First Order Logic : Predicates

Question

I have a small problem with the first order logic, in particular, predicate logic

Let us take this sentence as an example:

Each teacher has given a form to each student.

From this sentence, can we have different reading?

This is my try to solve such problem, I did not know if this is the answer for such question:

Every Teacher has given a form to each Student.

(∀x)Teacher(x)^(∀y)  Student(y)^(∃z)Form(z)^Give(x,y,z)

If X is a Student then he has received a form from a teacher

Student(x)→(∃y)  Teacher(y)^(∃z)Form(z)^Give(x,y,z)

If X is a Teacher then he has gave a from for all his students

Teacher(x)→(∀y)  Student(y)^(∃z)Form(z)^Give(x,y,z)

If X is a form then a teacher gave it to all student.

Form(x)→(∀y)  Employer(y)^(∃z)Teacher(z)^Give(x,y,z)

Answer 1

The sentence:

Each teacher has given a form to each student.

does indeed have some multiple readings, but not all of the ones you propose:

Every Teacher has given a form to each Student.

(∀x)Teacher(x)^(∀y) Student(y)^(∃z)Form(z)^Give(x,y,z)

I would say that this is the 'default' reading of the sentence, i.e. the one that is probably meant. Please note though that you should symbolize this as:

$$\forall x (Teacher(x) \to \forall y (Student(y) \to \exists z (Form(z) \land Give(x,y,z))))$$

In other words: you needed to use $\to$ instead of $\land$ in a few places (because of the $\forall$), and you needed to add parentheses to indicate the scope of the quantifiers.

If X is a Student then he has received a form from a teacher

Student(x)→(∃y) Teacher(y)^(∃z)Form(z)^Give(x,y,z)

This is not a good reading of the English sentence, since it does not specify that each teacher has given a form.

If X is a Teacher then he has gave a from for all his students

Teacher(x)→(∀y) Student(y)^(∃z)Form(z)^Give(x,y,z)

This statement requires $x$ to be quantified, and once you add the (universal, since it says each) quantifier, it is of course just the same sentence as the first one (of course, you also need to make the corrections I indicated earlier)

If X is a form then a teacher gave it to all student.

Form(x)→(∀y) Employer(y)^(∃z)Teacher(z)^Give(x,y,z)

.. I am not sure why you use 'Employer' as a predicate here instead of 'Student'., but more importantly, your reading is not a viable reading of the original sentence, since this reading states that all forms are given to students, but the original sentence makes no such claim. If not all forms are given to students, the original sentence can still be true, as long as each student receives some form.

A reading that is a viable alternative, is to interpret the original sentence as saying that it is the same form that each teacher has given to each student, which would be:

$$\exists z (Form(z) \land \forall x (Teacher(x) \to \forall y (Student(y) \to Give(x,y,z))))$$

And a third one would be kind of 'in between' the first and this last one: the form that each teacher hands to their students is the same one for that teacher, but different teachers may end up giving a different form:

$$\forall x (Teacher(x) \to \exists z (Form(z) \land \forall y (Student(y) \to Give(x,y,z))))$$