Using the following key, symbolize the following Using the following key, symbolize the following a = André C _ = _ is a cook P _ = _ is a philosopher W _ = _ is wise > (a) If all philosophers are cooks, then all cooks are philosophers. > > ∀x(Px↔Cx)→∀x(Cx∧Px) > > (b) If only philosophers are cooks, then all cooks are philosophers. > > ∀x(Px→Cx)→∀x(Cx→Px) > > (d) If only philosophers are wise, and John is a phil osopher then he is wise. > > ∀x(Px→Wx)→(Pa→Wa) is it correct? If it is wrong, please explain. ty

They're not correct.

* Seperate the question.



All philosophers are cooks: That is all $x$ who is a philosopher is also a cook. $\forall x (Px \implies Cx)$

All cooks are philosophers: That is all $x$ who is a cook is also a philosopher.$\forall x (Cx \implies Px)$

If all philosophers cooks, then all cooks are philosophers: $(\forall x (Px \implies Cx)) \implies (\forall y (Py \implies Cy))$

* Again seperate:



Only philosophers are cooks. That is, if there is noone who is cook but not philosopher. $\
eg\exists x(Cx \land \
eg Px)$

We made the second part in the previous question. Then the answer is:

$(\
eg\exists x(Cx \land \
eg Px)) \implies (\forall y (Py \implies Cy))$

* All philosophers are wise: $\forall x (Px \implies Wx)$



Andre (I suppose it's not John but Andre) is a philosopher: $Pa$

He (Andre) is wise: $Wa$

Then the answer is: $((\forall x (Px \implies Wx)) \land Pa )\implies Wa$