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Chapter 6. Subarguments A subargument is an argument offered to support a premise in another argument, sometimes called the main argument. Below is an example. 1.If P, then Q 2.P 2a. Either R or P 2b. Not R 2c. So, P 3.So, Q We follow here the convention of indenting th...

Chapter 6. Subarguments A subargument is an argument offered to support a premise in another argument, sometimes called the main argument. Below is an example. 1.If P, then Q 2.P 2a. Either R or P 2b. Not R 2c. So, P 3.So, Q We follow here the convention of indenting the subargument under the premise it’s meant to support and numbering the premises using the premise number from the main argument, with a letter added. Notice that the conclusion of the subargument (line 2c) is the premise the subargument is being used to prove. If we wanted to, we could provide a subargument for the first premise as well. This is how things look when we formally reconstruct an argument, but subarguments and main arguments can be and typically are in ordinary contexts written in paragraph form. The main argument above is the argument from (1) and (2) to (3). When we evaluate the main argument, we consider only these statements and ignore the subargument. A main argument could be sound even if its subarguments turn out to be unsound. Recall here a lesson from an earlier chapter: even if an argument is fallacious, its conclusion could still be true. So, even if a subargument fails, the premise it was supposed to prove could still be true, and the main argument could still be sound. A bad subargument could be embedded in a good main argument, and a good subargument could be embedded in a bad main argument. The best case scenario, of course, is having a good subargument within a good main argument, but it doesn’t always turn out this way. Consider the argument below. 1.If copper is a metal, then it conducts electricity. 2.Copper is a metal. 2a. All substances listed in the periodic table are metals. 2b. Copper is a substance listed in the periodic table. 2c. So, copper is a metal. 3.So, copper conducts electricity. The main argument is sound: it has the valid form modus ponens, and its premises are true. We attempted to support premise (2) with a subargument, but it turns out that the subargument we used is unsound, since premise (2a) is false. So, the main argument is a good one, but the subargument is flawed. All this means is that the subargument doesn’t manage to prove the truth of the second premise. But the second premise nevertheless is true, and the main argument is sound because it’s valid and has true premises. Matters in some cases are reversed: we can have a sound subargument embedded in an unsound main argument (perhaps because some other premise in the main argument is false, or because the main argument is invalid). Below is an example. 1.No octagons are hexagons. 1. No O are H 1a. No hexagons are octagons. 1a. No H are O 1b. So, no octagons are hexagons. 1b. So, no O are H Valid subargument 2.All hexagons are pentagons. 2. All H are P 3.So, no octagons are pentagons. 3. So, no O are P Invalid main argument The subargument is valid (it’s the conversion an E statement), and (1a) is true. Hence, the subargument is sound, and premise (1) in the main argument is true. But the main argument is unsound because it’s invalid (no cats are dogs, for example, and all dogs are animals; it doesn’t follow that no cats are animals). Here is a summary of the important points we’ve made: A subargument is an argument that’s supposed to support a premise in a main argument. If you offer premises A, B, and C in order to prove statement D, for example, if someone asks you to prove that premise B is true, and if you offer an argument in support of B, then you’re giving a subargument. A subargument might succeed in supporting the premise in question, or it might fail what’s true of any argument. What we want, of course, is a good subargument embedded in a good main argument, but strictly speaking we could end up with a fallacious subargument embedded in a main argument that just happens to be sound. The premise we tried to support could turn out to be true even if our subargument doesn’t manage to prove that it’s true. (Of course, if the only subargument we can find for a certain premise is flawed, then we might have to concede that the premise remains unproven and that we don’t know whether the main argument is sound or cogent, but it could nevertheless be sound or cogent, in principle.) The reverse also is true: we could have a perfectly good subargument that proves a premise in a main argument, and yet that main argument might have some other flaw (e.g., it might be invalid) that prevents it from being sound or cogent. Subarguments are used mainly when the truth of at least one of our main argument’s premises likely won’t be obvious to the argument’s intended audience. Hence, arguments don’t always require subarguments; it really depends on whether the audience can be expected to accept the premises of the main argument without the need for any subarguments. (If a subargument were required for every premise, then we’d need subarguments within subarguments, and so on without end. But see exercise 7, below.) Generally, you should not demand that arguers provide subarguments for their premises unless their premises seem dubious (not clearly true) to you. Exercises 1.What is a subargument? 2.Can a premise in a subargument also be given a subargument? 3.True or False: If the subargument for a premise is fallacious, then it follows that the premise is false and that the main argument is unsound or uncogent. 4.If a subargument for some premise in a main argument is sound, then the premise in question is true. Could the main argument overall be fallacious, however? 5.Can an inductive subargument be given for a premise in a deductive argument? 6.Can a deductive subargument be given for a premise in an inductive argument? 7.Generally speaking, you should not ask an arguer for a subargument unless the premise in question is one that’s not clearly true to you. With that in mind, could it ever be reasonable and fair to demand a subargument for a premise, and then demand a subargument for a premise in the subargument, and so on ad infinitum? Consider the argument below and try to imagine what might happen if 2b is challenged. 1.Psychic A says the world will end in 2030. 2.Psychic A is never wrong. 2a. Psychic B says Psychic A is never wrong. 2b. Psychic B is never wrong. 2c. So, Psychic A is never wrong. 3.So, the world will end in 2030. 8.Consider the argument below and answer the questions that follow. 1.P Q 2.~Q 2a. R ~Q 2b. ~R 2c. So, ~Q 3.So, ~P (a) Is the subargument for premise (2) valid? (b) Is the main argument valid? 9.Consider the argument below and answer the questions that follow. 1.English uses an alphabet. 1a. All languages use alphabets. 1b. English is a language. 1c. So, English uses an alphabet. 2.Alphabetic languages use letters to represent sounds. 3.So, English uses letters to represent sounds. (a) Is the main argument sound? (b) Is the subargument for (1) sound? 10. Consider the argument below and answer the questions that follow. 1.Some rocks are not igneous rocks. 1a. Some rocks (e.g., limestone) are formed by sedimentation. 1b. No igneous rocks are formed by sedimentation. 1c. So, some rocks are not igneous rocks. 2.So, some rocks are igneous rocks. (a) Is the main argument valid? (b) Is the subargument valid? 11. Consider the argument below and select the correct option. 1.All athletes are football players. 2.No football players are U.S. citizens. 2a. No U.S. citizens are football players. 2b. So, no football players are U.S. citizens. 3.So, no athletes are U.S. citizens. (a) The main argument and the subargument are both sound b.The main argument and the subargument are both unsound c. The main argument is sound, but the subargument is unsound d.The main argument is unsound, but the subargument is sound 12. Consider the argument below and select the correct option. 1.If this is a work by Picasso, then it’s valuable. 1a. If this is a work by Picasso, then it’s a masterpiece. 1b. If this is a masterpiece, then it’s valuable. 1c. So, if this is a work by Picasso, then it’s valuable. 2.This is not a work by Picasso. 3.So, this is not valuable. (a) The main argument and the subargument are both valid b.The main argument and the subargument are both invalid c. The main argument is valid, but the subargument is invalid d.The main argument is invalid, but the subargument is valid 13. Consider the argument below and select the correct option. 1.Canberra is the capital of Australia. 1a. If Canberra is the capital of Australia, then it’s a city down under. 1b. Canberra is a city down under. 1c. So, Canberra is the capital of Australia. 2.So, either Canberra or Sydney is the capital of Australia.  a. The main argument and the subargument are both sound b.The main argument and the subargument are both unsound c. The main argument is sound, but the subargument is unsound d.The main argument is unsound, but the subargument is sound 14. Consider the argument below and select the correct option. 1.Most stars are larger than Earth’s sun. 2.Rigel is a star. 2a. Jones, a neurologist, says that Rigel is a star. 2b. So, probably Rigel is a star. 3.So, probably Rigel is larger than Earth’s sun. a. The main argument and the subargument are both strong b.The main argument and the subargument are both weak c. The main argument is strong, but the subargument is weak d.The main argument is weak, but the subargument is strong 15. Evaluate the argument below in respect to validity and strength. 1.Smith has already been acquitted of this charge. 1a. The New York Times reports that Smith has already been acquitted of this charge. 1b. The New York Times is a reliable source of news. 1c. So, probably Smith has already been acquitted of this charge. 2.The guarantee against double jeopardy requires that no one be tried twice for the same charge having already been acquitted or convicted on that charge previously. 3.So, Smith cannot be tried on this charge again. Optional exercise: Suppose I want to justify my belief that P. According to a famous “trilemma” in philosophy, I have three choices: (1) I can treat my belief in P as self-justifying, or I can infer P from Q and treat my belief in Q as self-justifying, or I can infer Q from R and treat my belief in R as self-justifying, or…. And so on. In other words, I can treat some proposition, or my belief in that proposition, as so self-evidently true that no more proof is needed, thus halting any regress of justification. (2) I can reason in a circle, such as by inferring P from Q, inferring Q from R, and inferring R from P. (3) I can infer P from another statement Q, infer Q from R, infer R from S, and so on ad infinitum. The difference between this scenario and (1) is that in this case the regress of justification never halts. The difference between this scenario and (2) is that no proposition ever ends up being inferred from itself. Question: What is the connection between this trilemma and our discussion of subarguments? Chapter 7. Counterexamples Part A. Substitution Instances Consider the argument form (disjunctive syllogism) below. 1. Either p or q 2. Not p 3. So, q The italicized lowercase letters are variables representing statements. When you substitute specific statements for the letters, you get what’s called a substitution instance of the form. Below is a substitution instance of disjunctive syllogism. 1. Either Rolls Royce makes sports cars, or Ferrari makes sports cars. 2. Rolls Royce does not make sports cars. 3. So, Ferrari makes sports cars. How many substitution instances are there for disjunctive syllogism? Infinitely many. There are infinitely many substitution instances for any form of argument, whether that argument form is valid or invalid. Below is an invalid form of categorical argument (the uppercase letters represent classes). 1. All A are B 2. So, all B are A There are infinitely many classes you could substitute for A and B for example, you could substitute dogs for A and animals for B, or squares for A and shapes for B. Every form of argument has infinitely many substitution instances. We’re going to use the concept of a substitution instance to help define the key term in this chapter: counterexample. Part B. Counterexamples Consider the argument below. 1. All protons are hadrons. 2. So, all hadrons are protons. This argument has the form we used for illustration above. Using letters we find in the argument, we have 1. All P are H 2. So, all H are P We noted above that this form of argument is invalid. What makes a form an invalid one? From Chapter 4 we know that a form is invalid when it’s possible for an argument with that form to have true premises and a false conclusion at the same time. If an argument form is valid, then by definition this could never happen; for any valid form of argument it’s impossible to find any substitution instance of that form with true premises and a false conclusion. So, if you want to prove that an argument’s form is an invalid one, all you need to do is find a substitution instance of that form with true premises and a false conclusion. Let’s use a substitution instance we mentioned above, with dogs replacing P and animals replacing H: Argument Form 1. All P are H 2. So, all H are P Substitution Instance 1. All dogs are animals. (true) 2. So, all animals are dogs. (false) This substitution instance has a true premise and a false conclusion. Again, if the form were valid, then this would be impossible. Since it clearly is possible for an argument with the form above to have a true premise and a false conclusion, we’ve shown that the original argument about protons and hadrons has an invalid form. The method we used here is called the method of counterexample. A counterexample is a substitution instance that has true premises and a false conclusion (so, the substitution instance we used above is a counterexample). Valid argument forms have no counterexamples, but for any invalid form of argument there are infinitely many counterexamples. So, while all forms of argument have infinitely many substitution instances, only invalid argument forms have counterexamples. Here, in steps, is how we use the method of counterexample: (1) Reduce the argument you suspect is invalid to its logical form, using letters to represent the simple statements or classes. (2) Find a substitution instance of that form with true premises and a false conclusion. (3) Reject the original argument’s form as invalid. Consider the argument form below. 1. If p, then q  2. p 3. So, q Are there substitution instances of this argument? Yes, there are infinitely many of them, in fact: there are infinitely many statements you could substitute for p and q. Here’s another question: Does this argument have any counterexamples? No. Why? Because the argument form is valid (it’s modus ponens). No matter what statements you substitute for the letters in this form, if the premises turn out to be true, the conclusion will also be true; that’s exactly what makes this form a valid one. So, a substitution instance of this form with true premises and a false conclusion (i.e. a counterexample) does not exist. Now consider the form below. 1. If p, then q 2. q 3. So, p Does this argument have substitution instances? Yes, and infinitely many of them; all forms of argument, valid or not, have infinitely many substitution instances. Does it have any counterexamples? Yes, because it’s an invalid form. There are statements you can substitute for the letters in this form that will make the premises true and the conclusion false, which would be impossible for any valid form. (See exercise #7.) If you go on to study symbolic logic, you’ll learn more formal techniques for showing that an argument’s form is invalid techniques such as truth tables and Venn diagrams. Those techniques are beyond the scope of our informal introduction to logic, but the counterexample method is one that’s fairly intuitive and easy to use. It has a significant limitation, but we’ll reserve discussion of this for an exercise (#19). Exercises 1.What is a substitution instance of a form of argument? 2.True or False: All forms of argument, both valid and invalid, have infinitely many substitution instances. 3.What does it mean to say that a certain form of argument is an invalid form? 4.What is a counterexample, and how are counterexamples used to prove that an argument’s form is invalid? 5.If a form of argument is invalid, then how many counterexamples will it have? 6.Can there be any counterexample to the (categorical) argument form below? Why or why not? 1.Some A are B 2.So, some B are A 7.Indicate whether the argument below has an invalid form. If it does, then provide a counterexample. 1.If this is a Corvette, then it’s a sports car. 2.This is a sports car. 3.So, this is a Corvette. 8.Provide a counterexample to the invalid categorical argument below. 1.Some lackeys are toadies. 2.Some minions are lackeys. 3.So, some minions are toadies 9.Provide a counterexample to the invalid argument below. 1.If these berries contain arsenic, then they’re dangerous to eat. 2.These berries don’t contain arsenic. 3.So, these berries are not dangerous to eat. 10. Provide a counterexample to the invalid form of argument below (assume that A and B are events in this case). 1.B cannot happen without A. 2.So, whenever A happens, B happens. 11. Provide a counterexample to the invalid form of argument below, again assuming that A and B are events. 1.Whenever A happens, B happens. 2.A rarely happens. 3.So, B rarely happens. 12. Consider the two arguments below and select the correct option. Argument 1 1. Either Michelangelo was a sculptor, or Michelangelo was a painter. 2. Michelangelo was a sculptor. 3. So, Michelangelo was not a painter. Argument 2 1. Either Neil Armstrong was a test pilot, or Neil Armstrong was an astronaut. 2. Neil Armstrong was a test pilot. 3. So, Neil Armstrong was not an astronaut. a. Argument 2 has the same form as Argument 1, and it’s a counterexample to (1) b.Argument 2 has the same form as Argument 1, but it’s not a counterexample to (1) c. Argument 2 doesn’t have the same form as Argument 1, so it’s not a counterexample to (1) 13. Consider the two arguments below and select the correct option. Argument 1 1. If you won the lottery, then you’re rich. 2. You’re rich. 3. So, you won the lottery.  Argument 2 1. If your theory is correct, then gravitons exist. 2. Your theory is not correct. 3. So, gravitons do not exist. a. Argument 2 has the same form as Argument 1, and it’s a counterexample to (1) b.Argument 2 has the same form as Argument 1, but it’s not a counterexample to (1) c. Argument 2 doesn’t have the same form as Argument 1, so it’s not a counterexample to (1) 14. Consider the two arguments below and select the correct option. Argument 1 1. If this data is correct, then the sun is mostly hydrogen. 2. This data is correct. 3. So, the sun is mostly hydrogen. Argument 2 1. If lions are felines, then lions are cats. 2. Lions are felines. 3. So, lions are cats. a. Argument 2 has the same form as Argument 1, and it’s a counterexample to (1) b.Argument 2 has the same form as Argument 1, but it’s not a counterexample to (1) c. Argument 2 doesn’t have the same form as Argument 1, so it’s not a counterexample to (1) 15. Consider the argument below and select the correct option. 1.If you’re an oenophile, then you’re a wine-lover. 2.You’re not a wine-lover. 3.So, you’re not an oenophile. a. This argument’s form has no substitution instances b.This argument’s form has no counterexamples c. Both (a) and (b) d.Neither (a) nor (b) 16. Is the substitution instance below a counterexample to the argument form? Argument form 1.Something has the property of being A.  2.Something has the property of being B. 3.So something is both A and B. Substitution instance 1.Something has the property of being circular. 2.Something has the property of being square. 3.So, something is both circular and square. 17. Is the substitution instance below a counterexample to the argument form? Argument form 1.Something lacks the property of being A. 2.Something has the property of being B. 3.So, nothing is both A and B. Substitution instance 1.Something lacks the property of being square. 2.Something has the property of having a shape. 3.So, nothing is both square and has a shape. 18. In order to prove that a certain form of argument is invalid, how many counterexamples do you need to find? 19. If you can’t find any counterexamples to an argument, does this prove that the argument’s form is valid? 20. If an argument form is invalid, will every substitution instance of that form be a counterexample? Optional exercise 1: Can the counterexample method be used to show that an inductive form of argument is weak? Optional exercise 2: What’s the difference between a counterexample to an argument and a falsifying instance of a generalization? Optional exercise 3: We pointed out that not finding a counterexample to a form of argument does not prove that the form is valid. Can looking diligently for a counterexample and not finding one nevertheless constitute strong inductive evidence that the form is valid, even if it’s not airtight proof? Chapter 8. Conductive Arguments Part A. Linked versus Conductive Arguments Consider the two arguments below. 1. If P, then Q 1. Most As are Bs 2. If Q, then R 2. This is an A 3. It follows that if P, then R 3. So, probably this is a B Suppose, in the argument on the left, you were to remove a premise, either (1) or (2). It makes no difference which one you remove; in either case, the argument would be ruined: the conclusion would no longer follow from the remaining premise. Premise (1) does not mention R, so it doesn’t entail (3). Premise (2) doesn’t mention P, so it doesn’t entail (3). Removing either premise completely spoils the inference. If you examine the argument on the right, you’ll see that the same thing is true there: removing either premise would ruin the argument such that the conclusion would no longer follow probably from the remaining premise. That’s the nature of a linked argument. Linked argument: an argument in which the premises are supposed to work together in such a way that removing any of the premises is guaranteed to ruin the argument. Now consider the argument below. 1.When your lawn is untidy, it diminishes the value of your neighbor’s property. 2. It doesn’t take much time each week to keep your lawn tidy, and a tidy lawn looks nice. 3. The longer you let your lawn go without maintenance, the harder it will be to make it look nice when you try.  4. So, keeping your lawn tidy probably is a good idea. If you were to pick an arbitrary premise and remove it from the argument, would the argument be ruined? It would be weaker as a result, but could the remaining premises still constitute fairly good grounds for accepting the conclusion? The answer is yes. That makes the argument a conductive one. Conductive argument: an argument in which the premises are supposed to work together in such a way that removing a premise is not guaranteed to ruin the argument. In a conductive argument, which is a very common argument in everyday life and in specialized contexts such as the law or a formal debate, each premise is meant to give some support to the conclusion even in isolation from the other premises, although in combination they should give even greater support. The point is that, unlike the case with a linked argument, taking out an arbitrary premise (such as by finding that it’s actually false) won’t necessarily spoil the argument completely. In a prosecutor’s case against a defendant, the defense might be able to refute one particular piece of “evidence,” but the remaining pieces of evidence might still be enough for a conviction. Linked arguments are more common in formal logic and mathematics, such as when constructing a mathematical proof. Logicians find linked arguments much easier to evaluate by mechanical means such as Venn diagrams and truth tables, whereas the evaluation of conductive arguments relies more on intuition and experience. (Although conductive arguments typically are inductive, not all inductive arguments are conductive. The argument on the right near the start of this section, above, is inductive, but it’s a linked argument, since removing either premise would spoil the argument.) Part B. Formulating a Conductive Argument Formulating a conductive argument involves, as a first step, advancing the best considerations you have to support your conclusion. That much is obvious, but even here there are a couple of things to consider: Should you list literally every reason you know of that supports the conclusion, or should you mention only the strongest reasons and avoid mentioning anything you think is on less sure ground, if only to avoid creating any vulnerable targets for your opponents? Some think it’s best to stick to just your strongest points and avoid including anything about which you’re less sure. And even when it comes to the strongest points, you might, depending on the context (a public debate, say) keep the number down to just a few in order to avoid overtaxing the patience and the attention span of the audience. In the right context, there are practical considerations like these to consider. Should you lead off with your strongest point, or should you lead up to it gradually? Which one will have the biggest rhetorical impact i.e., which one will make your case more persuasive even without altering its intrinsic logic? There are different schools of  thought here. Some feel it’s best to make the strongest first impression, and some feel it’s best to end on your strongest point. Often when we formulate a conductive argument, we mention also what seem like relevant (or at least popular) counter-points—i.e., factors that might seem to weaken our case, and then we explain why we accept our argument’s conclusion nevertheless. Perhaps the countervailing factors (or “con” reasons, as distinguished from the “pro” considerations), despite their popularity, turn out on close inspection to be worthless; or perhaps they have some merit, but not enough to outweigh the “pro” considerations. Expanding on the example above about keeping one’s lawn tidy, the arguer might go on to say: Admittedly, there are factors that might seem to militate in the other direction in your own case. For example, you’re older, and it’s not as easy for an elderly person to keep his lawn tidy. And you live in a rural area, where the expectation that the lawn will be kept tidy is somewhat lower than it is in suburban areas. However, I don’t think these undermine my conclusion, because you’re not so old that you can’t get the job done, and you can cheaply hire someone to do it in any case. And even in rural areas there is some expectation that lawns won’t be too untidy. So, my conclusion still stands. From a rhetorical standpoint, it’s desirable to anticipate counter-considerations like these and dispose of them as part of your overall conductive case. It shows that you’ve thought the matter through, and that you’re not trying to hide anything that might seem damaging to your position; you’re not trying to cherry pick the facts by mentioning only those things that support your position and ignoring any that don’t. So, essentially, a conductive argument is a case for some claim, where each point in the case you’re building is supposed to lend some support to your conclusion, and where several in combination are meant to lend even more support. Part C. Attacking a Conductive Argument When you take on someone else’s conductive argument, you can attack it by any of these means: Claim that enough of the “pro” reasons are false or misleading to ruin the argument Claim that the “pro” reasons, although true, don’t really support the conclusion, or not as strongly as the arguer suggests Claim that the “pro” reasons support the conclusion, but that there are “cons” of sufficient number and/or importance to outweigh them (perhaps some were missed in the “con” portion of the arguer’s conductive case, or some that were mentioned were underestimated) Imagine the person who’s been told he should keep his lawn tidy responding to the conductive argument with the following: “Everything you said is true, but what you overlooked is that I only rent this property, and it’s the landlord’s responsibility to keep the lawn tidy, not mine. So, your claim that I should keep the lawn tidy is misguided.” To be fair, and to show that he’s not merely interested in “scoring points” or making the arguer look foolish, he might add, “I would concede, however, that you make a good case for why the landlord should keep the lawn tidy.” Suppose you manage to cast doubt on one of (say) four premises in a conductive argument for some conclusion; perhaps you can show that this premise is false. Does it follow at once that the overall argument is worthless? Not necessarily. It’s possible that the remaining premises are still enough, in combination, to show that the conclusion probably is true. Imagine that a prosecutor presents many forms of evidence for someone’s guilt at trial, and imagine that the defense manages to show that one piece of “evidence” is worthless. Does this mean the prosecutor’s overall conductive case has fallen apart? Not necessarily; the remaining pieces of evidence might still be enough to prove guilt beyond a reasonable doubt. In a linked argument, refuting one premise will ruin the argument (although, of course, it won’t necessarily falsify the conclusion) because all of the premises are needed to support the conclusion. This isn’t true of a conductive argument. If you manage to falsify a premise in a conductive argument, you now need to evaluate the argument with the remaining premises and determine whether it’s still a strong argument. Exercises 1.What is a linked argument? 2.What is a conductive argument, and how is such an argument evaluated? 3.True or False: A linked argument can be either deductive or inductive. 4.True or False: Conductive arguments typically are inductive. 5.Consider the evidence below that a prosecutor might present at trial to show that the defendant is guilty of murder. Two reliable witnesses testify that (from a fairly close distance under reasonable lighting conditions) they saw the defendant fleeing the scene after they heard the sound of a gunshot A gun was found in a dumpster nearby with the defendant’s fingerprints on it The gun found in the dumpster is registered to the defendant The gun found in the dumpster is a ballistics match for the slug found in the dead body The defendant’s wife was involved in an extramarital affair with the victim The defendant was heard threatening to kill the victim a day before the murder Hair found at the murder scene matches a hair sample taken from the defendant The defendant has no credible alibi regarding his whereabouts at the time of the murder The defendant failed a polygraph test Now, (1) Is any single piece of evidence in the prosecutor’s case enough by itself to prove guilt beyond a reasonable doubt?  (2) Could all these pieces of evidence in combination amount to proof beyond reasonable doubt?1 (3) Suppose the defense points out (correctly) that polygraph results are inadmissible as evidence at trial in this jurisdiction, and that the results actually were labeled “inconclusive,” anyway, thus refuting the last piece of evidence mentioned above. Has the defense then defeated the prosecutor’s overall case? (4) How does the point made in #3 illustrate an important difference between attacking a linked argument and attacking a conductive argument? 6.Is the argument below conductive, or linked? 1.All the band members are sick. 2.Jane is not sick. 3.So, Jane cannot be a band member. 7.Is the argument below conductive, or linked? 1. Tom says he hates the taste of walnuts. 2. Tom is allergic to walnuts. 3. Tom has always declined any offer of walnuts in the past. 4. So, probably Tom will decline any walnuts you offer him. 8.Is the argument below conductive, or linked? 1.The death penalty is expensive and requires considerable investment of prosecutor resources. 2.The death penalty appellate process is long and slow. 3.It can be difficult to obtain and maintain materials needed for executions. 4.So, probably we’d be better off without the death penalty. 9.Is the argument below conductive, or linked? 1.P implies Q 2.Q implies R 3.R implies S 4.So, P must imply S 10. What’s wrong with the conductive argument below? 1.Nixon helped forge better relations between the U.S. and China. 2.Nixon established the Environmental Protection Agency. 3.Nixon ended the U.S. involvement in the Vietnam War. 4.So, it’s reasonable to conclude that Nixon was a good president. 11. What’s wrong with the conductive argument below (other than the problem besetting the argument in the preceding problem)? 1.Nixon helped forge better relations between the U.S. and China. 2.Nixon created the U.S. Supreme Court. 3.So, it’s reasonable to conclude that Nixon was a good president. 1 Most of the evidence mentioned here is “circumstantial,” but this only means it’s not based on someone’s actually witnessing the murder. To call evidence circumstantial does not mean that it’s weak evidence, and contrary to popular belief juries are allowed to convict on purely circumstantial evidence. 12. What’s wrong with the conductive argument below? 1.Nixon was a good husband. 2.Nixon was kind to animals. 3.Nixon was born in a farmhouse. 4.So, it’s reasonable to conclude that Nixon was a good president. 13. Evaluate the conductive argument below. “Ladies and gentleman, I’m here to convince you that a college education is a waste of time and money. Here’s why: First, college results in a lot of student debt for most people. Second, you have to take a lot of courses in college that probably aren’t relevant to your intended career. Third, there are plenty of jobs you can get without a college degree. Now, I know that college can be fun, and you get to meet a lot of people and have some interesting times, but those experiences won’t last nearly as long as the debt. So, I think people ought to forget about college and go straight to the job market.” 14. Evaluate the conductive argument below. “Here’s why I don’t like tipping and don’t think anyone should be expected to do it: First, wait staff, cab drivers, porters, and so on already make a salary, just like teachers, cops, and plumbers, and you’re not expected to tip people in the last three categories. We all just do our job. Why should only particular people be tipped? Why not just stop tipping anyone? Second, adding a tip to a bill can add quite a bit to your cost, and that’s just going to make people less able to eat out, take a cab, or stay at a hotel. Third, being expected to tip makes people uncomfortable because they might not have any cash with them, or they might not be able to calculate the right amount or even know what percentage is expected. I realize that tipping can encourage better service, and that’s a point in favor of it. But it doesn’t stack up against the three considerations I’ve advanced, so let’s end this practice of tipping.” 15. Evaluate the conductive argument below. “I don’t see the point of voting in elections, for these reasons: All politicians are corrupt, and there’s not much point in choosing between corrupt candidates.  Your vote won’t sway the election one way or the other in a national, statewide, or even local election, most likely. Not voting will send a message that we’re fed up with the existing system. Now let me address the arguments in favor of voting. Well, there aren’t any worth mentioning. So, there you go. Case closed.” Optional exercise: As you probably can tell already, there is no calculus to use when deciding on the overall strength of a conductive argument after taking into consideration the pros and cons. Evaluating the argument is largely a matter of making a judgment call. Does this mean that there simply is no such thing as rational persuasion by means of conductive argument, or an objectively better case on one side or the other? Optional exercise 2: Consider the argument below. 1. P or Q 2. ~P 3. So, Q It’s clear that (2) by itself gives no support to the conclusion, Q. Does the first premise, however, support Q at least a little bit? You might suppose that if we know that either P or Q is true, then we know there’s a chance that Q is true. Is this right? In general, does simply making a statement one of the disjuncts of a disjunction and affirming the truth of the disjunction constitute evidence supporting the truth of this disjunct, where evidence is anything that makes it more reasonable than it otherwise would be to accept the statement? Chapter 9. Pragmatic and Normative Arguments Part A. Three Kinds of Argument Arguments have premises meant to supply good reasons for accepting a certain statement, the conclusion. But reasons come in different types. Imagine that your spouse is accused of a crime, and that I’m trying to persuade you to believe in his or her innocence. There are three kinds of appeal I might make here: I could point out that your spouse has a solid alibi and that the prosecution’s forensic evidence is weak. In this case, I’m appealing to evidence to persuade you that, in fact, your spouse is innocent. This would be an epistemic argument an appeal to evidence. I could suggest that if you don’t believe that your spouse is innocent, then you’ll go crazy, and you certainly won’t make a good witness if you’re called to the stand to testify. Notice that neither of these considerations even begin to show that your spouse really is innocent; they’re not evidence of innocence. Rather, I’m appealing to what’s in your best  interest for example, by pointing out that believing your spouse is innocent will help you avoid going crazy. This would be a pragmatic argument an appeal to what’s advantageous in some way. I could claim that as a spouse it’s your moral duty to stand by your partner until and unless guilt is proven beyond any reasonable doubt. This doesn’t constitute evidence of innocence, and I’m not appealing to what’s in your best interests; rather, I’m making a moral appeal. This would be a normative argument an appeal to moral considerations. The point here is that there are different kinds of reasons you can appeal to when you’re trying to persuade someone to adopt a certain belief or to undertake a certain action, and there are different kinds of argument corresponding to these different reasons. We define the relevant terms below. When you make an epistemic argument, you’re offering evidence for the truth of some proposition or other. This is the kind of argument in which logicians are mainly interested because it’s the kind the evaluation of which is most easily formalized. When you make a pragmatic argument, you’re trying to persuade someone that it’s in his or her best interest to accept a certain proposition (or to undertake a certain action). You can try to persuade someone to accept a proposition P without arguing that P is true, as in our second example above, and instead by claiming that the person stands to gain some benefit (or avoid some disadvantage) by accepting P. When you make a normative argument, you’re appealing to moral considerations in order to persuade someone to accept a certain proposition (or to undertake a certain action). You might argue that it’s this person’s moral duty to believe or do what you’re suggesting. Consider the argument below and see whether you can identify its type. If I were you, I’d just assume that lottery ticket you bought is a loser. You should just tell yourself it’s going to lose and let yourself be pleasantly surprised if it wins. Otherwise, you’ll just get your hopes up and almost certainly have them dashed. This argument is a. Epistemic b.Pragmatic c. Normative The correct answer is (b). The conclusion, in a nutshell, is that you should assume your lottery ticket is a loser. No evidence is offered to prove that the ticket really is a loser, such as by pointing out the long odds; that’s why the argument is not epistemic. And there is no mention of any moral obligations or ethical concepts in general; that’s why the argument is not normative. Instead, the arguer provides a pragmatic (i.e., practical) reason to believe the ticket is a loser: so that you can avoid having your hopes dashed. Part B. Evaluating Pragmatic and Normative Arguments Epistemic arguments, which are the ones most commonly dealt with in logic, are evaluated by all the means we’ve talked about in previous chapters. Evaluating pragmatic and normative arguments involves some additional issues. (1) Whenever you’re given a pragmatic argument for accepting some proposition, P, you should first point out that pragmatic considerations are not evidence of P’s truth. Confronted with our sample Argument 2, for example, you might first say “What you’ve given me may or may not show that it’s in my interest to believe in my spouse’s innocence, but it of course does not prove that my spouse is innocent. Whether a belief stands to benefit me somehow, and whether it’s true, are two different things.” (2) Even if the arguer is correct in saying that accepting P stands to benefit you somehow, it might be the case that there are some drawbacks to accepting P that outweigh the benefits. If the argument is formulated inductively, then the premises might have been cherry-picked by ignoring the drawbacks. (3) At a certain point, you have to decide whether you’re the kind of person who can or will accept a proposition, P, based simply on the potential benefits of doing so, and even if, in the extreme case, all the epistemic reasons are against P. Some people are willing to convince themselves of things the evidence suggests are false just to feel good or gain some other benefit. It’s not clear that there is a right or wrong answer about whether this is ever intellectually virtuous. (In the background to this is the issue of whether it’s even within our power to choose our beliefs. We’ll set this issue aside here. 2) Similar issues arise when evaluating normative arguments; we ought to consider, for example, whether there are countervailing moral considerations for not accepting P considerations ignored or underestimated by the arguer. But normative arguments raise a further issue a full exploration of which is well outside the scope of a logic text, and the issue is this: are any moral claims really true or false? Consider premise 1 from Argument 3, above: As a spouse, it’s your moral duty to stand by your partner until guilt is proven beyond a reasonable doubt. Now, is this a true statement? Indeed, is it a statement at all, such that it can be assigned a truth value? That’s a contentious issue in moral theory, and it’s not an issue we can settle here. 2 We should note here as well that there is a difference between believing P and making P your working assumption in the sense of just acting as if P were true; you can do the latter without doing the former. One who argues pragmatically should, strictly speaking, be clear about what he or she is recommending. But a few words are in order about the evaluation of arguments with normative premises and conclusions i.e. with statements (or “statements”) in them that express ethical judgments or make value claims. (1) If you and the arguer both accept certain deeper moral judgments, then if one side can show that the normative premise in question is inconsistent with those deeper commitments, this is a way to reject the premise. Suppose that both the arguer and you take your moral cues from the scriptures of some religion, and suppose you can show that the premise in question is inconsistent with the moral values expressed in those scriptures. Then you can point this out to the arguer. Or there might be some secular (non-religious) moral theory to which you both subscribe that can be used as a common point of moral reference in order to evaluate any moral claims in the argument. Indeed, even if you don’t share some of the deeper moral assumptions the arguer has, you might still claim that a premise in the argument is inconsistent with the author’s own fundamental commitments. This won’t necessarily falsify the premise (it’s really a form of what we’ll later call ad hominem reasoning), but it might compel the author to revise the argument to bring it into line with his or her own deeper beliefs. (2) It might be that you think a normative premise in someone’s argument is true generally, but that there are exceptions to it the arguer ignores and that are relevant to the case at hand. You might respond to the premise in Argument 3, for example, by claiming that although generally a spouse should stand by his or her partner, one is released from that obligation if the partner has a long history of wrongdoing and lying, or if the spouse has been unfaithful, or under some other circumstances that might obtain here. Or you might claim that there are other morally relevant factors in play here that the arguer does not mention. (3) By definition, arguments consist of statements, and statements, by definition, are sentences with a truth value. If a normative premise has no truth value, then the “argument” in which it features is no argument at all. Now, it might be that a statement such as “murder is wrong” does have a truth value, but that we cannot determine what the truth value is until the arguer spells out exactly what “wrong” means here. What looks like a subjective claim akin to “chocolate ice cream tastes better than vanilla” might turn out to be objectively correct or incorrect (as “a helium atom has more protons than a hydrogen atom” is objectively correct) when the key word “wrong” is given an explicit definition. So, one way in which to evaluate normative arguments (or “arguments”) is by asking the arguer to spell out explicitly what the normative terms in it (good, evil, right, wrong, duty, should, etc.) mean, so that you can determine whether the normative claims the arguer makes have a truth value at all, and what that truth value is. Remember the general lesson of this chapter: when it comes to why you ought to accept some statement or other, there are different sorts of reasons an arguer can give to you in the form of an argument. Some of them are about the evidence in favor of that statement, while other reasons are non-epistemic, focusing instead on what you stand to gain from adopting that statement as one of your beliefs, or on some (allegedly) morally important consideration. 3 Exercises 1.Define each term below. (a) Epistemic argument (b) Pragmatic argument (c) Normative argument 2.What type of argument is this? “You told me someone keeps calling and hanging up when you answer, that your husband gets defensive every time you come near his phone, and that he comes home late from work a lot lately. Those things are classic signs of an affair. So, there’s a good chance he’s cheating on you.” a. Epistemic b.Pragmatic c. Normative 3.What type of argument is this? “We’re not sure that this patient is brain dead. For all we know, there might still be some higher cognitive functioning going on. Now, if that were one of us, we’d want doctors to give us the benefit of the doubt and not pull the plug too soon. Since that’s what we would want, that’s what we should do for others. So, we should assume that this patient is not brain dead.” a. Epistemic b.Pragmatic c. Normative 4.What type of argument is this? “Animals have nervous systems much like ours, and they withdraw their limbs from harmful stimuli just as we do, sometimes crying out when their bodies are stung, burned, etc. We can feel pain, and given these similarities it’s reasonable to believe that animals can feel pain, too.” a. Epistemic b.Pragmatic c. Normative 5.The French philosopher and theologian Blaise Pascal once formulated an argument for why you should believe in God an argument now called “Pascal’s Wager.” It goes (roughly) like this: 1.If you believe in God and God exists, then you go to heaven – the best result you can get. 2.If you don’t believe in God and God exists, then you go to hell – the worst possible fate. 3.Whether you believe or don’t believe, if it turns out that God does not exist, then any gain or loss you receive will be small compared with the gains or losses mentioned in (1) and (2). 3 The premises of any argument are meant to be evidence supporting the truth of the argument’s conclusion, of course. But the conclusion of an argument can refer to another statement, as when the conclusion reads “So, you ought to believe that P.” The premises might present pragmatic reasons to believe that P.  4.So, you should believe in God. What type of argument is this? a. Epistemic b.Pragmatic c. Normative 6.Consider the exchange below and answer the questions that follow. Smith: Let’s take those pallets over there and break them up to use as kindling. Jones: I think we should assume that they belong to someone until we know for sure that they’ve been abandoned. That’s the only conscientious thing to do, since otherwise we’d be stealing. a. Does Jones present Smith with any evidence that the pallets don’t belong to anyone? b.What sort of argument is Jones offering? c. How should Jones’s argument be evaluated? d.Suppose Jones’s had said this: “I think we should assume they belong to someone until we’re sure they don’t. After all, we could get arrested for stealing or destruction of property otherwise.” What sort of argument would this be? e. How show the argument in (d) be evaluated? 7.Identify the type of argument below. 1.This liquid turned litmus paper red. 2.Only acids are known to turn litmus paper red. 3.So, this liquid is an acid. 8.Identify the type of argument below. 1.Our adversaries say they’re dismantling their nuclear weapons. 2.Of course, if they’re lying and we dismantle our weapons, then they’ll really have us over a barrel. 3.If we assume they’re lying, then we’ll keep our weapons, to the benefit of our national security. 4.So, it’s best to believe that our adversaries are not dismantling their nuclear weapons. 9.True or False: If the only reason you have for believing something is a pragmatic reason, it follows that you’re wrong. 10. True or False: If you have lots of epistemic reasons for believing something, it follows that you’re right. 11. True or False: It’s possible to combine epistemic, pragmatic, and normative considerations in one argument. 12. True or False: Epistemic, pragmatic, and normative reasons can sometimes be in conflict with one another. 13. True or False: Pragmatic arguments can be conductive arguments. 14. True or False: Normative arguments can be conductive arguments. 15. One of the things we can take from this chapter is that a person can have different sorts of reasons for holding a belief: A person can believe P because she has evidence suggesting P is true. This is an epistemic reason. A person can believe P because she stands to gain something from believing P. This is a pragmatic reason. A person can believe P because she feels morally bound to believe P. This is a normative reason. There is a fourth kind of reason a person can have: A person can believe P because some physical or psychological force compelled her to believe P for example, she might have been hypnotized or conditioned to believe P, or it might be the result of a drug’s influence. This is a causal reason. (Note that the word is causal, not casual; causal means pertaining to causes and effects.) One can reason about causes, to be sure (our last two chapters are about causal reasoning), but there are no causal arguments in the sense relevant to our discussion in this chapter; hypnosis, for example, is not a form of argument, because it doesn’t involve inference. Answer the following questions: (a) What in general does it mean to have a causal reason for believing a proposition, P? (b) If your only reason for believing P is a casual one, does it follow that P is false? (c) When teachers ask you to write an “argumentative” paper for some thesis, are they typically asking for your causal reasons for holding the view you hold, or some other type of reason(s)? Optional exercise 1: Suppose someone claims that all arguments are epistemic arguments because all arguments have premises that are supposed to be evidence supporting their conclusions. Is this right? If so, then what entitles us to say that some arguments are (for example) pragmatic rather than epistemic? Optional exercise 2: Suppose someone claims that every belief everyone holds is held for causal reasons because something or other even if it’s an argument has to cause us to hold the belief. Surely the state of affairs that involves my believing something has some cause or other. If that’s right, then why do we say (for example) that sometimes we hold beliefs for pragmatic rather than causal reasons? Chapter 10. Slippery Slope Arguments Part A. Chain Reactions Some pragmatic arguments counsel us against undertaking some action or other on the grounds that it’s apt to set off an unwanted chain of events. Consider the example below. If we support the enactment of this restriction on free speech, you can bet there will be more restrictions to come. Before you know it, there won’t be any real freedom of speech left. You don’t want that to happen. So, don’t support this restriction on free speech. This is what we call a slippery slope argument. Such arguments allege that doing something A will set off a chain of events (say, events B, C, and D) that involves the gradual worsening of basically the same undesired phenomenon. In our example, supporting a certain restriction on free speech allegedly will lead to more and worse restrictions down the road. Since that would be pragmatically (and perhaps normatively) undesirable, we shouldn’t, according to the arguer, let the government impose this first restriction. It would start us down a slippery slope. A similar sort of argument is sometimes called a domino argument. Consider the example below. If you let businesses pollute the air without any restrictions, then the result will be a climatic shift that will melt the polar ice caps, causing a rise in sea levels that will flood coastal areas, devastating their economies and forcing people to migrate further inland. So, restrictions must be placed on businesses that prevent them from polluting the air. Here again a certain action is predicted to set off an undesired sequence of events. In this case, however, the series involves different sorts of events (melting ice caps, inland migrations, etc.) instead of just the gradual worsening of the same sort of event (more pollution). The basic idea behind slippery slope and domino arguments, however, is the same: for pragmatic or normative reasons (or both), you should avoid undertaking a certain action, as it would start you down a road to a destination you’d rather avoid. How do we evaluate chain reaction arguments? There are three ways, the first two of which are the most common: (1) Ask whether the predicted chain of events is likely to happen. For example, if we let government officials place this restriction on free speech, is it likely that they’ll end up placing even tighter restrictions on speech in the future, or is the risk being exaggerated by the arguer? (Note that if the events in the predicted chain reaction are especially bad, then it might be wise to avoid risking them even if the probability of their happening is low.) (2) Ask whether not undertaking action A would itself have consequences we’d like to avoid, and whether they’d be worse than the events A might cause. (3) Ask whether the predicted chain of events is as bad as the arguer suggests whether we need to avoid them at all costs. Some people talk about slippery slope and domino arguments as if they were all fallacious, but not all of them are: there are some genuine slippery slopes out there, for example. Hence, we must distinguish between a slippery slope argument and a fallacious slippery slope argument. Fallacious slippery slope arguments describe a chain of events that’s not very plausible. There are many other names for chain reaction arguments, some of which lump together slippery slope and domino reasoning. For example, some refer to “camel’s nose” arguments: if you let a camel come part way into your tent, he’ll want to come all the way in. Part B. Consistency Arguments A consistency argument is somewhat different from a slippery slope or domino argument. The claim in this case is not that if you do A then probably certain other events will, in the natural course of things, happen as well. Rather, the claim is that if you do A, then you’d have to do B, C, and D in order to be consistent or fair. Consider the example below. 1.If you give full credit to this late paper, then you’ll have to give full credit to the other late papers you get in order to be consistent. 2.You don’t want to do that, or else your deadlines won’t mean anything anymore, and no one will turn in anything on time. 3.So, you shouldn’t give full credit to this late paper. Again, the claim is not that you would, just as a matter of fact, end up giving full credit to other papers. It’s that you’d have to do this in order to be consistent and fair. In other words, there would be no good reason not to treat the other late papers the same way. For the reasons mentioned in the second premise, you don’t want to start down this road, and so you shouldn’t take this first step. How do we evaluate a consistency argument? We could, as with chain reaction arguments, ask whether B, C, and D are as bad as the arguer implies (is giving full credit to all the late papers as bad as the arguer suggests?). But usually we focus on the claim about what consistency requires (the first premise in the argument above). The question is whether consistency really would require that A be followed by B, C, and D. Consider a hypothetical student response to the sample argument: Actually, I don’t think giving full credit to my paper would require you to give full credit to every other late paper. My circumstances are unique. No one else has the same urgent excuse that I have, and so you could deal with the others differently than you deal with me without sacrificing consistency and without being unfair. So the question is whether A on the one hand, and B, C, and D on the other hand, are relevantly similar enough that consistency requires anyone who does A to do B, C, and D as well. (We’ve mentioned three hypothetical consequences in each case, but a chain reaction or consistency argument can mention any number of them.) The student in our hypothetical reply will need to show that his own case is not merely different from others no two cases are exactly alike in every respect but different enough to merit different treatment. Exercises 1.What is the form of both slippery slope and domino arguments? 2.True or False: All slippery slope and domino arguments are fallacious. 3.How do we evaluate slippery slope and domino arguments? 4.What is the form of a consistency argument? 5.How do we evaluate consistency arguments? 6.True or False: All consistency arguments are fallacious. 7.Consider the argument below and answer the two questions that follow. 1.If you let your neighbor get away with having a raucous party today, he’ll have even more raucous parties later on. 2.You don’t want that to happen. 3.So, you should call the police about this party. a. What kind of argument is this? b.How would we evaluate this argument? 8.Consider the argument below and select the correct option. 1.If you increase Florida’s number of electoral votes without any change in the state’s population, you’d have to do the same for the other states in order to be fair. 2.You don’t want to increase the electoral votes for the other states. 3.So, you shouldn’t increase Florida’s electoral votes without any change in the state’s population. a. This is a slippery slope argument b.This is a consistency argument c. This is a domino argument d.None of the above 9.Consider the following argument and select the correct option. “If you let your daughter date a guy in an outlaw motorcycle gang, she’ll end up getting into the illegal drug trade. One day she’ll probably get busted and sent to prison on drug charges, and then she’ll either get stabbed during a prison fight or get out of prison and never be able to find work again. So, don’t let you daughter date a guy in an outlaw motorcycle gang.” a. This is a slippery slope argument b.This is a consistency argument c. This is a domino argument d.None of the above 10. Consider the following argument and answer the questions that follow. 1.Marijuana is a “gateway drug” that leads to the use of harder drugs such as club drugs, cocaine, and heroin. 2.You don’t want to end up addicted to heroin. 3.So, you shouldn’t start using marijuana. a. Is this a slippery slope or a domino argument? b.What main question would one ask in evaluating this argument? c. Suppose you’re convinced that marijuana use does not normally lead to the use of heroin, although it does lead to club drugs and cocaine. Could the argument still have force? 11. Consider the argument below and answer the questions that follow. 1.If we involve ourselves in this country’s affairs because we don’t like what’s going on, then we’ll have to get involved in the affairs of many other countries, too. There’d be no reason not to. 2.We don’t want to get involved in all these cases. 3.So, we should not get involved in this country’s affairs. a. What kind of argument is this? b.How could someone who rejects the conclusion of this argument attack the argument? 12. Consider the argument below and answer the questions that follow. “If we raise the speed limit on interstate highways to 80 mph, then it won’t be long before we raise the limit to 85, then 95, and probably 100 mph someday. At some point, we’ll end up doing away with speed limits altogether. And when that happens, the accidents that will result will be horrific. So, we should keep the speed limit where it is.” a. True or False: This argument combines elements of both slippery slope and domino argumentation. b.What is the main way in which to evaluate this argument? c. True or False: You might find this argument persuasive even if you don’t believe that eventually the speed limit would be abolished completely. 13. Our government has a policy of not negotiating with terrorists. Can the reasons why be understood in terms of the argument types mentioned in this chapter? 14. Would an argument against dropping out of high school likely be a slippery slope argument, or a domino argument? 15. Suppose someone argues that because of prison overcrowding we should parole some criminals who have served at least 75% of their sentences. Categorize and respond to this counterargument: “If we parole some convicts who have served at least three-quarters of their sentences, then we’ll have to do it for all of them in order to head off any accusation of unfairness. But we don’t want to let all of them out under these circumstances; some of them are dangerous. So, we shouldn’t let any of them out early.” Optional exercise: Respond to the argument below. Does the argument fit any of the patterns of argument discussed in this chapter? 1.Suppose you give a homeless person ten dollars. 2.The difference between $10 and $10.01 is only a penny, so surely there’s no good reason to limit your donation to $10. 3.The difference between $10.01 and $10.02 is only a penny, so surely there’s no good reason to stop at $10.01. 4.By repetition of this reasoning, we can see that if you give money to a homeless person, you’ll be obliged to give every penny you’ve got, which is absurd. 5.So, you shouldn’t start down this road at all: you shouldn’t give money to a homeless person.