Translating English to Predicate Logic

[ThunderBase]

Hi all,
I'm having trouble translating the following to predicate logic. I'm not sure as to what should be on the left or right side of the implication.

• Every student has a friend who is a teacher.
  • The answer is: ∀x student(x)→∃y teacher(y) ∧ friend(x,y)
• My questions is, why is it not the following: ∀x student(x) ∧ ∃y teacher(y)→ friend(x,y)

Similarly with the following question:

• Every student does not like some teacher.
  • Answer: ∀x ∃y student(x) ∧ teacher(y) → ~like(x,y)
• My question is, why is it not: ∀x student(x) → ∃y teacher(y) ∧ ~like(x,y)

The two questions seem so similar to each other, but I don't understand why the answers are different.
Thanks all.

 

[pka]

I'm having trouble translating the following to predicate logic. I'm not sure as to what should be on the left or right side of the implication.

• Every student has a friend who is a teacher.
  • The answer is: ∀x student(x)→∃y teacher(y) ∧ friend(x,y)
• My questions is, why is it not the following: ∀x student(x) ∧ ∃y teacher(y)→ friend(x,y)

Similarly with the following question:

• Every student does not like some teacher.
  • Answer: ∀x ∃y student(x) ∧ teacher(y) → ~like(x,y)
• My question is, why is it not: ∀x student(x) → ∃y teacher(y) ∧ ~like(x,y)

The two questions seem so similar to each other, but I don't understand why the answers are different.

@ThunderBase, these are excellence questions but also very difficult to answer on a board like this.
Hopefully, you have a good academic library to use. Two books: The Grammar of Mathematics, by LK Durst & Symbolic Logic by IM Copi. Both of those books have extensive discussions of your question. Have you studied scope of the quantifier? That is Copi's reply to your question.
Every student has a friend who is a teacher. Says if x is a student then some teacher, y, is his/her friend.
But I would say \(\displaystyle (\forall x)(\exists y)[S(x)\to T(y)\wedge F(x,y)]\). That is Durst's solution, i.e. using the quantifiers as a group.

 

[ThunderBase]

I only just started the course so I haven't studied scope of the quantifier. I'll be sure to look into those soon. How does the 1st question differ to the 2nd? "Every student does not like some teacher"

Thanks again!

 

[pka]

I only just started the course so I haven't studied scope of the quantifier. I'll be sure to look into those soon. How does the 1st question differ to the 2nd? "Every student does not like some teacher"

First I would say that this is : \(\displaystyle (\forall x)(\exists y)[S(x)\to T(y)\wedge \neg L(x,y)]\).
That translates, For ever x there exists a y such that if x is a student then there is a teacher y and x dislikes y.
Here is a useful table, known as the square of opposition:
\(\displaystyle \begin{align*}A.~\text{Universal positive }&(\forall x)[A(x)\to B(x)]\text{ All A is B } \\
E.~\text{Universal negative }&(\forall x)[A(x)\to \neg B(x)]\text{ No A is B }\\
I.~\text{Existential positive }&(\exists x)[A(x)\wedge B(x)]\text{ Some A is B }\\
O.~\text{Existential negative }&(\exists x)[A(x)\wedge \neg B(x)]\text{ Some A is not B }\\\end{align*}\)

\(\displaystyle A~\&~O\) are the negations of one another.
\(\displaystyle E~\&~I\) are the negations of one another.

 

[ThunderBase]

Sorry about the late reply. Thanks for that. Turns out I was correct about the 2nd statement. My course coordinator said that was a mistake!