Applications of the Tree Method

1. To test for validity:

Take the premises and the negation of the conclusion. Do a truth tree for this set of wffs. The argument is valid iff the tree closes.

Testing for validity with the truth tree method: An Example

2. To test for tautologyhood:

Take the given wff, negate it, and then do a tree for the negated wff. If the tree closes, the given wff is a tautology, otherwise not.

Testing for tautologyhood with the truth tree method: An Example

3. To test whether a wff is a contradiction:

Just do a tree for the wff, without negating it. If all paths close, there's no way the wff can be true. So it's a contradiction.

4. To test whether a wff is a contingency:

Test the wff to see whether it is a tautology or a contradiction. (Do a tree for the wff and another for it's negation.) If it is neither a tautology nor a contradiction, then it is a contingent wff.

Here's the reason we can only test for contradictoriness directly, and must use additional information to check for tautologyhood and contingency. I we simply do a tree for a wff, if the tree has open paths, we can't conclude whether it is a tautology or a contingency, since in both cases we have a consistent set of one wff.  Is it the case that if a wff is a tautology, then the truth tree for it will always have all open paths?  The answer is "no".  Consider the wff (P v ~P) v (P & ~P). It is a tautology. But it has a closed path.   We also noted in class that a wff can have all open paths and not be a tautology. (Try coming up with an example of this sort on your own.) So again, that means that you can't directly test to see whether a wff is a tautology or a contingency from a truth tree for that wff. All you can infer from a truth-tree which doesn't completely close is that the wff is a consistent set of one.  You can't infer that it is a tautlogy or a contingency. So you have to negate it to see whether its negation is a contradiction.

5. To test for logical equivalence of p and q:

Do a tree for {p, ~q} and another tree for { ~p, q}. p and q are logically equivalent iff both trees close.

Exercise: Is there another way to test {p,q} for logical equivalence? How?

The order in which the rules are applied to the given wffs is completely up to you. To make your trees as simple and as elegant as possible, however, the following rule may be applied: Apply non-branching rules before you apply any of the branching rules. this will minimize the number of branches that you have to keep track of. It is also important to realize that when you apply a rule to a wff, you must apply it on every open path which occurs below it.

Exercises:

• Use Truth Trees to test logical status
• Use Truth Trees to test for validity
• Translate and Test - This exercise combines translation and truth tree skills.

Also: Go back to the exercises you did using the truth table method and now use the truth tree method. Use the answers attained from the truth table method as a check against the correctness of your results.