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1.3 — Counterexamples and Invalidity, Exercises of English

English

The counterexample method (described below) is a method for showing that a given argument is formally invalid by constructing a good counterexample to its ...

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2022/2023

Uploaded on 03/01/2023

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1.3 — Counterexamples and Invalidity
In this section, we study a method for determining that some
invalid arguments are indeed invalid. This method will therefore
reveal cases of invalid argument forms:
Counterexamples
At first site this might seem to be an instance of modus tollens:
But in fact it’s an instance of the invalid argument form known as
the fallacy of denying the antecedent:
An invalid argument form is one that has an invalid
substitution instance.
So,
1. If Britney Spears is a philosopher, then Britney Spears is wise.
2. Britney Spears is not a philosopher.
3. Britney Spears is a not wise.
Modus Tollens
So,
1. If A, then B.
2. Not B.
3. Not A.
Fallacy of Denying the Antecedent
So,
1. If A, then B.
2. Not A.
3. Not B.
pf3
pf4
pf5
pf8
pf9
pfa

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1.3 — Counterexamples and Invalidity

In this section, we study a method for determining that some

invalid arguments are indeed invalid. This method will therefore

reveal cases of invalid argument forms :

Counterexamples

At first site this might seem to be an instance of modus tollens :

But in fact it’s an instance of the invalid argument form known as

the fallacy of denying the antecedent :

An invalid argument form is one that has an invalid

substitution instance.

So,

  1. If Britney Spears is a philosopher, then Britney Spears is wise.
  2. Britney Spears is not a philosopher.
  3. Britney Spears is a not wise. Modus Tollens So, 1. If A, then B. 2. Not B. 3. Not A. Fallacy of Denying the Antecedent So, 1. If A, then B. 2. Not A. 3. Not B.

It is easy to miss the invalidity of the argument above because

the conclusion is manifestly true.

But consider the following instance of the same argument form:

Unlike the argument above, this argument is clearly invalid

because most people (most Americans, at least) will recognize

that the premises are true and the conclusion false and hence as

a counterexample to the fallacious argument form above.

Not every counterexample is as effective as every other:

This may well be a counterexample; to the fallacy of denying the

antecedent, but it isn’t obvious unless you know who Dan is and

what the condition of his summer garden is.

So,

  1. If Britney Spears is an oil tycoon, then Britney Spears is rich.
  2. Britney Spears is not an oil tycoon.
  3. Britney Spears is not rich.

An counterexample to an argument form is a

substitution instance whose premises are true and

whose conclusion is false.

So,

  1. If there are Beefsteaks in Dan’s summer garden, then there are tomatoes in it.
  2. There are no Beefsteaks in Dan’s summer garden.
  3. There are no tomatoes in Dan’s summer garden.

Here is a good counterexample:

The Counterexample Method

  • A good counterexample to an argument form shows

vividly that that form is invalid.

  • An argument is formally invalid , recall, if it is an instance of

an invalid argument form.

  • The counterexample method (described below) is a method

for showing that a given argument is formally invalid by

constructing a good counterexample to its argument form.

Note : Recall that any argument whose conclusion cannot be false is valid, so there are valid instances of invalid argument forms. However, such arguments, in reality, are very rare. Hence, typically , a formally invalid argument is invalid outright.

Categorical Statements and Arguments (skipping pp. 40-42)

The counterexample method is most useful when applied to

arguments whose premises are categorical statements.

So,

  1. If lemons are red, then lemons have a color.
  2. Lemons have a color.
  3. Lemons are red.

An categorical statement is a statement that relates

two classes , or categories , of things.

Here is an argument composed of categorical statements:

  • The first premise says that the class P of presidents is included in the class H of human beings, i.e., that P is a subclass of H.
  • The second says that the class H of human beings is included in the class M of mammals, i.e., that H is a subclass of M.
  • The conclusion says that the class P of presidents is included in the class of mammals M, i.e., that P is a subclass of M.

Note, however, that this argument is an instance of the

following invalid form:

However, the argument above is clearly valid and, moreover, it is

so in virtue of its form. We just haven’t represented its form in

the most logically sensitive way.

To do so, we need to expand our use of variables to stand for

terms as well as statements:

So,

  1. All presidents are human beings.
  2. All human beings are mammals.
  3. All presidents are mammals. Form 5 (Invalid) So,

1. A.

2. B.

3. C.

A term is a word or phrase that stands for a class.

Further Valid Argument Involving Categorical Statements

This argument has the following form:

Diagramming, we have:

Some C A

B

The diagram clearly illustrates that argument form 7 (and hence

its instance above) is valid.

The “Some C” circle indicates simply that some members of the C class fall outside the B class — there might also be members of the C class that are also in the B class. But it won’t matter either way to the validity of the argument as long as there are at least some members of C that are outside B. So,

  1. All emeralds are gems.
  2. Some rocks are not gems.
  3. Some rocks are not emeralds. Form 7 (Valid) So, 1. All A are B. 2. Some C are not B. 3. Some C are not A.

And another:

This argument has the following form:

Diagramming, we have:

A some C B

Again, the diagram clearly illustrates that the argument form in

question (and hence its instance above) is valid.

So,

  1. Every sockeye is a member of Oncorhyncus.
  2. Some natives of the Copper River are sockeye.
  3. Some natives of the Copper River are members of Oncorhyncus. Form 8 (Valid) So, 1. Every A is a B. 2. Some C are A. 3. Some C are B.

Now all we have to do is see whether we can find a term for ‘B’

that will yield obviously false premises; clearly, we can:

One more example:

Replacing terms with ‘A’, ‘B’, and ‘C’ uniformly, we identify the

form:

Let’s start with the conclusion and find a counterexample:

So,

  1. All cheetas are animals.
  2. Some animals are not felines.
  3. Some cheetas are not felines. So,
  4. No swimming teachers are aquaphobes.
  5. Some bankers are not aquaphobes.
  6. Some bankers are not swimming teachers. Form 10 So,
  7. No A are B.
  8. Some C are not B.
  9. Some C are not A. So,
  10. No ________________ are ___________________.
  11. Some _______________ are not ________________.
  12. Some _______________ are not ________________.

A Final Limitation

We might fail to find a counterexample to an argument’s form

for one of two reasons

• The argument is formally valid

  • It is formally invalid, but we simply weren’t clever enough to find a counterexample.
  • Hence, if we fail to find a counterexample for an argument’s form, we cannot infer anything about the argument’s formal validity!

We will discover methods that overcome these

limitations in the next chapter!

The Counterexample Method

Step 1: Step 2: Step 3: Step 4: Step 5: Identify the most logically sensitive form of the argument. Use capital letters for variables to stand for statements or terms. Find English statements or terms that, if substituted for the variables in the conclusion of the argument, yield a well-known falsehood. Substitute these English statements or terms for the relevant variables uniformly throughout the argument form. Find English statements or terms that, if substituted uniformly for the remaining variables, produce premises that are well- known truths. Check your work. If you have succeeded, you have shown that the original argument is formally invalid (and, most likely, invalid outright).