Common Misunderstandings

Let us now consider some of the most common mistakes made with respect to the concept of validity. Consider the following questions and then take the true/false quiz.

 

Common mistakes people make when thinking about the concept of validity or invalidity.

  1. A valid argument must have a true conclusion.
  2. If all premises of a valid argument are false, then the conclusion must be false.
  3. A valid argument is an argument with all true premises and a true conclusion.
  4. The conjunction of all the premises in a valid argument must be true.

 

Value: 5

Which of the following statements about a valid argument are true?

 

1. A valid argument must have a true conclusion.

2. If all premises of a valid argument are false, then the conclusion must be false.

3. A valid argument is an argument with all true premises and a true conclusion.

4. The conjunction of all the premises in a valid argument must be true.

 
 
 
 
 
 
 
 
 

 

The first statement omits a very key part of the definition of validity that states "If all premises are true". Missing or forgetting key parts of definitions is perhaps the number one reason for making logical errors.

The second statement is not the definition of a valid inference or a statement implied by that definition. Review the visual representation of implication for now, paying attention to the last row. We will prove this later in Part 2.

The third statement is also incorrect. Re-read the definition of valid and take note of the differences between this statement and the definition.

The fourth statement is a bit more difficult. So we will demonstrate its falsity by example. Consider the following argument:

The Cullinan diamond weighs more than 100 grams.

The diamond was worn by Queen Victoria when she was crowned in 1964.

Therefore the Cullinan diamond weighs more than a 2 gram feather.

Clearly, this argument is valid (it is impossible to have a false conclusion with all true premises), but since Queen Victoria was not crowned in 1964, the conjunction of the premises can't be true (recall this is another way of saying at least one of the premises is false). The argument is valid because the truth of the first premise alone implies the truth of the conclusion, which means it is impossible to have a false conclusion and all true premises.

 We can now formally define arguments which are not valid. As you may have guessed these arguments are called "invalid".

Definition: If it is possible for an argument to have all true premises but still have a false conclusion, then the argument is invalid, which is to say it is not valid.

Equivalently, if the inference from the premises to the conclusion of an argument is not that of implication, then we call the argument invalid.

We use the term "invalid" to characterize the inference between the conjunction of the premises of an argument and the conclusion. As noted in the side-bar to this page, some logic books restrict this term to refer to arguments which claim to be proofs. In this textbook we simply call this restricted use by the name, "erroneous proof" and generalize the terms valid or invalid to apply to any argument.

Again, be careful not to add anything to the definition of an invalid argument that is not stated in the definition. By our classification, any argument where the premises do not imply the conclusion is an invalid argument. Invalid does not mean "false" or even "bad reasoning" - it means only that the possibility is open (no matter how remote) for the conclusion to be false even if all of the premises are true. This means that arguments with premises which are strongly relevant to the conclusion are still classified as being invalid.

All arguments can be classified under our classification scheme as being either valid or invalid. To help you keep the difference between these two classifications in mind, we have gathered together what this classification means with respect to the conclusion for arguments which are valid and invalid and when the premises are all true or when they are not all true.

 

Premises

Conclusion

Argument is valid

Argument is invalid

All true

Must be true

Need not be true (could be true or false)

Not all true (at least one is false)

Need not be true (could be true or false)

Need not be true (could be true or false)

 

Note that the table is divided into what is true of all premises and the corresponding result for the truth of the conclusion based on the inference type of the argument (valid or not). In every case except one, the conclusion need not be true. Keep this is mind when thinking about the implications of an argument being valid or not.