What is the fallacy in my reasoning of this sentence-to-first-order-logic-problem?

Question

Given predicates

• Occupation(p,o) (Person p has job o)
• Client(p1, p2) (Person p1 is a client of person p2)

And constants:

• Occupations: Lawyer, doctor, officer

• Persons: Emily, Joe

And sentence:

• There exists a lawyer of which all its clients are doctors.

I then translated the above sentence in first-order logic and came up with this:

∃x[Occupation(x,lawyer) ⟶ ∀y[Occupation(y,doctor) ∧ Client(y,x)]]

This reads (according to me): There exists an X such that if X is a lawyer, then all Y's are a customer of X and they're doctors.

However, peeking at the answers, I found that the only solution was:

∃x[Occupation(x,lawyer) ∧ ∀y[Client(y,x) ⟶ Occupation(y, doctor)]]

This reads (according to me): There exists an X such that if X is a lawyer and all Y's are a client of X, then they are all doctors. I feel like this doesn't express the sentence correctly.

I'm stumped as to why my answer is incorrect! I can't figure out the logical difference between those two sentences and English isn't my main language (so maybe that's where the problem lies?). What is the fallacy in my reasoning?

Answer 1

You are correct in your translation of your sentence, but this does not correspond to the initial plain English sentence.

This is how I would translate your sentence:

There exists X such that if X is a lawyer then for every Y, Y is doctor and a client of X

Pretty much the same as you translated it. What does this mean though? First of all notice that after there exist X we have an if. This means that the sentence is trivially true (since X can be not a lawyer, and the sentence is still true). So you want to start with a statement that X is a lawyer and then add more conditions. Furthermore, let's assume we have X being a lawyer, what does the rest of the sentence mean? It means that there is a lawyer that has all people as clients (including themselves) and all people are doctors. So if there are 10 people in our universe, then one is a lawyer and all are doctors and clients of his/hers. The initial plain English statement of course just meant that, for example, one lawyer just has 2 clients and both of them are doctors.

Furthermore you are not correct in your translation of the book sentence. Here's my translation:

There exist X such that X is a lawyer and for every Y, if Y is a client of X, then Y is a doctor.

This is the same as the initial plain English statement. See how it makes the definite statement that X is a lawyer and then continues with the conditional.