Multiple choice

Which of the following is a valid first order formula? (Here $\alpha$and $\beta$ are first order formulae with x as their only free variable)

  1. ((∀x)[α] ⇒ (∀x)[β]) ⇒ (∀x)[α ⇒ β]

  2. (∀x)[α] ⇒ (∃x)[α ∧ β]

  3. ((∀x)[α ∨ β] ⇒ (∃x)[α]) ⇒ (∀x)[α]

  4. (∀x)[α ⇒ β] ⇒ ((∀x)[α]) ⇒ (∀x)[β])

Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that, If the four tense operators P,F,H and G and following axioms G (Q $\rightarrow$R) $\rightarrow$(GQ $\rightarrow$GR) H (Q $\rightarrow$R) $\rightarrow$(HQ $\rightarrow$HR) G - HQ $\rightarrow$Q

  • H - GQ $\rightarrow$Q