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Please answer both questions that I listed at the end.

For question 37, B.

enter image description here

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"

enter image description here

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

enter image description here

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.

Chhangsreng P
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1 Answers1

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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.

user2661923
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  • Thank you, that's very insightful but would you mind clearing this question up:

    If we use "∧" as shown in quizlet, wouldn't the statement "∃x∀y((y≠Libya) ∧ P(x,y)" contradict itself since the domain of y includes all countries? And by using conjunction, we assert that y is not Libya even tho Libya is one of them?

     – Chhangsreng P Dec 17 '22 at 13:12
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    @ChhangsrengP Yes, it would contradict itself, and for just the reason that you have documented. The conjunction does imply that there is no country named Libya. So, the quizlet is wrong. My explanation is that the author of the solution made a mistake. Think about it in English. The author's interpretation is that "There exists a person that has visited all countries but not Libya". This implies that [1] The person has visited every country, and [2] The person has not visited Libya. The two implications do collectively imply that Libya does not exist. ...see next comment – user2661923 Dec 17 '22 at 15:18
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    @ChhangsrengP What the author should have initially said was: "There exists a person that has visited all countries $~\color{red}{\text{except}}~$ Libya." This would imply that if Libya exists, then the person has not visited every country. Apparently, the author of the quizlet intended the interpretation in this comment, but instead mis-worded it, as per my previous comment. Then, once the English statement was misworded, the author of the quizlet then went off the rails. – user2661923 Dec 17 '22 at 15:22