EXAMPLE 2.2.10

Select the statement that is a valid conclusion from the following premises,

if a valid conclusion is warranted.

If I win the Lotto, then I won't need a job.

If I have lots of bills, then I will need a job.

A. If I have lots of bills, then I didn't win the Lotto.

B. If I didn't win then Lotto, then I have lots of bills.

C. If I don't need a job, then I won the Lotto.

D. A valid conclusion is not warranted.

SOLUTION

Let p be the statement "I win the Lotto."

Let q be the statement "I won't need a job."

Let r be the statement "I have lots of bills."

The premises have this symbolic scheme:

1. p arrow q

2. r arrow not q.

In order to use Transitive Reasoning the antecedent of the second premise must agree with the consequent of the first premise. This can be achieved if we replace the second premise with its equivalent contrapositive:

1. p arrow q

2. q arrow not r.

Now we can form a valid argument:

p arrow q

q arrow not r.

Therefore, p arrow not r.

In words the valid conclusion is "If I win the Lotto, then I don't have lots of bills." This, however, is not among the listed choices. Note that our conclusion is equivalent to its contrapositive: "If I have lots of bills, then I didn't win the Lotto."

The correct choice is A.