Truth Trees

1
Truth Trees

• Intermediate Logic

2
Truth Table Method

• The truth table method systematically exhausts
  all possible truth value combinations of the
  statements involved.
• In the truth-table we look for a row that
  reflects a certain possibility, and that will
  tell us the answer to whatever question we had.
• E.g. If we want to know whether or not a certain
  statement is a tautology, we are interested in
  the possibility of that statement to be false
  the statement is a tautology iff it is not
  possible for that statement to be false
• For an argument Is it possible to have all true
  premises and a false conclusion? If so, invalid.
  If not, valid.

3
Drawback and Room for Solution

• A drawback of the truth table method is that the
  number of rows grows exponentially.
• Fortunately, there is room for a solution to this
  problem. Since all we are interested in, is the
  existence of a specific combination of truth
  values of the statements involved, all we need to
  find is one example of such a case. Once we have
  found such a case, there is no need to exhaust
  all other cases.

4
Simple Solution Stopping Early

• One solution to the problem of big truth tables
  is therefore to simply stop once you have found a
  row that represents the combination of truth
  values you are interested in.
• Thus, rather than working out a truth table
  column by column, you may want to do it row by
  row, so that you can stop as soon as you have
  found a row of the kind you are looking for.
• A big drawback of this approach is that if no row
  of the kind you are looking for exists, then you
  have to complete the whole truth table after all.

5
A More Focused Search

• Consider the following argument

P ? (Q ? R)
R ? ?Q
R

• We are interested in whether all premises can be
  true and the conclusion false
• In order for the conclusion to be false, R must
  be false.
• In order for the second premise to be true while
  R is false, Q must be false.
• In order for the first premise to be true while Q
  and R are both false, P must be false.

6
The Short Truth Table Method

• The Short Truth Table Method assigns truth values
  to the involved atomic and complex statements in
  order to try and obtain a certain combination of
  truth values

/?
P ? (Q ? R)
R
R ? ? Q
T
F
T
F
F
T
F
F
F
F

• The Short Truth Table Method thus tries to
  generate one row of the truth table that has the
  combination of truth values you are interested in.

7
Short Truth Table Method and Indirect Proof

• As you assign truth values to certain statements,
  the truth values of other statements can be
  forced.
• If you are forced to make a statement both true
  and false, then you know that the combination of
  truth values you are looking for does not exist

? Contradiction, so the statement is a tautology!
P ? (Q ? P)
F
F
T
F
T

• The short truth table method is therefore often a
  kind of proof by contradiction or indirect proof

8
Drawback of the Short Truth Table Method

• A drawback of the short truth table method is
  that you are not always forced to assign any
  further truth values

Q
R ? P
R ? (Q ? ?P)
(Q ? ?R) ? ?P
T
T
T
T
T
T

• At this point, you can choose to assign certain
  truth values, but if your choice does not lead to
  the row you are looking for, then you need to try
  a different option, and the short truth table
  method has no tools to do go through all of your
  options in a systematic way.

9
Truth Trees

• The obvious solution to the drawback of the short
  truth table method is to incorporate tools to
  systematically keep track of multiple options.
• One method that does so is the truth tree method
• The truth tree method tries to systematically
  derive a contradiction from the assumption that a
  certain set of statements is true.
• Like the short table method, it infers which
  other statements are forced to be true under this
  assumption.
• When nothing is forced, then the tree branches
  into the possible options.

10
Decomposition Rules for Truth Trees
P?Q
??P
?
?
?(P?Q)
?
P
P
?P
?Q
Q
P?Q
P?Q
?(P?Q)
?
?
?
P
Q
?P
P
Q
?Q
?(P?Q)
?
P?Q
?(P?Q)
?
?
P
P
?P
?P
?P
Q
Q
?Q
?Q
?Q
11
Truth Tree Example
?(((P?Q)?R) ? (P?(?Q?R)))
?
?((P?Q)?R)
(P?Q)?R
?
?
?(P?(?Q?R))
P?(?Q?R)
?
?
P
P?Q
?
All branches close ? the original statement
cannot be false ? tautology!
?(?Q?R)
?R
?
Q
P
Q
?R
?Q ? R
?
?P
?(P?Q)
R
?


?P
?Q
?Q
R




12
Further Rules for Truth Trees

• A decomposable statement is any statement that is
  not a literal.
• A sentence belongs to every branch below it.
• You can close a branch if an atomic statement and
  its negation both belong to that branch.
• Every undecomposed decomposable statements needs
  to be decomposed in every open branch it belongs
  to.
• A branch is finished if there are no undecomposed
  decomposable statements that belong to it.

13
How to Use Truth Trees

• At the root of the tree, write down all
  statements that you try and make true according
  to the combination of truth values you are
  interested in.
• Decompose according to the rules until you have a
  finished open branch or until all branches close.
• If there is a finished open branch, then that
  means that it is possible for all statements at
  the root of the tree to be true.
• If all branches close, then that means that it is
  not possible for all statements at the root of
  the tree to be true.
• It is up to you to draw the appropriate
  conclusion from this.

14
Example Testing for Validity

• To use a tree to test for validity
• 1. Write down at the root of the tree all
  premises and the negation of the conclusion
• 2. Work through the tree until you find an open
  and completed branch or all branches are closed
• 3a. If you found an open and completed branch,
  then that means that it is possible for all
  statements in the root of the tree to be true,
  which in turn means that it is possible for all
  premises to be true while the conclusion is
  false. Hence, the argument is invalid.
• 3b. If all branches closed, the opposite is true,
  i.e. the argument is valid.

15
How to Avoid Bushy Trees

• Since at any point there can be multiple
  undecomposed decomposable statements, the tree is
  going to look different based on which statement
  you choose to decompose.
• Since more branches means more work, you want to
  avoid branching as much as possible. So, as a
  heuristic
• Choose statements that dont branch
• If you have to branch, choose those that you know
  will quickly lead to a closed branch.
• If you dont know which one leads quickly to a
  closed branch, choose a large statement (why?)

16
Truth Trees for Predicate Logic

• Intermediate Logic

17
Running Examples
Valid Argument (13.24) ?x (Cube(x) ?
Small(x)) ??x Cube(x) ? ?x Small(x)
Invalid Argument (13.25) ?x Cube(x) ? ?x
Small(x) ??x (Cube(x) ? Small(x))
18
Truth-Functional Expansions

• Suppose that our Universe of Discourse (UD)
  contains only the objects a and b.
• Given this UD, the claim ?x Cube(x) is true iff
  Cube(a) ? Cube(b) is true.
• Similarly, the claim ?x Cube(x) is true iff
  Cube(a) ? Cube(b) is true.
• The truth-functional interpretation of the FO
  statements given a fixed UD is called the
  truth-functional expansion of the original FO
  statement with regard to that UD.

19
Truth-Functional Expansions and Proving FO
Invalidity

• Truth-Functional expansions can be used to prove
  FO invalidity. Example (13.25)

?x Cube(x) ? ?x Small(x) ??x (Cube(x) ? Small(x))
UD a,b
T
T
T
T
T
F
F
(Cube(a) ? Cube(b)) ? (Small(a) ? Small(b))
?(Cube(a) ? Small(a)) ? (Cube(b) ? Small(b))
T
T
F
F
F
F
F
This shows that there is a world in which the
premise is true and the conclusion false. Hence,
the original argument is FO invalid.
20
Truth-Functional Expansions and Proving FO
Validity

• If the truth-functional expansion of an FO
  argument in some UD is truth-functionally
  invalid, then the original argument is FO
  invalid, but if it is truth-functionally valid,
  then that does not mean that the original
  argument is FO valid.
• For example, with UD a, the expansion of the
  argument would be truth-functionally valid. In
  general, it is always possible that adding one
  more object to the UD makes the expansion
  invalid.
• Thus, we cant prove validity using the expansion
  method, as we would have to show the expansion to
  be valid in every possible UD, and there are
  infinitely many UDs.
• The expansion method is therefore only good for
  proving invalidity. Indeed, it searches for
  countermodels.

21
The Expansion Method as a Systematic Procedure

• Still, the expansion method can be made into a
  systematic procedure to test for FO invalidity
• Step 1 Expand FO argument (which can be done
  systematically) in UD a.
• Step 2 Use some systematic procedure (e.g.
  truth-table method or truth-tree method) to test
  whether the expansion is TF invalid. If it is TF
  invalid, then stop the FO argument is FO
  invalid. Otherwise, expand FO argument in UD
  a,b, and repeat step 2.

22
A More Focused Search

• A further drawback of the expansion method is
  that the search for a counterexample is very
  inefficient.
• A focused search for a counterexample is more
  efficient
• (13.25) I want there to be at least one cube, and
  at least one small object, but no small cubes.
  So, if we have a cube, a, then a cannot be small,
  so I need a second object, b, which is small, but
  not a cube. Counterexample, so the argument is
  invalid.

23
Advantage of a Focused Search

• The focused search method is like the indirect
  truth-table method.
• Indeed, like the indirect truth-table method, the
  focused search method can prove validity
• (13.24) I want there to be at least one small
  cube. Let us call this small cube a. How, I dont
  want it to be true that there is at least one
  cube and at least one small object. However, a is
  both a cube and small. Contradiction, so I cant
  generate a counterexample.

24
Truth-Trees for Predicate Logic

• Like the direct method, the focused search method
  needs to be systematized, especially since the
  search often involves making choices.
• Fortunately, the truth-tree method, which
  systematized the indirect truth-table method in
  truth-functional logic, can be extended for
  predicate logic.

25
Truth-Tree Rules for Quantifiers
??x ?(x)
??x ?(x)
?
?
?x ??(x)
?x ??(x)
?x ?(x)
?
?x ?(x)
?(c)
?(c)
with c a new constant in that branch
with c any constant
26
Truth-Tree Rules for Identity
?(c)
a ? a
c d (or d c)

?(d)
(where ?(d) is the result of replacing any
number of cs with ds in ?(c))
27
Truth-Tree Example I
?x Cube(x) ? ?x Small(x) ??x (Cube(x) ? Small(x))
?
?
?x Cube(x)
?
?x Small(x)
?
?x ?(Cube(x) ? Small(x))
Cube(a)
Small(b)
?(Cube(a) ? Small(a))
?
?(Cube(b) ? Small(b))
?
?Cube(a)
?Small(a)

?Cube(b)
?Small(b)
Open branch, so its invalid

28
Truth-Tree Example II
?x (Cube(x) ? Small(x)) ?(?x Cube(x) ? ?x
Small(x))
?
?
Cube(a) ? Small(a)
?
Cube(a)
Small(a)
??x Cube(x)
??x Small(x)
?
?
?x ?Cube(x)
?x ?Small(x)
?Small(a)
?Cube(a)


All branches close, so its valid
29
Modifications on Tree Method

• Since universal statements never get checked off,
  you dont have to instantiate a universal in
  every branch that it belongs to.
• An open branch is finished if every statements in
  that branch that has not been decomposed is
  either a literal or a universal that has been
  instantiated for every constant in that branch.

30
Rules of KE Calculus
P ? Q
??P
?(P ? Q)
?(P ? Q)
P
P
?P
P
Q
?Q
?
??
?Q
DN
Branch
Alpha
P ? Q
P ? Q
P ? Q
?(P ? Q)
P
P
?P
P
Q
Q
Q
?Q
?(P ? Q)
?(P ? Q)
P ? Q
P ? Q
P
?P
?P
?Q
Q
?P
?Q
?Q
Eta
Beta
31
HW 4

• 1. Demonstrate that 8.47 from LPL is invalid
  using
• a. the short truth-table method
• b. the truth tree method
• 2. 4.4.1.c from Bostock
• 3. Use the expansion method to show that (4) from
  3.6.3.b from Bostock is not valid
• 4. 4.4.1.i from Bostock
• 5. Show that KE Calculus can do whatever the
  truth tree method can do