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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...

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

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For a, neither one is correct. Your first says there are two lawyers (who might be the same person), one of whom respects the other. You are trying to say that there is a lawyer and everybody he respects is a lawyer.

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  • $\begingroup$ 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? $\endgroup$ – dyingatmidnight Feb 9 '15 at 22:00
  • $\begingroup$ 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. $\endgroup$ – Ross Millikan Feb 10 '15 at 1:52
  • $\begingroup$ That makes sense thanks! $\endgroup$ – dyingatmidnight Feb 10 '15 at 13:14

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