I'm trying to figure out the following translation from English sentence to FOL, and I have some questions:
Let the binary predicate $LeftOf(x,y)$ be $x$ is left of $y$
Let the ternary predicate $Between(x,y,z)$ mean $x$ is in between $y$ and $z$ where $x,y,z$ are variables.
Let the binary predicate $BackOf(x,y)$ mean $x$ is in the back of $y$
Question 1: "Every small block is between two others"
For this question, I would assume that a block involves all Tet, Cube, Dodec in the Blocks Language. And this sentences seems impossible since the last and first cube can't be between anything. So, the translation should be: $\forall x [((Cube(x) \vee Tet(x) \vee Dodec(x)) \wedge Small(x)) \rightarrow \exists y \exists z(Between(x,y,z) \wedge x \neq y \wedge x\neq z \wedge y \neq z \wedge ((Cube(y) \vee Tet(y) \vee Dodec(y))\wedge((Cube(z) \vee Tet(z) \vee Dodec(z)) \wedge Small(y) \wedge Small(z))]$ The reason I used inequality is to show that $x,y,z$ are not the same object. However, this doesn't look right, because the English sentence says that the small block must be between two other blocks with the same property, but the above FOL could be that a Tet is between a Cube and Dodec, or a Dodec is between two Tet's, which seems vague. I am not sure about this. Maybe if we only assume a block as a Cube, then:
$\forall x [(Cube(x) \wedge Small(x)) \rightarrow \exists y \exists z (Cube(y) \wedge Cube(z) \wedge Small(y) \wedge Small(z) \wedge Between(x,y,z) \wedge y \neq z \wedge x \neq y \wedge x \neq z)]$
Question 2: "Every cube in back of a dodecahedron is also smaller than it."
Translates to: $\forall x [Cube(x) \rightarrow \exists y(Dodec(y) \wedge BackOf(x,y) \wedge Smaller(x,y))] $
Question 3: "Every large cube is in back of some small tetrahedron." Translates to:
$\forall x[(Cube(x) \wedge Large(x)) \rightarrow \exists y(BackOf(x,y) \wedge Tet(y) \wedge Small(y))]$
Question 4: "No Dodecahedron is the same size as any tetrahedron." Translates to:
$\forall x \neg (Dodec(x) \rightarrow \forall y(Tet(y) \rightarrow SameSize(x,y)))$
Question 5: "If a tetrahedron has something to its left, then it is large." Translates to:
$\exists x(Tet(x) \wedge \exists y LeftOf(y,x)) \rightarrow \exists xSmall(x)$
I will appreciate it very much if anyone can verify my solutions and help me with my confusion on question #1.
Thank you.