# quantificational translation

symbolize the following in Lg using the the interruption function below.

U: {o|o is a person} L2: { , m loves n} H1: { p|o, o is exaited} C1: {o|o, o is a kid b: Bob

"if any loves Bob, then Bob loves everyone." if theres a x (Lxb --> Lcx) right?

"happy people and kids love Bob." For any x[Hx ^ If therex a x Hbx) .... im not sure how to finish this one. does anyone know?

"there is some kid that everyone loves" if theres an X, if theres a y(Lyx)

looks something like this... ]x]yLyx

"there is a happy kid who loves all unhappy kids"

"at least two kids love Bob"

]x]y(Lxb ^ Lyb)

"the kid who loves Bob loves everyone" ]x(Lxb -->VyLxy

any feedback. if i am wrong please correct me.

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## migrated from cstheory.stackexchange.comMar 3 '11 at 10:30

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@Lulluu: Here we have LaTeX support, you can insert a pair of dollar sign to latexify your formulas like this: $U = \{o \mid o \textrm{ is a person}\}$. – Hsien-Chih Chang 張顯之 Mar 3 '11 at 8:39

First of all some general remarks: We use the existential quantifier ($\exists$) when we want to talk about the existence of something. If the phrase is talking about a specific individual then (most probably) the existential quantifier is what you need. On the other hand the universal quantifier ($\forall$) talks about everyone. If the phrase refers to everyone, or a group of people with a property or about people generally then chances are that you need to use the universal quantifier. For example if we say "If there is a kid who is happy then everyone else is happy" then the first part of the phrase refers to a specific kid. Not to groups of kids or kids in general but about a specific individual. The second part of the phrase refers to everyone. So the first part will require an existential quantifier while the second a universal.

When we say "Anyone that has a property A also has property B", this is translated with quantifiers as $(\forall x)[A(x)\to B(x)]$. On the other hand when we want to say "If there exists someone with property A then something else (unrelated to who that someone is)" then we write $[(\exists x)A(x)]\to\ldots$. In this case observe that what goes after the "$\to$" isn't in the range of the existential quantifier. That means that it's independent of the $x$. Finally when we want to say that "There exists someone with property A and property B" then (as you may have guessed) we write $(\exists x)[A(x)\land B(x)]$. Returning now to the example I gave in the first paragraph: "If there is a kid who is happy then everyone is happy". Observe that what goes after the mention of kid isn't related to the kid at all. So this will be written using quantifiers as $[(\exists x)(C(x)\land H(x))]\to(\forall x)H(x)$. If I say "If someone is a kid then he is happy" then observe that I am not referring to a specific kid but rather to kids in general. Thus I will use a universal quantifier: $(\forall x)(C(x)\to H(x))$.

So when you try to turn English phrases into ones with quantifiers you have to first understand the meaning of the phrase and based on that, decide the type of the quantifier that you should use as well as its range.

1. "If any loves Bob, then Bob loves everyone." In this case that someone is specific. A certain person that loves Bob. After that it is a simple $p\to q$, so to figure this one out write using quantifiers the $p$ and $q$. "someone loves Bob" is $(\exists x)L(x,b)$ and "Bob loves everyone" is $(\forall y)L(b,y)$. So this one should be $(\exists x)L(x,b)\to(\forall y)L(b,y)$.
2. "Happy people and kids love Bob." There are two ways to see this. A first simpler approach is to write it as "happy people love Bob and kids love Bob". This would be $[(\forall x)(U(x)\land H(x)\to L(x,b))]\land[(\forall y)(C(y)\to L(y,b))]$. But there is another way to approach this, namely: "If someone is a happy person or he is a kid then he loves Bob". That would be $(\forall x)[(U(x)\land H(x))\lor C(x)\to L(x,b)]$.
3. "There is some kid that everyone loves". In this one your answer is not correct. What you wrote says "there is a kid that is loved by someone" but we want a kid that is loved by everyone. This would be $(\exists x)(\forall y)(L(y,x))$.
4. "There is a happy kid who loves all unhappy kids". Again this happy kid is specific. So this would be $(\exists x)[H(x)\land C(x)\land (\forall y)(\lnot H(y)\to L(x,y)]$. In this one since this kid is related the unhappy kids observe that the range of the existential quantifier is the whole phrase.
5. "At least two kids love Bob". Without use of the equality relation you cannot properly write this unless you assume something more to be true (for example that one person cannot love himself and that every kid loves another kid). Your phrase is almost correct except for the fact that it doesn't specify that these $y$ and $x$ are two different individuals. To do this simply add $x\neq y$. So this would be $\exists x\exists y(L(x,b)\land L(y,b)\land x\neq y)$.
6. "The kid who loves Bob loves everyone". This appears a bit ambiguous, since it makes it sound like there exists only one kid that loves Bob. But I will assume that the phrase means "If a kid loves Bob then he loves everyone". This phrase doesn't refer to a specific kid but to any kid that loves Bob (if you have a hard time seeing this observe that the phrase as I wrote it means the same with the phrase "Kids that love Bob love everyone"). This would be $(\forall x)[(C(x)\land L(x,b))\to(\forall y)(L(x,y)]$.
P.S.: Regarding the last one if the phrase means "There is a unique kid that loves Bob and that kid loves everyone" then you have to convey the information that this kid is unique as well. This would be $(\exists x)[C(x)\land L(x,b)\land(\forall y)((C(y)\land L(y,b))\to y=x)]$. What this says is that if anyone is a kid and loves Bob then he is $x$. So the whole phrase would be: $(\exists x)[C(x)\land L(x,b)\land(\forall y)((C(y)\land L(y,b))\to y=x)\land(\forall z)L(x,z)]$.