The tree method of The Logic Book was designed to ensure that truth flows down along at least
one branch of a tree. Supposing that all the members in a set of sentences are true, the
decomposition rules only allow you to write down component or negated component sentences
that are true on at least one branch.
But the devil always lies and is very concerned that no truths be derived from his lies. He would
therefore prefer to have a tree method that starts off with the assumption that all the sentences
in a set are false and that proceeds to decompose these sentences into sentences that must be
false along at least one branch. Your job is to design an alternative tree method for the devil to
use. As part of this job, you should do the following:
1. Design an alternative set of decomposition rules for the devil to use when constructing
trees — rules that will ensure that if a sentence is false, then the sentence or sentences
it is decomposed into will be false along at least one branch. As part of this, you should
state the truth value assignments that would be made to negated and unnegated literals
on an open tree branch, and the conditions under which a tree branch would close. Do
not forget to include a rule for negated negations.
2. Formulate an account of what the tests for our concepts of truth-functional truth, falsity,
indeterminacy, equivalence, consistency, entailment, and validity would have to be like
were we to use devil trees rather than the truth trees we have studied. For example,
when testing for t-f equivalence, what sentence or sentences do we put on top of a devil
tree and should we expect the tree to be open or closed?
3. Do the unstarred exercises in section 4.5E #3 using the devil tree rules you have
designed. Compare your results with those in the answer book for these same
questions. What does this tell you?
4. The devil does not like our concepts of truth functional equivalence, consistency,
entailment, and validity. He prefers i