# FOL to English translation questions.

I am trying to translate the following FOL Tarski's World Block language to English sentence, and some of the sentences seems quite complicated. I will really appreciate it if anyone can help verify that my attempted solutions are correct.

Question 1: $\exists x [Dodec(x) \wedge \neg \exists y (Cube(y) \wedge BackOf(y, x))]$

First, I change the above to: $\exists x [Dodec(x) \wedge \forall y \neg(Cube(y) \wedge BackOf(y, x))]$ I then translate this to:

"There exists some dodecahedron such that no cubes are to the back of it". I am not sure if this sentence correctly describes $\forall y \neg(Cube(y) \wedge BackOf(y, x))$, because if I try to distribute the negation into the parenthesis, then it becomes: $\forall y (\neg Cube(y) \vee \neg BackOf(y, x))$ which I think the translate changes to: "There exists some dodecahedron such that there are either no cubes at all or $x$ is not to the back of $y$". I think it sounds a little weird.

I am uncertain if any of these translations make sense based on the given FOL.

Question 2: $\forall x((Cube(x) \wedge \exists y BackOf(y,x)) \rightarrow Small(x))$

Translates to: "For every cube where some object $y$ is behind it, then all the cubes are small"

I am confused with the meaning of the antecedent $\forall x((Cube(x) \wedge \exists y BackOf(y,x))$, does this mean that if we arrange the objects, then each individual cube must have some $y$ behind it, or does it mean that there must be some $y$ that is/are behind all cubes, for example if we arrange 2 cubes and 2 $y$'s from left(front of) to right(back of):

1) Cube > $y$ > Cube >> $y$ Or:

2) Cube > Cube > $y$ > $y$, if this is true then shouldn't it be: "For all cube where some object $y$ is behind all of the cubes, then all the cubes are small"

Question 3: $\forall x ((Dodec(x) \wedge \exists y (Cube(y) \wedge LeftOf(x, y)) \rightarrow Large(x))$ translates to:

"If every dodecahedron is to the left of some cube, then all dodecahedrons are large."

Question 4: $\forall x(Tet(x) \rightarrow (\forall y( Cube(y) \rightarrow FrontOf(x, y)))$ Translates to:

"All tetrahedrons are in front of all cubes"

Question 5: $\neg \exists x(Dodec(x) \wedge \exists y BackOf(y, x))$

I changed the above to: $\forall x \neg (Dodec(x) \wedge \exists y BackOf(y, x))$ then this translates to:

"No dodecahedrons are to the back of some object $y$" Or

"Either there are no dodecahedrons or no $y$'s are in the back of any dodecahedrons"

Sorry my post is a bit long, and thank you for taking the time to read this post.

1. Your first translation is fine ... and much more natural than the second one!

2. No. It is that any cube with some object behind it is small. Your translation is the translation of the following FOL sentence:

$\forall x(Cube(x) \wedge \exists y BackOf(y,x)) \rightarrow \forall x \ Small(x))$

Do you see the difference?

1. You make the same mistake as with 2. Now that you know what 2 translates to, can you try 3 again?

2. Correct. You could also say: Any tetrahedron is in front of any cube

3. Look closely at the arguments of the $BackOf$ predicate: which object is in the back of which object?

• Thank you, and excellent example you have shown. I understand it now. For #3, the correct translation should be "Any dodec that is to the left of some cube is large". And for #5 it should be "There exists no y that's behind any dodecahedrons". Mar 26, 2018 at 22:56
• @IhavelowIQ yes, both are right! Maybe a little more natural for 5: there are no dodecahedrons with anything to the back of them Mar 26, 2018 at 22:58
• Thanks, that looks much better. It's gonna take sometime for me to get use to the translations and find fitting words for them. Mar 26, 2018 at 23:47