Write the proposition in words - $\urcorner\left(\forall x P\left(x\right)\right)$

Question

Hi here is the problem and my answer attempt.

$\urcorner\left(\forall x P\left(x\right)\right)$

Let P(x) denote, "x is taking a math science course". Domain is the set of all students. Write the proposition in words.

Here is my work breakdown and attempt at writing this.

D = {all students}
For every x $\in$ D then P(x); true.
For every x $\ni$ D then P(x); false.
Word answer: Not every student x is taking a science course.

How does this breakdown and word answer look?

Thanks

Answer 1

Literally, we read $$\neg\left(\forall xP(x)\right)$$ as: "It is not the case ($\neg$) that all ($\forall$) students ($x$) are taking a math science course ($P(x)$)".

What does it mean? If not all students are taking a math science course, then at least one student is taking another course. In logic, we say that "there exists one student who is not taking a math science course", and we translate it as: $$\exists x\neg P(x)$$ This is a fundamental relation between the universal ($\forall$) and the existential ($\exists$) quantifiers. In fact, we can write (for any domain and predicate): $$\forall x P(x)\equiv\neg\exists x\neg P(x)$$ And this means that "if everything is $P$, then nothing exists that is not $P$".