First order logic : Predicates

[radouani]

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)

 

[andrewkirk]

There are really only two reasonable interpretations of the natural language sentence:

(1) Every teacher has given a form to every student, so that if there are N teachers then every student has received N forms

or

(2) Every student has received a form from a different teacher. For this to work, the number of teachers must be the same as the number of students.

I think (1) sounds more plausible.

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

This says: everybody is a teacher and everybody is also a student and there is a single form that every person has given to every other person. That doesn't match either of the possible interpretations.

 

[radouani]

Thanks for your reply.

I think that the form is in logical dependency, either of student, either of teacher, or both, so we can find three other readings, right?

And How can I transform this sentence then?

Each teacher has given a form to each student.

 

[WWGD]

This is more a rule of thumb than an actual full answer. Try different (Logical) worlds/universes, different predicates and see if you get what you would expect to get. Then tinkering may hopefully lead you to the right set up.