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[Solved]: How can I translate this quantified logical expression into english

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Problem Detail: 

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?

Asked By : Mahesha999

Answered By : David Richerby

The negation is that every woman has an airline that she has never flown with. This is equivalent to saying that, for every woman there's an airline with the following property:

  • for every flight in the whole world, either the woman was not on that flight (regardless of what airline operated it) or the flight was not operated by that airline (regardless of whether the woman was on it or not).

That is, there was no flight by the airline in question that the woman was on.

This is not quite the same as what you have.

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