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June 1st, 2017, 03:32 AM   #1
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First order logic

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 whether this is the answer for such question:
Quote:
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 form 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)

Last edited by skipjack; June 1st, 2017 at 07:28 AM.
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January 18th, 2018, 07:34 AM   #2
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Quote:
Originally Posted by radouani View Post
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 whether this is the answer for such question:
When you say something like:

There exists a teacher such that there exists a form such that they gave the form to student x it should be written as:

$\exists_{y \in T} (\exists_{z \in F} (Give(x,y,z)))$, where $T$ is the set of all teachers and $F$ is the set of all forms.

Note: T and F can be taken from predefined terms. I'm just using that portion to clarify my meaning.

This is because $\exists$ and $\forall$ are aggregate operators similar to the summation operator or large product operator. If you don't give it a statement, then it isn't valid notation. This is how I learned it, but it looks better in my opinion. Another way to write it would be to say this.

$(y \in T) \land (z \in F) \land (Give(x,y,z))$

It merely states that $y$ is in the set and $z$ is in the set. However, this is mildly flawed as $y$ and $z$ haven't been created yet! So definitely go with the first one assuming $y$ and $z$ are not predefined things that may or may not be forms or teachers and you wish to say are the form and teacher mentioned there.
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