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   Q / R F T F F F F T T 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:
P  (Q  P)
 Contradiction, so the
statement is a tautology! T F T F F
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:
R  (Q  P)
(Q  R)  P
R  P Q 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Q)  P P P Q Q
PQ 
PQ 
(PQ)  P P Q 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) 
All branches close
 the original
statement cannot be
false  tautology!
(QR)  R Q P Q R
(PQ)  R 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 don’t branch
If you have to branch, choose those that you know will quickly lead to a closed branch.
If you don’t 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 F T F T T
(Cube(a)  Cube(b))  (Small(a)  Small(b))
(Cube(a)  Small(a))  (Cube(b)  Small(b)) T F F F F F T
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 can’t 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 UD’s.
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 don’t 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 can’t 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 c’s with d’s 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 it’s 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)
Cube(a)
Small(a) × ×
All branches close,
so it’s valid

29. Modifications on Tree Method
Since universal statements never get checked off, you don’t 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 (P  Q) P  Q (P  Q) P P P P Q Q   QDN
Alpha
Branch
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 Q P P Q P Q Q
Beta
Eta

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