# Multiple quantifiers translation - The thought process

I'm having a great deal of difficulty in doing translations from English sentences to first order logic with multiple quantifiers. The process of breaking them apart is killing me. I'm looking to see if you could please help me in some of the thought processes that you go through in doing these translations.

Here are a few example sentences:

1. Every dodec is as large as every tet.
2. If a tet is right of a cube but not in back of it, then it is as large as the cube.
3. No dodec with nothing to its left is between two dodecs.
4. The only large tets are b and c.
5. At most b and c are large tets.

I was able to get the fifth one down:

∀x((Large(x) ∧ Tet(x)) → (x = b ∨ x = c))

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1. You want to say that if $x$ is a dodec and $y$ is a tec, then $x\geq y$. This should hold for all dodecs and all tecs, so this suggests two universal quantifiers, one for $x$ and one for $y$. note also the "if... then...", which suggests an implication.

2. Again, you want to say that if $x$ is a tet, $y$ is a cube, and $x$ is "right of" $y$ and not "back of" $y$, then $x\geq y$. This is for all tets and all cubes, suggesting another double universal quantification.

3. If $y$ is a dodec, and for all $x$, $x$ is not to the left of $y$, then there do not exist dodecs $z$ and $w$ such that $y$ is between $z$ and $w$. This is again something that is true for all $y$...

4. This is an assertion about $b$ and $c$, so it should simply be a declarative statement giving a property for $b$ and a property for $c$. This makes two assertions: one is a declarative assertion about $b$ and $c$; the second is an assertion that if $x$ is any tet, then $x$ must be either $b$ or $c$. So you'll want a conjunction of the declarative statement, and another formula which has a universal quantifier and an implication...

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Doesn't 4 also claim that no other x's are large tets? –  Doug Chatham May 9 '11 at 0:03
@Doug: Yes, it does. Thanks. –  Arturo Magidin May 9 '11 at 0:54
thank you again for a great explanation. –  KerxPhilo May 9 '11 at 23:15

I think "transcription" is a more accurate word here than translation, since the two languiages--every day "informal" --and formal language, are not similar-enough for a translation; you would not translate between, say Java+ and English, right?

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