Truth Trees for Propositional Logic
A Truth Tree (TT) is a branching set of formulae to be constructed in accordance with rules laid out
below to test the consistency of any set of formulae.
When you have a set of formulae (including the singleton set, containing one formula only), you can test
the consistency of this set by decomposing the formulae in the set according to the following rules:
http://legacy.earlham.edu/~peters/courses/log/treeprop.htm
As you can see, the rules of decomposition strictly correspond to the equivalence rules of natural
deduction with the following provisos: non-branching rules yield the conjuncts of a conjunction listed
one after the other on the same branch, while branching rules yield the disjuncts of a disjunction on
separate branches.
The reason why a truth-tree is a reliable method for testing validity is that it decomposes the set of
initial formulae into an equivalent disjunction of conjunctions of the atomic propositions or their
negations occurring in the original formulae. Thus, the original propositions can be true together (they
are consistent) just in case one branch is open; that is to say, one disjunct of the equivalent disjunction is
free of contradiction. This enables TT’s to serve as testers both for validity and invalidity of arguments.
For when a complete TT has an open branch, i.e., the corresponding disjunction of conjunctions has a
contradiction-free member, then the evaluation of the atomic propositions of that disjunct (assigning
truth to affirmative and falsity to negative atomic propositions) will verify all formulae of the original
propositions together (proving their consistency), whereas if all of the branches of the TT are closed, i.e.,
all disjuncts of the disjunction of the conjunctions of atomic propositions or their negations
corresponding to the TT are contradictory, then there is no evaluation of the atomic propositions that
would render all the original formulae true, i.e., they are inconsistent.
Truth Trees for Monadic Predicate Logic
These truth trees are constructed in the same way as those for propositional logic, with the addition of
the following rules:
(1) Quantifier Negation (QN): ~x(Ax) :: x~(Ax) and ~x(Ax) :: x~(Ax)
(2) Universal Instantiation (UI): x(Ax)  A(x/a)
(3) Existential Instantiation (EI): x(Ax)  A(x/a)
In (2) and (3) ‘A(x/a)’ is the formula you get by replacing all occurrences of ‘x’ in ‘A’ bound by ‘x’ with
‘a’, with the proviso that if you use (EI) once (UI) has been used, you have to introduce a new individual
name, say ‘b’, that has not occurred in the tree before, whereas if you use (AI) after (EI) has been used
(which introduced a new individual name) you will have to use all the individual names that have
occurred in the tree before.
That’s it, this is all you need to understand about TT’s; the rest is a matter of practice.