Translating a particular statement into quantifiers

Question

I have the generalized statement: $$\text{It is } \textbf{not}\text{ necessarily the case that }P(x,y).$$ The translation appropriate is, for me, hard to come by. I have attempted this: $$\forall x\exists y\left[\neg P(x,y)\right]$$

I am nonetheless led to believe that I am wrong; what is the proper way of translating this statement?

(Original: deleted because the wrong wording was put)

Answer 1

It depends on what you mean by "not necessarily".

Some will say "not necessarily" means "may or may not be the case". In other words, the truth is indefinite.

$$\forall x\forall y :(P(x,y)\vee \neg P(x,y))$$

However, that is not a very useful statement.

Then there is the interpretation that if something is "not necessary" then it is "possibly not", which means "there is a way it can not happen".

$$\exists x \exists y : \neg P(x,y)$$

---

If you're using modal logic, we might say: $\neg \Box \, P(x,y)$ or equivalently. $\Diamond \neg P(x,y)$

Answer 2

How about $ \exists x, y [\neg P(x, y)] $ ?

This means that there are values of x and y for which P might not hold, which is another way to word what you want.

Answer 3

It is modal logic and can be written as $\neg\square p(x,y)$.