Proving Invalidity: The Counterexample Method

(Hurley, Section 1.5)

The validity of a deductive argument is determined by its form. Yet, some arguments do not have a clearly identifiable form. Consider:

Geese are migratory waterfowl, so they fly south for the winter.

This argument is missing a premise, viz., Migratory waterfowl fly south for the winter.

Thus, we add the premise:

All Geese are migratory waterfowl

All migratory waterfowl are birds that fly south for the winter

Therefore, all geese are birds that fly south for the winter.

This argument has the form:

All A are B

All B are C

All A are C

Which is valid. The argument about geese migration is a substitution instance of the above form. Any argument that has this valid form is also a valid argument.

Consider this invalid form:

All A are B

All C are B

All A are C

Why is this form invalid?

Note the substitution instance:

All women are featherless bipeds All men are featherless bipeds All women are men

True True False

Note: While every substitution instance of a valid form is itself valid, it is not the case the every substitution instance of an invalid form is invalid. Look at Hurley, page 56, especially the note.

To recap, consider the following:

All adlers are bobkins.

All bobkins are crockers.

Therefore, all adlers are crockers.

This form:

All A are B

All B are C

All A are C

Is a valid form, and, again, any uniform substitution instance of this form results in a valid argument.

The Counterexample Method: A substitution instance having true premises and a false conclusion is called a counterexample. We can use this method in a systematic fashion to demonstrate and argument’s invalidity. To use the method, we first isolate the form of the argument, and then construct a substitution instance having true premises and a false conclusion, which, as we recall, is the very definition of invalidity.

For categorical syllogisms. Consider:

All adlers are bobkins.

All crockers are bobkins.

Therefore, all adlers are crockers.

The form of this argument is the same as the previous one:

All A are B

All C are B

All A are C

This form is invalid, and any uniform substitution instance of that form results in an invalid argument (with the caveat from above!). Notice the following counterexample:

All cats (A) are animals (B).

All dogs (C) are animals (B).

All cats (A) are dogs (C).

This argument has true premises and a false conclusion. Thus, this is proven invalid.

Note: For categorical syllogisms, use dogs, cats, mammals, fish, and animals to substitute for the terms. Also remember that in logic, “some” means “at least one.” So, “some dogs are mammals” is a true statement. We cannot infer from “some dogs are mammals” that some of them are not mammals.

But, the categorical syllogism is but one form of deductive arguments. Consider:

If the government imposes import restrictions, the price of automobiles will rise. Therefore, since the government will not impose import restrictions, it follows that the price of automobiles will not rise.

To represent this form, we need a different approach. We use letters to represent entire statements rather than terms. In this case, we get:

If G, then P.

Not G.

Therefore, not P.

If we give the following substitution instance:

G = Abraham Lincoln committed suicide.

P = Abraham Lincoln is dead.

We get:

If Abraham Lincoln committed suicide, then Abraham Lincoln is dead.

Abraham Lincoln did not commit suicide.

Therefore, Abraham Lincoln is not dead.

We have true premises and a false conclusion. Yes, that first premise is true. Check Hurley, bottom of 58-59.

Identifying basic argument forms requires that we be familiar with basic deductive forms. First, write the argument with premises first and conclusion last. Then identity the form by noting the "form words." Leave the form words as they are and substitute the non-form words with letters. For categorical syllogisms, form words are "all," "no", "are", and "not." For hypothetical syllogisms, "if," "then," and "not" are form words. For other arguments, we have "either", "or", "both", and "and."

Remember: The counterexample method cannot prove validity, only invalidity.