Determining Invalidity

Refutation by Logical Analogy

Sandra LaFave



A logically correct deductive argument is said to be valid.

A valid argument is one in which the conclusion must be true ó canít be false ó if the premises are true. (Note that the premises donít have to be actually true for an argument to be valid.)

Some argument forms ó ways of constructing arguments ó guarantee validity. Arguments that are put together correctly have valid argument forms, and are guaranteed to be valid. Modus Ponens, Modus Tollens, Hypothetical Syllogism, Disjunctive Syllogism, are examples of valid forms. If an argument has a valid form, it is valid (logically correct). You can prove that an argument is valid simply by showing that it has a valid form.

On the other hand, some argument forms guarantee invalidity. Arguments that have invalid forms are guaranteed to be invalid. If the form is invalid, the conclusion does not follow from the premises, even if the premises and conclusion are all true. Affirming the Consequent and Denying the Antecedent are examples of invalid forms. If an argument has an invalid form, itís invalid. You can prove that an argument is invalid simply by showing it has an invalid form.

Suppose, however, that youíre an ordinary person on the street. You† havenít studied valid and invalid forms, or techniques of symbolic logic (like truth tables or truth trees). Is there any intuitive way you as a non-specialist can determine if an argument is invalid (not logically correct)?

The answer is YES. You can show an argument is invalid by showing that its form is invalid. You can show its form is invalid by showing that the form can lead to an obviously false conclusion when the premises are obviously true.

To show invalidity, just do the following:

1.      Determine the form of the argument whose validity is in question.

2.      Attempt to construct another argument of the same form with obviously true premises and an obviously false conclusion.

3.      If you succeed, you have shown the original argument invalid.

This method works because a valid argument form guarantees validity, and validity means itís impossible for the conclusion to the false if the premises are true. If the premises had been true, and the form valid, the conclusion could not have been false. (The argument would have been sound: valid, with all true premises.) So the very fact that the form allows true premises and a false conclusion shows it canít be a valid form. And if the form of an argument is invalid, the argument is invalid.

Example:

You are confronted with the following argument:

ďIf I were the President, Iíd be famous. So Iím not famous, since Iím not the President.Ē

The conclusion indicator ďsoĒ tells you the conclusion is ďIím not famousĒ.

So the argument is:

Premise:††††††† If I were President, Iíd be famous.

Premise:†††† †††Iím not President.

Conclusion:††††Iím not famous.

The argument obviously is an instance of the invalid form Denying the Antecedent, but weíre supposing you donít know that.

Step 1: Determine the form of the argument whose validity is in question.

Premise:††††††† If A, then B.

Premise:††††††† Not A (or ďA is not the caseĒ).

Conclusion:††† Not B (or ďB is not the caseĒ).

Step 2: Now that youíve extracted the form, try to construct an argument of the very same form with all true premises and a false conclusion. For example:

Premise:††††††† If Robert Redford were† President, heíd be famous. (true)

Premise:††††††† Robert Redford is not President. (true)

Conclusion:††† Robert Redford is not famous. (false)

Note that the premises are true, but the conclusion is false (Robert Redford is famous).

The following argument would work too:

Premise:††††††† If I am decapitated, Iíll die. (true)

Premise:††††††† I wonít be decapitated. (very probably true)

Conclusion:††† I wonít die. (alas, false)

This method of showing invalidity is called the Refutation by Logical Analogy, and people use it all the time (ďThatís just like arguing ÖĒ). The neat thing is, it really does show invalidity!

Try it yourself.

Use the method of Refutation by Logical Analogy to show the invalidity of the following arguments:

1.      All Presidents live in the White House. Bill Clinton lives in the White house, so Bill Clinton is the President.

2.      If thereís a government conspiracy to cover up the existence of extraterrestrial visitors, then the government would deny any knowledge of UFOs. The government does deny any knowledge of UFOs. This proves there is a conspiracy to cover up the existence of extraterrestrial visitors.

3.      All persons have skin. Howdy Doody isnít a person, though, so Howdy Doody doesnít have skin.


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