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Artificial Intelligence and First Order Logic

I am undecided when to use universal quantifier or existential quantifier.
Here is my example:
Anyone passing his history exams and winning the lottery is happy.
In first order logic :
∀x Pass(x,history) ^ win(x,lottery) -> happy(x)
or
∃x Pass(x, history) ^ win(x,lottery) -> happy(x)
I would like to know if the logic is correct.

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2

Answers


  1. 0

    The existential quantifier would mean that “there is a person, X, such that …”, not “any person, X, such that”.

    Not sure why you have a “prolog” tag. But the corresponding clause in Prolog would be:

    happy(X) :- pass(X, history), win(X, lottery).
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  2. 0

    You should understand that both existential and universal quantifiers are complements of each other (similar to plus/minus and multiplication/division). Any can be achieved by negating the other one.

    Anyone passing his history exams and winning the lottery is happy.

    The answers would be:

    1. ∀x {Pass(x,history) ^ win(x,lottery) -> happy(x)}
      Read as: For all x, if x passes history and x wins lottery, then x is happy.
    2. ~∃x {Pass(x, history) ^ win(x,lottery) -> ~happy(x)}
      Read as: It is not the case (or it is impossible) that if a person passes history and wins lottery, then he is not happy.
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