Testing for consistency with the truth-tree method: A First Example

We will use the truth-tree method to see whether the following set of wffs is consistent:

S ⊃ T
S & ~T
A v (B ⊃ S)

 

          S ⊃ T          
          S & ~T          
          A v (B  ⊃ S)          
          S         begin by applying rule tot he conjunction.
          ~T        
        /   \       Now the conditional
      ~S       T    
      X       X     all paths are closed.
                     

We began by applying the conjunction rule to (S & ~T). (You can apply rules in any order, but your trees will be simplest if you apply non-branching rules first.) Then we applied the rule for the conditional to (S ⊃ T). It is a branching rule. But then we noticed that on each open path, we have a sentence letter and its negation. That means that each path closes. Each is an impossible path. So our tree is finished, and the conclusion we reach is that the set of wffs is inconsistent.