A successful counterexample must have the same argument form as the argument to which it is put forward as a counterexample. It is impossible for a valid argument form to allow a substitution instance with all true premises and a false conclusion. Therefore, if the proposed counterexample has the same argument form, and if it has all true premises and a false conclusion, then the counterexample is sufficient to show that the original argument's form is invalid. Consider the following argument and argument form: Given Argument If the Bernoulli Principle holds, then airplanes can fly. And if Earth has an atmosphere, then airplanes can fly. Therefore, if the Bernoulli Principle holds, then Earth has an atmosphere. Given Argument's Form P1: If B, then A. P2: If E, then A If B, then E. C: Below are three proposed counterexamples that purport to prove that the given argument's form is invalid. Determine the truth values of each sentence in the counterexample. Put a "T" or "P" in the spaces provided to indicate the truth value of each statement. Then, based on these truth values, indicate whether each proposed counterexample succeeds in proving that the given argument and its argument form are invalid. Truth Value? P1: Proposed Counterexample A If tigers are felines, then tigers are animals. If tigers are mammals, then tigers are animals. If tigers are felines, then tigers are mammals. P2: C: Based on these truth values, is Proposed Counterexample A a successful counterexample to the given argument form? O Yes O No Truth Value? P1: Proposed Counterexample B If this fruit is a "Red Delicious," then this fruit is a pancake. If this fruit is an apple, then this fruit is a pancake. If this fruit is a "red Delicious," then this fruit is an apple. P2: C: Based on these truth values, is proposed Counterexample B a successful counterexample to the given argument form? O Yes O No Truth Value? P1: Proposed Counterexample C If William McKinley was assassinated, then William McKinley died. If William McKinley committed suicide, then William McKinley died. If William McKinley was assassinated, then William McKinley committed suicide. P2: C: Based on these truth values, is Proposed Counterexample C a successful counterexample to the given argument form? O Yes No A successful counterexample must have the same argument form as the argument to which it is put forward as a counterexample. It is impossible for a valid argument form to allow a substitution instance with all true premises and a false conclusion. Therefore, if the proposed counterexample has the same argument form, and if it has all true premises and a false conclusion, then the counterexample is sufficient to show that the original argument's form is invalid. Consider the following argument and argument form: Given Argument If the Bernoulli Principle holds, then airplanes can fly. And if Earth has an atmosphere, then airplanes can fly. Therefore, if the Bernoulli Principle holds, then Earth has an atmosphere. Given Argument's Form P1: If B, then A. P2: If E, then A If B, then E. C: Below are three proposed counterexamples that purport to prove that the given argument's form is invalid. Determine the truth values of each sentence in the counterexample. Put a "T" or "P" in the spaces provided to indicate the truth value of each statement. Then, based on these truth values, indicate whether each proposed counterexample succeeds in proving that the given argument and its argument form are invalid. Truth Value? P1: Proposed Counterexample A If tigers are felines, then tigers are animals. If tigers are mammals, then tigers are animals. If tigers are felines, then tigers are mammals. P2: C: Based on these truth values, is Proposed Counterexample A a successful counterexample to the given argument form? O Yes O No Truth Value? P1: Proposed Counterexample B If this fruit is a "Red Delicious," then this fruit is a pancake. If this fruit is an apple, then this fruit is a pancake. If this fruit is a "red Delicious," then this fruit is an apple. P2: C: Based on these truth values, is proposed Counterexample B a successful counterexample to the given argument form? O Yes O No Truth Value? P1: Proposed Counterexample C If William McKinley was assassinated, then William McKinley died. If William McKinley committed suicide, then William McKinley died. If William McKinley was assassinated, then William McKinley committed suicide. P2: C: Based on these truth values, is Proposed Counterexample C a successful counterexample to the given argument form? O Yes No
