The example we have just completed contains all the ideas of the truth tree method. But explaining two more points will help in understanding the rest of the rules.
First, why did I write a stack when I worked on line 3 and a two-legged branch when I worked on lines 1 and 2? Here is the guiding idea: If there are two alternative ways of making a sentence true I must list the two ways separately. Only in this way will the method nose out all the possible different combinations of ways of making the original sentences true. I can make line I true by making 'A' true and, separately, by making 'B' true. I just list these two alternative ways separately, that is, on separate branches. This will enable me to combine the alternatives separately with all possible combinations of ways of making the other lines true.
But when I tried to make line 3 true there was only one way to do it. I can make ' ~(AvC)' true only by making both '~A' and '~C' true. Because there is only one way of making line 3 true, line 3 does not generate two branches. It generates only one thing: the extension of all existing open branches with the stack composed of '~A' followed by '~C'.
I have just explained how I decide whether to write a branch or a stack when working on a sentence. (I'll call the sentence we are working on the "target" sentence.) But how do I decide what new sentence to write on the resulting branches or the resulting stack? Since each path represents a way of making all the sentences along it true, I should put enough sentences to ensure the truth of the target sentence along each branched or stacked path. But I also want to be sure that I don't miss any of the possible ways of making the target sentence true.
It turns out that I can achieve this objective most efficiently by writing the least amount on a branch which gives one way of making the target sentence true. I must then write, along separate branches, all the different ways in which I can thus make my target sentence true with as few components as possible. I will express this idea by saying that, when working on a sentence, I must write, along separate branches, each minimally sufficient sentence or stack of sentences which ensures the truth of my target sentence. This will make sure that no avoidable inconsistency will arise. And in this way the method will be sure to find a way of making all the initial sentences true if there is a way.
In sum, in working on a target sentence, I do the following. I figure out all the minimally sufficient ways of making my target sentence true. If there is only one such way, I write it at the bottom of every open path on which the target sentence appears. If there is more than one, I write each separate minimally sufficient way on a separate branch at the bottom of every open path on which the target sentence appears.
This reasoning works to explain all the further rules.
8-1. Use the truth tree method to show that the following arguments are valid. Show your trees, following the models in the text, complete with annotations to the right indicating at each stage which rule you have used and what line you have worked on.
a) D b) ~(Mv~N) c) ~(FvP) d) Hvl
J N ~Fv~P ~IvJ
8-2. If you ran into a conjunction on a tree, what would you do? If you don't see right away, try to figure this out on the basis of the description in this section of how the rules work and the two rules you have seen. If you succeed in stating the rule for a conjunction, try to state the rules for negated conjunctions, conditionals, and negated conditionals. If you succeed again, try stating the rules for biconditionals and negated biconditionals, which are a little harder.