SOME COMMENTS ON TRUTH-TREES

The tree method of proof has two interesting features. First, it is entirely mechanical. No creativity or ingenuity is required in order to complete a tree. Second, the tree method, implemented correctly, is guaranteed to determine whether a given sequent of elementary logic is valid or invalid.

Our natural deduction system has neither of these features. It can take insight to see how to prove a valid sequent using our inference rules. And our natural deduction system can only show a valid sequent to be valid. It cannot prove an invalid sequent to be invalid. But the tree method can conclusively show an invalid sequent to be invalid.

These noteworthy features of the tree method come at a cost. First, using the tree method we can easily determine whether a given sequent is valid or invalid, but we get no sense of why it is valid or invalid. In contrast, natural deduction proofs allow us to see, step by step, that the conclusion of a valid sequent follows from its premises, and why. Second, tree proofs can become very unwieldy, especially in the case of more complex sequents. Natural deduction proofs, in contrast, are relatively compact and easy to follow.