Logic & Critical Reasoning
Truth-Tree Analysis
Truth-Tree Analysis
A tree-test is a consistency test for a given input of formulas. The stack is
the input, and by applying the rules for tree construction one breaks
down all complex formulas into either affirmations or negations of
atomic statement letters.
The main rule of the tree system is the closed path rule:
Closed Path Rule: if a formula and its negation appear on the same path,
then there is a contradiction on the path, and thus the path is closed. If
there is no contradiction on a given path, the path remains open.
The tree-test can be used to determine the answer to a number of
questions that can also be determined via truth-table analysis
Consistency and Inconsistency
The tree system is a method for testing whether a stack of formulas
given as input is consistent or inconsistent.
Consistent: A stack is consistent when there is at least one open
path.
Inconsistent: A stack is inconsistent when there are no open
paths
Validity
Validity: to test whether an argument is valid negate the
conclusion of the argument, and check to see whether the stack
containing the original premises and the negation of the
conclusion is consistent. If the negation of the conclusion and the
original set of premises leads to at least one open path, then the
original argument is invalid. However, if the original premises, and
the negation of the conclusion lead to all closed paths, then the
original argument is valid.
The key idea is that an argument is valid on a tree-test when the
premises and the negation of the conclusion are inconsistent.
Tautology & Logical Equivalence
Tautology: to test whether a formula is a tautology, negate the formula
and check to see if the negation is a contradiction. If the negation of the
formula is a contradiction, then the original formula is a tautology.
However, if there is at least one path that is open, then the negation of
the formula is consistent, and so the original formula is not a tautology.
Logical Equivalence: to test whether two formulas are logically equivalent
create a biconditional between the two formulas and test to see whether
the biconditional is a tautology. In particular, negate the biconditional,
check to see whether every path closes. If every path closes, then the
original biconditional formula is a tautology, and the two formulas in
particular are logically equivalent.
Rules for Tree Construction
1. Any time a formula occurs in a tree and the negation (if it is a negation, its
affirmation) of the formula at some line below it on a connected branch one places
an X underneath the formula, signifying that the branch is closed.


X
2. A formula with two negations applying directly to it can be replaced on a line by
the formula itself.


Rules for Tree Construction
3. A negated conditional is broken down into a single trunk with the affirmation of the
antecedent followed by the negation of the consequent.
 (  )


4. A conditional is broken down into two separate trunks, one trunk with the negation
of the antecedent, and one with the affirmation of the consequent.
(  )


Rules for Tree Construction
5. A conjunction is broken down into a single trunk with the affirmation of both
conjuncts stacked.
(  )


6. A disjunction is broken down into two trunks, one trunk for each disjunct.
(  )


Rules for Tree Construction
7. A biconditional is broken down into two trunks, one trunk has both the
affirmation of the antecedent and the consequent, and the second trunk has the
negation of both the antecedent and consequent.
(  )




8. A negated conjunction is broken down into two trunks; each trunk has a
negation of one of the conjuncts.
(  )
 
Rules for Tree Construction
9. A negated disjunction is broken down into a single trunk with each disjunct negated.
(  )


10. A negated biconditional is broken down into two trunks, one trunk contains the
negation of the antecedent, and the affirmation of the consequent, the other trunk
contains the affirmation of the antecedent, and the negation of the consequent.
(  )




General Procedure for Constructing a Tree
Step 1: Determine what question is being asked. For example, are you
trying to determine whether a formula is a tautology, an argument is
valid, or a set of formulas is consistent.
Step 2: Given the question being asked alter the stack appropriately prior
to building the tree. For example, if the question is whether the stack is
consistent do not change any of the formulas since a tree-test is a
consistency test. However, if the question is whether the argument is
valid, then negate the conclusion prior to building the tree.
Step 3: Follow the rules for tree construction, until every compound
formula has been broken down.
Check to see whether all paths close, or if at least one path is open.
Tips for Tree Construction
Always trunk before you branch. If you have the option of breaking down two
or more formulas you should always choose to break down a formula that
has a trunk rule over breaking down any formulas with a branch rule.
Trunk rules go straight down, branch rules split.
To close a path all that matters is that there is some formula and it’s opposite on
the path. When looking to see if you can close a path all you need is some
formula and it’s opposite. It does not have to be in any order, nor does it
have to be the opposite of the formula at the top of the path, the path just
needs some formula and it’s opposite.
When breaking down formulas look for contradictions along the way. There are
two ways to build a tree. One way is to break down all formulas and then
look for contradictions. The other way is to look for contradictions as you
build. The second way is superior because you trim your tree as you search
for the answer.
Cases
(P  Q)  R)
Apply rule for 
(P  Q)
Apply rule for 
No rule required
P
R
Q
R
Cases
(P  Q)  R)
Apply rule for 
Apply rule for ,
followed by 
No rules required
(P  Q)
P
R
Q
R
Cases
(P  Q)  R)
Apply rule for 
(P  Q)
Apply rule for 
No rules required
P
R
Q
R