# What is the difference between the given English Sentences in Predicate Logic?

What is the difference between the given English Sentences in Predicate Logic ?

• All Americans are Nice
• Only Americans are Nice

• All Americans are Nice --> This means If a Person is an American, then the person is Nice, but If a person is not American does not mean that he is not nice.
• Only Americans are Nice-->This means that If a person is an American, then only he is Nice but all non Americans are not Nice.

Now, I think that in Predicate Logic, the two sentences would mean something :

• All Americans are Nice -->

Suppose, American(x) = x is an American

Nice(x) = x is Nice

$\forall x \left ( American(x)\rightarrow Nice(x)\right )$

• Only Americans are Nice -->

1).

$\exists x \left ( American(x)\ \Lambda\ Nice(x)\right )$ --> If I write it like this, then it is wrong because this would mean

There exists a Nice American

which is not what I want

2).

$\exists x \left ( American(x)\ \rightarrow Nice(x)\right )$ --> I guess this is right, because this would mean

There exists a person $x$ such that If $x$ is an american, then $x$ is Nice

3).

What IF I write the sentence as

All those who are Nice are Americans

I can write this as:

$\forall x \left ( Nice(x) \ \rightarrow American(x)\right )$ --> I again think this is right, but not sure.

Am I right with my understanding or Do I have totally stumbled myself in this subject ?

• The 3. is right, but 2. is not. – user202729 Dec 16 '16 at 4:19
• @user202729 why not 2 ? – Jon Garrick Dec 16 '16 at 4:20
• Because false imply everything. – user202729 Dec 16 '16 at 4:21
• @user202729 Am I right with the meaning of English Statement "Only Americans are Nice" ? – Jon Garrick Dec 16 '16 at 4:23
• I told you that 3, is right, but 2. is not. Because $\exists x (not(American(x)))$. Let $x_0$ such that $False = American(x_0)$. So, $American(x_0)\rightarrow Nice(x_0)$ is true $\Rightarrow \exists x (American(x)\rightarrow Nice(x))$. – user202729 Dec 16 '16 at 4:26

As you suspected yourself, 1 is indeed not correct, since 1 says that there is definitely some American that is nice, which is not implied by Ónly Amercians are nice', and that is because Ónly Americans are nice'would be true if they are no nice people at all, and in that case 1 is false.

1. is also not correct, since if there would be, say, a nice Frenhman, then statement 2 is true (both because the ''íf' part is false as well as because the 'then'part is true), but obviously it would not be true that ónly Americans are nice.

2. is indeed the correct translation of ''Only Americans are nice"'. Something else you could is:

$\forall x (\neg American(x) \rightarrow \neg Nice(x))$

This is equivalent to 3, since it's just the contrapositive. But maybe it is a little more intuitive: anyone who is not American is not nice ... so the only ones that are nice would have to be Americans ... but it does not mean that all Americans are nice.

• anyone who is not American is not nice. This can be written as ∀x(¬American(x)→¬Nice(x)) , right ? Have you made a typo :) – Jon Garrick Dec 16 '16 at 15:05
• Good catch! ... I could say ''just seeing if you were paying attention'', but no, that was a typo. But you were paying attention! :) – Bram28 Dec 16 '16 at 19:18
• Thanks !!! For bearing with me for my every question and now its all crystal cleared :) – Jon Garrick Dec 17 '16 at 3:30