EXAMPLE 2.2.14

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

People who dislike cats are degenerates. All pirates own parrots. People who like cats never own parrots.

A. If you are a pirate, then you are a degenerate.

B. All degenerates own parrots.

C. All cats lick parrots.

D. None of these is warranted.

SOLUTION

Let "cat" represent "____ likes cats."

Let "degenerate" represent "____ is a degenerate."

Let "pirate" represent "_____ is a pirate."

Let "parrot" represent "______ owns a parrot."

The premise scheme as this form:

1. not cat arrow degenerate

2. pirate arrow parrot.

3. cat arrow not parrot.

We want the first premise to begin with a symbol that appears only once. Since "pirate" appears only once, we can take the second premise and make it first.

2. pirate arrow parrot.

To continue the chain of reasoning, we need to find another premise whose antecedent is "parrot." We can use the contrapositive of the original third premise:

2. pirate arrow parrot.

3. parrot arrow not cat.

To finish the chain of reasoning we need another premise whose antecedent is "not cat." We can use the original first premise:

2. pirate arrow parrot.

3. parrot arrow not cat.

1. not cat arrow degenerate

Now we can form a valid argument:

pirate arrow parrot.

parrot arrow not cat.

not cat arrow degenerate.

Therefore, pirate arrow degnerate.

In words, the valid conclusion is "If one is a pirate, then one is a degenerate." This is the same as "All pirates are degenerates." It is also the same as "If one isn't a degenerate, then one isn't a pirate." We see that choice A is correct.