How to translate "Only dogs and dolphins jump if petted" into predicate logic?

Question

The Logic2010 software provides exercises for the symbolization of English sentences. But i'm stuck with one symbolization, which closely resembles another one I solved correctly.

The first one concerns the following sentence in English:

1. Dogs and dolphins jump if petted.

Which I correctly symbolized as:

2. ∀x (Hx → ((Ix ∨ Kx) → Fx))

However, the second sentence is this one:

3. Only dogs and dolphins jump if petted.

Which I symbolized as

4. ∀x (Hx → (Fx → (Ix ∨ Kx)))

My reasoning was as follows: (1) and (2) are both universally quantified conditionals. Their antecedent ("if petted") is identical. They only differ with respect to the consequent: In (1) the consequent is "Dogs and dolphins jump" while in (2) the consequent is "Only dogs and dolphins jump". Both the consequent of (1) and (2) are again a conditional. As in (1) it could be symbolized correctly to "((Ix ∨ Kx) → Fx)", I believed (2) could be symbolized as "(Fx → (Ix ∨ Kx))". The reason for my belief was that "If A then B" translates as "A → B" and "Only if A then B" translates as "B → A", so analagous reasoning should apply to the consequents of (1) and (2). However, my reasoning must be flawed because the solution Logic2010 provides differs from my own. This is it:

5. ∼∃x ((∼Kx ∧ ∼Ix) ∧ (Hx → Fx))

I also checked if 4 and 5 could be equivalent but this is not the case. Now i'm struggling to see the reason for this. Why is (5) the correct symbolization of (3) and why is (4) not correct? Is there any intuitive difference between what (4) and (5) say? Or is (3) ambiguous so that both (4) and (5) can be correct?

Thanks for your help!

Answer 1

It may be easier to think of 'jump if petted' as a single predicate, expressed by (Hx → Fx). Next, notice that (as you yourself reasoned) "Only dogs and dolphins are A" should be translated as (Ax → (Ix ∨ Kx)). Then the solution is:

∀x ((Hx → Fx) → (Ix ∨ Kx)))

which is equivalent to (5).

Your own solution (4) is not right because it is equivalent to ∀x ((Hx ∧ Fx) → (Ix ∨ Kx))), which can be translated as "only dogs and dolphins both jump and are petted".