EXAMPLE 2.2.12

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

All poodles love noodles. All pooches love noodles.

A. All pooches are poodles.

B. All poodles are pooches.

C. No poodles are pooches.

D. None of these is warranted.

SOLUTION

We should recognize instantly that the correct choice is D, because this is a false chain. In case we don't see that however, first note that the premises can be rewritten:

If one is a poodle, then one loves noodles.

If one is a pooch, then one loves noodles.

Let p be "One is a poodle."

Let q be "One loves noodles."

Let r be "One is a pooch."

The premise scheme has this symbolic form:

Again, we should recognize this as the premise scheme for a False Chain, and be done with it. The first premise already begins with a variable (p) that appears only one time, so there is no reason to rearrange the order in which the premises are listed. The antecedent of the second premise doesn't agree with the consequent of the first premise, and we can't do anything to make it do so; if we replaced the second premise with its contrapositive, then the second premise would beginning with ~q rather than q; furthermore, we could replace any statement with its converse or inverse, because those things aren't equivalent to the original statement. Thus, nothing we can do will yield a non-trivial valid conclusion, so the correct choice is D.