1. Let us build the tree for the argument Q>-P | P>Q.
An Invalid Argument
Let us build the tree for the argument Q>-P | P>Q.

2. Negate the Conclusion To check and argument for validity ...
Begin by negating the conclusion.
Q>-P
-(P>Q)
Q>-P | P>Q

3. Apply the (->) Rule
Q>-P
-(P>Q)
Q>-P | P>Q

4. Apply the (->) Rule The (->) Rule: -(A>B) Q>-P
-(P>Q)
Q>-P | P>Q
-(A>B) =
A&-B

5. Apply the (->) Rule The (->) Rule: -(A>B) Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
-(A>B) =
A&-B

6. Apply the (>) Rule
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1

7. Apply the (>) Rule Q>-P Q>-P | P>Q -(P>Q)
A>B
-A B 1
A>B =
-AvB

8. Apply the (>) Rule 2 Q>-P -(P>Q) P -Q Q>-P | P>Q
A>B
-A B 1
-Q P
A>B =
-AvB

9. Check for Validity 2
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
-Q P

10. contains P and -P, which is
Check for Validity 2
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
The right hand branch
contains P and -P, which is
impossible,
so the branch is
closed (*).
-Q P *

11. contains no contradiction. It is open and indicates
Check for Validity 2
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
The left hand branch
contains no contradiction.
It is open and indicates
that the argument has
a counterexample.
-Q P *

12. contains no contradiction. It is open and indicates
Check for Validity 2
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
The left hand branch
contains no contradiction.
It is open and indicates
that the argument has
a counterexample.
-Q P *
P is T
Q is F

13. contains no contradiction. It is open and indicates
Check for Validity 2
Q>-P
-(P>Q) P -Q
Q>-P | P>Q 1
The left hand branch
contains no contradiction.
It is open and indicates
that the argument has
a counterexample.
-Q P * P T
F T F
QT T F F
Q>-P | P>Q F
T T T T
T F T
P is T
Q is F
Counterexample

14. Check for Validity Counterexample 2 Q>-P -(P>Q) P -Q1
The left hand branch
contains no contradiction.
It is open and indicates
that the argument has
a counterexample.
-Q P * P T
F T F
QT T F F
Q>-P | P>Q F
T T T T
T F T
P is T
Q is F
Counterexample
So the Argument is INVALID.

15. Check for Validity 2 Q>-P -(P>Q) P -Q Q>-P | P>Q 1
The left hand branch
contains no contradiction.
It is open and indicates
that the argument has
a counterexample.
-Q P *
P is T
Q is F
To calculate the counterexample
for an open branch. Make single
letters (like P) = T and negated letters
(like Q) = F.

16. For more click here Checking for Validity THE BOTTOM LINE
If the tree is OPEN there is a counterexample.
So the argument is INVALID.
If the tree is CLOSED (all branches are closed),
there is no counterexample.
So the argument is VALID.
For more click here