Here is a clear procedure for working truth trees:

• 1. Put the argument in vertical form.
• 2. Deny the conclusion, noting ‘DC' to the extreme right of the denied conclusion.
• 3. Search the sentences of the argument for one or more which, when a rule has been applied to it/them, the "trunk" is extended down (rather than making branches).
• 4. Apply a rule to sentence. (It doesn't matter where you start, although simpler trees usually result when you can extend the trunk rather than branch. Remember, when you "apply a rule", what you are doing is applying the correct rule for the sentence with which you are working. For example, if the sentence you are working on is a negated conditional, then you apply the rule for negated conditional.)
• 5. Check and number the sentence to which you just applied the rule
• 6. Check for contradictions.
• 7. Close each branch on which you find a contradiction.
• 8. If all branches close, designate the argument valid.
• 9. If there remains one or more open branches and if there is one or more sentence(s) which may be checked but aren't, choose one of those sentences and apply a rule to it. (At this point, you are back to #4 above. Continue on.)
• 10. Check for contradictions immediately after you check and number the sentence to which you just applied a rule.
• 11. If all sentences which can be checked have been checked, and if there is one or more open branches, designate the argument invalid.

I have discovered that it is impossible for me to achieve consistent formatting which would allow me to put a finished, and readable truth tree on the web. So, barring that, I'll run through, in English, the truth tree for the argument above. I will presume you will make each of the moves as correspond with 'a', 'b', 'c', etc., below.

• a. Deny the conclusion, to get '-(F · -I)'; then write 'DC' to the right of the denied conclusion.
• b. Note that each of the sentences we have in the argument will branch out when we apply a rule to them (rather than extending the trunk). Let's start with the 2nd premise, the negated biconditional.
• c. Directly under '-(F · -I)', make two branches. After consulting the truth tree rule for negated biconditionals, put 'I' and '- -F' on the left hand branch and put '-I' and '-F' on the right hand branch.
• d. Check and number the second premise. Give it 'ck 1'.
• e. Look for contradictions. Go from the bottom of the left branch all the way up to the first premise. No contradictions there. Now go from the bottom of the right branch up to the first premise. Again, no contradictions. We're looking for sentences such as 'I' and '-I' on the same branch. OK. We now have two open branches.
• f. Let's get rid of the doubly negated 'F' on the left open branch. Write 'F' under '- -F'; check '- -F' and give it number 2. Still no contradictions.
• g. Next, apply the rule for negated conjunction to the denied conclusion. To do this, make branches under both the open branches we now have. This gives us 4 open branches now. Put '-F' on both left branches; put '- -I' on both right branches.
• h. Put a check next to the negated conclusion, with the number 3.
• i. Check for contradictions. The extreme left branch has a contradiction on it. It has 'F' and it has '-F'. Write an 'X' under '-F'. That branch is closed. We will write nothing below this 'X'.
• j. There is also a contradiction on the extreme right branch in the form of '-I' and '- -I'. There is no need to apply a rule to '- -I' here because we see immediately that it contradicts another sentence. We don't need to break it down any further. So, write an 'X' directly under '- -I'. That branch is closed.
• k. Note there remain 2 open branches.
• l. Apply a rule to '- -I' on the left open branch, writing 'I' directly below it. Put "check4" next to '- -I'.
• m. There is one sentence that is not checked which can be checked, i.e., the first premise. This sentence is a disjunction and we'll have to make two sets of branches for it. Again, when applying a rule to this sentence, we do not extend the trunk.
• n. Make those branches. Now we should have 4 open branches again.
• o. On each of the left branches in the sets, write '(B ® F)'.
• p. On each of the right branches in the sets, write '-I'.
• q. Put "ck5" next to the first premise.
• r. Check for contradictions. There is a contradiction on the second branch from the left, or the first right hand branch, of the form '-I' and 'I'. Write an 'X' directly under '-I'. That branch is closed.
• s. Note that there are no contradictions, at this point, on any of the other branches. However, we still have two sentences which are "checkable" but haven't been checked, namely '(B ® F)' on the two left branches. Normally we would continue to apply rules to these sentences. We don't have to do that in this case, because the extreme right open branch will never close. Notice that every sentence on that branch which can be checked has been checked and no contradiction was found. Since no other sentences on that branch will be checked and since that remains an open branch, we write a ring ('O') directly below the last sentence on that branch, i.e., '-I'. You can now designate this argument Invalid.
• t. But, what happens to the two occurrences of '(B ® F)' on the branches to the left? You may either apply rules to them, or leave them untouched. If you clearly identify the one branch as open by writing the ring, then even if the other two branches were to close, that wouldn't effect the open branch; you'd still have that one open branch, and that one open branch informs us that the argument is invalid. No need to go any further.

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Text: First Logic, 2/e, Michael F. Goodman

Michael F. Goodman
Department of Philosophy
Humboldt State University