The Problem with Truth Tables
• The problem with standard truth tables is that they
grow exponentially as the number of sentence letters
grows, so…
• Most of our work is wasted because most of the Ts and
Fs we plug in don’t show anything!
• But indirect truth table only work effectively for rigged
examples
• We need something better: i.e. Truth Trees!
Truth Tree Tests
• Improved version of Short-Cut Truth Tables
– We assign truth values to whole sentences (here, by putting
them on a branch of the tree)
– And work to smaller parts to see if we can get a coherent truth
value assignment that makes them have those truth values: this
‘grows’ the tree.
• Consistency: we assign true to each sentence.
• Validity: we assign true to the premises and the negation of the
conclusion.
– Note: this is an indirect proof (‘reducio’, ‘proof by contradiction’)
method!
Short-Cut Truth Tables and Truth Trees
• Both methods assign truth values to whole sentences and
then figure out what truth values of their components
produce the assigned truth value—we are, in effect,
“decomposing” the sentences in to their parts.
• Both methods test to see whether it is possible to produce a
correct truth value assignment to the sentence letters that
gets the assigned truth value for the whole sentences
• Recall the short cut truth table test for consistency…
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
T
T
T
T
Write the sentences on one line with slashes
between them
Assign true to each sentence by writing ‘T’
under its main connective
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
T
T
T
T F
Since  A is true,
A must be false
so this truth
value is “forced”
on A
Assign “forced” truth values.
We start with the last sentence because
assigning true to the other sentences doesn’t
“force” truth values on their parts.
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T
T
F
T
T F
Now that we’ve assigned a truth value to A,
other truth values are forced by that:
All the other A’s must be false too!
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F
T
T
T F
This forces more truth values:
Since A is false, to make the first sentence true
we have to assign true to B—which makes all
the B’s true.
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F
T
F T
Since B is true,  B must be false—so yet
another truth value is forced
T F
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F
F
F
T
F T
T F
Since  B is false, C must be false in order to
make the conditional, C   B, true--so we
have another forced truth value: all C’s have to
be false
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F F
F
F
T
F T
T F
Now we can complete the truth value
assignment—and there’s only one way to do
it: by assigning false to C  A, since both of its
parts are false.
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F F
F
F
T
F T
T F
But this isn’t a possible truth value assignment
because it says that the conditional,
B  (C  A), is true even though its
antecedent is true and its consequent false.
And there’s no way to avoid this since all
truth values were forced!
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
F F
F
F
T
F T
This shows that there’s no truth value
assignment that makes all sentences true
Therefore that this set of sentences is
inconsistent.
T F
Short-Cut Truth Tables: Consistency
A  B / B  (C  A) / C   B /  A
F T T
T
T
T T
F
T
T
F T
T F
Note: if you assigned truth values in a different
order the problem will pop up in a different
place (see Hurley p. 40)—but it will pop up
somewhere, like a lump under the carpet!
Truth Trees
• In doing a truth tree we start in the same way: by assigning
truth values to whole sentences and then working backward
until we’ve assigned truth values to all sentence letters.
• We do this by “growing” a tree-structure according to tree
rules which “decompose” sentences into their constituent
sentence letters.
• The tree rules represent the ways in which the sentence
forms to which they apply are made true—so, e.g.
p  q is made true by either p’s being true or q’s being true
p • q is made true by both p and q being true
Why?
• p  q is made true by either p’s being true or q’s being true
• In both cases where p is true, p v q is true.
• In both cases where q is true, p v q is true.
Why?
• p • q is made true by both p’p and q being true
• We can construct tree rules from the characteristic truth
tables for the connectives in this way!
Conjunction and Disjunction
p•q
p
q
pvq
p
q
Truth
• To make p • q true, Truth
has to flow through both p
and q
• To make p v q true, all we
need is truth flowing
through one of its parts
• So we represent
conjunction by a nonbranching rule
• So we represent disjunction
by a branching rule
Tree Rules
DN
p
p
• The rule for Double Negation is:
rewrite, erasing two ’s
OR
p q
• Sentences that are basically OR’s are
represented as branching structures
p
• Sentences that are basically AND’s are
represented by non-branching
structures.
q
AND
p•q
p
q
• We understand conditionals and
biconditionals as basically OR’s and
AND’s
Conditional and Biconditional
• Conditional and biconditional are actually “extras” in our
language: we can say everything they say just in terms of
conjunction, disjunction and negation.
• p  q is equivalent to either  p OR q so we formulate the tree
rule for conditional as a branching OR rule
• p  q is equivalent to either (p AND q) OR ( p AND  q) so we
formulate the tree rule for biconditional as a branching OR
rule with ANDs on both branches.
• To see why this is so, consider the truth tables for conditional
and for biconditional
Truth Tables for the Connectives
p
p
p
q
p• q
pq
pq
pq
T
F
T
T
T
T
T
T
F
T
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
F
T
T
• p  q is true if either p is false or q is true so it’s logically
equivalent to  p v q
• You can prove this by testing the two sentences for
equivalence!
Truth Tables for the Connectives
p
p
p
q
p•q
pq
pq
pq
T
F
T
T
T
T
T
T
F
T
T
F
F
T
F
F
F
T
F
T
T
F
F
F
F
F
T
T
• p  q is true if either both p and q are true or both p and q are false.
• So it’s equivalent to (p • q) v ( p •  q)
• Note: we’re helping ourselves to the idea that saying p is false is the
same thing as saying  p—which is ok given the truth table for 
Conditional and Biconditional
p  q
p
q
pq
p
q
p
q
• To make p  q true, Truth
has to flow through either
 p or q
• p  q says either p • q or
 p •  q so it’s a branching
rule with conjunctions on both
branches
•  p says p is false so this
says what makes p  q is p
being false or q being true
• Truth has to either flow
through both p and q or
through both  p and  q
Negation of Conditional and Biconditional
 (p  q)
p
q
 (p  q)
p
q
p
q
Truth
•  (p  q) says p  q is false
• What makes a conditional
false is true antecedent, false
consequent
• So we represent this as a
conjunction of p and  q
•  (p  q) says that p and q
have opposite truth value
• Truth has to either flow
through p and  q or
through  p and q
Negation of Disjunction and Conjunction
 (p v q)
p
q
 (p • q)
p
q
Truth
•  (p v q) is equivalent to
 p •  q by DeMorgan’s
Law
•  (p • q) is equivalent to
 p v  q by DeMorgan’s
Law
• So we represent  (p  q) by
this non-branching rule
• So we represent  (p • q) by
this branching rule
Double Negation
  p
p
Truth
• The double negation rule is
obvious!
• p is equivalent to   p so, a
fortiori, p makes   p true.
Truth trees are upside down
To represent the
truth value
assignment that
makes a sentence
true we want to
show truth flowing
up the tree—like sap
from the roots
T
Except in this case
truth flows upward
from the branches!
How to Grow a Truth Tree
• We use the rules to grow the tree downward.
• We apply the tree rules to each sentence successively to
“decompose” it into simpler sentences that make it true…
• …and we decompose those sentences into even simpler
sentences…
• …until we get down to sentences that can’t be
decomposed any further, that is
• Sentence letters and negations of sentence
letters
• Then the tree is complete.
Growing a Truth Tree to Test Consistency
PQ
P•Q
Q
• Write the sentences to be tested in a vertical column: these
are the initial sentences
• We’re looking for a truth value assignment that will make all
of them true (if there is one)
• So we start by considering truth value assignments that make
each of them true individually
• And see if we can put them together
Growing a Truth Tree to Test Consistency
√
~P
Q
• Apply tree rules to each sentence to which they apply, checking
sentences when they’ve had rules applied to them
• We start with non-branching rules to keep the tree from getting too
big.
Growing a Truth Tree to Test Consistency
√
√
~P
Q
~P
Q
• Now we apply the rule for conditional to P  Q writing the
result at the bottom the tree
• The tree stops growing because no further rules can be
applied.
Growing a Truth Tree to Test Consistency
√
√
~P
Q
~P
Q
• A “branch” or “path” is the result of tracing from each
sentence at the bottom of the tree all the way up to the top
• There are 2 (overlapping) branches on this tree: the initial
sentences are on both branches.
Growing a Truth Tree to Test Consistency
√
√
~P
Q
~P
Q
• Each branch wants to represent a truth value assignment to the
initial sentences which we can read off as follows:
• If a sentence letter occurs on a branch, TRUE is assigned to that
sentence letter; if the negation of a sentence letter occurs, FALSE is
assigned to that sentence letter.
Growing a Truth Tree to Test Consistency
√
√
~P
Q
~P
Q
• On this tree, both branches assign FALSE to P and TRUE to Q
• So each branch represents the same truth value assignment, viz.
• The truth value assignment represented by the row of the truth
table in which all sentences got true, remember…
Testing Sets of Sentences for Consistency
P 
P is FALSE;
Q is TRUE
Q
/  P
•
Q / Q
T
T
T
F
T
F
T
T
T
F
F
F
T
F
F
F
F
T
T
T
F
T
T
T
F
T
F
T
F
F
F
F
Consistent or inconsistent? Consistent
We constructed this row of the truth table on the
truth tree without wasting time doing the other rows
that didn’t matter!
But what if things were different?
√PQ
√P• Q
P
Q
~P
Q
X
• The left branch doesn’t represent a truth value assignment
because it assigns both TRUE and FALSE to P!
• So we say that branch is “closed” and indicate that by putting
an X at the bottom
Open and Closed Trees
• A completed tree is open if it has at least one open
branch.
• A completed tree is closed if it has no open branches,
i.e. if all of its branches are closed.
• Consistency only requires the some (i.e. at least one)
truth value assignment make all the sentences true so
– If the tree is open, then the initial sentences are
consistent
– If the tree is closed, then the initial sentences are
inconsistent
Summing up so far
• So now we can do two things:
– We can determine whether a set of sentences is
consistent or inconsistent
• Open tree – consistent
• Closed tree – inconsistent
– And if the sentences are consistent we can
determine which truth value assignment(s) makes
them all true by reading the the open branch(es)
• But what if a set of sentences is inconsistent?
But what if things were different?
√
PQ
P
~Q
Initial
Sentences
~P
Q
X
X
• This tree is closed so the initial sentences are inconsistent.
• There is no truth value assignment that makes all initial
sentences true.
So what should I be able to do?
• Know the tree rules and how how they are derived
• Be able to invent a tree rule for a symbol if given its characteristic
truth table
• Grow a truth tree
• Determine what a completed truth tree tells you about the
consistency or inconsistency of initial sentences
• If the initial sentences are consistent, determine which truth
value assignment makes them all true
• Given a completed tree, determine what its initial sentences are.
Growing a Truth Tree to Test Validity
(P  Q)  R
R
P
• Write out the argument vertically, premises first and then
conclusion
• The truth tree test for validity is an indirect proof method (aka
reductio, proof by contradiction): we want to show that it’s
not possible for all the premises to be true and the conclusion
false.
• So we ask: “What if the premises were true and the
conclusion were false?”
Growing a Truth Tree to Test Validity
(P  Q)  R
R
P
• To ask that question, we negate the conclusion, grow a tree,
and see what happens.
• When we test an argument for validity, we call the premises +
the negation of the conclusion, the sentences above, the initial
sentences.
• We then test these initial sentences for consistency by growing
a truth tree from them.
Growing a Truth Tree to Test Validity
(P  Q)  R
R
P
• We know that:
– If the premises + negation of conclusion are consistent the
argument is invalid.
– If the premises + negation of conclusion are inconsistent
the argument is valid.
• So by testing these sentences for consistency, we can
determine whether the argument is valid or invalid!
How does this show validity or invalidity?
P1 / P2 / . . . Pn / ~ C
T
T
T
When we say that the premises +
the negation of the conclusion are
consistent we’re saying that
there’s a truth value assignment
(row of truth table) in which all
these sentences are true.
T
Please
run this
by me
again
How does this show validity or invalidity?
P1 / P2 / . . . Pn // ~ C
T
T
T
If there’s a row in which all the
premises and the negation of the
conclusion are true then in that very
row all the premises are true and
the conclusion itself is false.
So the argument is invalid!
T F
Please
run this
by me
again
How does this show validity or invalidity?
P1 / P2 / . . . Pn / ~ C
T
T
T
T
Now suppose that the premises +
the negation of the conclusion are
inconsistent.
This means that there’s no row in
which the premises and the
negation of the conclusion are all
true.
Please
run this
by me
again
How does this show validity or invalidity?
P1 / P2 / . . . Pn // ~ C
T
T
T
T F
So there’s no row in which all the
premises are true and the
conclusion itself is false.
So the argument is valid!
Please
run this
by me
again
Summing Up: Testing for Validity
• Using the tree method, we test for validity by testing the
initial sentences—premises + negation of conclusion for
consistency.
• If the initial sentences are consistent the argument is invalid.
• If the initial sentences are inconsistent the argument is valid.
• So now let’s try it!
Growing a Truth Tree to Test Validity
(P  Q)  R
R
negation of the
conclusion
P
• We’re going to test these initial sentences for consistency.
• If the tree closes, they’re inconsistent, so the argument is
valid.
• If the tree is open, they’re consistent, so the argument is
invalid.
Growing a Truth Tree to Test Validity
(P  Q)  R
R
√P
P
We apply the double negation
rule to this sentence, check it,
and write the result at the
bottom of the tree
Growing a Truth Tree to Test Validity
√(P  Q)  R
R
√P
P
 (P  Q)
R
Are there any problems? We check both branches to see whether either of
them includes a sentence and its negation.
Note: a sentence is “included” on a branch if it occurs on a line by itself—not
just as part of a longer sentence.
Growing a Truth Tree to Test Validity
√(P  Q)  R
R
√P
P
 (P  Q)
R
X
We’ve got a problem: R and  R are on the same branch, so that branch stops
growing and closes.
We show that the branch is closed by putting an X at the bottom.
Growing a Truth Tree to Test Validity
√(P  Q)  R
R
√P
P
Now we apply the
negation of a
disjunction rule to
this sentence
√ (P  Q)
P
Q
R
X
The tree is now finished growing because each sentence to which a rule could
be applied has been checked—showing that the appropriate rule has been
applied to it.
Is there a problem?
Growing a Truth Tree to Test Validity
√(P  Q)  R
R
√P
P
√ (P  Q)
P
Q
X
R
X
Yes! The remaining branch includes P and  P so it closes, and we show that
by putting an X at the bottom of the branch.
The tree is now complete and it is closed—so the argument is valid!
What would an invalid argument look like?
√(P • Q)  R
R
√P
P
√ (P • Q)
P
Q
R
X
This tree has finished growing but is open so the argument is invalid.
We can also determine some more things about this argument by “reading” its
truth tree…
What’s the conclusion?
√(P • Q)  R
R
√P
P
√ (P • Q)
P
Q
R
X
• Reading from the bottom up, we look for the first sentence which
wasn’t the result of applying a tree rule.
• That sentence is the negation of the conclusion.
• So the conclusion of this argument is  P
Are the initial sentences consistent or inconsistent?
√(P • Q)  R
R
√P
P
√ (P • Q)
P
Q
R
X
• The initial sentences (the premises + negation of the
conclusion of the argument) are consistent.
• The open path represents a truth value assignment that
makes all the initial sentences true.
What truth value assignment makes all initial sentences true?
√(P • Q)  R
R
√P
P
√ (P • Q)
P
Q
R
X
• If a sentence letter appears on an open path, that truth value
assignment assigns TRUE to that sentence letter.
• If the negation of a sentence letter appears, it assigns FALSE to
that sentence letter
What truth value assignment makes all initial sentences true?
√(P • Q)  R
R
√P
P
√ (P • Q)
P
Q
R
X
• So, the truth value assignment that makes all initial sentences
true is…
• P – TRUE; Q – TRUE; R – FALSE
The initial sentences are consistent
(P
•
Q)

R

R


P
T
T
T
T
T
F
T
T
F
T
T
T
T
T
F
T
F
T
F
T
T
F
F
T
T
F
T
T
F
T
T
F
F
F
F
T
F
T
F
T
F
F
T
T
T
F
T
F
T
F
F
F
T
F
F
T
F
F
T
F
F
F
F
T
T
F
T
F
T
F
F
F
F
F
F
T
T
F
T
F
/
/
So the argument is invalid
(P
•
Q)

R

R

P
T
T
T
T
T
F
T
F
T
T
T
T
T
F
T
F
F
T
T
F
F
T
T
F
T
F
T
T
F
F
F
F
T
F
F
T
F
F
T
T
T
F
T
T
F
F
F
T
F
F
T
F
T
F
F
F
F
T
T
F
T
T
F
F
F
F
F
F
T
T
T
F
/
//
Cheers!