The summary statement tells you that you may be apply the rules in any order you like. Indeed, the order does not matter in getting the right answer. But the order can make a practical difference as to how quickly you get the right answer. In all the examples you have seen so far, I started with the first sentence on the tree and worked downward. To show you that order can make a practical difference I am going to redo the first example and work on line 3 first.
Compare this tree with the first way I worked the problem, and you will see that this one is a good bit shorter. I worked our the problem the longer way the first time because I wanted to illustrate how branched get stacked on top of several other branches. And I wanted to show you how working on th sentences in a different order can make a difference as to how quickly you can do the problem.
But how can you tell what the shortest way will be? I have no surefire formula which will work for you all the time. In general, you have to "look ahead: and try to see how the problem will work our to decide on an order in which to work the lines. Your objective is to get as many branches to close as quickly as possible. If you don't make the best choice, the worst will happen is that your problem will take a little longer to do.
There are several practical rules of thumb to help you out. First,
Practical guide: Work on lines that produce stacks before lines that produce branches.
Branches make trees messy. Stacks line up more sentences along a path and improve your chances of getting a path to close. In the problem which I redid I knew I was likely to get a shorter tree by working on line 3 before working on lines 1 and 2. I knew this because line 3 produces a stack while both 1 and 2 produce branches.
A second suggestion:
Practical guide: When you have as sentences on which to work only ones which are going to produce branches, work first on one which will produce at least some closed paths.
If the first. two suggestions don't apply, here is a final piece of advice:
Practical guide: If the first two guides don't apply, then work on the longest sentence.
If you put off working on a long sentence, you may have to copy the results of working on it at the bottom of many open branches. By working on a long sentence early on, you may get away with its long pieces at the bottom of relatively few branches, making your tree a little less complicated.
Now you should sharpen your understanding of the rules by working the following problems. The easiest way to remember the rules is to understand how they work. Once you understand the reasoning which led us to the rules, you will very quickly be able to reconstruct any rule you forget. But you may also refer to the rule summary on the inside back cover. You should understand the boxes and circles in this way: Whenever you find a sentence with the form of what you see in a box occurring as the entire sentence at a point along a tree, you should write what you see in the circle at the bottom of every open path on which the first sentence occu'rred. Then check the first sentence. I have written the abbreviated name of the rule above each rule.
8-4. Use the truth tree method to determine whether or not the following arguments are valid. In each case show your tree, indicating which paths are closed. Say whether the argument is valid or invalid, and if invalid give all the counterexamples provided by your tree.
a) A b) ~C⊃H c) K⊃F d) P⊃M e) FvG f) J⊃D
B ~H⊃C F M⊃B ~GvH K⊃D
A&B K P⊃B F&H J&K
g) (KvS) h) J⊃D i) M≡N j) T≡I k) ~(F&~L)
~(K&H) K⊃D M=~N J&A ~(L&~C)
K⊃H JvK ~Tv~A F≡L
l) F≡G m) C⊃N n) ~(Q⊃D) o) R≡~A p) O≡~F
G≡H ~(I⊃C) ~(Q&B) ~(R=T) ~(F≡K)
F≡H ~(Nv~I) ~(B&~R) R&~S O⊃K
8-5. The rules, as I have stated them, specify an order for the branches and an order for sentences in a stack. For example, the rule for a negated conditional, ~(X⊃Y) instructs you to write a stack
But could you just as well write the stack
at the bottom of every open path on which ~(X⊃Y) appears? If so, why? If not, why not? Likewise, the rule for the conditional, X⊃Y, instructs you to write the branches
But could you just as well write the branches in the other order, writing
at the bottom of every open path on which X⊃Y appears? If so, why? If not, why not? Comment on the order of branches and the order within stacks in the other rules as well. I
8-6. If we allow ourselves to use certain logical equivalences the truth tree method needs fewer rules. For example, we know from 1 chapter 4 that, for any sentences X and Y, X⊃Y is logically equivalent to ~XvY. Now suppose we find a sentence of the form X⊃Y on a tree. We reason as follows: Our objective is to make this sentence true by making other (in general shorter) sentences true. But since ~XvY is logically equivalent to X⊃Y, we can make X⊃Y true by making ~XvY true. So I will write ~-XvY at the bottom of every open branch on which X⊃Y appears, check X⊃Y, and then apply the rule for disjunctions to ~XvY. In this way we can avoid the need for a special rule for conditional sentences.
Apply this kind of reasoning to show that, by appealing to de Morgan's rules, we can do without the rules for negated conjunctions and negated disjunctions, using the rules for disjunctions and conjunctions in their place. Also show that we could equally well do without the rules for conjunctions and disjunctions, using the rules for negated disjunctions and negated conjunctions in their place. 8-7. In chapter 3 I extended the definition of conjunctionsand disjunctions to include sentences with three or more conjuncts and sentences with three or more disjuncts. But we have not yet stated truth tree rules for such sentences.
a) State truth tree rules for conjunctions of the form X&Y&Z and for disjunctions of the form XvYvZ.
b) State truth tree rules for conjunctions and disjunctions of arbitrary length.
8-8. Write a truth tree rule for the Sheffer stroke, defined in section 3-5.
chapter summary Exercise
Here are this chapter's important new ideas. Write a short explanation for each in your notebook.
a) Truth Tree
c) Branching Rule
d) Nonbranching Rule
e) Closed Branch (or Path)
f) Open Branch (or Path)
g) Rule ~~
h) Rule &
i) Rule ~&
j) Rule v
k) Rule ~v
1) Rule ⊃
m) Rule -⊃
n) Rule ≡
0) Rule -≡