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 axions are $G (Q\rightarrow R)\rightarrow (GQ \rightarrow GR)$ $H (Q\rightarrow R) \rightarrow (HQ\rightarrow HR)$ $G - HQ \rightarrow Q$ $- H - GQ\rightarrow Q$

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