SOLUTION EXAMPLE 2.2.9

We rely on the fact that statements of the form "All A are B" can be written as conditional statements having the form "If A, then B." Thus, the premises given above can be rephrased as:

If one gets many tickets, then one is uninsurable.

If one is a careless driver, then one gets many tickets.

If one is uninsurable, then one has a bad credit rating.

 

Let p be the statement "One gets many tickets."

Let q be the statement "One is uninsurable."

Let r be the statement "One is a careless driver."

Let s be the statement "One has a bad credit rating."

 

The premise arrangement has this form:

p arrow q. r arrow p. q arrow s

Although this looks similar to the premise arrangement for transitive reasoning, it doesn't quite conform to that pattern. For example, in order to link the "if...then" statements to form a valid conclusion, the antecedent of the second premise must be exactly the same as the consequent of the first premise, and the antecedent of the third premise must be exactly the same as the consequent of the second premise.

We may be able to form a valid conclusion if we rearrange the order in which the three premises are listed:

r arrow p. p arrow q. q arrow s.

Notice that we haven't changed any of the premises; we've just arranged them in a different order, and after performing this rearrangement, we are able to link the premises to form a valid conclusion:

r arrow p. p arrow q. q arrow s. Therefore, r arrow s.

In words, the valid conclusion is "If one is a careless driver, then one has a bad credit rating." In natrual language, this is "All careless drivers have bad credit ratings," which is choice A.

Note: another valid conclusion would be the contrapositve of "If one is a careless driver, then one has a bad credit rating," which is "If one doesn't have a bad credit rating, then one isn't a careless driver." In natural language, this corresponds to the awkward-sounding "All people who don't have bad credit ratings are people who aren't careless drivers."