Validity & Invalidity

1. Validity & Invalidity (Exercises) December 23, 2005

2. 1. All dogs have two heads. 2. All tigers are dogs.--------------------------------- 3. All tigers have two heads. Valid If all members of the class of dogs are two-headed, and if all members of the class of tigers are also members of the class of dogs, then all members of the class of tigers must be two-headed. 2-Headed Things Dogs Tigers

3. 1. All A is B. 2. All C is B. ------------------ 3. All C is A. Invalid Just because the members of A and the members of C are all members of B does not mean that all members of C are members of A. B A C

4. 1. All cats are animals. 2. All tigers are cats. -------------------------------- 3. All tigers are animals. Valid If all members of the class of cats are also members of the class of animals (as they are), and if all members of the class of tigers are also members of the class of cats (as they are), then all members of the class of tigers must be members of the class of animals (right?).

5. 1. If Polly is a cat, then Polly is an animal. • 2. Polly is an animal. • ----------------------------------------------3. Polly is a cat. • Invalid • All cats are animals, but not all animals are cats. It's true that if Polly is a cat, then she is an animal (1st premise). However, the fact that Polly is an animal (2nd premise) does not prove that she is also a cat (conclusion).

6. 1. Either it is raining or the sun is shining. 2. It is raining. ---------------------------------------------- 3. The sun is not shining. Invalid "Either p or q" (which is the logical form of the 1st premise) means "either p is true or q is true, and perhaps they are both true." It could be raining while the sun is shining. Thus, the fact that it is raining (2nd premise) does not rule out the possibility that the sun is shining. The conclusion does not follow necessarily from the premises.

7. 1. If it rains, then Mary carries an umbrella. 2. It is raining now. ---------------------------------------------- 3. Mary is carrying an umbrella. Valid Assuming that Mary carries an umbrella whenever it rains (1st premise), and assuming that it is raining now (2nd premise), it follows necessarily that Mary is carrying an umbrella (conclusion). Do you see that?

8. 1. Either John is a lawyer or Mary is a doctor. 2. John is not a lawyer. ----------------------------------------------- 3. Mary is a doctor. Valid "Either p or q" (the logical form of the 1st premise) means "either p is true or q is true, and perhaps they are both true." In this argument, the 1st premise says that either "John is a lawyer" is true or "Mary is a doctor" is true. One or the other is true, or they are both true, according to the 1st premise. The 2nd premise says that it is NOT TRUE that "John is a lawyer." So it follows necessarily that Mary is a doctor. Right?

9. 1. All cats are animals. 2. Some cats are tigers. -------------------------------- 3. Some animals are tigers. Valid If all cats are animals (as they are), and if some of them are tigers (which is also the case), then some animals must be tigers (namely, the animals that are cats that are tigers).

10. 1. All dogs are animals. 2. Some dogs are tigers. ------------------------------ 3. Some animals are tigers. Valid If all dogs are animals (which is true), and if some dogs are tigers (which is false), then some animals must be tigers (which is true). Remember, it is the logical FORM of an argument that makes it valid or invalid, not its truth content. In this argument, the second premise is false. However, *IF* it were true that some dogs are tigers, then it would follow from the two premises that some animals are tigers.

11. 1. If Polly is a tiger, then Polly is a cat. 2. If Polly is a cat, then Polly is an animal. ------------------------------------------------- 3. If Polly is a tiger, then Polly is an animal. Valid If T, then C; if C, then A; therefore, if T, then A. Right? If tiger, then cat; if cat, then animal; therefore, if tiger, then animal.

12. 1. If Polly is alive, then Polly is an animal. 2. If Polly is an animal, then Polly is a tiger. ------------------------------------------------ 3. If Polly is alive, then Polly is a tiger. Valid All of the statements in this argument (premises and conclusion) are false. Something that is alive might be a plant rather than an animal (contrary to the 1st premise); and something that is an animal might be other than a tiger (contrary to the 2nd premise). The conclusion is also false: there are living things that are not tigers. However, *IF* it were true that Polly's being alive made her an animal, and *IF* it were true that Polly's being an animal made her a tiger, then it would follow necessarily that Polly's being alive would make her a tiger. The argument is valid (but unsound).

13. 1. If Polly is a tiger, then Polly is a cat. 2. If Polly is a cat, then Polly is an animal. ------------------------------------------------- 3. If Polly is an animal, then Polly is a tiger. Invalid The conclusion does not follow from the (true) premises. If tiger, then cat; and if cat, then animal; but there are animals that are not tigers (e.g., dogs, pigeons, etc.).

14. 1. All cats are animals. 2. Some animals are not dogs. ----------------------------------- 3. Some animals are not cats. Invalid All three statements in this argument are true, but the inference in the argument is invalid. The 1st premise (all cats are animals) leaves open the possibility that all animals are cats (which, of course, we know from our background experience to be false). The 1st premise does not tell us whether or not there are animals that are NOT cats. Similarly, the 2nd premise, which states that there are animals that are not dogs, does not tell us whether or not there are also animals that are not cats (conclusion). The 2nd premise does not mention cats at all. So we cannot draw the conclusion from the stated premises.

15. 1. Some cats are black. 2. Some animals are not cats. ---------------------------------- 3. Some animals are not black. Invalid All three statements in this argument are true, but the inference in the argument is invalid. The premises leave open the possibility that all animals are black. The 1st premise (some cats are black) does not tell us whether or not there are animals that are NOT black. The 2nd premise (some animals are not cats) does not tell us anything about the colors animals might or might not have. So we cannot validly deduce the conclusion (some animals are not black) from the stated premises. (Hint: Any two-premise argument with both premises beginning with "some" must be invalid. Can you figure out [or find out] why?)

16. 1. If p, then q. 2. p. ------------------- 3. q. Valid This argument *form* says that if p is true, then q is true (1st premise). Then it goes on to say that p is true (2nd premise). Now, if p is true, as the 2nd premise says, and if q must be true if p is true, as the 1st premise says, then it follows necessarily that q is true (conclusion). Do you see that? (Notice that the actual truth or falsity of the statements in the argument is irrelevant on the question of whether the argument *FORM* is valid or invalid.)

17. 1. If p, then q. 2. Not p. -------------------- 3. Not q. Invalid This argument *form* says that if p is true, then q is true (1st premise). Then it goes on to say that p is NOT true (2nd premise). Now, the 1st premise leaves open the logical possibility that q might be true even if p is false. So, on the basis of the premises, we cannot validly conclude that q is false.

18. 1. If it rains, then Mary carries an umbrella. 2. Mary is not carrying an umbrella. ---------------------------------------------- 3. It is not raining. Valid Assuming that Mary carries an umbrella whenever it rains (1st premise), and assuming that she is not carrying an umbrella now (2nd premise), it follows necessarily that it is not raining now. The 1st premise asserts that if "it rains" is true, then "Mary carries an umbrella" must also be true. So if Mary is NOT carrying an umbrella, we can be sure that it is not raining.

19. 1. If it rains, then Mary carries an umbrella. 2. Mary is carrying an umbrella now. ---------------------------------------------- 3. It is raining. Invalid Assuming that Mary carries an umbrella whenever it rains (1st premise), and assuming that she is carrying an umbrella now (2nd premise), it does NOT follow necessarily that it is raining now. The 1st premise asserts that if "it rains" is true, then "Mary carries an umbrella" must also be true; but the 1st premise does not rule out the logical possibility that Mary carries an umbrella even when it is NOT raining. It just says IF it rains, then she will carry an umbrella; it does NOT say what will happen if it is not raining. So if all we know is that Mary is carrying an umbrella, we cannot tell whether it is raining or not.

20. 1. If pain exists, then God does not exist. 2. Pain exists. ------------------------------------------- 3. God does not exist. Valid Valid, right? (1) If p is true, then q is true. (2) p is true. Therefore, (3) q is true. It does not matter what p and q stand for; and it does not matter whether the statements p and q stand for are either true or false (or unconvincing). It is the *FORM* of a deductive argument that is either valid or invalid (regardless of the argument's truth content).

21. 1. Either p or q. 2. q. ----------------------- 3. Not p. Invalid The 1st premise says that either p is true or q is true, and that maybe they are both true. So if q is true (2nd premise), we cannot conclude that p is false (conclusion), since they might both be true, according to the 1st premise.