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Suppose that x is over the domain of all things and I have the following predicates:

H(x) = x is a person, J(x) = x is named John, F(x,y) = x is happier than y, a = John Smith

My interpretation of this sentence is that it's equivalent to saying "there exists a person who is happier than every person, and their name is NOT named John"

My attempt at translating this sentence is: ∃x(H(x) ^ ∀y((H(y) → F(x,y)) → ¬J(x)))

but the programme I'm using is telling me that it's not correct, can someone please guide me on how to correct my symbolic statement? Thank you so much.

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  • Why not H(y) ^ F(x, y)? The implication is vacuously true if H(y) is false.
    – Frank
    Commented Mar 12, 2023 at 23:02
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    @Frank, if there are any non-humans in the domain, your suggestion would make that subformula unsatisfiable. Commented Mar 12, 2023 at 23:18
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    Is F(x,y) reflexive? If not, then F(x,x) for the happiest person might be false. In that case you need to qualify that y=x or F(x,y). Commented Mar 12, 2023 at 23:20
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    There are many assumptions at play here, and it is moot whether you wish to make them explicit. That all persons are comparable in their happiness. That there is a unique person who is happier than anyone else. That nobody is happier than themselves. That things other than persons might be named John. If you are not worried about making these explicit, then you might try the following: For any x such that x is a person and x is named John, there exists at least one person happier than x.
    – Bumble
    Commented Mar 12, 2023 at 23:50
  • @DavidGudeman True. Let's skip my suggestion.
    – Frank
    Commented Mar 12, 2023 at 23:52

3 Answers 3

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Step by step.

"x is the happiest person" translates to: H(x) ^ ∀y ((H(y) ^ y != x) -> F(x, y)). In English, "x is a person, and if y is a person, and y is not x, then x is happier than y."

"The happiest person is not named John" translates to "There is a person who is the happiest person, and that person is not named John." ∃x ~J(x) ^ H(x) ^ ∀y ((H(y) ^ y != x) -> F(x, y))

You might alternatively translate it as "All the people who are the happiest person are not named John," which, in contrast to the previous interpretation, would be true if there is no happiest person. That would be, ∀x (H(x) ^ ∀y ((H(y) ^ y != x) -> F(x, y))) -> ~J(x)

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Youre not satisfying the task at hand, whatyou're saying is basically "if someone is happier than the other person, then this other person is not John".

I think this cannot be completed without introducing some extra predicate, because can a person be happier than him/herself? :D

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    You can be happier now than you were. You can also change your name.
    – Scott Rowe
    Commented Jul 22, 2023 at 0:36
  • true, and you can change the domain of discourse, but the task's not mentioning temporal logic nor an object being a person that used to be named John Commented Jul 23, 2023 at 11:35
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My, perhaps sorry, attempt.

S: The happiest person is John: John is a person and John is happy, and for all x, if x is a person and x is not John then, John is happier than x

(Pj & Hj) & (Ax)((Px & ~x=j) --> Hjx). [side note: How do I use logical symbols?]

~S: The happiest person is not John: ~((Pj & Hj) & (Ax)((Px & ~x=j) --> Hjx)).

This can be further worked on ...
~(Pj & Hj) v ~(Ax)((Px & ~x=j) --> Hjx)

(~Pj v ~Hj) v (Ex)~((Px & ~x=j) --> Hjx)

(~Pj v ~Hj) v (Ex)~(~(Px & ~x=j) v Hjx)

(~Pj v ~Hj) v (Ex)(Px & ~x=j & ~ Hjx)

Legend: j is John; Px: x is a person; Hx: x is happy; Hxy: x is happier than y

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