EXAMPLE 2.2.16

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

Sylvestor isn't a parakeet. Elephants never squawk. All parakeets squawk. No elephants are tiny.

A. Sylvester is an elephant.

B. Sylvestor isn't tiny.

C. All parakeets are tiny.

D. None of these is warranted.

SOLUTION

Let "Sylvestor" represent "____ is Sylvestor."

Let "parakeet" represent "____ is a parakeet."

Let "elephant" represent "____ is an elephant."

Let "squawk" represent "____ squawks."

Let "tiny" represent "____ is tiny."

The premise scheme has this symbolic form:

In order to use Transitive Reasoning, we want the antecedent of the first premise to be a symbol that appears only once. Since "Sylvestor" appears only once, the current first premise is a suitble starting point.

In order to continue the chain, we must find another premise that begins with "~parakeet." We see that this is impossible, because we cannot replace the original third premise with its (nonequivalent) inverse. Thus, we have reach a dead end; this tells us that the correct choice is D. We cannot form a non-trivial valid conclusion.

What if we have chosen the original fourth premise, in its contrapositive form, for our first premise (this is a legitimate move, since "tiny" appears only once in the premise arrangement)?

In order to continue the chain, we must find another premise that begins with "~elephant." Once again, this cannot be done, so we have hit a dead end, confirming the fact that the correct choice is D.