Testing for tautologyhood with the truth-tree method: A Third Example

We will use the truth-tree method to see whether the following wff is a tautology:

(A v ~B) ⊃ (~A ⊃ B)

 

        ~[(A v ~B) ⊃ (~A ⊃ B)]       negation of the wff we are testing;
          (A v ~B)         use rule for negation of a conditional;
          ~(~A ⊃ B)        
          ~A         again use rule for the negation of a conditional.
          ~B        
        /   \       rule for disjunction.
      A       ~B    
      X             left path closes. Right is finished and open.
                     
                     
                     
                     
                     
                     
                     
                     

There is a finished open path, the right path. So that means that there's a way of making the negation of our original wff true. Which means there is a way of making the original wff false. Which means that our original wff is most definitely not a tautology!