tree method question

Question

When using the tree method for testing the consistency of a set of sentences, if I have the set of equivalent sentences {(∃x)¬Fx , ¬(∀x)Fx)}, can one fully develop a tree with only the constant a, or do I have to extend it using three constants a, b and c? In other words, on the picture, is the last line necessary to fully develop the tree?

Answer 1

First of all, even if do you decompose the second $\exists x \neg Fx$, there is no need for branching ... You would just need to get $\neg Fb$ right under the $\neg Fa$.

Ok, but the main question is: do you need to decompose both statements, if they are the exact same statement (and occur in the same branch)? Well, if you decompose one, then there is really no need to decompose the other one given that it is the exact same statement (or, put a different way, the other one can be checked off at the same time, since it will be satisfied by the statements that you have decomposed the first statement to). So, in practice, the answer is that no, you don't have to decompose both.

But, the algorithm that's given to you probably does not recognize a case like this, and a computer program will probably indeed end up decomposing both, unless it has an explicit checking mechanism for duplicate statements. So, a purist may force you to decompose both statements.

But for all practical matters, I would just check off both statements at the same time.