As you know from working truth table for sentences, given the Law of Excluded Middle, all sentences are either true or false (not both and not neither). Now sometimes you can tell when a sentence is true. For instance, if someone says there will be a full moon tonight, there are ways you can find out whether what that person said is true. You might go to a meteorologist, or a current tide book, or watch the weather channel. Of course maybe the meteorologist is crazy or not very good at her/his job; and the tide book may have a misprint; and the weather channel may have inadvertantly been given the wrong data from its source. All right, perhaps you can't be absolutely certain about it, but you can come very close to knowing in the commonsense way of using that term. Sometimes you can discover the truth value (whether true or false) of a sentence using logic alone. Logical truths (tautologies) are this way and so are logical falsehoods (otherwise known as self-contradictions). A good example of a tautology is that "every sentence implies itself". Take the sentence "If Mary goes to Italy, then Mary will go to Italy." This sentence is true no matter whether Mary does or does not go to Italy and you can prove it so by either tables or trees, without knowing whether Mary goes or not. The sentence is true in every possible world. Self contradictions are false in every possible world. Every sentence which is neither L-true nor L-false is L-indeterminate (the 'L' stands for 'logically').
To determine whether some sentence is L-true via truth trees, it suffices to construct a tree for the denial of the sentence itself. If all branches close, you know the sentence is L-true. This method is not intuitive, but some reflection will help. Take any sentence, say
(R v -R)
[let it stand for 'It will either rain on Monday or it won't.']; if this sentence is L-true and if we deny it, then the denied sentence, which would be
-(R v -R)
would be L-false, or self contradictory. Since a self contradictory sentence is contradictory in all possible worlds, or under every interpretation (or, in tables, in each row), then the truth tree for this sentence should yield contradictions on every branch. That is, every branch should close. When all the branches close in a tree for a sentence that has been denied, we know the original sentence is L-true.
If you deny a sentence and do a tree for it and you have one or more open branches, then you know the original sentence is not L-true. That's all you know about it's logical status; you still don't know whether it is L-false. To determine whether some sentence is L-false via truth trees, it sufficies to construct a tree for the sentence itself. If the sentence is L-false, all the branches will close. If you do the tree and see that you have one or more open branches you know the sentence is not L-false.
If neither tree shows the sentence to be L-true or L-false, you know the sentence is L-indeterminate. This is worth remembering: To determine whether some sentence is L-indeterminate, it sufficies to construct two trees, one showing that it is not L-true and one showing that it is not L-false.
Set up the tree for a sentence in the same way you set them up for arguments, that is, vertically. In place of the premises, write "Zero premises', draw a line under this and the sentence itself under the line. If you suspect that the sentence you're working on is a tautology, work the tree to see if this is so (i.e., construct a tree for the denial of the sentence). If you suspect that the sentence is contradictory, construct a tree for the sentence itself. If you have no suspicion about the sentence, then you're on your own. That is, choose what you want to find out first, whether the sentence is L-true or L-false. It doesn't matter in the least which tree you start with. What does matter is that you don't stop until you have the full answer for the sentence. For example, if you work a tree for the sentence itself and the tree doesn't close (that is, not all branches close), then you know the sentence is not L-false. But, you don't know, yet, whether it is L-true. Hence, you'll have to construct a second tree (to see whether it is indeed L-true). If the second tree doesn't close, you know the sentence is L-indeterminate, because you've proven that it is neither L-false nor L-true.
Denying Sentences
To deny any simple sentence, it suffices to affix a negation sign ( the bar, '-') at the beginning of the sentence. Hence, the denial of the sentence 'S' is '-S'. Similarly, the denial of '-S' is '- -S' (which, of course, is logically equivalent to 'S').
To deny any compound sentence, one must first locate the primary/major connective. Take the sentence '[R v (D & -E)]'. Since the major connective there is the wedge, you deny this sentence by placing a bar directly before the left bracket, since the wedge ranges over all symbols from the left to the right bracket (Note how the ampersand (&) ranges only from the right parenthesis to the left parenthesis.). So, '[R v (D & -E)]' denied becomes '-[R v (D & -E)]'. Below are a few examples of sentences (on the left) and their negations/denials (on the right):