Testing for validity with the truth-tree method: A Second Example

We will use the truth-tree method to see whether the following argument is valid:

A v (B & C)
~A & E
C  ⊃ D
D

 

          A v (B &C)          
          ~A & E          
          C  ⊃ D          
          ~D         we write down the premises and the negation of the conclusion.
          ~A         we begin with the conjunction.
          E          
        /   \       apply disjunction rule to first wff.
left path closes.     A       B&C    
    X       B     apply conjunction rule to B&C
              C    
            /   \   apply rule for conditional
          ~C       D
          X       X  
                     
                     
                     

We start by placing the premises and the negation of the conlusion at the upsidedown root of the tree. Then we apply our rules. We apply the rule to the conjunction first, because it is non-branching. Then we applied the rule for disjunctions to the first wff, and we noticed that the left path contains a sentence letter and its negation. So we close that path. We apply the rule for the conjunction B&C on the right path, and then the rule for the conditional. All paths close. So what conclusion should we draw about the original argument? It is valid. Review our reasons for drawing this conclusion!