Truth Trees

1
Truth Trees
2
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 dont show anything!
• But indirect truth table only work effectively
  for rigged examples
• We need something better i.e. Truth Trees!

3
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!

4
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
  valuewe 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
• \\

5
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
Write the sentences on one line with slashes
between them Assign true to each sentence by
writing T under its main connective
6
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 doesnt force truth values on their
parts.
7
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
F
F
F
Now that weve assigned a truth value to A, other
truth values are forced by that All the other
As must be false too!
8
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
This forces more truth values Since A is false,
to make the first sentence true we have to assign
true to Bwhich makes all the Bs true.
9
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
Since B is true, ? B must be falseso yet another
truth value is forced
10
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
F
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 Cs have to be
false
11
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
F
F
F
Now we can complete the truth value
assignmentand theres only one way to do it by
assigning false to C ? A, since both of its parts
are false.
12
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
F
F
F
But this isnt 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 theres no way to
avoid this since all truth values were forced!
13
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
F
F
F
This shows that theres no truth value assignment
that makes all sentences true Therefore that this
set of sentences is inconsistent.
14
Short-Cut Truth Tables Consistency
A ? B / B ? (C ? A) / C ? ? B / ? A
T
T
T
T
T
T
F
F
F
T
F
T
T
T
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!
15
Short-Cut Truth Tables Validity

• An argument is valid if there is no truth value
  assignment that makes all its premises true and
  its conclusion false.
• To test for validity we write the argument on a
  single line with slashes between the premises and
  a double slash between the last premise and the
  conclusion
• We assign true to each of the premises by writing
  T under its main connective, and false to the
  conclusion by writing F under its main
  connective
• And attempt to construct a truth value assignment
  that gets that result
• If thats possible, the argument is invalid
• If its not possible, the argument is valid

16
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
We assign true to each of the premises by writing
T under its main connective and assign false to
the conclusion by writing F under its main
connective. Were seeing if we can show
invalidity.
17
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
T
F
Making the conclusion, C ? A, false forces C to
be true and A to be false since thats the only
case in which a conditional is false.
18
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
T
F
T
F
This forces truth values on all the other Cs and
As all the Cs get true and and the As get
false
19
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
T
F
T
F
F
T
F
There are more forced truth values since ? B is
true, B must be false, so we assign F to all
the Bs And now that we know A is false, ? A must
be true.
20
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
T
F
T
F
F
T
F
T
Now we can complete the table by filling in the
truth value for the first premise. So the first
premise is a true conditional with a true
antecedent and true consequentand thats ok. The
other sentences are ok too.
21
Short-Cut Truth Tables Validity
? A ? (B ? C) / ? B // C ? A
T
F
T
T
F
T
F
F
T
F
T
Since everythings ok, this is a possible truth
value assignment Since this truth value
assignment makes all the premises true and the
conclusion false the argument is shown to be
invalid.
22
Short-Cut Truth Tables Summary

• Short-cut truth tables are a method for
  constructing the rows of a regular truth table
  that matter
• We assign truth values to whole sentences and
  work backward until weve assigned truth values
  to all their parts
• But there is a problem if truth values arent
  forced we may have to do moreand moreand more
  rows (see Hurley 6.5!) so this method only works
  neatly on rigged problems!
• Thats why we prefer the truth tree methoda
  fancy version of the short-cut truth table method
  that works better.

23
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 weve 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
  trueso, e.g.
• p ? q is made true by either ps being true or
  qs being true
• p q is made true by both p and q being true

24
Why?

• p ? q is made true by either ps being true or
  qs 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.

25
Why?

• p q is made true by both pp and q being true
• We can construct tree rules from the
  characteristic truth tables for the connectives
  in this way!

26
Conjunction and Disjunction

• p v q
• p q

• p qpq

Truth
Truth
Truth

• To make p v q true, all we need is truth flowing
  through one of its parts
• So we represent disjunction by a branching rule

• To make p q true, Truth has to flow through
  both p and q
• So we represent conjunction by a non-branching
  rule

27
Tree Rules
DN ? ? p p

• The rule for Double Negation is rewrite, erasing
  two ?s
• Sentences that are basically ORs are represented
  as branching structures
• Sentences that are basically ANDs are
  represented by non-branching structures.
• We understand conditionals and biconditionals as
  basically ORs and ANDs

AND p q p q
28
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

29
Truth Tables for the Connectives
p ? p
T F
F T
p q p q p ? q p ? q p ? q
T T T T T 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 its logically equivalent to ? p v q
• You can prove this by testing the two sentences
  for equivalence!

30
Truth Tables for the Connectives
p ? p
T F
F T
p q p q p ? q p ? q p ? q
T T T T T 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 its equivalent to (p q) v (? p ? q)
• Note were helping ourselves to the idea that
  saying p is false is the same thing as saying ?
  pwhich is ok given the truth table for ?

31
Conditional and Biconditional

• p ? q
• ? p q

• p ? qp ? pq ? q

Truth
Truth
Truth
Truth

• p ? q says either p q or ? p ? q so its a
  branching rule with conjunctions on both branches
• Truth has to either flow through both p and q or
  through both ? p and ? q

• To make p ? q true, Truth has to flow through
  either ? p or q
• ? p says p is false so this says what makes p ?
  q is p being false or q being true

32
Negation of Conditional and Biconditional

• ? (p ? q)p? q

• ? (p ? q) p ? p? q q

Truth
Truth
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

33
Negation of Disjunction and Conjunction

• ? (p v q)? p? q

• ? (p q)
• ? p ? q

Truth
Truth
Truth

• ? (p v q) is equivalent to ? p ? q by
  DeMorgans Law
• So we represent ? (p ? q) by this non-branching
  rule

• ? (p q) is equivalent to ? p v ? q by
  DeMorgans Law
• So we represent ? (p q) by this branching rule

34
Double Negation

• ? ? pp

Truth

• The double negation rule is obvious!
• p is equivalent to ? ? p so, a fortiori, p makes
  ? ? p true.

35
Truth trees are upside down
To represent the truth value assignment that
makes a sentence true we want to show truth
flowing up the treelike sap from the roots
Except in this case truth flows upward from the
branches!
T
T
T
36
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 cant be
  decomposed any further, that is
• Sentence letters and negations of
  sentenceletters
• Then the tree is complete.

37
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
• Were 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

38
Growing a Truth Tree to Test Consistency
v
P Q

• Apply tree rules to each sentence to which they
  apply, checking sentences when theyve had rules
  applied to them
• We start with non-branching rules to keep the
  tree from getting too big.

39
Growing a Truth Tree to Test Consistency
v
v
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.

40
Growing a Truth Tree to Test Consistency
v
v
P Q
P
Q

• A branch 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.

41
Growing a Truth Tree to Test Consistency
v
v
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.

42
Growing a Truth Tree to Test Consistency
v
v
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

43
Testing Sets of Sentences for Consistency
P ? Q / ? P Q / Q
T
F
T
T
T
T
F
T
T
F
F
F
F
T
F
F
P is FALSEQ is TRUE
F
T
T
T
F
T
T
T
F
T
F
F
F
F
F
T
Consistent or inconsistent? Consistent
We constructed this row of the truth table on the
truth tree without wasting time doing the other
rows that didnt matter!
44
But what if things were different?
v P ? Q v P Q
P Q
P
Q
X

• The left branch doesnt 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

45
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

46
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?

47
But what if things were different?
InitialSentences
v
P ? Q
P Q
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.

48
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.

49
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 its 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?

50
Growing a Truth Tree to Test Validity
(P ? Q) ? R
negation of the conclusion
? 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.

51
Growing a Truth Tree to Test Validity
(P ? Q) ? R
negation of the conclusion
? 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!

52
How does this show validity or invalidity?
P1 / P2 / . . . Pn // C
T T T T
When we say that the premises the negation of
the conclusion are consistent were saying that
theres a truth value assignment (row of truth
table) in which all these sentences are true.
Please run this by me again
53
How does this show validity or invalidity?
P1 / P2 / . . . Pn // C
T T T T F
If theres 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!
Please run this by me again
54
How does this show validity or invalidity?
P1 / P2 / . . . Pn / C
T T T T
Inconsistent theres no row like this
Now suppose that the premises the negation of
the conclusion are inconsistent. This means that
theres no row in which the premises and the
negation of the conclusion are all true.
Please run this by me again
55
How does this show validity or invalidity?
P1 / P2 / . . . Pn // C
T T T T F
Valid theres no row like this
So theres 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
56
Summing Up Testing for Validity

• Using the tree method, we test for validity by
  testing the initial sentencespremises 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 lets try it!

57
Growing a Truth Tree to Test Validity
(P ? Q) ? R
? R
negation of the conclusion
? ? P

• Were going to test these initial sentences for
  consistency.
• If the tree closes, theyre inconsistent, so the
  argument is valid.
• If the tree is open, theyre consistent, so the
  argument is invalid.

58
Growing a Truth Tree to Test Validity
(P ? Q) ? R
We apply the double negation rule to this
sentence, check it, and write the result at the
bottom of the tree
? R
v ? ? P
P
59
Growing a Truth Tree to Test Validity
v(P ? Q) ? R
We apply the rule for conditional to this
sentence, check it, and write the result at the
bottom of the tree
? R
v ? ? 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 on a branch
if it occurs on a line by itselfnot just as part
of a longer sentence.
60
Growing a Truth Tree to Test Validity
v(P ? Q) ? R
? R
v ? ? P
P
? (P ? Q)
R
X
Weve 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.
61
Growing a Truth Tree to Test Validity
v(P ? Q) ? R
? R
v ? ? P
P
Now we apply the negation of a disjunction rule
to this sentence
v? (P ? Q)
R
X

• P
• Q

The tree is now finished growing because each
sentence to which a rule could be applied has
been checkedshowing that the appropriate rule
has been applied to it. Is there a problem?
62
Growing a Truth Tree to Test Validity
v(P ? Q) ? R
? R
v ? ? P
P
v? (P ? Q)
R
X

• P
• Q

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 closedso the argument is
valid!
63
What would an invalid argument look like?
v(P Q) ? R
? R
v ? ? P
P
v (P Q)
R
X
P Q
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
64
Whats the conclusion?
v(P Q) ? R
? R
v ? ? P
P
v (P Q)
R
X
P Q

• Reading from top down, we look for the last
  sentence that wasnt the result of applying a
  tree rule.
• That sentence is the negation of the conclusion,
  viz. ? ? P
• So the conclusion of this argument is ? P

65
Are the initial sentences consistent or
inconsistent?
v(P Q) ? R
? R
v ? ? P
P
v (P Q)
R
X
P Q

• 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.

66
What truth value assignment makes all initial
sentences true?
v(P Q) ? R
? R
v ? ? P
P
v (P Q)
R
X
P Q

• 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

67
What truth value assignment makes all initial
sentences true?
v(P Q) ? R
? R
v ? ? P
P
v (P Q)
R
X
P Q

• So, the truth value assignment that makes all
  initial sentences true is
• P TRUE Q TRUE R FALSE

68
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
69
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
70
So weve saved ourselveslots of work
And can go home and relax!