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We were asked to express some functions using quantifiers then find the negation of the statement, so that no negation is left of a quantifier then express it in simple english.

we were given this statement: "Everyone is rich, but unhappy" and i'm not sure how to express it i thought of these:

∀x(R(x) ∧ F(x)) or ∀x(R(x) → F(x))

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    $\begingroup$ A "but" is really an "and." $\endgroup$
    – Randall
    Oct 3 '17 at 15:55
  • $\begingroup$ but can we use "and" with ∀x. my friend told me we can't use it for some reason :/ $\endgroup$
    – Mon
    Oct 3 '17 at 15:57
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    $\begingroup$ Logically "and" and "but are the same. The only difference in "natural language" is purely emotional. I think you misunderstood what your friend meant when s/he said you can't use "and" with all. $\endgroup$
    – fleablood
    Oct 3 '17 at 16:21
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    $\begingroup$ @fleablood It's hard for me to tell how literal you're being. "But" isn't emotional. "But" is a conjunction, just like "and", but they are very different. In my native language, "but" is classified as an "adversative conjunction". It just so happens that predicate calculus' conjunction "$\land$" isn't strong enough to model the natural language's "but", and that's OK in practice, predicate calculus is powerful enough as it is. $\endgroup$
    – Git Gud
    Oct 3 '17 at 21:19
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    $\begingroup$ @fleablood "Math logic" is a very broad term. There is no difference in predicate calculus between "but" and "and", but that's a limitation of predicate calculus. There are other kinds of calculi (see for instance modal logic in which one can model stuff such as "I might be hungry", which is something you can't do in predicate calculus), there might be one in which one can properly model "but". And even if there isn't, it is possible that it can be done. $\endgroup$
    – Git Gud
    Oct 3 '17 at 22:11
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You can reword the sentence as "Everyone is unhappy, but rich." Unlike implication, conjunction is a commutative operator.

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Assuming $F(x)$ stands for '$x$ is unhappy', you should use

$$\forall x (R(x) \land F(x))$$ because saying that 'everyone is rich but unhappy' is ssaying that everyone is rich and unhappy.

You were told that "the $\land$ doesn;t go with the $\forall$", because often you indeed want to use the $\rightarrow$ with the $\forall$. For example, if we want to say that 'Everyone who is rich is unhappy', we would use $$\forall x (R(x) \rightarrow F(x))$$

Notice that this statement is syaing that everyone who is rich is indeed unhappy, but it does not say that everyone is rich. Indeed, while those who are rich are unhappy, this statement says nothing about those are not rich, because the conditional is trivially true for anyone who is not rich.

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