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If $F_1, F_2$ and $F_3$ are propositional formulae/expressions, over same set of propositional variables, such that $(F_1\rightarrow F_2)\rightarrow F_3$ is a contradiction, then which of the following is/are necessarily true?

  1. It is not possible that $F_1$ is Tautology and $F_2$ is Contingency.
  2. It is not possible that $F_1$ is Tautology and $F_2$ is Contradiction.
  3. $F_3$ is a contradiction.
  4. It is not possible that $F_1$ is Contingency and $F_2$ is Contradiction.
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Answer : A, B, C, D

Given :   (f1 → f2) → f3 is contradiction (false for all rows), therefore 

  1. f3 must be false for all rows 
  2. f1 → f2 must be true for all rows; therefore 
  3. (f1,f2) can’t be (T,F) for any row.

option A :  Suppose f1 is tautology (all rows true) and f2 is contingency (at least one row true and at least one row false) then we will invalidate above mentioned 3rd point because (f1,f2) will be (T,F) for at least one row. Thus, it is not possible that f1 is tautology and f2 is contingency.

option B :  Suppose f1 is tautology (all rows true) and f2 is contradiction (all rows false) then we will invalidate above mentioned 3rd point because (f1,f2) will be (T,F) for all rows. Thus, it is not possible that f1 is tautology and f2 is contradiction.

option C :  From above mentioned 1st point. f3 is contradiction.

option D :  Suppose f1 is contingency (at least one row true and at least one row false) and f2 is contradiction (all rows false) then we will invalidate above mentioned 3rd point because (f1,f2) will be (T,F) for at least one row. Thus, it is not possible that f1 is contingency and f2 is contradiction.

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