Rules of distribution of quantifiers over conditional and biconditional

Question

Which of the following propositional logic statements are true and why?

1. $(∀x(P(x)⟹Q(x)))⟹((∀xP(x))⟹(∀xQ(x)))$
2. $(∀x(P(x))⟹∀x(Q(x)))⟹(∀x(P(x)⟹Q(x)))$
3. $(∀x(P(x))⇔(∀x(Q(x))))⟹(∀x(P(x)⇔Q(x)))$

4. $(∀x(P(x)⇔Q(x)))⟹(∀x(P(x))⇔(∀x(Q(x))))$

5. Are their any standard laws/rules of distribution of universal quantifier over conditional and binconditional that can help me solve this?

6. Also rules for distribution of existential quantifier over conditional and binconditional?

Recently I came across distribution of quantifiers over $\vee$ and $\wedge$, which gave set theoretic interpretation of them as follows:

• $((∀x)G(x)∨ (∀x)H(x))→ (∀x)(F(x)∨ G(x))$

  In set theoretic terms, if we have that $(f(G) = D ∨ f(H) = D)$, then we have $(f(G) ∪ f(H)) = D$

• $(∃x)(G(x)∧ H(x))→((∃x)G(x)∧ (∃x)H(x))$

  In set theoretic terms, if we have that $(f(G) ∩ f(H)) ≥ 1$, then we have $(f(G) ≥ 1 ∧ f(H) ≥ 1)$

Can we say similar for distribution of quantifiers over conditional and biconditional (just to bring in more clarity)?

Answer 1

Formulas (1) and (4) are valid, i.e. they are true in every first-order $\mathcal{L}$-structure.

Formulas (2) and (3) are not valid, i.e. there exists a $\mathcal{L}$-structure in which they are not true. For instance, take the $\mathcal{L}$-structure $\mathcal{N}$ whose domain is $\mathbb{N}$ and whose interpretation of $P$ is $2\mathbb{N}$ (the set of even natural numbers), and whose interpretation of $Q$ is $\mathbb{N} \smallsetminus 2\mathbb{N}$ (the set of odd natural numbers). You have that the formula $\forall xP(x) \Rightarrow \forall x Q(x)$ is vacuously true in $\mathcal{N}$ (it claims that "if every natural number is even then every natural number is odd"), but the formula $\forall x(P(x) \Rightarrow Q(x))$ is false in $\mathcal{N}$ (it claims that "for every natural number, if it is even then it is odd"), therefore your formula (2) is false in $\mathcal{N}$. Similarly for the formula (3), since $A \Leftrightarrow B$ is equivalent to $(A \Rightarrow B) \land (B \Rightarrow A)$.

In general, when one talks about distributivity of something over something else (for instance, distributivity of $\land$ over $\lor$), one means that two formulas are logically equivalent. With this meaning, the answer to your question "Does the universal quantifier distribute over conditional or biconditional?" is negative since the formula $\forall xP(x) \Rightarrow \forall x Q(x)$ is not logically equivalent to the formula $\forall x(P(x) \Rightarrow Q(x))$ (your formula (1) is valid, but your formula (2) is not valid), and similarly the formula $\forall xP(x) \Leftrightarrow \forall x Q(x)$ is not logically equivalent to formula $\forall x(P(x) \Leftrightarrow Q(x))$ (your formula (4) is valid, but your formula (3) is not valid).