct 11 '14 at 10:34
• There's a reason we use formal logic rather than English for stating complex mathematical propositions: English is imprecise and not a great tool for that purpose. – D.W. Oct 11 '14 at 18:57