Review

1
Review

• We can use truth tables to evaluate the validity
  of arguments.
• Example
• P1 B?(C v A)
• P2 B v C
• C A
• B?(C v A) / B v C // A

2
Like This

• B?(C v A) / B v C / A
• T T T T T T T T T
• T T T T F T T T F
• T T F T T T T F T
• T F F F F T T F F
• F T T T T F T T T
• F T T T F F T T F
• F T F T T F F F T
• F T F F F F F F F
• Row 2 proves that the argument is invalid (so
  does row 6)

3
Indirect Truth Tables

• Consider an argument symbolized thusly
• P1. (B D)?C
• P2. D v B
• C1. C?B
• If we were to use a full truth table to evaluate
  its validity, we would need 8 rows

4
Shorter Method

• There is a shorter way to test validity
• Start by assuming the argument is invalid
• Represent this by putting Ts under the main
  logical operator of the premises and an F under
  the main logical operator of the conclusion

5
Like This

• (B D)?C / D v B // C?B
• T T F
• Next, determine the truth values of the other
  statements, given the truth values of the
  premises and conclusion
• (B D)?C / D v B // C?B
• F F T TT F T T F TF F F

6
Notice

• We have a consistent assignment of truth values
  that makes the premises true and the conclusion
  false
• Therefore, the argument is invalid, as assumed

7
Another Example

• Consider this argument
• P1 C?(B?D)
• P2 (B?D)
• C1 C
• C?(B?D) / (B?D) // C
• T T F

8
A Contradiction!

• C?(B?D) / (B?D) // C
• T T TF F T T F F FT
• Notice that we cannot give a consistent
  assignment of truth values. The first premise,
  if we assume it to be true, is false. Because we
  get a contradiction, the argument is valid.

9
A Complication

• Sometimes there is more than one way for the
  premises to be true and the conclusion false
• Example
• A v B / B?A / A v B // B A
• There are three ways the conclusion can be false!

10
See?

• A v B / B?A / A v B // B A
• T T T TF F F
• T T T FT F T
• T T T FT F F
• To test the validity of the argument, we need to
  see if we can find a consistent set of truth
  value assignments on at least one of the three
  rows

11
Start with the First Row

• A v B / B?A / A v B // B A
• F T F TF F F
• FT F T
• FT F F
• Notice that we have a contradiction in the first
  premise we assumed it is true, but that
  assumption is inconsistent with the truth values
  of A and B on that row

12
Can we Stop?

• No! That there is a contradiction on the first
  row does not prove the argument is valid, for
  there may be a consistent assignment of truth
  values on the other two rows. Remember, we are
  looking for at least one consistent assignment of
  truth values to prove the argument invalid

13
Go To Second Row

• A v B / B?A / A v B // B A
• F T F TF F F
• FT T FT FT F T
• FT F F
• We have a contradiction in the third premise on
  the second row. So me must look at the third row

14
Go to Third Row

• A v B / B?A / A v B // B A
• F T F TF F F
• FT T FT FT F T
• T T F FT F F
• We have a contradiction in the second premise.
  Because there is a contradiction on every row, we
  know the argument is valid (there are no rows
  with a consistent assignment of truth values that
  makes the premises true and the conclusion false)

15
A Good Procedure to Follow

• Determine the number of ways the conclusion can
  be false and represent them with a truth table
  under the conclusion
• Start with the first row and check if a
  contradiction is produced. If so, go to the
  second row. If not, stop the argument is
  invalid
• Repeat the second step until there are no more
  rows. If a contradiction is produced on every
  row, the argument is valid.

16
Testing Consistency with Indirect Truth Tables

• We can determine whether a set of statements is
  consistent by first assuming that all the
  statements are true
• Example
• A?B/ B?(C?A)/ B?C/ A
• T T T T
• Next, compute the truth values of the other
  statements, based on this assumption

17
Like This

• A?B/ B?(C?A)/ B?C/ A
• TF TT T T TFF F T T TF TF
• Notice that there is a contradiction in the
  second statement.
• Also notice that we need only one row, because
  there is only one way for the last statement to
  be true (and thus the assumption that all of them
  are true).
• Because we derived a contradiction, the
  statements cannot all be truethus, the
  statements are inconsistent.

18
A Complication

• Sometimes there is more than one way for a set of
  statements to come out true
• Example
• A v (B C)/ C v A/ B?A
• T T T
• In this case, there are three ways each of the
  statements can be true (to see this, recall the
  truth tables for disjunction and conditional)

19
What to do?

• Select one of the statements and represent the
  three ways it can be true. Lets select the last
  statement
• A v (B C)/ C v A/ B?A
• T T FT T F
• FT T T
• TF T T

20
Look at the First Row

• A v (B C)/ C v A/ B?A
• TF T T T TF FT T F
• FT T T
• TF T T
• Notice that we do not have to assign a truth
  value to C in the first row, because the first
  two statements will come out true regardless.
  Since there is a consistent assignment of truth
  values on at least one row, the statements are
  consistent.

21
The Procedure for Testing Consistency

• Look at the statements and pick the one that has
  the fewest number of ways it can come out true.
• Represent those ways with a truth table under the
  statement
• Start with the first row and determine the truth
  values of the other statements.
• If no contradiction is produced, stop. The
  statements are consistent. If not, go to the
  second row and repeat the previous step
• If a contradiction is produced on every row, the
  statements are inconsistent.

22
Common Valid and Invalid Argument Forms

• An argument form is valid just in case every
  substitution instance of that form is such that
  it is impossible for the premises to be true and
  the conclusion false
• In the language of truth tables, there are no
  rows where the premises are true and the
  conclusion false

23
Continued

• How shall we define an invalid argument form?
• Should we say that an argument form is invalid
  just in case every substitution instance of that
  form is invalid (e.g. there is at least one row
  on the truth table where the premises are true
  and the conclusion false?)

24
No

• Heres why consider this argument form
• p?q
• q
• p
• It is in an invalid form, but there are
  substitution instances of the form that are
  valid!

25
Huh?

• See? Consider this substitution instance
• (A v A)?B
• B
• (A v A)
• It is impossible for the premises to be true and
  the conclusion false, because the conclusion is a
  tautology (it is true on every row of the truth
  table)

26
So

• Lets define an invalid argument form this way
• An argument form is invalid just in case there is
  at least one substitution instance of that form
  that is invalid.
• In the language of truth tables, there is at
  least one substitution instance where at least
  one row has true premises and a false conclusion