# Predicate logic sentence translation help?

I have an assignment on predicate logic and while I understand my notes when I'm reading them, applying those notes to the questions I'm being asked isn't working so well. I've got a couple different solutions to this question and I don't know if it's right.

Formulate the following sentences into predicate logic:

a) There are lawyers who only respect lawyers. L(x) is a lawyer and R(x,y) x respects y.

I've come up with two different answers but I'm not sure which one is right.

$\exists x \exists y(L(x) \land L(y) \land R(x,y))$

$\exists x \exists y ((L(y) \land R(x,y)) \implies L(x))$

And this one I'm not even sure how to start.

e) Everyone's mother is respected, if she is a strong woman. Use mother(x) = x's mother, not mother(x,y). R(x,y) x respects y, W(x) x is a woman, S(x) x is strong. I had an answer but it doesn't look right...

$\forall x \exists y(W(y) \land S(y) \land mother(x)) \implies R(x,y)$

• so the second option was more on track? If I change it like this: $$\exists x \forall y ((L(x) \land R(x,y)) \implies L(y)$$ does that make more sense? – dyingatmidnight Feb 9 '15 at 22:00
• Not quite. The correct one is $\exists x (L(x) \land (\forall y (R(x,y) \implies L(y)))$ or, if you need the quantifiers out front $\exists x \forall y (L(x) \land ((R(x,y) \implies L(y)))$ In your version, if $L(x)$ is false for all $x$ the sentence is true because the antecedent is false. – Ross Millikan Feb 10 '15 at 1:52