1. Predicate logic - Translate?

Where:

$Ax = x$ is diamond
$Gx = x$ shines
$Lxy = x$ likes $y$

1. Not everything that shines is diamonds
2. Someone likes diamonds
3. Everyone who likes diamonds likes everything that shines
4. Someone likes only thing that shines
5. Everything that shines are liked by someone
6. There is something that is neither diamond or shines, but are liked by everyone

How would this logic expression look like....

Appreciate any help I can get!

2. Predicate Logic

Hello jokke22
Originally Posted by jokke22
Where:

$Ax = x$ is diamond
$Gx = x$ shines
$Lxy = x$ likes $y$

1. Not everything that shines is diamonds
You can do this in two ways: Either re-write it as 'It is not true that if x shines, then x is a diamond'; or as 'There exists an x, x shines and x is not a diamond'

The first gives you: $\neg (\forall x, Gx \Rightarrow Ax)$

and the second: $\exists x, (Gx \wedge \neg Ax)$

Either of these is equally correct.

2. Someone likes diamonds
Re-write this as 'There exists an x, such that for all y, if y is a diamond then x likes y'

So this symbolises as: $\exists x, \forall y, Ay \Rightarrow Lxy$

3. Everyone who likes diamonds likes everything that shines
This can be re-written 'If x is a diamond, and y likes x then if z shines, then y likes z'. I'll start you off:

$\forall x, y, (Ax \wedge Lyx) \Rightarrow \dots$

Can you finish it?

4. Someone likes only thing that shines
Re-write this as 'There exists an x, such that if x likes y, then y shines'

5. Everything that shines are liked by someone
This is 'There exists an x, such that if y shines, then x likes y.'

6. There is something that is neither diamond or shines, but are liked by everyone
'There exists an x, such that x is not a diamond and x does not shine and for all y, y likes x.'

Can you complete these now?

Though, I have given it some tries now but im pretty green at the area here is what I've come up with...

4. Someone likes only thing that shines

$\exists x, (Ax \wedge Gx) \Rightarrow Lxy$

5. Everything that shines is are liked by someone

$\exists x, (\forall x, Gx) \Rightarrow Lxy$

6. There is something that is neither diamond or shines, but are liked by everyone.

$\exists x, \neg (\forall x \wedge Gx) \Rightarrow Lxy$

I think i'm way off but... If you/anyone have the time please elaborate!

4. Predicate Logic

Hello jokke22
Originally Posted by jokke22

Though, I have given it some tries now but im pretty green at the area here is what I've come up with...

4. Someone likes only thing that shines

$\exists x, (Ax \wedge Gx) \Rightarrow Lxy$
You have this the wrong way round. y is the thing that shines; and this is if x likes y. So it is:

$\exists x, \forall y, Lxy \Rightarrow Gy$

5. Everything that shines is are liked by someone
$\exists x, (\forall x, Gx) \Rightarrow Lxy$
Again, y is the thing that shines, so it is:

$\exists x, \forall y, Gy \Rightarrow Lxy$

6.
There is something that is neither diamond or shines, but are liked by everyone.

$\exists x, \neg (\forall x \wedge Gx) \Rightarrow Lxy$
Note there are three propositions connected by 'and'. They are:

• x is not a diamond
• x does not shine
• y likes x

So it is:

$\exists x, \forall y, (\neg Ax) \wedge (\neg Gx) \wedge (Lyx)$

Hello jokke22You have this the wrong way round. y is the thing that shines; and this is if x likes y. So it is:

$\exists x, \forall y, Lxy \Rightarrow Gy$

Again, y is the thing that shines, so it is:

$\exists x, \forall y, Gy \Rightarrow Lxy$

Note there are three propositions connected by 'and'. They are:

• x is not a diamond
• x does not shine
• y likes x

So it is:

$\exists x, \forall y, (\neg Ax) \wedge (\neg Gx) \wedge (Lyx)$