Routledge

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Chapter 3: Logic – deductive validity

Students’ material: solution

A

1 valid; 2 valid; 3 invalid; 4 invalid ; 5 valid; 6 invalid; 7 invalid; 8 invalid; 9 valid; 10 invalid
Notes
The invalid arguments in questions 3, 8 and 10 are examples of the fallacy of denying the antecedent, which is discussed in detail in Chapter 7. Notice that the conclusion doesn’t follow from the premises; that is, if the premises were true, the conclusion would not have to be true. If, for example, it were true that If it’s raining, we need umbrellas, and if it were also true that it’s not raining, it would not have to be true that we don’t need umbrellas. P1) only tells us a single condition under which we need umbrellas, there could be other conditions under which we need umbrellas.
The invalid arguments in questions 4 and 7 are examples of the fallacy of affirming the consequent, which is discussed in detail in Chapter 7. Notice that the conclusion doesn’t follow from the premises; that is, if the premises were true, the conclusion would not have to be true. If, for example, it were true that if he’s listening to the radio, he’s not at work, and if it were also true that he’s not at work, it would not have to be true that he’s listening to the radio. The conditional in P1) tells us a condition under which he’s not at work, but it doesn’t tell us a condition under which he’s listening to the radio, so we cannot legitimately infer from P2), which affirms the consequent of the conditional, that he’s listening to the radio.

B

1 valid; 2 valid; 3 invalid; 4 valid; 5 valid; 6 valid

C

1 yes; 2 yes; 3 yes; 4 yes; 5 yes; 6 yes; 7 no; 8 yes; 9 no; 10 yes
Hint: notice that the instruction asks you whether an argument whose premises and conclusion has these truth values could be valid. Whether or not an actual argument is valid will depend not on the actual truth values of the premises, but on whether or not if they were true, the conclusion would have to be true (i.e. whether the conclusion follows from the conclusion). The exercise demonstrates the only case in which you can tell from the actual truth-values of an argument that it is invalid. The extended argument in question 9 is invalid because, although the first inference is valid, the second is invalid, this makes the whole argument invalid.

D

1 sound; 2 unsound*; 3 sound; 4 sound; 5 unsound; 6 sound; 7 sound; 8 unsound**; 9 unsound; 10 unsound
Notes
* This argument is unsound because it is invalid. We cannot legitimately infer from the claim that all sheep are herbivores and the claim that Shrek is a herbivore that Shrek is a sheep. Shrek may be a herbivore, but some other kind of animal. The class of herbivores does not include all and only sheep. This argument is very similar to the fallacy of affirming the consequent (discussed in detail in Chapter 7). The proposition in P1) could be expressed as a conditional – if something is a sheep, it is a herbivore. Now it should be easy to see that P2) affirms the consequent of that conditional.
** This argument is also unsound because it is invalid. It is an example of the fallacy of denying the antecedent (discussed in more detail in Chapter 7). From the claim that if Knut is a polar bear, Knut eats seals and the claim that Knut is not a polar bear, we cannot legitimately infer that Knut does not eat seals. Knut could be some other seal-eating non-polar bear creature.

E

1 yes; 2 yes; 3 yes; 4 no*; 5 yes; 6 yes; 7 no; 8 yes; 9 yes; 10 yes**
Notes
* ‘if P then Q ’ is not equivalent ‘to unless Q then P’ This can be seen when we consider that the conditional’s antecedent does not state the only condition under which your children will not drive you insane – it just gives one such conditional – whereas ‘unless Q then P’ says that this is the only condition under which you will avoid them driving you insane. If you pay close attention to the relevant questions, you will see that ‘P only if Q’ is equivalent to ‘unless Q then P’
** Notice that ‘when’ has the same meaning as ‘if’ in this proposition.

1 a; 2 b; 3 a; 4 a; 5 b; 6 a; 7 a; 8 a; 9 a; 10 b

G.

1. Invalid. The qualifier “almost” means that there are some days in which my llama is having a bad hair day, but the air isn’t humid.

2. Valid. This argument has the valid argument form affirming the antecedent.

3. Valid. Here is the same argument reconstructed to make the valid form affirming the antecedent more visible:

P1)       If something is an apple, then it doesn’t fall far from the tree.
P2)       This is an apple.
C)        This apple will not fall far from the tree.

4. Invalid. If people often resemble their parents, then sometimes they do not.

5. Invalid. This argument has the invalid form affirming the consequent.

6. Invalid. Same as no. 5.

7. Invalid. The fact that a particular gene is necessary for speech in humans does not mean that it is sufficient for speech wherever it occurs.

8. Valid. Here gene x is taken as sufficient for language (or sufficient for us to conclude that language exists wherever it exists).

9. Invalid. These reasons provide some support for the conclusion, but they don’t make it necessarily true. Perhaps the hazards involved in police work is to high for this person’s preferences, or perhaps she is disabled, and will not be accepted by the police.

10. Valid. Try reconstructing this argument into a clearer affirming the antecedent.

11. Valid.

12. Invalid. The fact that a person doesn’t believe he is being lied to doesn’t mean that it is in fact the case.

13. Invalid. Although advertising on TV is very often inaccurate and untruthful, it isn’t always so. This is implied by P2.

14. Valid. Since P2 tells us that we can’t trust advertising, there is no way to be absolutely sure, based on P1 and P2 alone, if this news program is going live up to the expectations or not. Both options are logically possible.

15. Invalid. Compare with no. 16.

16. Valid. When judging validity, we must assume temporarily that the premises are true. So we can’t object that perhaps the meat industry could be changed to be humane. This would contradict P1. Similarly, we can’t object to P2 or P3.

17. Valid. If P1 and P2 were true, C would have to be true.

18. Invalid. “Not sufficiently supported” does not equal “false.”

19. Valid. Reconstruct as affirming the antecedent.

20. Valid. See answer to no. 19.

21. Invalid. There is no premise saying that everything that the Bible says is true.

I.

  1. John Campbell’s argument is invalid.
  2. John Campbell’s argument is valid.   
  3. What most people believe is true.                  
  4. What almost everyone does can’t be morally wrong.            
  5. If you do something then it is okay for me to do it.  
  6. Someone who raised eight kids successfully must be an expert in raising kids/is an authority on smacking.
  7. John Smith lost the election.
  8. Manchester Utd are top of the table only if Liverpool lost against Arsenal.

J.

1. This argument is valid. It takes the valid form affirming the antecedent:

P1) If p then q
P2) p               
C) q

The first premise is a conditional that states hypothetically that if one thing is true (p) then another thing will also be true (q). The second premise then asserts that the first thing is indeed true. If we assume both premises to be correct we are forced to conclude that the second thing (q) is also true.

It is highly likely that P2 is false. If you didn’t know that, you could have said that it is controversial, and so can’t be treated as true. The argument is therefore deductively unsound. (There is no problem with P1.)

2. This argument is invalid. If we assume the premises to be true we could still imagine a situation where the conclusion will be false (e.g. God isn’t telling the truth). Since this argument is invalid it can’t be deductively sound. If you thought that the argument is valid you should have still doubted the truth of both premises – both are controversial, the second one highly so.

K.

1.
The only correct answer is ‘e’. The rest are common mistakes which you should be wary of. Option ‘a’ is incorrect because even bad argument can have true conclusions – it’s just that we can’t tell if the conclusion is true based on a bad argument. Option ‘b’ is incorrect because there are good kinds of arguments which are not deductive, and so do not need to be deductively valid. This is also why ‘c’ is false. Option ‘d’ focuses on deductive arguments. These are the kinds of arguments for which the test of validity is important. The problem is that ‘d’ makes the same mistake as ‘a’ – an invalid deductive argument can still have a true conclusion.

2. The only correct answer is ‘d’. The rest are common mistakes which you should be wary of. Option ‘a’ is false because imagining situations in which the conclusion is false helps in judging validity only if these situations do not contradict the temporarily assumed truth of the premises. This is why ‘d’ is correct. Options ‘b’, ‘c’ and ‘e’ are all incorrect because of the same reason: There can be valid arguments which have false premises, a false conclusion, or both.

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