# Predicate Logic to English

I need to take the following predicate logic statements and write them in plain English. I'm just wondering if these look right.

L(x,y) = "x loves y." H(x) = "x is handsome." M(x) = "x is a man." W(x) is "x is a woman."

j and k are variables jack and katy.

• H(j)∧L(k,j) -> "Jack is handsome, and Katy loves Jack"
• ∀x[M(x) ⇒ H(x)] -> "All men are handsome"
• ∀x [W(x) ⇒ ¬(∀y [L(x,y) ⇒ M(y)∧H(y)])] -> I'm unsure here. Maybe "All women don't love all handsome men"?

You're right about the first two. Let's look at the last one. Yes, it says something about all women $x$. "All women... something"; let's figure out what something is. First, the part inside the negation: $$\forall y (L(x,y) \to (M(y) \wedge H(y))$$ In English, there are a few ways to say it: "whoever $x$ loves is a handsome man", or "$x$ only loves handsome men". Negating it, however, gives something harder to express in English without transforming the formula. Let's move the negation past the quantifier and find a more comprehensible way to write it: \begin{align} \neg (\forall y (L(x,y) \to (M(y) \wedge H(y))) &\iff \exists y (L(x,y) \wedge \neg(M(y) \wedge H(y))) \\ &\iff \exists y (L(x,y) \wedge (\neg M(y) \vee \neg H(y))) \end{align} This says, "$x$ loves someone ($y$) who either isn't a man or isn't handsome".