Consider the following well-formed formulae:

$${\rm I}.$$ $$\,\,\neg \forall x\left( {P\left( x \right)} \right)$$
$${\rm I}{\rm I}.\,\,\,\,\,\,\neg \exists x\left( {P\left( x \right)} \right)$$
$${\rm I}{\rm I}{\rm I}.\,\,\,\,\,\,\neg \exists x\left( {\neg P\left( x \right)} \right)$$
$${\rm I}V.\,\,\,\,\,\,\exists x\left( {\neg P\left( x \right)} \right)$$

Which of the above are equivalent?

The binary operation ◻ is defined as follows:

Which one of the following is equivalence to $$P \vee Q$$?

Which of the following is the negation of $$$\left[ {\forall x,\alpha \to \left( {\exists y,\beta \to \left( {\forall u,\exists v,\gamma } \right)} \right)} \right]?$$$

Which of the following first order formulae is logically valid? Here $$\alpha \left( x \right)$$ is a first order formulae with $$x$$ as a free variable, and $$\beta $$ is a first order formula with no free variable.