# Help with a question: do these predicate logic formulas represent the truth-conditional meaning?

I'm current learning about predicate logic, compositional semantics, quantifier raising and scope ambiguity in my semantics class and I'm having trouble answering some questions. I've attempted to answer the questions below, but i'm not sure if i'm answering the questions in the correct way.

(1) If Jake owns a donkey, he beats it

(a) The following predicate logic formula does not correctly represent the truth-conditional meaning of (1). Why not? Explain your answer.

(2) ∃x[donkey(x) ∧ own(j, x)] → beat(j, x)

(b) The following predicate logic formula does not correctly represent the truth-conditional meaning of (1) either. Why not? Explain your answer.

(3) ∃x[[donkey(x) ∧ own(j, x)] → beat(j, x)]

(c) Provide a predicate logic formula that most closely represents the truth-conditional meaning of (1). Explain your answer.

a) The formula does not represent the truth conditional meaning because the existential quantifier does not scope over (x) in beat(j,x), meaning that (x) is free. The donkey that Jake owns in own(j,x), is not the same donkey that was beaten in beat(j,x). The truth conditions of the entire sentence would be false, because (x) is not an element of both own(j,x) and beat(j,x).

I'm a bit confused about b) and c), because I thought that b) was the correct representation, as the existential quantifier bounds both (x), thus it is an element of both own(j,x) and beat(j,x)

If I could get any help with these questions, I would be very grateful!

(1) If Jake owns a donkey, he beats it

(2) ∃x[donkey(x) ∧ own(j, x)] → beat(j, x)

(3) ∃x[[donkey(x) ∧ own(j, x)] → beat(j, x)]

Formula (2) is incorrect because $x$ appears both free and bound in the formula. You are correct, it may not be the same donkey that is beaten in beat$(j,x)$.

Formula (3) is a bit more cryptic. The only thing wrong with this formula is the existential quantifier at the start rather than the for all quantifier.

Notice how the sentence (1) is worded "If Jack owns a donkey, he beats it."

• This suggests that it doesn't matter which donkey he owns, it will still be beaten. This is the first indication that a $\forall$ quantifier should be used.

• The second is that the structure of sentence (1) is written as if then. Notice that given the way it's written in English, the sentence will still be true even when there are no donkeys on earth. That is, they don't exist.

As a general rule of thumb, when the sentence is in an if then structure, consider using a $\forall$ quantifier first. When the sentence looks like two statements joined together, use an existential one.

A correct translation of the sentence would be:

$\forall x [\ (donkey(x)\land Own(j,x)\ )\rightarrow beat(j, x)]$