How to solve the given Premises using quantifiers?

Question

I have a question that

I have tried as but i am not getting the actual one. I tried by chaging the consequent formula to ( or ) formula. Still The premise 2 is not equal as the consequent of premise 1. What I am thinking is ,I tried by changing the consequent to negation , but the (not W(y)) is not getting for me. Can anyone tried to solve this.Please.

Answer 1

Hint

Try by contradiction, assuming the negation of the conclusion :

$\lnot (\forall x)(Px \to \lnot Qx)$.

It is equivalent to :

$(\exists x)(Px \land Qx)$.

Now with this we can apply modus ponens to 1st premise to derive : $ (\forall y)(My \to Wy)$ and this will give us a contradiction with the 2nd premise : $(\exists y)(My \land \lnot Wy)$.

Answer 2

More hints:

line 2: you can't do existential specification on 1, since 1 is not an existential ... It is a conditional whose antecedent is an existential. So it looks like an existential sicne the first symbol is $\exists$, but it really is not. Its main connective is the $\rightarrow$. So, if you want to use 1, you will probably at some point have to get the antecedent (the existential!) so you can do Modus Ponens.

Line 3: Likewise you can't do universal specification on 2, since 2 is not a universal statement. This is a lot easier to see, since the universal is in the middle of the statement. As a general rule, you can never do any specification of any quantifier that occurs somewhere in the middle of a statement!