Is anyone able to articulate the difference between the properties of soundness and completeness insofar as they relate to the validity of the tree test?
Some informal definitions first:
Soundness is the property of only being able to prove "true" things.
Completeness is the property of being able to prove all true things.
So a given logical system is sound if and only if the inference rules of the system admit only valid formulas. Or another way, if we start with valid premises, the inference rules do not allow an invalid conclusion to be drawn.
A system is complete if and only if all valid formula can be derived from the axioms and the inference rules. So there are no valid formula that we can't prove.
Truth trees seem to be a method of evaluating a logical proposition (they are new to me, so this is a first glance assessment), however the rules are really just graphical versions of the normal rules of inference, so should be sound as long as the inference rules are. Of course completeness depends on the logical system. So for propositional logic, you're fine.
Taken from Restall's Logic:
- The tree method is sound. If X├A then X╞A. That is, if an argument is valid according to trees, it is valid according to truth tables too.
- The tree method is complete. If X╞A then X├A. That is, if an argument is valid according to truth tables, it is also valid according to trees.
If all branches of a tableau are closed, the formula represented by the tableau is unsatisfiable; therefore, the original set is unsatisfiable as well.
Now, I think, but could be wrong, that if the set is unsatisfiable that would mean that the set is unsound.