Expressing "Someone has visited every country in the world except Libya" using quantifiers

Question

Please answer both questions that I listed at the end.

For question 37, B.

Here's what I did.

Let P(x,y) denote "x has visited y", 
The domain of x be {everyone in the world}, and
The domain of y be {every country in the world}

Rewrite it in the logical form:
∃x∀y((y≠Libya) → P(x,y)       (The Mistake) 

Thus the negation of it is:
¬∃x∀y((y≠Libya) → P(x,y))     
∀x∃y¬((y≠Libya) → P(x,y))     De Morgan's law.
∀x∃y¬(¬(y≠Libya) ∨ P(x,y))    p → q ≡ ¬p ∨ q
∀x∃y((y≠Libya) ∧ ¬P(x,y))     De Morgan's law and double negation law.

Which I interpret in English as:
Everyone in the world has not visited at least a country that is not Libya. 

But when I check the answer at the back of the book: It seems that I missed the part that says "Everyone has visited Libya"

Then I went on to search for an answer on Quizlet, and this is what I found:

It seems that the difference is right at the beginning, they uses conjunction(∧) while I use implication(→). My questions are:

• Why should we use "∧" and not "→" here?
• If we use "∧", wouldn't the statement "∃x∀y((y≠Libya) ∧ P(x,y)" contradict itself since the domain of y include all countries. And by using conjunction, we assert that y is not Libya even tho Libya is one of them?

Please enlighten me.

Edited: I uploaded the wrong photo.

Answer 1

Let P(x,y) denote "x has visited y", The domain of x be {everyone in the world}, and The domain of y be {every country in the world}

Rewrite it in the logical form:

∃x∀y((y≠Libya) → P(x,y) (The Mistake)

$$ ∃x∀y ~\left[ ~(y \neq ~\text{Libya}) ~~\color{red}{\iff} ~~P(x,y) ~\right]. \tag1 $$

In effect, if you have two assertions, $R$ and $S$, then the assertion $~~R ~~\color{red}{\text{except}}~~ S$ is saying two things:

• $R \implies \neg S.$
• $\neg R \implies S.$
  This second statement is equivalent to $~\neg S \implies R$.

Edit
In effect, the second statement is equivalent to saying:
$R$ is true whenever $S$ is not true.
This second statement leaves open the possibility that $R$ and $S$ might both be true. That possibility is negated by the first statement. I (subjectively) regard it as reasonable that the statement $~~R ~~\text{except}~~ S~~$ be construed to exclude the possibility that $~R~$ and $~S~$ are both true.

---

Edit
Note that (1) above allows for the possibilities that:

• The country known as Libya does not exist.
• There is no country other than the country known as Libya.
• There are no countries.

The above three (bullet-pointed) possibilities do not represent the intent of the statement, when the statement is not spoken by a logician. However, to a logician, the above three possibilities have not been eliminated by the statement.