invalid, or inductively unforceful, is the method of refutation by counter-example. This involves giving an argument that employs the same type of inference as the one in question, but which is obviously invalid or unforceful.
It is important to avoid the who is to say? criticism. One form this takes is to complain that the conclusion of an inductively forceful argument has not been 'proved', that it 'might' be false. Another is to object to an argument on the grounds that it contains a value-laden term. It should be clear that these are not genuine criticisms from a rational point of view.
EXERCISES 1 For questions a-i, consider this argument:
P1) If Rangers won the match then the pub will sell pints for £1 tonight.
P2) Whenever Rangers are ahead at halftime of a Scottish Premiership match, it is highly probable that they win the match.
P3) Rangers were ahead at halftime, and this is a Scottish Premiership match.
C) (Probably) The pub will sell pints for £1 tonight.
Suppose that the premises are true. Suppose that both Andrew and James know that the premises are true (they have both seen the sign in the pub window about the £1 pints, they both saw the halftime score of 1 -0 in favour of Rangers announced on television, and both know that Rangers have won every Scottish Premiership match in the past three years when leading at halftime). Andrew has no other information relevant to the truth of the conclusion; but unlike Andrew, James saw on television that the opposing side scored twice in the second half to win the match 2-1.
a Is the argument deductively valid?
b Is the argument deductively sound?
c Is the argument inductively forceful?
d Is the argument inductively sound?
e Is the argument rationally persuasive for Andrew?
f Is the argument rationally persuasive for James?
g Should Andrew be persuaded by the argument?
h Should James be persuaded by the argument?
i If the argument is not rationally persuasive for either Andrew or James, explain why.
2 Now consider this argument:
P1) The majority of well-educated Germans speak English. P2) Jacob is a well-educated German.
C) (Probably) Jacob speaks English.
Assume this time that P1 is true and that the conclusion is true; but assume that P2 is false: Jacob is Austrian, and not very well educated either (still, he did learn English quite well in secondary school). Assume that Catherine, Jane, Mary and Anna all know that P1 is true. Catherine believes P2 is true because her friend David told her it is; David is a reliable person who knows Jacob, and Catherine has no reason to doubt him. In fact David's mistake was quite reasonable: Jacob lied to him, telling him he completed a university degree in history; further, it was reasonable for David to infer, from Jacob's accent, that Jacob is German. For, although German is the national language of Austria, the vast majority of native German speakers are German. Catherine knows nothing else about Jacob, and accepts C. Jane also believes P2, but for different reasons: she fancies Jacob, having seen him, at a distance, at a party, and thinks he's well educated because she saw Jacob wearing glasses and a tie. This belief is clearly not well supported. Jane also accepts C, but has no other information relevant to the truth of C. Mary believes both P1 and P2, for the same reasons as Catherine; but she does not accept C, because she heard Jacob speaking German to David at the party, and inferred, quite reasonably, that Jacob does not speak English. She has no other information relevant to the truth of C. Anna does not believe P2. She heard what David said to Catherine, and believes that David is always sincere and well informed; but Jacob is athletic and handsome and she has a stupid irrational prejudice according to which athletic, handsome men are almost always stupid, and therefore not well educated. She has no other information relevant to the truth of C.
a Is the argument inductively forceful?
b Is the argument inductively sound?
c Is the argument rationally persuasive for Catherine? Why or why not?
d Is the argument rationally persuasive for Jane? Why or why not?
e Is the argument rationally persuasive for Mary? Why or why not?
f Is the argument rationally persuasive for Anna? Why or why not?
3 Reconstruct the following arguments. Then, using the techniques discussed earlier, explain why the argument is, or is not, valid.
a If this man is not a spy, then he is a detective. But all detectives wear trench coats, and he is not wearing one. If he is a spy, then either he is Russian or American. But no American spy knows how to order dessert wines, unless he is from New York. But all spies from New York wear trench coats. Therefore, if this man knows how to order dessert wine, then he is a Russian spy.
b If the ancient Allemani had been both vicious and loyal, then the Romans would have purchased their allegiance. If they had done so, then the Franks would never have challenged the Allemani. But they certainly did. Therefore, if the Allemani were vicious, they were not loyal.
c Some ancient Visigoths were literate. But all Roman citizens were literate. Therefore some ancient Visigoths were not Roman citizens.
d If most educated Romans learned Greek, then most read Homer. But anyone who reads Homer knows the story of Achilles. Therefore most Romans knew the story of Achilles.
e If all ancient Visigoths were pagans, then so were all ancient Picts. Some ancient Picts were non-pagans, or some ancient Saxons were. But some ancient Saxons were Christian, and no Christian is a pagan. Therefore some ancient Visigoths were non-pagan.
4 The conclusions of the first two arguments in exercise 3 are conditionals. Split them into two statements, antecedent and consequent. Now add the antecedent to the premises, and regard the consequent as the conclusion of the argument. Using the method of supposing the conclusion false, explain why the resulting argument is valid or not valid.
5 Reconstruct the following arguments, then refute them by counterexample - that is, by giving arguments that embody the same pattern, but which have true premises and false conclusions.
a A significant increase in the rabbit population would bring about an increase in the number of foxes. And sure enough, the number of foxes has increased lately. This must, therefore, be due to an increase in the rabbit population.
b If free will is impossible, then the concept of responsibility is nonsense. If so, then the justice system is based upon a confused idea. Therefore if free will is possible, then the justice system is not based upon a confused idea.
c You and others have tried for years to prove that Nessie does not exist. But you have failed. You should by now admit it: Nessie is real.
d You admit that the Loch Ness monster has not been seen in recent years. If so, then it must be very secretive. But if it is secretive, then, obviously, it exists. Therefore you have admitted that Nessie exists.
e Why don't we do what we know will stop these evils? We should give drug-dealers and paedophiles life prison sentences.
6 Go back through the arguments in question 3, and draw tree diagrams for them.
7 List the explicit premises and the conclusions of the arguments in questions 13, 14 and 15 of Chapter 5. Draw tree diagrams for these, including only the explicit premises. Now add the needed implicit premises to the list, and draw new tree diagrams for the complete arguments.
8 Could one give an argument that (i) one knows to be sound (ii) is rationally persuasive for its audience, and (iii) is not rationally persuasive for oneself? If not, why not? If so, would that be deceitful, or otherwise naughty in any way? Why or why not? Make up an example.
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