# Math Help - Logical representation of sentence

1. ## Logical representation of sentence

Hi,

I have to represent the following sentences in logic.

1. Juliet loves Romeo, her father and her mother.
2. Romeo loves Juliet and also everybody loved by Juliet.

Lets say we have the predicate .loves.
and the function theFatherOf. and theMotherOf.

Would this be a correct representation of the 1. sentence?
(I couldnt find the mathematical symbols here in the forum ...)

there exists x. there exists y loves(x,y) AND there exists x loves (x, theFatherOf(x)) AND there exists x loves (x, theMotherOf(x))

For the 2nd sentence:

there exists x there exists y loves(x,y) AND there exists x there exists y for all z loves(x,loves(y,z))

Am I on the right track?

2. Hello,
Originally Posted by sanv
Hi,

I have to represent the following sentences in logic.

1. Juliet loves Romeo, her father and her mother.
2. Romeo loves Juliet and also everybody loved by Juliet.

Lets say we have the predicate .loves.
and the function theFatherOf. and theMotherOf.
Okay

Would this be a correct representation of the 1. sentence?
(I couldnt find the mathematical symbols here in the forum ...)
Go to the latex subforum (you can click on the pictures I'll produce to see the codes)

there exists x. there exists y loves(x,y) AND there exists x loves (x, theFatherOf(x)) AND there exists x loves (x, theMotherOf(x))
Not really. In the 1st sentence, "x" remains the same (Juliet). In what you wrote, it can be different "x".
In each coloured part, the x is a bound variable. So you can rename it like you want. Whereas it has to be the same. I'm sorry if it's not clear...
So keep quantifiers for the whole sentence !

This would rather be :
$\exists x,~ \exists y,~ [\text{loves(x,y)} ~ \wedge ~ \text{loves(x,theFatherOf(x))} ~ \wedge ~ \text{loves(x, theMotherOf(x))}]$

do you understand better ? :s

For the 2nd sentence:

there exists x there exists y loves(x,y) AND there exists x there exists y for all z loves(x,loves(y,z))

Am I on the right track?

Same thing here.

But you defined the predicates well;

3. Thank you very much for the help.
It makes more sense now.

So for the second one it would be:

Romeo loves Juliet and also everybody loved by Juliet.

$
\exists x, ~ \exists y, ~\forall z, ~ [\text{loves(x,y)} ~\wedge ~ \text{loves(x,loves(y,z))}]
$

4. Originally Posted by sanv
Thank you very much for the help.
It makes more sense now.

So for the second one it would be:

Romeo loves Juliet and also everybody loved by Juliet.

$
\exists x, ~ \exists y, ~\forall z, ~ [\text{loves(x,y)} ~\wedge ~ \text{loves(x,loves(y,z))}]
$
Looks good !

(I'm sorry, in general, there is no comma between quantifiers... so do it the way you know, just don't copy and past what you have done here )

and I hope I didn't put you in a wrong way

5. Originally Posted by sanv
1. Juliet loves Romeo, her father and her mother.
2. Romeo loves Juliet and also everybody loved by Juliet.
Put another way,

You first need to define your constants, functions, and relations.

Constants: Juliet, Romeo
Variables : x
Functions: father(), mother() [example: father(Juliet)]
Predicates: Loves(x, y) [x loves y], Person(x) [x is a person], LovedBy(x,y).

I don't think you need a quantifier for 1.

1. $\text{Loves(Juliet, Romeo)} \wedge \text{Loves(Juliet, father(Juliet))} \wedge \text{Loves(Juliet, mother(Juliet))}$

2. $\text{Loves(Romeo, Juliet)} \wedge \forall x((Person(x) \wedge (x \neg = \text{Juliet})) \rightarrow \text{LovedBy(x, Juliet)})$

6. Originally Posted by sanv
Hi,

I have to represent the following sentences in logic.

1. Juliet loves Romeo, her father and her mother.
2. Romeo loves Juliet and also everybody loved by Juliet.

Lets say we have the predicate .loves.
and the function theFatherOf. and theMotherOf.

Would this be a correct representation of the 1. sentence?
(I couldnt find the mathematical symbols here in the forum ...)

there exists x. there exists y loves(x,y) AND there exists x loves (x, theFatherOf(x)) AND there exists x loves (x, theMotherOf(x))

For the 2nd sentence:

there exists x there exists y loves(x,y) AND there exists x there exists y for all z loves(x,loves(y,z))

Am I on the right track?

sanv ,things are not so difficult as they look like.

To learn formalization of sentences,look at the pages 116 to 121 in the book called logic of schaum's outline series

There on page 119 there is the solution to your questions,with the only difference that the book uses the names Bob and Cathy instead of Juliet and Romeo
For example the author writes and i quote:"the sentence Cathy loves Bob is formalized as :Lcb"

There you will find all the "love combinations"

7. thanks for the help.

just to verify:

1. Juliet loves Romeo, her father and her mother.

$
\text{loves(Juliet,Romeo)} \wedge \text{loves(Juliet,fatherOf(Juliet)} \wedge \text{loves(Juliet,motherOf(Juliet)}
$

2. Romeo loves Juliet and also everybody loved by Juliet.

$
\text{loves(Romeo,Juliet)} \wedge \forall x (\text{(Person(x)} \wedge (x \neg = \text{Juliet})) \rightarrow \text{lovedBy(x,Juliet)})
$

3. The father of Romeo loves Romeo, but does not love the father of Juliet.

$
\text{loves(fatherOf(Romeo),Romeo)} \wedge \neg \text{loves(fatherOf(Romeo),fatherOf(Juliet))}
$

4. The father of Romeo loves everybody loved by Romeo.

$
\forall x (\text{loves(fatherOf(Romeo),x)} \rightarrow (\text{Person(x)} \rightarrow \text{lovedBy(x,Romeo)}) )
$