# 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"?

Thanks in advance

## 1 Answer

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".

So the entire statement says,

Every woman loves someone who either isn't a man or isn't handsome.

You could even drop "either".

• That transformation is certainly valid, but your claim that it's hard to express in English is not: "For every woman, not all of the people she loves are handsome men." – CR Drost Oct 16 '15 at 21:54
• Maybe. "not all of the people etc" works, but not so well as part of the larger sentence. In ordinary English people don't go around saying, "For all women, blah-blah" -- we say "All women..." or "Every woman ...". Thus, "Every woman is such that not all of the people she loves are handsome men." Again, "such that" doesn't occur much in ordinary usage. It's English all right, but pretty formal/stilted, not ordinary conversational English. – BrianO Oct 16 '15 at 22:00