I'm working my way through the proof above, and would just like some feedback as to whether my proposed proof is complete. I think I've got everything, but I just wanted to make sure. I'm preparing for a final, and this is a question in my logic text that the prof says we will probably see reappear on the final. I also recently posted this in the philosophy stack exchange, but I figured this was a better place for it. Anyway, the question suggests approaching this in 3 parts. A) Prove that a tree with a finite set of quantifier free sentences will finish, B) If there is an open path in the tree, then the set of sentences S is unsatisfiable, C) If there is no open path, then the set of sentences is unsatisfiable
A) If we are given a finite set of quantifier free sentences with which to start a tree, then we can process them one by one. Since they are quantifier free, we will apply rules until the sentence we are working on is broken down into its atomic constituents. At this point, we are done applying logical rules to the sentence, and can start applying rules to the next sentence (call it s*) by attaching s* to any open branches on our tree, and proceeding as above. If we continue in this way, we will have eventually have attached every sentence of S to our tree, and broken down each sentence of S until no more logical rules can be applied. At this point, the tree will be finished since no more rules can be applied
B) If there is an open path in our tree, we can construct a model in which S is satisfied by ensuring that each atomic constituent along the open path is made true in an interpretation (assigning meanings to predicate symbols, and extending our domain as appropriate).
C) If the tree does not have an open path, then on every path down the tree, we must have reached a contradiction and closed the path. By the completeness of the tree method, this means that the set of sentences S must not have a satisfiable model.Thus, S must be unsatisfiable.