1. Hurley … Chapter 6.5 Indirect Truth Tables
Propositional Logic
Hurley … Chapter 6.5
Indirect Truth Tables

2. Indirect Truth Table Method
Rather than creating a full truth table and looking for a line of the table on which the premises are true and the conclusion false (proof that the argument is invalid), we can simply assign truth to the premises and falsehood to the conclusion and fill in the required values. If it turns out that we cannot fill in the required values without violating one of the basic truth tables (a truth table for triple bar, horseshoe, dot, wedge, or tilde), the argument is valid (meaning, it is not possible for the premises to be true while the conclusion is false). Clever!

3. Consider a Classic, Invalid Argument…
P  Q Q P Rewrite the argument in a single line: P  Q / Q // P

4. Invalid Argument Example ….
P  Q / Q // P F T T T F
Since there is no conflict with the truth table for the conditional operator, this argument form has no trouble instantiating true premises and a false conclusion … in other words, it’s a bad argument form … an invalid argument form. It goes by this famous name: affirming the consequent.
On the next slide, consider an equally famous valid form…

5. Valid Argument Example ….
P  Q / P // Q T T F T F
Since a conflict occurs when assigning truth to the premises and falsehood to the conclusion of this argument form (this is a famous valid argument form called “Modus Ponens”), the argument is a good argument, a valid argument form. There is no way for the premises to be true while the conclusion is false.

6. Another Valid Argument Example ….
P  Q / ~Q // ~P T T F T F F T
Again, a conflict occurs when trying to have true premises with a false conclusion. This famous valid argument form is called “Modus Tollens.”

7. Yet Another Valid Argument Example …
P  Q / Q  R // P  R T T F F T F T F F
Again, a conflict occurs when trying to have true premises with a false conclusion. This famous valid argument form is called “Hypothetical Syllogism.”

8. Practicing Indirect Truth Tables
The greatest number of lines an indirect truth table can have, is 3. That is because, no truth table for any logical operator (dot, wedge, etc.) is true or false in more than 3 ways. For simplicity on your quizzes and final exam, I will only give you arguments that require one line … arguments in which at least one premise can be true in only one way, or in which the conclusion can be false in only one way. You’re welcome! 