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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. January 18th, 2018, 07:34 AM   #2
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Math Focus: Algebraic Number Theory / Differential Fork Theory
Quote:
 Originally Posted by 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 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. Tags logic, order Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post logicigol Applied Math 1 April 8th, 2013 02:44 PM rhymin Computer Science 4 March 20th, 2013 10:39 AM jetro Applied Math 1 January 24th, 2013 06:18 PM jaysnanavati Applied Math 0 May 1st, 2012 03:19 PM payam Applied Math 1 January 2nd, 2011 08:51 AM

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