Deductive Validity In this tutorial you will learn how to determine whether deductive arguments are valid or invalid. Chapter 3.b

Perhaps the most important concept in logic is the concept of deductive validity. Deductive arguments are either valid or invalid. A valid argument is a deductive argument in which the conclusion follows logically (i.e., with strict logical necessity) from the premises. In other words, a valid argument is a deductive argument in which it would be contradictory to assert all the premises as true and yet deny the conclusion. An invalid argument is a deductive argument in which the conclusion does not follow logically from the premises.

Chapter 3 introduces you to a helpful (but not foolproof) method for testing arguments for validity called the "The C's Test." The Three C's Test involves three steps: 1. Check to see whether the premises are actually true and the conclusion is actually false. If they are, then the argument is invalid. (By definition, no valid argument can have all true premises and a false conclusion.) If they are not, or if you don't know whether the premises are true and the conclusion is false, then go on to step 2.

2. See if you can conceive a possible scenario in which the premises would be true and the conclusion false. If you can, then the argument is invalid. If you can't, and it is not obvious that the conclusion follows necessarily from the premises, then go on to step Try to construct a counterexample--a special kind of parallel argument--that proves that the argument is invalid. Constructing a counterexample involves two steps: (1) Determine the logical form of the argument you are testing for invalidity, using letters (A, B, C, etc.) to represent the various terms in the argument. (2) Construct a parallel argument that has exactly the same logical pattern as the argument you are testing but that has premises that are clearly true and a conclusion that is clearly false. If you can successfully construct such a counterexample, then the argument is invalid. If, after repeated attempts, you cannot construct such a counterexample, then the argument is probably valid.

If Michelangelo painted the Mona Lisa, then he's a great painter. Michelangelo is a great painter. So, Michelangelo painted the Mona Lisa. Is this argument valid or invalid? How can we use the Three C's Test to determine if it is valid or invalid?

If Michelangelo painted the Mona Lisa, then he's a great painter. Michelangelo is a great painter. So, Michelangelo painted the Mona Lisa. This argument is invalid. We can most readily see that the argument is invalid by applying the first step of the Three C's Test. The premises of the argument are, in fact, true, and the conclusion of the argument is, in fact, false. Since no valid argument can have true premises and a false conclusion, we know straight away that the argument is invalid.

If Bill Clinton is president, then he lives in the White House. Bill Clinton is not president. So, Bill Clinton doesn't live in the White House. Is this argument valid or invalid? How can we use the Three C's Test to determine whether it is valid or invalid?

If Bill Clinton is president, then he lives in the White House. Bill Clinton is not president. So, Bill Clinton doesn't live in the White House. This argument is invalid. The first step of the Three C's Test is not applicable here, because both the premises and the conclusion are actually true. However, the second step of the Three C's Test shows that the argument is invalid. It's easy to conceive of circumstances in which the premises and the conclusion is false. This would be the case, for example, if Clinton became an advisor who lived in the White House. Because we can imagine circumstances in which the premises could be true and the conclusion false, the conclusion does not follow from the premises with strict logical necessity. This shows that the argument is invalid.

All Alphans are Betans. Some Betans are Deltans. So, some Deltans are Alphans. Is this argument valid or invalid? How can we use the Three C's Test to determine whether it is valid or invalid?

All Alphans are Betans. Some Betans are Deltans. So, some Deltans are Alphans. With this argument, the first step of the Three C's Test is useless, because the terms are just made up, and thus the statements are neither true nor false. It's also difficult to apply the second test, since the logic of the argument is complex. Thus, let's apply the third test: the counterexample method of proving invalidity.

All Alphans are Betans. Some Betans are Deltans. So, some Deltans are Alphans. To apply the counterexample method, we first must determine the logical pattern, or form, of the argument, using letters (A, B, C, etc.) to represent the various terms. What is the logical form of this argument?

1. All A's are B's. 2. Some B's are D's. 3. So, some D's are A's. The second step in the counterexample method is to try to construct a second argument--one that has exactly the same logical form as the argument we are testing for validity but that has premises that are obviously true and a conclusion that is obviously false. If we can successfully construct such an argument, that will show that our first argument, the argument being tested for validity, is invalid. For if any argument with a certain logical form is invalid, then all arguments with that form are invalid. Can you construct a counterexample to the argument form given at the top of this page?

All dogs are carnivores. Some carnivores are cats. So, some cats are dogs. [This is the end of the tutorial] X Bingo! This argument has the same logical form as the first argument, but in this argument the premises are both clearly true and the conclusion is clearly false. This shows that arguments with this pattern of reasoning are not guaranteed to have true conclusions if the premises are true. And this shows that all arguments that have that pattern of reasoning are invalid.