Truth Trees in
Sentence Logic
Truth trees are a
purely mechanical method for testing the validity of arguments in sentence logic. They work on a combination of principles
from the method of indirect proof and truth tables. The essence of the idea behind them is that of the distinction
between validity and invalidity. A
valid argument is one in which, if the premises are true, the conclusion must
be true. An invalid argument is one
such that if the premises are true, the conclusion may not be true. Another way to put the point is that in a
valid argument, the assumption of the falsehood of the conclusion is
inconsistent with the truth of the premises.
In an invalid argument, the assumption of the falsehood of the
conclusion is consistent with the truth of the premises.
On the basis of this
distinction (between validity and invalidity), it is possible to find a method
by which to determine whether an argument is valid or invalid by assuming that
the conclusion is true and discovering what the quality of the premises must be
on this assumption. If all the premises
can be true while the conclusion is false, the argument is invalid. But if all the premises cannot be true while
the conclusion is false, the argument is valid.
The best means by
which to understand both the method and the principle behind truth trees is to
do one. Consider the following
argument:
A
v B
-A/
B
The first step is to
add the negation of the conclusion to the set of premises. Once this is done, you have formed the
"trunk" of the truth tree.
The "branches" of the truth tree are formed by
"decomposed" statements.
Below is a completed truth tree.
Following that is an explanation of each step in its construction and
what the completed tree shows and why that it is the case.
A
v B
~A/
B
~B
A B
(x) (x)
(The (x) under each "branch" of the truth tree indicates that
the branch is closed.)
Explanation: The first step is
to negate the conclusion. Then, once
the conclusion has been negated, "decompose" each statement in the
argument that is a compound sentence so that you may see all the atomic
sentence components. -A is already as
"simple" as it can be, as is -B.
But "A v B" is a compound sentence that must be decomposed
into its smallest atomic sentence components.
Every disjunction is true when at least one of the disjuncts is
true. Since it is not necessary that
both disjuncts be true, the truth tree "branches" on a
disjunction. Once the branch has been
produced, check the resulting branches back through the trunk to see whether
there are any logical contradictions derived on the basis of the falsehood of
the conclusion. There are. In the branch on the left, A is contradicted
by the negation of A. In the branch on
the right, B is contradicted by the negation of B. This means that the branches of the truth tree are all
closed. When all the branches of a
truth tree are closed, it indicates that the argument is valid. If at least one branch remains open, it
indicates that the argument is invalid.
In other words, when all the branches are closed, it indicates that the
falsehood of the conclusion is inconsistent with the truth of the premises (or,
put another way, it indicates that the premises cannot all be true while the
conclusion is false). When at least one
branch of the truth tree is open, it indicates that the falsehood of the
conclusion is consistent with the truth of the premises (or, in other words, it
indicates that the premises can all be true while the conclusion is false).
Consider the next
argument:
A
® B
A
× B/~B
B
A
B
-A B
(x) open
Since at least one
branch of the truth tree is open, the argument in question is invalid.
The first step in
constructing the truth tree is to assume that the conclusion is false. (Note that if the conclusion were a compound
sentence, such as "A v B", the negation would be ~(A v B). If the conclusion were a compound such as
that given in the note above, it would also need to be decomposed according to
the rules for decomposition that appear below.) Once you have negated the conclusion, check for
contradictions. It sometimes happens
that the truth tree runs itself into contradictions before you have decomposed
all the sentences. If that happens, and
all the branches close, it is not necessary to continue to decompose any
statements at all. Simply indicate that
all the branches are closed and write that the argument is valid. The second step is to decompose the
statement that is a conjunction. It is
always easier to do that, since every statement that remains to be decomposed
must be decomposed on every available open branch. So, for example, if
you were to decompose the disjunction (every implication turns into a
disjunction), you would have to decompose the conjunction twice in this truth
tree. Decomposing the conjunction first
eliminates that problem. Check for
contradictions after you have decomposed the conjunction. Since there are none, decompose the
disjunction. Once that is done (and you
have now decomposed all the statements in the argument), check for
contradictions. You note that there is
only one branch that shows a contradiction.
The other one does not.
Therefore, the argument in question has been shown to be invalid.
Rules
for Decomposition of Statements in SL Truth Trees
p ® q ~p q |
p v q p q |
p . q p q |
p « q p ~p q ~q |
~(p ® q) p ~q |
~(p v q) ~p ~q |
~(p × q) ~p ~q |
p ~p ~q q |
The rules for
decomposition are very simple. If you
were to check each one on a truth table, you would find that all of them are
logically equivalent. That is, consider
the first rule:
(p ® q) « (~p v q). All
that the decomposition shows is this rule without the logical symbols being all
present. The truth tree shows
schematically what the rule "material implication" shows with
symbols. The same is the case for all
the other rules.
One more truth tree
should suffice to explain how they are done:
[A
v (B × C)]
C
® A
A
× D / B
v D
~(B
v D)
~B
~D
A
D
(x)
The truth tree closes
even before all the statements are decomposed since D is contradicted by -D. The argument is shown to be valid.