How to formulate these statements I have a few logic statements and am unsure how to formulate them


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*A student is excellent if he can solve all the logic questions

*No student can solve all the logic questions

*there is a logic problem no student can solve 

*No student is excellent


I was thinking of formulating it the following way:


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*forall_yS(x,y)->E(x) (for all logic questions y if solve student x questions, then x is excellent)

*~exsit_xforall_yS(x,y) (not exist student x for all questions y student solve questions)

*exist_yforall_x~S(x,y)  (exist question y for all student x student not solve question)

*~exist_xE(x)  (not exist x such that x is excellent)


How can i be sure if i'm wrong or right ?

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*You should always give your translation key in answering this sort of exercise. What is the domain of quantification? That needs to be made clear, as Mauro points out. And what does $S$ mean? Does $S(x, y)$ say e.g. that (a) $x$ can solve $y$, or that (b) $x$ is a student and $y$ is a logic problem and $x$ can solve $y$, for example? This sort of thing should be made explicit too.

*The first sentence is intended to be read as generalization, but the translation has a dangling free variable. You want at least
$$\forall x(\forall yS(x,y) \to E(x))$$
or maybe something more like
$$\forall x([T(x) \land \{\forall y(Ly \to S(x,y)\}] \to E(x))$$
(with $T(x)$ saying that $x$ is a student, etc.), depending whether you are giving interpretation (a) or (b) to $S$.
