Hurley

1
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

T
T
F
F
T
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, its 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
T
F
T
F
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.
• Youre welcome! ?